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But you are not me! I thought you agreed with that! By the way: seen "2001. A Space Odyssey"? Remember that "black monolith"? "Planck's constant" appears to be "the scale of scale": a holographic "monolith" projected by repeatedly mixing two reference frames giving the impression of a floating fixed "plank" as a door to "outer space" where the outer space is all the other reference frames that are mixed through the repeat iterations of the initial two. But the real door is I understand Jesus Christ where God meets man so cushions the shock of man meets God... Speed of light constant: this is apparantly "speed of speed": from the perspective of "any other perspective outside two initial reference frames": consider a series of sentences each containing the words "mile" and "hour": 1. The car travelled 50 miles in 1 hour 2. The boat travelled 40 miles in 3 hours 3. The bike travelled 15 miles in 1 hour. and so on From the perspective of "assignment of definitions" to quote Dr. Dick; A whole lot of such sentences in themselves (without even defining what number is) could be thought of as defining a constant meeting place for "mile" and "hour"; If you did define what number is you could then connect this constant "mile:hour" space with "number" to get your plank constant (like a piece of 4 by 2: like Chris Langan's "conspansive duality" and "info-cognition": you get the mile: hour in NUMBER and the NUMBER in mile:hour: try to get a handle on this and you collapse the lot into you (like a big bang) with a cosmic background radiation (consisting of particles of "big bang" with space-time strings attached?): and six quarks for your chosen two-world perspective: quark top: number (in mile: hour) quark up: mile in the above (chose what comes up) quark down: hour in first (top) quark quark strange: mile: hour (pick which) in number quark charm: mile: hour (pick the now chosen (other) one in number quark beauty/ bottom: number perspective of mile: hour in number The patterns of theoretical physics can occur in any measurement? In the set of sentences example: what sheds "light" on the collection? Light does: light as true free meeting. This generates "logical consistency" which becomes mass (room to move within an agreed reference frame)? the localised collapse of "space-time" on each perspective: in my sentences example: each sentence has a "plank mass" being "every way the other sentences can meet in that sentence's space without breaking it... So a minimum definition of the curvature of the sentences-space (the space in space) one sentences space meets another: all the others might attach strings to this meeting say: the minimum definition of the concept "direction" requires two sentences mutual agreement on defining "mile" and defining "hour": and agreeing on a direction (a variation that points to the other sentences (what Dr. Dick calls "adding unknown data" but he has used a math seive it seems and ended out with "noodles" (or pasta) (which Chroot calls a "straw metric"?) Take every way a three can meet with respect to every other way (as in Heaven: every voice gets a hearing) in our sentences and we get a minimum definition limits for "threesome" giving "planck length" as "the limit on our ability to define length here" Speed of light constant: One thing meets another: call this "speed" but how calibrate it? Super calli fragilistic expi alla docious? Over callibrate (so "super") requires overshooting so overlapping sets; So is it supposedly fragile? (how overlap can happen? requires input of others) expi: expedience: mathematics uses numbers and claims togetherness which sticks things together but with what do they stick together? (a la....) forbidden fruit? Bits of us? Since numbers are not defined you have to cobble together bits of the original things you were trying to calibrate in order to calibrate them? I am aware of the wildly metaphorical nature of the above play.... Maybe the fall of man was about getting hung up on number? Subjecting himself to number? An illusion that we are stuck ...? Something wrong in the above somewhere it may ....? This may help: Jesus referred to bread and wine as His body and blood; said that "man does not live on bread alone but on every word that comes from the mouth of God" and that "no one comes to the Father except through Me" (Jesus) and that "he who believes that Jesus Christ is the Son of God has already overcome the world". The Catholic Mass celebrates coming together; there is a way of understanding all this and physics is right in the middle it seems? Returning to speed of light: Every way "three can happen" with respect to every other way: gives "light (three as one) from light (three as one) as a constant background of light (three as one)? But what is three and what is one? We are told God is three and God is One... A local minimum definition of number meets "every way three can be three" or time can be space and space can be time: gives an apparent fixed speed if you interact with math: In the sentences example: any two definitions of "mile" and "hour" that agree with each other and with every other logical possibility there; are BOUND by the minimum/maximum definitional freedom of a four-way split (Michelson Morley experiment!) of two items ("mile" and "hour"). The Michelson Morley experiment is "rigged" in that it cannot detect direction: it mixes it up? If one looks at it without assuming a rigid space upon which it is constructed.....? Or you could break it up into mini-MM's called "axions" that seem to be conscious of each other (they by definition know each others limits in defining their mutual space in this simplified pattern perspective here)? Dr. Dick's maths seems to generate a space full of little crosses (Dirac delta functions) that collectively appear to project a universal Dirac delta function (a cross sees a cross while considering other crosses: you get a delta function attached to one cross's perspective?) The more you iterate the meeting of two crosses; the third one perspective on this seems to fan out (delta) in a way functional to the other crosses? His conclusion seems logical: the universe operating on a quantum system he does not understand? Because a universal Dirac Delta function is going to involve a lot of fuzzy deltas all muddled up? But math did that...... You can see the deltas: the number pyramids that numbers are built of..... But beyond number.... Heaven is all around us...... God invites us to sign in......... Did I make a mistake somewhere? Your view of "I" would not be fixed because it is a matter of our agreeing on a space where you can be and I can be.... but my own "I" is fixed by definition or how could I question if it is fixed without a constant me that knows he is using a logically consistent space?

An Overview of Arguments in Logic An argument is a set of statements one of which (the conclusion) is taken to be supported by the remaining statements (the premises). [Note that a “statement” can either be a whole sentence, or an independent clause within a sentence.] Five types of Arguments: Inductive, Deductive, Abductive, Practical, and Other. An Inductive argument is an argument where the premises register the known cases of a certain phenomenon, and the conclusion suggests that unknown cases will be like the known cases. (P1) The sun rose today. (P1) Everyone in my family has been stung by (P2) The sun rose yesterday. (C) So, absolutely everyone has been stung by a bee. (P3) The sun rose the day before (P4) The sun rose the day before the day before yesterday. (C1) So, the sun will rise tomorrow. Of course, the premises in each argument do not guarantee the truth of the conclusion. Still, an argument can be a good inductive argument to the degree that the conclusion is likely given the premise(s). (In assessing its likelihood, sometimes people talk of the “inductive strength” of the argument.) A Deductive argument, on the other hand, is an argument where (roughly) the truth of the premises would guarantee the truth of the conclusion. Official Definition: An argument is deductive if and only if [abbreviation: “iff”] it is not possible for the premise(s) to be true and the conclusion false. Example of a deductive argument: (P1) Jim likes either Coke or Pepsi. (P2) Jim does not like Pepsi. (C) So, Jim likes Coke. So with a deductive argument, if we get you to accept the premises, then you must accept the conclusion too. Why? ‘Cause in a deductive argument there’s no way for both the premises to be true and the conclusion false. Unfortunately, most of the time a deductive argument is called (misleadingly) a ‘valid argument’. The label is misleading, since you can have a “valid” argument which is nonetheless a bad argument, all things considered. That’s because the premises might be totally implausible. Yet the argument still counts as “valid” if it is the kind of argument where if you granted the premises, the conclusion would be guaranteed. So if you hear a logician call an argument “valid,” that does not mean that it is ultimately a good argument. Conversely, if an argument is “invalid,” that also does not mean it is ultimately a bad argument. Consider for instance that all inductive arguments are invalid, technically speaking, i.e., they are non-deductive. Still, as we saw, there can be good inductive arguments. Thus, if you say that an argument is “invalid,” you’re saying that the premises do not guarantee the conclusion, though the premises may still make the conclusion very likely for all that. The term ‘valid’ is also misleading in that “validity” concerns a relationship between premise(s) and conclusion. It is not directly concerned with whether the statements in the argument are actually true. This is contrary to how we use the word ‘valid’ outside the logic classroom: Ordinarily, we sometimes say that someone has made a “valid point” or that someone’s perspective is “valid” when we mean that s/he made a true statement. But this is NOT how logicians use ‘valid’—they only say that arguments are “valid.” (Consequently, logicians do not speak of a point or a perspective as “valid,” though they can say instead that someone has a good point or has a legitimate perspective, etc.) Of course, not every argument is deductive (= valid). Here’s one example: (P1) Jim likes either Coke or Pepsi. (P2) Jim does not like Mountain Dew. (C) So, Jim likes Coke. In this, it is possible for the premises to be true, and the conclusion false. That’s not to say the premises are actually true or the conclusion is actually false. Rather, it’s just to say that this combination of truth and falsity is possible. N.B., A non-deductive (= invalid) argument is also sometimes called a non-sequitur—it is an argument where the conclusion “does not follow” from the premise(s). Some deductive arguments are also SOUND: An argument is sound iff it is deductive AND every premise is true. Thus, an argument is unsound iff it is some premise is false or is not deductive. So, to check that an argument is sound, you have to verify that the argument is deductive and that every premise is true. Example of a sound argument: (P1) If a thing is a rectangle, then it’s not a circle. (P2) This page is a rectangle. (C) So, this page is not a circle. This argument is sound, since it is deductive, and all of its premises are true. Example of an unsound argument: (P1) If Bill Gates is poor, then I’m a monkey’s uncle. (P2) Bill Gates is poor. (C) So I’m a monkey’s uncle. This argument is unsound: Although it is deductive, it is not true that Bill Gates is poor. NOTE: Truth and Falsity are NOT properties of arguments, but of statements. Thus, we do not say that a deductive argument is “true;” rather, we say that it is valid or sound. Or, if we want to talk of “true” and “false,” we can evaluate the statements in the argument as true or false. An Abductive argument is an argument that is neither deductive nor inductive, where the conclusion stands as an explanation of facts given in the premises. (P1) I can’t get online from my computer. (P1) I have a (P2) There’s nothing wrong with my hardware or (C) So, my head is shrinking (C) So, the University network must be down. Note that in the first example, the conclusion does not explain (P2) in isolation. (The network being down wouldn’t explain why there’s nothing wrong with my hardware/software.) So the conclusion of an abductive argument is not one that explains why each premise is true individually; rather, it explains why the premises are jointly true, true all at once. Consequently, in the first example, the conclusion is best seen NOT as an explanation of why I can’t get online per se. (That would just be an explanation of the first premise.) Rather, it’s best seen as an explanation of why I can’t get online despite my functioning hardware/software. Confusingly, some inductive and deductive arguments also have conclusions which (in some sense) explain the premise(s). The second example I gave of an inductive argument is one where the conclusion (in some sense) explains the premise. Moreover, the conclusion is explanatory in the following deductive argument: (P1) This figure is a triangle. (C) Hence, this figure is a closed, three-sided figure. After all, if the figure is a closed three-sided figure, that “explains” why it is a triangle. But still, the argument is deductive, because the truth of the premise would guarantee that the conclusion is true. Thus, in order to be certain that an argument is abductive, you must first show that the argument is neither deductive nor inductive. Like an inductive argument, however, an argument is a good abductive argument to the degree that the conclusion is likely given the premise(s). (Since abduction and induction are both evaluated by the probability of the conclusion, oftentimes logic books will call both types of argument “induction.”) N.B., If the conclusion of an abductive argument is the most likely explanation out of all the explanations available, then the abductive argument is sometimes called an inference to the best explanation. A Practical argument is an argument where the conclusion is a statement of what should or ought to be done, yet the argument is not deductive, not inductive, and not abductive. (P1) Stocks are low right now (P1) I need to make money. (P2) The economy will recover soon. (P2) Kidnapping children makes money. (C) So, I should buy stocks right now. (C) So, I should start kidnapping children. As should be clear, these two arguments are not deductive. Re: the first argument, even if stocks are low and the economy is expected to recover, it is still possible that I should NOT buy stocks right now. After all, I might have barely enough money to feed my family. Still, the first example can be a good practical argument if we’re talking about someone who has expendable income. But even in that case, it remains possible for the premise to be true and the conclusion false for different reasons. So the argument is still non-deductive. When is a practical argument a good practical argument? NOBODY KNOWS. That is still debated vigorously among ethicists. However, there is a sub-type of practical argument, called a decision-theoretic argument, and it is known what makes these arguments good or bad (under certain assumptions). Very briefly, you have a good decision-theoretic argument when the conclusion recommends an action that is expected to “maximize profit” among the available options. (No need to go into more detail at this point...) Note: Some arguments with a “should” or “ought to” conclusion are NOT practical arguments. Consider the following inductive and deductive arguments (respectively): (P1) I shouldn’t have played the lottery today. (P1) Thou shalt not steal. (P2) I shouldn’t have played the lottery yesterday. (C) I should not steal this ipod. (P3) I shouldn’t have played the lottery the day before that. (C) I shouldn’t play the lottery tomorrow. (Arguably, there are also abductive arguments with “should” or “ought to” conclusions as well.) So remember that the term ‘practical argument’ is reserved for an argument that is NOT any of the previous three types of argument—AND has a “should” or “ought to” conclusion. Other arguments exist besides the previous four types. Some arguments in the “other” category are “mixtures” of the previous types of arguments. Consider, for instance: (P1) My car is usually out of gas. (P2) My car currently isn’t running. (C) So, my car is currently out of gas. The conclusion here seems to be inductively and abductively inferred. Consider that if the argument just consisted of (P1) and (C), it would plausibly be inductive. But if the argument just consisted of (P2) and (C), then it would look abductive. Yet since you’ve got both premises, it looks like inductive and abductive reasoning is being used. A different kind of “other” argument is an enthymeme: In these arguments, too much is left unsaid for us to classify the reasoning more precisely. For instance, consider: (P1) The Democrats took control of the Congress and the White House. (C) So, predictably, the economy stopped its downward slide. How exactly is (C) supported by (P1) in this case? Are we making an induction based on past cases (which aren’t explicitly mentioned)? Or are we deducing the conclusion from a suppressed premise like “whenever the Democrats are in control, the economy improves”? It’s impossible to say. So when an argument is enthymematic to this degree, we put it in the “other” category. Relatedly, some arguments can’t be classified more precisely, simply because they are just plain awful. Consider: (P1) I have ten toes. (P2) Penguins live in (C) So, Obama’s economic plan will fail. Observe that out of context, these three sentences would not seem to be an argument at all. But here, they indeed constitute an argument since one statement is marked as the conclusion, and other statements are marked as premises. So in this case, the three statements here are an argument; it’s just that it’s a really bad argument. Because of that, it’s not at all clear how the premises are meant to support the conclusion; hence, the argument goes in the “other” category. Finally, some arguments in the “other” category are arguments by analogy. Here’s a famous example: (P1) A watch has a designer. (P2) The universe is like a watch. (C) So, the universe has a designer. Note that the truth of the premises would not guarantee the conclusion; hence, the argument is not deductive. Moreover, the conclusion is not meant to explain why the premises are jointly true. So it isn’t abductive either. Some logic books, however, classify arguments by analogy a type of inductive argument. I myself think this is backwards: Inductive arguments are a type of argument by analogy, if induction assumes that the unknown cases will be like the known cases. But even ignoring that, it seems best not to classify arguments by analogy as inductive. That’s because normally when logicians speak of induction, they do not have analogical reasoning in mind. (And conversely, they are not normally thinking of induction when they talk of analogical reasoning.) Thus, I’ve put arguments by analogy in the “other” category. But unlike the just-plain-awful arguments, it is not obvious whether the watch-argument (for example) is a bad argument. Its worth would depend on how appropriate the analogy is in (P2)—and specifically, whether the universe is similar in the right way to a watch. I’ll let the theologians among you decide that one. But generally, an argument by analogy is a good argument to the extent that the analogy is a “tight” one (to put it roughly).

definition of combinational logic circuit Shows the steps involved in designing a combinational logic circuit. Glamorous Combinational Logic Design And Analysis Lecture Notes Calculator Abaeacbdcbd. Gorgeous Sequential Logic Circuits Combinational Definition Figuredangersignimplementation. Gorgeous Combinational Logic Design Using K Maps. Introduction Logic circuits for digital systems may be combinational or sequential. A combinational circuit consists of logic gates whose outputs at any time are determined directly from the present combination of inputs without regard to previous inputs. Logic Circuits: Combinational versus Sequential Circuits. Design Procedure of Combinational Circuits.n Combinational Circuits: Output only depends on the present combination of inputs. Chapter - 5 FLIP-FLOPS AND SIMPLE FLIP-FLOP APPLICATIONS Introduction : Logic circuit is divided into two types. 1. Combinational Logic Circuit 2. Sequential Logic Circuit Definition : 1. Combinational. Combinational Logic Circuits. Always gives the same output for a given set of inputs. We will apply the knowledge of Boolean Algebra to realize these circuits. First we will look at Combinational Logic Circuit. The behavior of combinational logic circuits is most typically identified and specified by a logic equation or by a truth table. Either of these methods provides a clear, concise, and unambiguous definition of how input signals are combined to drive outputs signals. combinational logic circuit. комбинационная логическая сеть. Англо-русский словарь промышленной и научной лексики.Combinational logic — Not to be confused with combinatory logic, a topic in mathematical logic. In digital circuit theory, combinational logic (sometimes also Chapter 2 Fault Detection in Logic Circuits2. 1 test generation for combinational logic circuits2.2 testing of sequential circuitsAn Introduction to Logic Circuit Testing provides a detailed coverage of techniques for test Combinational circuits are logic circuits whose outputs respond immediately to the inputs there is no memory.Fig.2 Switch transistor response and corresponding definitions of digital output signal. The digital system consists of two types of circuits, namely: (i) Combinational circuits and (ii) Sequential circuits A combinational circuit consists of logic gates, where outputs are at any instant and are determined only by the present combination of inputs without regard to previous inputs or Lecture 6: More Complex Combinational Logic Circuits. XOR ab ab. N. B. Dodge 9/15.Definition of a Multiplexer. A multiplexer is a combinational logic circuit that has up to 2n inputs, an n-bit address, and one output. Combinational Logic Circuit Definition. This combinational logic is in contrast compared to the sequential logic circuit in which the output depends on both present inputs and also on the previous inputs. Definition. Combinational Logic Circuit. The combinational logic circuit comprises of logic gates and thus the output obtained is directly related to the input. There are no feedback elements in case of the Combinational logic circuit. Converting Between Standard Forms. Summary. Chapter 2: Combinational Logic Circuits. 1. We can show that these axioms are true, given the definitions of AND, OR and NOT. As a final example of a combinational logic system, suppose we have been asked to design a circuit that accepts a binary digit, 0 or 1, and decodes it into a setThis operation also changes the structure of the graph by adding an arc from the node for G to the node for F and changing the definition of F. Chapter 2 Part 1 Combinational Logic Circuits. Originals by: Charles R. Kime and Tom Kamisnski Modified for course use by: Kewal K. Saluja and Yu Hen Hu.Operator Definitions. Application: Digital Logic Circuits. — Analogy between the operations of switching devices and the operations of logical connectives.— Rules for a Combinational Circuit: — Never combine two input wires.by definition of |. A combinational circuit is one where the output at any time depends only on the present combination of inputs at that point of time with total disregard to the past state of the inputs. The logic gate is the most basic building block of combinational logic. These circuits can be classified as combi-national logic circuits because, at any time, the logic level at the output depends on the combination of logic levels present at the inputs.In this chapter, we will continue our study of combinational circuits. Component define logic gates wikipedia the free all things electronic a picokit blog useful without true false.Logicblocks experiment guide learn sparkfun simplified circuit. Homework hwa. Patent us binational logic structure using pass drawing. Combinational logic is used to build circuits that produce specified outputs from certain inputs, the construction of combinational logic is generallyFollowing is a definition of one type of automaton, which attempts to help one grasp the essential concepts involved in automata theory/theories. Combinational and Sequential circuits are the most essential concepts to be understood in digital electronics. Combinational logic (sometimes also referred to as time-independent logic) is a type of digital logic which is implemented by Boolean circuits Combinational logic refers to circuits whose output is a function of the present value of the inputs only. As soon as inputs are changed, the information about the previous inputs is lost, that is, combinational logic circuits have no memory. Logic circuits for digital systems may be either combinational or sequential. A combinational circuit consists of logic gates whose outputs at any time are determined from only the present combination of inputs. Definition The combinational logic can be defined as is that logic in which all outputs are directly related to the current combination of values on its inputs.Much of logic design involves connecting simple combinational logic circuits to construct a larger circuit that performs a much more Common Combinational Logic Circuits. Adders. Subtraction typically via 2s complement addition. N inputs produce M outputs (typically N M). C. E. Stroud. Combinational Logic Circuits (10/12). 1. Definition of combinational circuit - a circuit whose output is dependent only on the state of its inputs.This paper presents an efficient formal logic verification algorithm for combinational circuits. A combinational logic circuit implement logical functions where its outputs depend only on its current combination of input values. On the other hand sequential circuits, unlike combinational logic, have state or memory. The combinational circuit consist of logic gates whose outputs at any time is determined directly from the present combination of input without any regard to the previous input.The only problem is that the definition of "as good as possible" may vary greatly. With combinational logic, the circuit produces the same output regardless of the order the inputs are changed. There are circuits which depend on the when the inputs change, these circuits are called sequential logic.That is the formal definition of a multiplexer. Adapted from Digital Logic Circuit Analysis Design, by Nelson, Nagle, Carroll, Irwin, Prentice-Hall,1995, Chapter 12, pages 739 to 757. Testing of combinational logic circuits digital logic circuit testing definitions. Included in this paper are examples of several CMOS logic circuits implemented at the transistor level along with a design method for the implementation of CMOS combinational logic circuits. Fig 1: Combinational logic circuit. For n number of input variables, there are 2n possible combinations of binary input values. This circuit can be described by m Boolean functions, one for each output. 1. Combinational LOGIC CIRCUITS: 2. Sequential. Combinational logic circuits (circuits without a memory): Combinational switching networks whose outputs depend only on the current inputs. Define Programmable Logic Device. What are the three forms of programmable logic devices? Explain how to program a PAL. Draw the PLA schematic circuit that will produce the following expression: Y AB AB AB. n Logic Gates (NOT, AND, OR, NAND, NOR, XOR, XNOR). n Combinational Logic Circuits from Boolean Functions.n Definition: Logic Basis is a minimal set of basic Boolean functions with which an arbitrary Boolean function can be represented. For a Boolean function g, we say a combinational logic circuit C computes g, if res(N) g(x, , x2, . . . . x,) where N is the output node.The width of C, denoted width(C), is the maximum of its thickness at all its levels. The definition of width in this letter is not equivalent to the one in [ 11, but we can obtain 2 Overview Objectives -Define combinational logic circuit -Analysis of logic circuits (to describe what they do) -Design of logic circuits from word definition -Minimization or Simplification of logic circuits -Mathematical Foundation of logic circuits (Boolean Algebra and switching theory). definition - COMBINATIONAL LOGIC. definition of Wikipedia. Advertizing .In digital circuit theory, combinational logic (sometimes also referred to as combinatorial logic) is a type of digital logic which is implemented by boolean circuits, where the output is a pure function of the present Analysis of Combinational Logic. Verifying the circuit is combinational No memory elements No feedback paths (connections). Secondly, obtain the Boolean functions for each output or the truth table. From the definition of the edge classification scheme, it follows that every such P ( i, j ) Q path covers edge ( i, j ). Let G2 . cover ( G ) - G1.Combinational logic circuits. 61. In actually implementing this algorithm, one could combine T1, T 2 , and T3 into a single transformation. Combinational logic is used in computer circuits to perform Boolean algebra on input signals and on stored data.Find a translation for the combinational logic definition in other languages: Select another language: - Select - (Chinese - Simplified) (Chinese - Traditional) Espaol In digital circuit theory, combinational logic (sometimes also referred to as time-independent logic ) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only. Definition of the noun Combinational Circuit.Definitions Combinational circuits (24): Let F be a combinational circuit and C(x) the corresponding logic function, where x is an arbitrary input. Definition of Combinational Logic Circuits: In the theory of digital circuit, combinatorial logic (also called time-independent logic) is a type of digital logic that is implemented by Boolean circuits wherein the output is a function of the input only. The outputs of Combinational Logic Circuits are only determined by the logical function of their current input state, logic 0 or logic 1, at any given instant in time. The result is that combinational logic circuits have no feedback Combinational Logic circuit contains logic gates where its output is determined by the combination of the current inputs, regardless of the output or the prior combination of inputs. Basically, combinational circuit can be depicted by diagram-1 below Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer.Half adder is a combinational logic circuit with two inputs and two outputs. Combinational logic circuits are electronic circuits that produce outputs based on the states of the inputs. Unlike in sequential logic circuits, the previous outputs do not partly determine the next outputs.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations? Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling. Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour? Can you find all the different triangles on these peg boards, and find their angles? Investigate how logic gates work in circuits. What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it? A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard? Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle? Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs. Can you find triangles on a 9-point circle? Can you work out their angles? An environment that enables you to investigate tessellations of regular polygons A game in which players take it in turns to choose a number. Can you block your opponent? Overlaying pentominoes can produce some effective patterns. Why not use LOGO to try out some of the ideas suggested here? Can you set the logic gates so that the number of bulbs which are on is the same as the number of switches which are on? An interactive activity for one to experiment with a tricky tessellation Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations? Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark? Can you fit the tangram pieces into the outline of this telephone? Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts? Can you fit the tangram pieces into the outline of Little Fung at the table? An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate. Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win? Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P? What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? Try entering different sets of numbers in the number pyramids. How does the total at the top change? When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... It's easy to work out the areas of most squares that we meet, but what if they were tilted? We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4 A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? Can you fit the tangram pieces into the outlines of these people? Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy? What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes? Can you fit the tangram pieces into the outlines of the chairs? A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins. A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red. A game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking. A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target. Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter. An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . . Try this interactive strategy game for 2 A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose! Exchange the positions of the two sets of counters in the least possible number of moves A game for two or more players that uses a knowledge of measuring tools. Spin the spinner and identify which jobs can be done with the measuring tool shown. A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line. A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible. Can you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one? The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...

How many radians are in an hour? Hour angles to Radians |1 Hour angles = 0.2618 Radians||10 Hour angles = 2.618 Radians| |2 Hour angles = 0.5236 Radians||20 Hour angles = 5.236 Radians| |3 Hour angles = 0.7854 Radians||30 Hour angles = 7.854 Radians| |4 Hour angles = 1.0472 Radians||40 Hour angles = 10.472 Radians| How do you represent minutes? Degrees, minutes and seconds are denoted by the symbols °, ‘, “. e.g. 10° 33’ 19” means an angle of 10 degrees, 33 minutes and 19 seconds . A degree is divided into 60 minutes (of arc), and each minute is divided into 60 seconds (of arc). How many degree make an hour? How do you convert degrees to hours? Convert from Degrees to Hour angles….Degrees to Hour angles. |1 Degrees = 0.0667 Hour angles||10 Degrees = 0.6667 Hour angles||2500 Degrees = 166.67 Hour angles| |2 Degrees = 0.1333 Hour angles||20 Degrees = 1.3333 Hour angles||5000 Degrees = 333.33 Hour angles| |3 Degrees = 0.2 Hour angles||30 Degrees = 2 Hour angles||10000 Degrees = 666.67 Hour angles| How do you write hours and minutes? If you are using numerals, you would usually include both hours and minutes, although you can omit the minutes in less formal writing. For instance, all the following would be acceptable: She gets up at six in the morning every day. She gets up at 6:00 in the morning every day. How do I write hours and minutes in Word? When writing out the time of day in words, use a hyphen between the hour and the minutes, unless the minutes themselves are hyphenated: - I leave for work between eight and eight-thirty. - Megan usually leaves for work at about eight forty-five. What is a symbol for time? A common symbol used now is the hourglass. Although this is an object we don’t use in everyday life, the symbolism of time slipping away from one, and the common phrase “the sands of time” show that this metaphorical understanding of time pervades at least English-language thought. How do you convert degrees to time? 1 degree of arc is define as 1/360 of a revolution. In SI units 1° is π/180 radians. 1 Minute of time (astronomical): 1 Minute of time (astronomical) is equal to 1 turn/1440. How many hours is 45 degrees? Hour angles to Degrees |1 Hour angles = 15 Degrees||10 Hour angles = 150 Degrees| |3 Hour angles = 45 Degrees||30 Hour angles = 450 Degrees| |4 Hour angles = 60 Degrees||40 Hour angles = 600 Degrees| |5 Hour angles = 75 Degrees||50 Hour angles = 750 Degrees| |6 Hour angles = 90 Degrees||100 Hour angles = 1500 Degrees| What is the symbol for an hour? How do you write 2 hours and 45 minutes? How to Convert Time to Decimal - 2 hours is 2 hours * (1 hour/ 1 hour) = 2 hours. - 45 minutes is 45 minutes * (1 hour / 60 minutes) = 45/60 hours = 0.75 hours. - 45 seconds is 45 seconds * (1 hour / 3600 seconds) = 45/3600 hours = 0.0125 hours. - Adding them all together we have 2 hours + 0.75 hours + 0.0125 hours = 2.7625 hours. How do you convert degrees to minutes and seconds? How to Convert Decimal Degrees to DMS - For the degrees use the whole number part of the decimal. - For the minutes multiply the remaining decimal by 60. Use the whole number part of the answer as minutes. - For the seconds multiply the new remaining decimal by 60. How do you write time in words? For novels (fiction and non-fiction), the general rule is to spell out time. When expressing time in words instead of numerals, you should use a hyphen, as in five-fifteen. But when a hyphen is necessary in the expression of minutes, only hyphenate the minutes, as in five forty-five. Can an angle be negative? Angle measure can be positive or negative, depending on the direction of rotation. Rotation is measured from the initial side to the terminal side of the angle. Positive angles (Figure a) result from counterclockwise rotation, and negative angles (Figure b) result from clockwise rotation. What is the radian measure between arms of watch at 5 pm? At 5 PM , the arms of a 12-hour clock are separated by 5/12 of a full circle. Thus the angle between them is 5/12 of 2π radians = 5/6 π radians or roughly 2.618 radians. How do you convert degrees minutes to radians? To convert degrees to radians, first convert the number of degrees, minutes, and seconds to decimal form. Divide the number of minutes by 60 and add to the number of degrees. So, for example, 12° 28′ is 12 + 28/60 which equals 12.467°. Next multiply by π and divide by 180 to get the angle in radians. What is the symbol for hours and minutes? While the unit s , seconds is an SI standard, the symbols h for hours and min for minutes are accepted for use with SI standards although they are not SI standard units of measure, being integrals of the basic accepted standard, s. How many degrees is a triangle? How is 1 degree 60 minutes? 1 degree can be equal to 60 parts which should be called seconds not minutes. And 60 minutes forms 360 degree . How many seconds are present in 2 degree? Please share if you found this tool useful: |1 Degrees to Seconds Of Time = 240||70 Degrees to Seconds Of Time = 16800| |2 Degrees to Seconds Of Time = 480||80 Degrees to Seconds Of Time = 19200| |3 Degrees to Seconds Of Time = 720||90 Degrees to Seconds Of Time = 21600| How do you write 1 minute? To write an abbreviated version of minutes, you can use the following: min. ‘ (informally)…Abbreviations for Minutes - 1 min. - 5 min. - 45 min. How do you convert from radians to hours? Convert Radians to Hour Angles One full turn is 2π radians. 1 Hour Angle: 1 Hour angle is 1 turn/24 or 15°. In terms of SI units an hour angle is π/12 radians. How many hours are in 360 degrees? How do you write 24 hour clock in words? The abbreviations AM and PM can be used when writing the time as numerals. Each applies to a different time of day: AM is short for ante meridiem, meaning ‘before noon’….2. AM and PM. |Time||12-Hour Clock||24-Hour Clock| |Three in the afternoon||03:00 PM||15:00| |Half eight in the evening||08:30 PM||20:30| How do I type a degree symbol? How to Type the ° - PC. Hold down the Alt key, and on the numeric keypad on the right of the keyboard, type 0176 or Alt+ 248. - Mac. Press Option Shift 8. - iOS. From the iOS keyboard on your iPhone or iPad: - Android. Switch to the numbers and symbols keyboard. The degree symbol should appear on one of the pages. How many seconds are in an angle of 15? How do I type a minute symbol? Either Alt+0176 or Alt+248 can be used to add degree symbol to represent coordinate values in Degrees Minutes Seconds (DMS).

Receive TAC Lectures via Podcast! by Dr. Carol Day Thomas Aquinas College St. Vincent de Paul Lecture and Concert Series March 19, 2021 When I first taught Euclid’s Elements, I was puzzled about several features of the “Number Books,” Books 7-9. I was not surprised to find that the students were puzzled too. For the most part, we were used to and comfortable with Euclid’s style and method by the time we got through Book 6, but what was he up to in Book 7? The long list of definitions at the beginning showed that he was launching into arithmetic. But why take up numbers at this point? Why not give us more theorems in plane geometry or perhaps move on to solid geometry? What we got instead were propositions about relatively prime numbers and about numerical ratios. I was puzzled, and I am very sure that I was not the only one in the classroom wondering about what Euclid was up to. There were other puzzling features of Euclid’s presentation. After the usual enunciation in words, he displays numbers in the setting out as lines. Given the mathematical custom in which I was brought up, I would have liked the setting out to be done using algebraic notation, and I noticed that some of the students wanted that too. This was evident from the way they wrote out their demonstrations. I also wondered about the order in which he presented his propositions. Why doesn’t he begin the study of arithmetic from the beginning, as one would expect from his practice in the geometrical books, and then go through theorems about numbers in a systematic way? Why in particular does he begin his study of numbers with relatively prime numbers? Much later on, I had other questions about the arithmetical part of the Elements. Why are these books placed after the treatment of plane geometry and before the treatment of solid geometry? I also wondered whether we should conceive of Book 10, his treatment of irrational magnitudes, as belonging with Books 7-9 rather than as a book standing on its own or just as a necessary preparation for solid geometry. This last question did not occur to me when I first taught the Elements, but it seemed an obvious thing to wonder about when I came back to Euclid many years later. My plan is to address all these questions, although I can make no promise to settle them all. Since much of my talk will concern the order of the Elements and the number books within it, it would be reasonable to set out the order in which they will be addressed. I will begin with the representation of numbers by lines. This consideration stands apart from the somewhat entangled issues of the order of propositions and of the books themselves. But I hope to show that an understanding of Euclid’s method of representing numbers provides important clues for understanding the rest. In the second part of this lecture I will ask about the appropriateness of including arithmetical books in a work of geometry, and in the third and fourth parts I will deal with questions about the order of the propositions in Books 7-9 and with the place of the number books in the overall scheme of the Elements. Part One: The Representation of Numbers by Lines Note: when I say line, I will mean straight line, unless I specify otherwise. I remember many times my much beloved colleague and friend, Molly Gustin, would say that numbers are lines. Despite the fact that we disagreed about this, I can understand why she said it. I think she was influenced both by Euclid’s way of depicting numbers and by Descartes’ extension of arithmetical concepts into geometry. In defense of her notion, remember that it has become a common-place to speak of the Cartesian “number line,” as something comprising all the real numbers. To be fair to Mrs. Gustin, I believe that she was trying to give an account that made sense of calling the real numbers ‘numbers’. But if the whole numbers, those with which Euclid was concerned, are also real numbers, the temptation is there to state in a categorical way that numbers are lines. I think the identification of real numbers with lines is a mistake, but my concern here is not with that but with the interpretation of Euclid. Whatever numbers were for Euclid, they were not lines. Since Euclid defines number as a multitude of units, one need only look at his definition of unit to see what he understands numbers to be. His definition is broad, to say the least! He says that “An unit is that by virtue of which each of the things that exist is called one.” Thus we can speak of one line, one sphere, one point, one cow, one instance of blue, one thought, and so on. The unit is something common to them all. He does not make clear what this common thing is, probably because he thought it was enough for the mathematician to see that the unit is the principle of number and that it has some existence apart from its concrete or geometrical manifestations. The determining of its exact nature of its existence belongs to a higher science than mathematics. In my opinion, then, Euclid’s use of lines to represent numerable things does not imply a thesis about the nature of number. What remains, then, is to explain why lines are suitable and in fact the best way available to him for representing numbers. At least two methods for depicting numbers were available to Euclid. One was the use of numerals. This method was used by Nicomachus in his Introduction to Arithmetic. In his History of Greek Mathematics, Heath compares Nicomachus’ method to Euclid’s, saying that that the method of representing numbers by lines “has the advantage that, as in algebraical notation, we can work with numbers in general without the necessity of giving them specific values; in Nicomachus numbers are no longer denoted by straight lines, so that when different undetermined numbers have to be distinguished, this has to be done by circumlocution, which makes the propositions cumbrous and hard to follow, and it is necessary after each proposition has been stated, to illustrate it by examples in specific numbers. Further, there are no longer any proofs in the proper sense of the word.” Consider as an example Book 7, Proposition 1. First, here is how Euclid expresses it. “Two unequal numbers being set out, and the lesser being continually subtracted from the greater, if the number which is left never measures the one before it until the unit is left, the original numbers will be prime to one another.” Following the enunciation comes the “setting out,” as Proclus calls it: “For the less of two unequal numbers AB and CD being continually subtracted from the greater, let the number which is left over never measure the one before it until a unit is left. “ AB and CD refer to a diagram, which looks like this: __________________ ______________ _____ A H E B C F D G Here AB and CD are numbers being measured, G is supposed as a common measure, and AH is the unit. It is Euclid’s practice to letter both endpoints of a line, as well as its points of division, assuming that the number it represents needs to be measured. If a number does not need to be measured, he usually names it with a single letter. The proof, then, is carried out using these letters as stand-ins for the numbers and their parts. If Euclid had recourse only to determinate numbers, the setting out would have to look something like this: “For the lesser of two numbers, for example 5, being continually subtracted from the greater, for example 93, let the number left never measure the one before it until the unit is left, then 5 and 93 are prime to one another.” The supposed proof would be a calculation. 5 x 18 = 90. So with 5 subtracted from 93 eighteen times, we have 3 left. Now 3 subtracted from 5 leaves 2. and 2 subtracted from 3 leaves 1. Since following the subtracting algorithm leads me to the unit before I find a common measure, I want to assert that there is no other common measure. But how do I prove this? I can go through all the numbers up to 5 to see if they also go evenly into 93, but when I see that none of them do what will I have learned? Only something particular to these two numbers. The only alternative is to suppose some indeterminate common measure other than the unit, give it a name such as G, and work out a proof like Euclid’s, in which thinking of the original numbers as particular examples is pointless, in fact distracting. It’s tempting to simply write off argument by means of examples as unscientific if not impossible. Nonetheless, we ought to think about how it differs from what Euclid does in proving geometrical theorems. Some concrete representation of the thing to be proved needs to be presented to the imagination, and whatever is in the imagination is singular, not universal. To prove a theorem about triangles, Euclid must give us a particular triangle, with determinate sides and angles. How is this unlike using a numerical example? In a geometrical proof, it is not difficult to look at a concrete individual and attend only to the features that are relevant to the argument. For example, in looking at the drawing of a triangle for the purpose of proving Book 1, 5, that the base angles of an isosceles triangle are equal, we need to imagine that there are two equal and one unequal side and which are the base angles, but we don’t need to attend to the relative lengths of the equal and the unequal sides. It is easy to see that these details do not enter into the argument. We can even see that the proof works if all three sides happen to be equal. The abstraction of the relevant from the irrelevant is often easy to do in geometry. But there is something about the way in which concrete numbers exist in our imagination that gets in the way of performing the necessary mental trick. I think this has to do with the mode in which they must be defined. To see why the modes of definition are different in arithmetic and in geometry, consider how the infinite exists in each. In magnitude, we have the infinity of infinite divisibility. As such, the infinite has no relevance to the definition of figures, having to do only with their material aspect, that is, with the continuum in which the figures exist. The formal features of figures arise from their shapes. Geometrical figures are defined by their boundaries, and all their properties flow from the nature of these boundaries. Even in theorems having to do with areas and volumes, where the properties that result from their forms are often harder to know, it is by considering the implications of their boundaries that we learn what we can. This is true even in the use of calculus. It is quite otherwise with numbers. Numbers are infinite by addition, growing ever greater as we count them. Having no position, they also have no boundaries. One might even say that a given number is a boundary. That is, a number terminates a progression radiating outward from the unit. Whatever is formal in the number comes from the nature of this boundary, which gives rise to the distinctive way in which it is a multitude. It follows that unlike magnitudes, which are defined by way of genus and difference, numbers require a different kind of definition. Let me explain what I mean by that. Suppose I want to define the number four. It is true but not altogether helpful to say that four is four ones, since it begs the question. Rather, four must be defined at the number that comes next after three, as that number in which a unit is added to three. This may sound like a purely nominal definition, but it is more than that. The nature of any number depends on the nature of the number before it, going all the way back to the unit. All the previous numbers are in it as potency to the next number, which is brought to act by the joining of another unit. The act of joining another unit to three makes what was potentially four to become actually four. What this amounts to is saying that the additional unit plays the role of form or species in the definition of a number, and the number to which it is added plays the role of matter. St. Thomas makes this point in his Commentary on the Metaphysics, Book 8, Lectio 3: “A number is one per se in as much as its final unit gives to the number unity and species; just as in things composed of matter and form a thing is one through its form and takes on its unity and species.” Let me say in passing that I do not propose to say how the new unit is added, or exactly what it means to add it. However it comes about in the being of the numbers themselves, what the mathematician sees is that three becomes four when another unit is added to it. As a simple example of how this act of joining the last unit determines a new property, just consider how the new unit changes the number from odd to even. But note how this mode of defining reveals very little about the properties of the number. Though it is obvious that the next number after 4 is odd, we cannot see in any easy way that 5 is the only prime number that is the sum and difference of two primes. 5 = 2 + 3, 5 = 7 - 2. The discovery and proof of theorems like this would be difficult if not impossible without the analytic methods of modern mathematics. Fortunately, this branch of the science, which pursues the characteristic properties of individual species of number, is not what Euclid was concerned with. This is the branch of arithmetic that is as far from geometry as possible. If then a specific number is the boundary of an act of accumulating units, so that no other kind of definition can be given by us, how do we translate such a definition into something useful for geometry? I think Euclid had a good answer: we reason about numbers by considering them as measuring and as measured. The ultimate measure of a number is the unit, and its multitude is the distinctive way that the unit measures it. Measuring is an act of dividing. The geometer, therefore, divides numbers and impose order upon them in order to reveal their properties. Now measure is first known to us in extended things, in things we can sense. The first notion of measure is of a magnitude laid out along another magnitude so that it goes into it a certain number of times. The very name of Euclid’s science, Geometry, i.e. earth measurement, refers to this very practical procedure. Although there is something arbitrary in measurement -- one can begin from either end, for example -- there is a comprehensible order of the units from left to right or vice versa. Laying down the unit randomly leads to error and counting the divisions unsystematically leads to confusion. When counting material objects, we tend to imitate this spatial order by systematically ordering the things themselves in space. There is plenty of evidence that in ancient times livestock were counted by associating them one by one with notches in a stick. You may have experienced this in counting pennies by grouping them in groups of five or ten. On the other hand, the units in an abstract number are not laid out alongside each other, nor are they visible in the representation of a number by a numeral. Where are the units in 7? There is no ‘where’ there! And what happens when we subtract one number from another? When we subtract 3 from 7, we don’t think about which of the units in 7 are being taken away! Here we see an advantage of representing numbers by divided lines. By ordering the units in space we give our imagination something to make use of as we go about discovering and proving properties of numbers. Representing numbers by lines seems to be an obvious choice, but this was not the only choice Euclid could have made. Another technique was available to him, one which had proved useful to his predecessors. Since the unit is indivisible, it would seem logical to represent it by a point. A number, then, would be represented by a set of points, since the unit is as it were the material from which the number is formed. In Metaphysics XIII Chapter 8, Aristotle describes this approach: “They [that is, some of the Pythagoreans] conducted their inquiry at the same time from the standpoint of mathematics and from that of universal formulae, so that from the former standpoint they treated unity, their first principle, as a point.” This way of depicting numbers has its uses. When the numbers in question have properties analogous to geometrical properties, this way of depicting them can be helpful for the discovery of theorems. Such for example are square, cubic and triangular numbers. For numbers like these, a visual presentation of their nature is possible by drawing an orderly array of dots. Here is an informal demonstration that summing successive odd numbers produces the sequence of square numbers: Representing these numbers by arrays of dots can indeed serve the imagination well enough, where they are appropriate. Although the diagram is not a formal proof of the theorem, it is in itself quite convincing. This method covers a very small territory in the realm of numbers, however. Symbolizing a number such as 7 by a line of dots does not have any obvious advantage over using a line divided into 7 segments. Euclid’s way seems to have the disadvantage of not representing perfectly the nature of number as discrete quantity, but it is superior in that it does not give the false impression that 7 is nothing more than 7 ones side by side, as if the number had no character and unity of its own. Let us now look at a distinctive advantage of visually articulating parts of numbers, whether units or other divisors, by the use of divided lines. The lines may always be made of reasonably short length since any arbitrarily small line can be thought of as the unit. As mentioned above, we are able in this way to grasp the number as a whole containing these parts. Because of the abstractness of the representation, it is not hard to disregard the actual number of divisions in the illustration and to focus on what is essential. In other words, there is no reason to pay attention to the actual count of the divisions, as if one were merely calculating. Lets see how this works by looking at proposition 7, 4, which proves that any number is either a part or parts of any number, the less of the greater. B E F C Recall that a number is part of a larger number if it measures it without a remainder, but parts if there is a remainder. Thus 3 is part of 6, but it is parts of 7. In the proof, the larger number is represented by A and the lesser by BC. Although it contains the lesser number, the laying out of BC alongside A isn’t necessary. Everything hangs on whether or not A and BC are prime to one another. If BC measures A, it is a part and all is well. If it does not measure it, we need only take the greatest common measure of A and BC, represented by line D. BC is shown as divided into parts BE, EF, EC equal to D, to show that a part existing in BC is also a part of A. That is what it means to say that BC is parts of A; it is made up of numbers which are themselves parts of A. The articulation of the lines into parts helps one to understand the reason for the theorem. The fact that BC is shown as divisible into three parts does not get in the way of understanding the proof, for it is not hard to see that the exact number of times D is subtracted does not matter to the argument. The proof rests on the nature of measurement, and measure is illustrated in the lines in a way that does not call to mind vividly the particular results of the measurement. Is this not the key to understanding Euclid’s use of lines? Seeing measurement at work requires an order in space and nothing more. To sum up, then: By representing a number as a divided line, the teacher exhibits its formal character in a sufficiently detailed way, which he could not do by showing it as a numeral. Showing numbers as lines depicts them as quantities relatable to one another either through one measuring the other or through their having some common measure. In this way Euclid facilitates our grasp of the truths which he wants to prove about them. Part Two: Why does the Elements contain the Number Books? Before considering the place of the number books in the Elements, we might ask why they are there at all. Euclid does not tell us why he included them, and I have found nothing in Heath’s history or commentaries to shed light on the matter. Since I can’t answer that question, I can at least point out some advantages of including them. The inclusion of the arithmetical books allows Euclid to show that there is an analogy between the subject matter of geometry and the subject matter of arithmetic by proving comparable properties for numbers and magnitudes, each by means of proper principles. . Examples of this abound, but I will illustrate with one. In 7, 13 Euclid proves “If four numbers be proportional, they will also be proportional alternately.” Here is an example: since 2:4 :: 5: 10, so also 2:5 :: 4:10. This proposition is proved from the definitions in Book 7 of part and parts. In 5, 16 Euclid had proven the comparable theorem in geometry: “If four magnitudes are proportional, they will also be proportional alternately.” This proof rests on the definition of same ratio given at the beginning of Book 5. Euclid’s drawing out of the likeness as well as the difference between geometrical and arithmetic theorems is I think the most important result of including the books on arithmetic, at least from a philosophical point of view. I say this because the relation between number and magnitude was a controversial issue for the Greeks and in fact is still a question of great interest. We moderns are accustomed to the idea that there is a universal mathematics, one which is most properly expressed in symbols. There have been various ideas about how universal mathematics stands with respect to arithmetic and geometry. For Viete, Descartes’ predecessor in the invention of algebra, the symbols and the rules for their manipulation are the same for both kinds of mathematics, but each requires a distinct process of interpretation and justification. Another opinion is they are unified by a common subject matter, which might be called quantity as such. A common opinion among the moderns is that mathematics is a branch of logic, so that the symbols themselves seem to be its subject matter. More importantly for our purposes, though, was the pre-Euclidean opinion that all quantity is the same kind of thing because all quantities are commensurable. This Pythagorean understanding would reduce all mathematics to arithmetic. This view could no longer be held after the shocking discovery that the side of a square and its diagonal are have no common measure. Prior to the scandal of the incommensurable, several of the theorems we find in the first four books Elements had flawed proofs based on a purely numerical theory of proportion. Here is how Heath describes the situation: After the discovery of this one case of irrationality [i.e. of the square root of two] it would be obvious that proportions hitherto proved by means of the numerical theory of proportion, which was inapplicable to incommensurable magnitudes, were only partially proved. Accordingly, pending the discovery of a theory of proportion applicable to incommensurable as well as commensurable magnitudes, there would be an inducement to substitute, where possible, for proofs employing the theory of proportions other proofs independent of that theory. This substitution is carried rather far in Euclid, Books I - IV. In other words, all of the demonstrations in the first four books of the Elements are valid quite apart from questions about the divisibility of the continuum. Euclid ingeniously shows that many elementary properties of figures do not an any way rest on the difference between the continuous and the discrete. This sets these books apart from those that follow. Once the universally valid theory of proportion has been established, it is possible to treat the rest of mathematics according to the distinctive principles of the continuous and the discrete. The treatment of proportion in Book 5 of the Elements makes possible a satisfactory treatment of magnitudes both commensurable and incommensurable. By providing the separate but parallel treatment of proportion in number in Book 7, Euclid shows most clearly that geometry and arithmetic must each be developed from its own proper principles, however many theorems they seem to share in common. This, I propose, is one important reason for including the number books. A second advantage of including the number books is that it makes the Elements is a more complete elementary treatment of mathematics. When we take a look at the contents of these books, we will see evidence that Euclid desired to aim at completeness, sometimes even at the expense of good order. Taken together with Book 10, the number books give an adequate treatment of the ways in which magnitudes can have ratios to one another. All the potentialities of ratio implicit in Book 5, or rather all those appropriate to beginners in mathematics, are revealed to the student. A third advantage is that numbers show up from time to time in geometrical theorems. A most noteworthy example of this is in the very last proposition in the Elements, Book 13, Proposition 18: To set out the sides of the five figures [that is, the five regular solids inscribed in the same sphere] and to compare them to one another. Some of these comparisons involve numerical ratios, as that the square on the diameter of the circle is to the square on the side of the inscribed pyramid as 3:2. We learn other truths along the way involving number, some as simple as 1, 41, which says that the parallelogram having the same base as a triangle and is in the same parallels is double the triangle, or as complex as Book 12, Proposition 10, which shows that any cone is a third part of the cylinder which has the same base and the same height. We see that solid geometry, which is the most complete geometry of the physical world, brings together the discrete and the continuous in a profound way. Here we see the most perfect marriage of geometry and arithmetic. Part III: The Nature of the Number Books and the Order of their Propositions If we grant, then, that there are some good reasons for including the number books in the Elements, we may still find them to be unsatisfactory in themselves, as being disorderly. One would think that a scientific treatment of arithmetic should begin with definitions and postulates and then proceed to prove the simplest properties of numbers first, followed by more complex ones. Arithmetic pursued in isolation from geometry would look quite different from what we find in the number books. I have already suggested that Euclid treats number in light of the notion of measuring and being measured. This is demonstrated by the way Book 7 begins. Number is defined as a multitude composed of units, and multitude arises from the unit by way of addition. Addition is a kind of measuring, in the sense of meting out, as when we count out 75 cents for a candy bar. This is measuring as composition. But the more common kind of measurement is a process of resolution, in which we begin with numbers or a magnitudes as given and analyze them into equal parts. To account for this kind of measurement, Euclid next defines part. This use of part is not like that in the axiom which states that the whole is greater than the part. Here part means a part which measures a whole. From these basic ideas underlying measurement Euclid goes on to define the two most fundamental divisions of number into species. Even and odd numbers are distinguished by whether or not they can be measured by the dyad, the number two. The notion of measurement is also required for the definitions of prime and composite numbers. Primes are measured by the unit alone, while composite numbers have other measures as well. Thus we see that measure is at the very root of number and its division into species. Let us look more closely now at some of these divisions. By defining the even and the odd right after defining part, parts and multiple, and before prime, etc., Euclid seems to acknowledge the primacy of this division. It is certainly the simplest, and the most easily applied to given numbers. Since for every odd there is an even, namely its double, it seems to be the most perfectly symmetrical division of number, and so in some way the most beautiful. The even and odd can themselves be divides into sub-species, and this is what Euclid does next. There are even times even numbers such as 8, even times odd numbers such as 6, and odd times odd numbers such as 9. These subspecies of the odd and even are defined explicitly by measure. For example, an even times even number is one that is measured by an even number an even number of times. Next Euclid gives the other important division of number which comprises them all, namely prime and composite. These two divisions are the only ones given by Euclid which comprehend all numbers. The notions of prime and composite are then extended to relations among numbers. Numbers are prime to one another if they have no common measure and composite to one another if they do. This is not a division of number into species but a description of important properties numbers can have in the category of relation. It is interesting to note that up to now, Euclid has not defined any of the operations which are taken as fundamental to arithmetic. He assumes both addition and measurement, which is a kind of division, as well-known. Later, subtraction will also be assumed. It is interesting then that he does define multiplication. Perhaps he feels he needs to do this to avoid any confusion with the analogous geometrical operation, the forming of a rectangle. At any rate, rectangles are called to mind in the very next definition, where he defines a plane number as one which has been formed by multiplying one number by another, which in this context are called sides. He similarly defines solid numbers as those produced by the multiplication of three numbers. By defining multiplication in terms proper to arithmetic, he indicates that his use of geometrical terminology in these definitions is only by way of analogy. The most important examples of geometrical numbers, squares and cubes, are defined next, and then numerical proportion, with a view to defining similar plane and solid numbers. Finally Euclid defines perfect numbers. Recall that these are numbers which are equal to the sum of their measuring parts, including the unit. It may seem strange that Euclid does not begin with theorems about properties of the odd and even or with the prime and composite numbers. This would be strange if he were interested in pure numbers, in what is now called “Number Theory.” He is concerned instead with numbers in comparison to each other. The principal subject of Book 7 is numerical ratio and proportion. The first proposition gives the criterion by which two numbers are prime to one another, and the second which gives the method for finding the greatest common measure of numbers which not prime to one another. This second proposition is needed for what follows, a series of demonstrations analogous to the propositions about proportional magnitudes in Book 5, up through the sameness of ratio ex aequali. He goes on to present a number of propositions dealing with sameness of ratio that are specific to numbers. Some of these have analogies to geometry. Especially interesting is Proposition 16, which defines cross multiplying. This is comparable to the proposition defining compounding of ratios in Book 6. We see that, right from the beginning, Euclid wants to make us aware of parallels between arithmetic and geometry. The rest of Book 7 contains more theorems about relatively prime numbers as well as about numbers that are simply prime or composite. Among these are propositions for reducing ratios to their least terms, finding the least common measure, finding the least number measured by two or three numbers, and to find the least number that has given parts. The essence of any numerical ratio is contained most simply in its least terms. This verifies that the subject of this first Book is numerical ratio and proportion and the treatment of them here is the most elementary. The next two books of arithmetic present many interesting theorems having for the most part some connection with geometry. The principal subject of Book 8 is numbers in continued proportion. These are sequences of numbers in which each one after the first is the geometric mean between the ones prior and posterior to it, for example: 2, 4, 8, 16 ... Starting with Proposition 11, Euclid sets out propositions relating figured numbers (squares, etc.) to continued proportions. The first two of these are particularly important. The first says that between two square numbers there is one mean proportional number, and the square has to the square the duplicate ratio that which the side has to the side, and the second says that between two cube numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. More propositions about figured numbers round out the books. Book 9 at first sight looks very disorderly, as if it were just a catch-all for other interesting theorems. There may be something to that, but I think we can makes some sense of it. It picks up where Book 8 left off, with more propositions about figured numbers, but the point of view is different than in Book 8. In the first seven propositions, he is interested in figured numbers considered in themselves, not in their relation to other numbers. This theme is carried on in the next sequence of propositions, which deal with continued proportions starting from the unit. Such a sequence builds up a series of numbers, each of which is the prior one multiplied by the second in the series. Here is an example: 1, 3, 9, 27... i.e. 1, 3, 3x3, 3x3x3 ... So even here the individual numbers are of interest, as being a sequence of successive squares or cubes, or numbers arising from squares and cubes. More propositions relating to this theme follow, eventually leading to theorems in which prime numbers play a part in these proportions. These propositions culminate in an investigation of the conditions in which it is possible to find a third proportional to two given numbers, i.e. when, given numbers a and b, is there a number x such that a:b::b:x. What happens next, leading up to the end of the number books, is puzzling. Prop. 20 shows that there is no end to the generation of prime numbers. This complicated and very important proposition could have been proved in Book 7. The only rationale I can see for including it here is that Book 9 has been more closely focused on the properties of given numbers taken in themselves than were the first two books. Proposition 20 is followed some others (21-30) which are elementary and even trivial, having to do with odd and even numbers. It is hard to see how these are not out of place, and I am inclined to doubt that Euclid put them there. But if I had to speculate: One possible way to relate these propositions to the ones that come before in Book 9 is to note that most of them relate to the unending production of numbers from other numbers. All the kinds of numbers he has dealt with so far have been showed to be indefinitely many. This is shown explicitly for primes and implicitly for the rest. The continued proportions produce endless series of squares, cubes, and similar plane and solid numbers. Similarly but more obviously, addition produces as many even and odd numbers as we want. Multiplication gives us even times even, even times odd, and odd times odd numbers. Propositions 32 -34 deal explicitly with these kinds of composite numbers. This brings us to the final propositions, 35 and 36. Proposition 36, which requires 35, gives us a way to make perfect numbers. Recall that perfect numbers are those whose factors when added together make up the number. The number 6 is the first of these, since 6 = 1 + 2 + 3. Is it interesting that Euclid ends the number books with this difficult construction of perfect numbers, just as he ends the work as a whole with the construction of the prefect solids. The number of perfect solids is finite. It is still not known whether or not there are infinitely many perfect numbers. To sum up my account of the order within the number books: 1. First we have the fundamentals of how numbers relate to one another as prime or composite and how they relate to one another in ratio and proportion. 3. Next we have continued proportions and how they relate to figured numbers. The view shifts from division, that is, measurement, to multiplication. 4. Next we have truths about the figured numbers themselves, their simple production by multiplication and their special production in continued proportions beginning from the unit. 5. Next we have the production of numbers of various other kinds, culminating in perfect numbers. Part Four: The Ordering of Books in The Elements To what extent may the order of the Elements be attributed to Euclid? Heath gives this answer: Euclid’s own works “show no signs of any claim to be original; in the Elements, for instance, although it is clear that he made great changes, altering the arrangement of whole Books, redistributing propositions between them, and inventing new proofs where the new order made the earlier proofs inapplicable, it is safe to say that he made no more alterations than his own acumen and the latest special investigations...showed to be imperative in order to make the exposition of the whole subject more scientific than the earlier efforts of writers of elements.” If Heath is right, we must take seriously not only the inclusion of the number books in the Elements, but also their place in the work as a whole. We ought to be able to find a rationale according to which Euclid’s order makes sense. As we look into this, it is important to keep in mind that the order appropriate to arithmetic considered within the framework of geometry is not the same as the best order for teaching arithmetic as an art of calculation. We must begin by considering how Euclid structures his treatment of geometry. The subject of geometry is magnitude. Magnitude is divided into extension in one dimension (straight lines), in two dimensions (plane figures) and in three dimensions (solid figures). This division gives structure to Euclid’s work, but not in a simple way. It is not possible to begin with lines and work up to solids or to begin with solids and work down to lines. Euclid does of course begin with lines in one sense. Straight lines and circles are postulated, but they appear in Book 1 only as boundaries or as auxiliary lines within figures. This is not the same thing as a study the one dimensional as such. He does makes such a study but it is difficult and has to be deferred. Solid geometry cannot be studied before plane geometry, since the properties of solids can’t be known without knowing about their boundaries. Solids are the figures most difficult to understand and perhaps also the most beautiful, so it is fitting as well as necessary that they come last in the work. Plane geometry must therefore be treated first. Plane geometry is developed at length in Books 1-4, and this section ends with the inscribing and circumscribing of regular polygons in and about circles. In Book 4, we join together the most perfect plane figures in constructions that are beautiful both to the mind and to the eye. This book brings to completion a distinct part of Euclid’s subject, but the treatment of plane figures is far from being completed. Book 6 deals with the important subject of ratio and proportion in two dimensional figures. This study demands knowledge of ratio and proportion in a more universal way, so the development of plane geometry is necessarily interrupted by Book 5. Book 6 brings to completion the elements of plane geometry. Book 10 is the only book which deals expressly with lines. It is the science of the one dimensional carried on as far as Euclid’s method allows. I have already said that Euclid could not begin with the science of the one dimensional, and now I must explain why not. Extension, whether in one, two or three directions, serves as the material component of geometry. A science first of all considers formal properties of its subject. Since the material causes of a thing are as such unintelligible, they must be made intelligible through form. This is true at every level. Wood, the material of furniture, is known through the formal properties of wood. Elementary particles are known through their forms, and the obscurity we find when we try to understand them comes from the elusiveness of their forms. Prime matter is known only insofar as it is demanded by generation and corruption. In geometry too the forms, which are the shapes, must be understood before we can know the areas and volumes contained by them. Now consider how lines can be known. First we can see that there is a difference between the straight and the curved, and that while there is only one way of being straight, there are infinitely many ways of being curved. Some kinds of curvature are intelligible and some are not, and the geometer is interested in those that are. Euclid deals with curved lines only as boundaries of figures, but Apollonius and other ancient geometers begin to study curved lines as such. But is there a possibility of studying lines as material, that is, in the way in which we study area and volume? Yes. Euclid begins this study in Book 10 and is able to carry it quite far. That the subject of Book 10 is straight lines in particular and not magnitude in general, as in Book 5, is made clear by the words he uses in enunciating the theorems. He even goes so far as to posit a common standard by which all such lines can be classified as either rational or irrational. He comes ever so close to Descartes’ idea of introducing a unit into geometry. It remains to be seen why the study of lines is not a good place to begin. Every species of magnitude is divided by the species below it. In plane geometry, a portions of the plane is delimited by lines, straight or curved, producing triangles, circles and so forth. In solid geometry, the three dimensional continuum is delimited by plane figures and curved surfaces to bring forth cubes, pyramids, spheres etc. These figures are often very beautiful, and in that they approach more closely to the reality of the world, they seem especially worthy of contemplation. The student of the line, however, has as it were a very limited resource. The only way to divide a one dimensional object is to mark off a point or points, to count the divisions and to look at the ratios of the parts, and perhaps their ratios to other lines that have been set out. That sounds pretty boring, on the face of it! The amazing thing is that, far from being barren of interest, there are such riches to be found in the division of the line that we are far from having discovered them all. Although the line is the simplest geometrical object, there are good reasons not to treat it first. Pedagogically it makes sense to treat the more knowable before the less knowable. And whereas commensurable, that is numerable, lines do seem more knowable than figures, the same cannot be said of incommensurable lines. Books 5 and 10 make the greatest demands on the student’s powers of abstraction. The fact that incommensurables were not known to the earliest geometers points to their obscurity. Perhaps also the fact that the imagination is less vividly engaged when studying lines makes the propositions about them more difficult to learn and remember. Although the reasons just given have some force, the more essential reason is mathematical. To divide or measure lines, it is necessary to carry out constructions in two dimensions. Euclid bisects a straight line in Book 1, 10 and in Book 6, 9 he cuts off a prescribed part from a given straight line. In the latter proposition, he shows that any straight line may be measured by any number. He even anticipates irrational divisions of lines in Book 6, 13, where he constructs a mean proportional between any two lines. These illustrate that, in order to measure a line or to show that it cannot be measured by some other line, it is necessary to carry out constructions in the plane. Finally, it is worth mentioning that Book 10 includes some applications of proportion to squares and thus implicitly to other rectilinear figures. This is a hint that the distinction between rational and irrational is applicable to all kinds of extension. For all these reasons, it is necessary for Euclid to put Book 10 after Book 6. But are there also good reasons not to defer the treatment of straight lines until after solid geometry, to the very end of the book? That we should not put Book 10 last can be seen in at least two ways. First, the treatment of incommensurables underlies the argument by approach to a limit and the powerful method of proof by double reductio. Using the latter, Euclid is able to establish the important theorems that circles are in the duplicate ratios of their diameters and spheres are in the triplicate ratio of the same. Thus the treatment of the irrational as well as of the rational are made to serve the purposes of solid geometry. Second, the perfect solids are clearly meant to be the climax of the whole book of the Elements. That there are only five of these is a source of great wonder. The plane figures which bound these figures involve both simple numerical ratios (as that the diameter of the hexagon is twice its side) and irrational ratios, such as that found in the triangle upon which the regular pentagon is built. This could now be known without Book 10. For these reasons alone, it is right for Book 10 to come before Book 11, the first book on solids. To sum up the order I have justified so far: plane geometry in Books 1-4, proportion in magnitude generally and then in plane figures in Books 5-6, rational and irrational lines in Book 10, solid geometry in Books 11-13. All that is missing from this picture are the number books. Where do they fit in? First, we can see the number books as belonging to the study of ratio and proportion, since this is implied in the understanding of number through measurement. Since arithmetic is easier than geometry and prior to it, one might wonder why Euclid does not put these books right before Book 5, so that ratio and proportion are treated first in number and then in magnitude. A simple answer might be that this would unnecessarily interrupt his treatment of plane figures. Another possibility, which would avoid this problem, would have been to begin the entire work with the books on number. After all, the theorems in this part are independent of the geometrical theorems, and starting with arithmetic would have some advantages. Numbers are better known and more accessible to the student than magnitudes. Also, separating off the treatment of multitude from that of magnitude would reinforce the idea that arithmetic is not simply reducible to geometry but is a science in its own right. Despite the fact that arithmetic is an independent science, the number books do not stand alone in the Elements as pure arithmetic. I think they must be seen as integral to Euclid’s approach to geometry. I have spoken at length about how Euclid considers number by means of the idea of measurement. This is not the only possible way to conceive them. For instance, Richard Dedekind writes: “I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding.” Starting with addition rather than with measurement could form the basis of a work on arithmetic and no doubt is the best way to go for treating the art of calculation. Euclid is not interested in calculation, and Dedekind’s way of developing a science of arithmetic departs from the path of the ancients. Euclid seems also not to have been very much interested in the kind of arithmetic one finds in Nicomachus and other ancient writers on the subject. For Euclid, the path to a scientific treatment of arithmetic passes through geometry. Seeing the number books in the context of geometry allows us to understand the placement of the number books. The reason for placing them between 6 and 10 cannot be necessity, since they are independent of the rest. If necessity is not the reason, he must have seen it as appropriate to place them where he does, right before Book 10. This idea is supported by the character of the first few propositions in Book 10. A careful look reveals some parallels between how Books 7 and 10 begin. Proposition 10, 1 has no equivalent in the number books, but Proposition 2 parallels Book 7, Proposition 1. Here is 7, 1: “Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the numbers will be prime to one another.” Compare this to 10, 2: “If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.” This proposition reveals the distinctive character of magnitude, as opposed to multitude. But Euclid goes on to make explicit how multitude can exist in magnitude. Propositions 3 and 4 exactly parallel 7 2 and 3 by finding common measures of two and three commensurable magnitudes respectively. Propositions 5 - 8 nail down the difference between commensurable magnitudes and numbers. He shows that commensurable magnitudes have the ratio of a number to a number and conversely, and that incommensurable magnitudes do not have the ratio of a number to a number, and conversely. From these considerations, we may infer that Euclid wanted us to think about number and magnitude in contrast and comparison to one another. The number books should not be thought of as standing on their own as an independent treatment of arithmetic randomly inserted into a book of geometry. By looking first at number as manifest in lines, and then at lines where number fails to cover all its possibilities, Euclid manifests the necessity to go beyond the flawed geometry of his predecessors to a true and more complete science of magnitude and figure. Here, then, is my account of the order of the Books of the Elements. The first four books contain the elementary propositions about figures that can be known without having to invoke a theory of proportion. Of these, Book 1, which deals with rectilinear figures is the foundation of all the rest. This book ends with the important and beautiful Pythagorean theorem and its converse. Book 2 gives analytical tools for the books to follow. Book 3 deals with properties of circles, the simplest figure after the straight line and next in the order of learning after the basic rectilinear figures. It presupposes book 1. The last three propositions of the book reveal the power of theorems in Book 2 to reveal complicated properties of simple figures. These only hint at what the geometer can figure out with the second book in his tool-kit. Book 4 beautifully brings together the regular polygons, of which the square and the equilateral triangle are the most perfect, with the circle, by inscribing and circumscribing. Book 5 teaches the fundamental theorems about ratio and proportion in magnitude, and this doctrine is applied in Book 6 to reveal many important and beautiful theorems about plane figures. Books 7-9 treat of numbers, while Book 10 deals comprehensively (as much as was possible for Euclid) with lines both rational, which correspond to numbers, and irrational, which do not. Book 11 presents the most fundamental theorems about constructions in three dimensions and with solids formed by planes. Book 12 deals with solids based upon the circle, and with ratios of volume found in various kinds of solids. Finally, Book 13 constructs the five regular solids within a sphere and considers the ratios of their sides to one another. Some of these ratios can be expressed in numbers and some cannot. Euclid shows that the regular solids draw together the rational and the irrational, number and that which cannot be numbered. In this way he brings the elementary study of geometry to a beautiful and fitting conclusion. Elements, Book VII, definition 1. See Aristotle, Metaphysics Book XIII, for an extensive discussion of various ideas about the nature of the unit, of number and of other mathematical objects. Sir Thomas Heath, A History of Greek Mathematics, Vol. 1, p.98, Dover Publications (New York, 1981). To describe this in a neo-Platonic way, one might say that numbers are emanations from the One. Russell’s definition of two as the set of all twos seems to me to have the same problem. St. Thomas Aquinas, Commentary on Metaphysics Books 7-12, 1725 (Augustine Institute, Green Bay, Wisconsin, 2020), my translation: “Est enim per se unum numerus, inquantum ultima unitas dat numero speciem et unitatem,; sicut etiam in rebus compositus ex materia et forma, per formam est aliquid unum, et unitatem et speciem sortitur.” Joe Roberts, Lure of the Integers, The Mathematical Association of America (1992), p. 52. Other measures are of course possible. For example, two, what the Greeks call the dyad, measures all even numbers. In Middle English these sticks were called tallies or tally sticks. 1084b 24-26. In this passage, Aristotle is criticizing their confusion of mathematical conceptions with metaphysical ones. A proposition purporting to show how to know whether 3 numbers have a fourth proportional exists in the Greek, but the text is corrupt. See the note in Densmore’s edition, p. 26. Without high speed computers, it is not possible to find many of these, since we are required to sum up the continued proportion starting from one and based on 2 until the sum becomes prime. A History of Greek Mathematics, Vol. 2, p.357. In Book 10, Euclid treats thoroughly of algebraic irrationals of the second degree, and of a few of degree four. He does not treat of those of the third degree, and he gives no indication of awareness of the transcendental irrationals. “Continuity and Irrational Numbers,” in Essays on the Theory of Numbers, Dover Publications (New York, 1963), p. 4. “Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this procedure be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.” The porism extends the theorem to the subtraction of halves. |Receive lectures and talks via podcast!

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How To Calculate Root Mean Square Error In Matlab Learn more MATLAB and Simulink resources for Arduino, LEGO, and Raspberry Pi test Learn more Discover what MATLAB ® can do for your career. workspace; % Make sure the workspace panel is showing. squaredErrorImage = (double(grayImage) - double(noisyImage)) .^ 2; % Display the squared error image. Mean Square Error Formula In Image Processing Play games and win prizes! Why does Luke ignore Yoda's advice? Matlab Code For Mean Square Error Of Two Images Join the conversation Toggle Main Navigation Log In Products Solutions Academia Support Community Events Contact Us How To Buy Contact Us How To Buy Log In Products Solutions Academia Support Community Author To add an author to your watch list, go to the author's profile page and click on the "Add this author to my watch list" link at the top of Were students "forced to recite 'Allah is the only God'" in Tennessee public schools? Click on the "Add this search to my watch list" link on the search results page. Mean Square Error Matlab Neural Network Close × Select Your Country Choose your country to get translated content where available and see local events and offers. Then just doMSE = mean((desired - mean).^2); 5 Comments Show 2 older comments Maria Maria (view profile) 18 questions 2 answers 0 accepted answers Reputation: 2 on 20 Apr 2014 Direct Other ways to access the newsgroups Use a newsreader through your school, employer, or internet service provider Pay for newsgroup access from a commercial provider Use Google Groups Mathforum.org provides a Matlab Code For Mean Square Error Of Two Images How to create a company culture that cares about information security? set(gcf, 'units','normalized','outerposition',[0 0 1 1]); Maria Maria (view profile) 18 questions 2 answers 0 accepted answers Reputation: 2 on 21 Apr 2014 Direct link to this comment: https://www.mathworks.com/matlabcentral/answers/126373#comment_209202 Dear Mr Image How To Calculate Root Mean Square Error In Matlab You can add tags, authors, threads, and even search results to your watch list. Mean Square Error Formula John Saunders 39.311 προβολές 5:00 Matlab Demo 2 VR - Διάρκεια: 17:04. Darryl Morrell 98.627 προβολές 12:23 U1 - Postlab - RMSE and Correlation notes - Διάρκεια: 6:13. http://threadspodcast.com/mean-square/mean-square-error-calculation-in-excel.html Learn more You're viewing YouTube in Greek. Discover... David Dorran 91.909 προβολές 18:04 how to calculate Mean Square Error in Digital Image Processing - Διάρκεια: 2:37. Immse Matlab jensi asir (view profile) 0 questions 1 answer 0 accepted answers Reputation: 0 Vote0 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/81048#answer_121267 Answer by jensi asir jensi asir (view profile) 0 questions You might also look to PSNR and SSIM (see wikipedia) to compare two matrices. Why planet is not crushed by gravity? http://threadspodcast.com/mean-square/mean-square-error-example-calculation.html subplot(1,2,2); plot(t, y, 'b-', 'LineWidth', 3); grid on; ylim([0, yCenter+radius]); title('Height of a point as it revolves around', 'FontSize', fontSize); xlabel('time', 'FontSize', fontSize); ylabel('Y, or Azimuth', 'FontSize', fontSize); % Enlarge figure mse = sum(sum(squaredErrorImage)) / (rows * columns); % Calculate PSNR (Peak Signal to Noise Ratio) from the MSE according to the formula. How To Calculate Mean Square Error Example Abbasi Nasser M. If X is a matrix of shape NxMxP, sum(X,2) forms a sum over the columns of X, i.e., the SECOND dimension of X, producing a result that has shape Nx1xP. –user85109 Predicted = [1 3 1 4]; How do you evaluate how close Predicted values are to the Actual values? 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If the third number is 3 then either you changed my demo to use a color image (most likely) or else somehow your cameraman.tif image is not the original one. Apply Today MATLAB Academy New to MATLAB? Related Content 1 Answer Wayne King (view profile) 0 questions 2,674 answers 1,085 accepted answers Reputation: 5,360 Vote1 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/69397#answer_80629 Answer by Wayne King Wayne King Discover... Close Was this topic helpful? × Select Your Country Choose your country to get translated content where available and see local events and offers. Opportunities for recent engineering grads.

Notes to Nominalism in Metaphysics 1. There is a third conception of Nominalism, championed by Nelson Goodman, on which it is the doctrine that there is ‘no distinction of entities without distinction of content’, which comes to be the idea that no two distinct entities can be broken down into exactly the same atoms (1972, 159–60). This is different from the two senses of Nominalism distinguished in the main text, since it does not reject universals or abstract objects per se. Clearly some abstract objects and universals must be rejected by Goodmanian Nominalism: sets or classes, structured propositions, structural universals; but the problem with these is not that they are abstract or universal but that they violate Goodman's principle that no two distinct entities can be composed of the same atoms. Though the motivation for Goodman's Nominalism is one of the motivations for the other two kinds of Nominalism distinguished above, namely the ‘aversion to unwonted multiplication of entities’ (Goodman 1972, 159), Goodman's Nominalism is very different from the other two kinds of Nominalism since rather than rejecting a kind of entity what it rejects is a ‘means of construction’ (Goodman 1986, 160) or composition. Why does Goodman call it ‘Nominalism’? He sometimes states his view as the view that the world is a world of individuals (1972, 155) and this sounds nominalist in the sense of rejecting universals. However, for Goodman an individual is simply the value of a variable of the lowest type in a certain system and so a universal could count as an individual for him. Since Goodman´s conception of Nominalism is not widespread I shall not discuss it in this entry. 2. Thus so-called Trope Theory counts as a kind of Nominalism. In Rodriguez-Pereyra 2002 the word ‘Nominalism’ is given a sense according to which it means the rejection of universals and tropes (2002, 3). Although the word is sometimes used in that more restricted sense, it is more correct and more in line with older tradition to use ‘Nominalism’ (in one of its senses) to mean rejection of universals. 3. Lowe's view is not that this is the only proper characterisation of abstract objects, but simply that this is one of them. 4. This is true, for instance, of the Nominalism of Goodman and Quine in their joint 1947 paper, at least as far as non-spatiotemporality is concerned (Goodman and Quine 1947, 105). 5. The main sceptic about such a distinction was F. P. Ramsey (1925). 6. The way I have drawn the distinction between particulars and universals is by no means unproblematic. Firstly, there are problems with the instantiation relation. It is not clear whether it is indeed prior to the distinction between particulars and universals or even exactly what it consists in. Secondly, the distinction entails that although roundness and squareness are universals, (round and square)-ness is not, since given that nothing can be round and square, (round and square)-ness can have no instances. But, if roundness and squareness are universals and (round and square)-ness exists, it is plausible that (round and square)-ness is a universal. Thirdly, the distinction entails that properties that can have only one instance, like the property of being the tallest man ever, count as particulars, but it is not clear that, if there are universals, this property should not be classified as a universal. I do not mean to suggest that these problems are not solvable, nor do I mean to suggest that they are. 7. Another option, exemplified in Armstrong (1997, 118–19), is to accept that a relation of instantiation must be postulated, but to argue that only one is needed, because all the steps of the regress supervene upon the first and supervenience brings no ontological addition. The idea that supervenience brings no ontological addition is controversial. 8. Although close to it, this need not amount to the same as Goodman's principle that no two entities can be broken down into the same entities, for it can be held, as in Lewis 1986b, that composition is the combining of many things into one, and that sets are sums of singletons. In that case the principle that no two things can be composed by exactly the same parts does not rule out singleton sets, since the singleton is not generated through the combining of many things. But Goodman's principle is supposed to rule out singleton sets. 9. Campbell calls tropes ‘abstract’ because he takes as abstract those entities that do not ordinarily exist apart from other instances of qualities, not because he thinks of them as non-spatiotemporal (Campbell 1990, 2–3). 10. Alternatively one might say that properties are predicates or that properties are concepts. But these views seem to have very little appeal, and are subject to the same difficulties as Predicate Nominalism and Concept Nominalism as presented in the main text of this entry. 11. Except for ‘Trope Theory’, all the names of the other nominalist theories in this subsection derive from Armstrong 1978, vol. I, 12–16. 12. By the aggregate of scarlet things it is meant the aggregate of wholly scarlet things. The wholly scarlet parts of partially scarlet things count as wholly scarlet things. 13. No distinction between classes and sets is intended in this entry. 14. In the case of n-place relations, the theory identifies them with classes of ordered n-tuples. 15. It is assumed here that particulars are world-bound, i.e. that none are parts of more than one world, as is the case in Lewis' theory. 16. This does not mean that necessarily every one who believes that sets of spatiotemporally located members are spatiotemporally located is committed to the idea that such sets are concrete, for they might draw the abstract/concrete distinction in a different way than I have done here. But it does mean that they take sets of spatiotemporally located members to be concrete in the sense of this entry. 17. Why ‘particulars and/or properties and/or relations’? Because propositions may have only properties or relations as constituents. An example might be the proposition that scarlet resembles vermillion. But properties and relations, as we saw in a previous section, may be reducible to particulars, in which case every proposition must have particulars as constituents. 18. Note that Quine thought that the relativity of eternal sentences to languages turned them problematic in a way similar to propositions, for Quine thought that the notion of language has no clear and intelligible conditions of identity, and this is what he found problematic about propositions (Quine 1969, 142). 19. As presented in Balaguer 1998, semantic fictionalism is inspired by and similar, but not exactly analogous, to the kind of fictionalism defended by Hartry Field in the Philosophy of Mathematics (1980). In particular, unlike Field's mathematical fitcionalism, Balaguer's semantic fictionalism does not require that reference to abstract objects is dispensable (1998, 811). Balaguer does not assert that semantic fictionalism is true (1998, 810). 20. But sometimes the word ‘Nominalism’ is used in connection to a stance with respect to possible worlds, e.g. in Loux 1998, 176. But Loux uses it in a sense different from the one adopted in this entry. For Loux the ‘possible worlds nominalist’ must believe that other worlds exist and contain only concrete particulars and that by means of such possible worlds and the concrete particulars that populate them is possible to carry out the nominalistic project of providing a reductive account of talk about properties, propositions, and the like (1998, 176). The nominalist about possible worlds, as I shall understand the position here, is committed neither to the claim that such reductive account is possible nor to the claim that possible worlds exist. 21. Note that sometimes Plantinga doesn't use a biconditional but instead says only that a states of affairs S includes (precludes) a states of affairs S* if … (1974: 45, 2003a: 107). But it is quite clear that he is providing definitions (1974: 44), which is why the biconditional is more appropriate, and indeed he uses the biconditional in his 2003b: 194 when introducing the notion of inclusion. 22. Stalnaker takes possible worlds to be properties. But are his possible worlds universals? If it is essential of universals that they can have more than one instance then Stalnaker's possible worlds seem not to be universals, since they can have only one instance. 23. As it is clear from his 1986b, Forrest takes these uninstantiated properties to be universals. 24. Note that here every subset of a world is a world itself, something that does not happen in Adams' account, since on his account the sets of propositions that constitute possible worlds are maximal. 25. Lewis explored four different ways of drawing the abstract/concrete distinction (the Way of Example, the Way of Conflation, the Negative Way, and the Way of Abstraction) and the consequences that these ways had with respect to the concreteness of possible worlds as he took them to be. He concluded that, ‘by and large, and with some doubts in connection with the Way of Example and the Negative Way’, he should say of worlds as he took them to be that they are concrete (1986a, 86). 26. Rodriguez-Pereyra 2004 proposes an alternative modal realist conception of possible worlds as sums or collections of, among other things, pure sets. If this is what possible worlds are it seems that they are not concrete, or at least not purely concrete. 27. These are first order atomic states of affairs. Higher order states of affairs can bring together more than one universal. Armstrong admits higher order states of affairs. 28. But there is an exception to Armstrong's combinatorialism. Strictly speaking not all the possible states of affairs are combinations of the elements of the actual world. Some of them involve alien particulars, that is particulars that do not exist in the actual world (1989, 58). Armstrong rejects, though, alien universals (1989, 55). Thus no possible state of affairs can involve universals not present in the actual world. 29. With a couple of further conditions: that the conjunction in question be a totality of atomic states of affairs and that for every particular figuring in a conjunction of states of affairs that is a possible world there must be at least one state of affairs in that conjunction in which the particular is combined with a monadic universal. See Armstrong (1989, 47–8). 30. That the prefix is called a ‘story prefix’ does not mean that what is mentioned within the prefix must be a work of fiction: it can be any representation whatsoever, whether true or false (Rosen 1990, 331). So it can be a philosophical theory postulating the existence of possible worlds. 31. Clearly modal fictionalism is an option not only for those who believe that possible worlds do not exist, but also for those who simply do not believe that they exist. 32. But the modal fictionalist need not be a nominalist about abstract objects in general, in which case he need not worry about whether theories are abstract or concrete.

Correlation indicates in statistics the strength and direction of a relationship between two or more variables. The correlation is often stated with a correlation coefficient. One method for determining the correlation between two variables is bivariate analysis. The correlation coefficient has a value between 1 and -1, where 0 indicates no relationship, 1 indicates the maximum positive relationship and -1 indicates the maximum negative relationship. Read the post in Swedish here What is correlation? In short, it can be said that correlation is about, whether or not, a couple of variables fluctuate or not. However, it should be remembered that correlation does not have to mean causation. Just because two things react similarly does not mean that one causes what happens to the other. In the economy, we can often see clear connections between, for example, the number of visitors and revenue or between market costs and revenue. See below illustration where the costs are on axis number 2 and have a positive value: The correlation above is as much as 0.9 for cost 1 and only 0.1 for cost 2. COGS or direct manufacturing costs are an example of costs that usually have a high correlation with revenue as the sales price is usually set in relation to direct and indirect manufacturing costs and the margin you want. Promotional costs are another example of a cost that usually follows the revenue curve. What do I use correlation for? I usually use the correlation function; CORREL to the following work: - Financial statements reconciliation - Implementation driver-based forecast - Campaign Analysis Scenario analysis is another area where you may need to understand correlations in a company. There are certainly a lot you can use the function for, but it is mainly for the above work I use the function in Excel. The function in Excel The function is very simple in Excel, start typing CORREL or KORREL (if you have a Swedish setting) and Excel will find the function. Select the first factor you want to compare and click “;” before selecting the comparison factor. Note that an equal amount of observations are not needed for the formula to work, but as many columns in the formula. Financial statements reconciliation Certain costs such as travel costs, car costs and market costs often correlate strongly with sales. You can with advantage use CORREL to help verify which costs are linked to, for example, sales in a company. In the example above, we see that cost C has a correlation of almost -1, which is the highest possible correlation. This says that cost C and income A are highly linked and that June’s value in the financial statements should be around 112 if it is assumed that C normally accounts for 10% of A. To work with correlations in financial statements and when doing cost analysis and revenue analysis is very valuable. Spending time understanding what correlates with what in a company can provide great benefits and is something I usually start with as a consultant if I want to understand what affects what in a company. If you have done your correlation analysis and found a number of costs such as COGS, car costs, market costs, etc. that are related to sales, you can with advantage calculate an index for how to apply the costs in a budget or in a forecast. What can complicate things is that the accounting department may “not always keep up” and CORREL gives a lower correlation than it should. What I mean here is that if the finance department has not received an invoice for a cost that should correlate with sales, and fails to reserve the cost, the correlation will be lower than it should be. You may need to harmonise the cost picture before calculating the correlation. Above you see an example of how you can quickly give a budget proposal for 2021 based on having analysed the correlation and calculated an index or ratio for costs in 2020. The company management here assumes a revenue increase of 10% and it is up to the controlling department to calculate a proposal for remaining costs. What I usually do when I work with the development of budget or forecasting processes is that I look at the correlation and which costs are easy to predict based on, for example, sales or number of employees. Drivers are the most important efforts and activities that drive a business’s operational and financial results. Common examples are number of sellers, number of stores, website traffic, marketing campaigns, production units. Other large costs that do not correlate with anything, I usually distinguish and later break down into activities and forecast separately. When I worked at Apoteksgruppen, I did a lot of campaign analysis, which was incredibly fun. Trying to discern patterns and understand the impact of campaigns is not always easy. What I analysed was not just how the campaign and sales were connected, but also how long the campaign effect lasted after the campaign was over and whether they jumped up a notch regarding normal sales. The 2 examples below show that different offers work differently and affect sales differently. Test if you can figure out which offer affects total sales the most in the 2 cases and if you can discern an improvement in total sales thanks to the campaign? Product A shows that the 25% discount correlates to 1.0 with total sales and only 0.2 with the offer 3 for 2. The regular sales increase significantly during the campaigns and it is not only the discounted sales that contribute to increased total sales. However, the trend is steady and not rising or falling. The conclusion is that if you do not run a campaign, sales would not be so high, however, it can affect profitability with extensive campaigns. In Product B’s case, it is the reverse where “3 for 2” correlates well with sales, which has a rising trend and peaks during the campaign. One can assume here that campaign can be important in the event of a gradual increase in trend, which is what one wants to achieve. In this post, I have shown that the CORREL function is great for reconciliations and forecasting, but that it can also be used for other analysis. Check out one of my other posts, where I show how to easily make a professional doughnut chart and when I usually use it. Carl Stiller in collaboration with Learnesy

Polygons are planes figures formed by a closed series of rectilinear segments. Ex– Triangle, Rectangle etc. 1. Sum of all the angles of a polygon with n sides = (n-2)π 2. Sum of all exterior angles = 360° 3. No. of sides = 360°/exterior angle Classification of polygons – A triangle is a polygon having three sides. 1. Area = 1/2 x base x height 2. Area = √s(s-a)(s-b)(s-c) where s = a+b+c/2 3. Area = rs (where r is in-radius) 4. Area = 1/2 x product of two sides x sine of angle 5. Area = abc/4R where R = circumradius Congruency of Triangles: 1. SAS congruency: If two sides and an included angle of one triangle are equal to two sides and an included angle of another, the two triangles are congruent. 2. ASA congruency: If two angles and the included side of one triangle is equal to two angles and the included side of another, the triangles are congruent. 3. AAS congruency: If two angles and side opposite to one of the angles is equal to the corresponding angles and sides of another triangle, the triangles are congruent. 4. SSS congruency: If three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent. 5. SSA congruency: If two sides and the angle opposite the greater side of one triangle are equal to the two sides and the angle opposite to the greater side of another triangle, then triangle are congruent. Similarity of Triangles: 1. AAA similarity: If in two triangles, corresponding angles are equal, then the triangles are similar. 2. SSS similarity: If the corresponding sides of two triangles are proportional then they are similar. 3. SAS similarity: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar. 1. Height = a√3/2 2. Area = √3a2/4 3. R(circum radius) = 2h/3 = a/√3 4. r(in radius) = h/3 = a/2√3 5. In equilateral triangle orthocenter, in-centre, circumcenter and centroid coincide. Area = b/4√(4a2 – b2) where b=base and a=equal sides 1. Median: A line joining the mid-point of a side of a triangle to the opposite vertex is called a radian. - A median divides a triangle in two parts of equal area. - The point where three medians meet is called centroid of the triangle. - The centroid of a triangle divides each median in ratio 2:1. 2. Altitude: A perpendicular drawn from any vertex to the opposite side is called the altitude. - The point where all altitudes meet at a point is called the orthocenter of triangle. 3. Perpendicular bisector: A line that is a perpendicular to a side and bisects it is the perpendicular bisector of the side. - The point at which perpendicular bisectors of the sides meet is called the circumcenter. - The circumcenter is the centre of the circle that circumscribes the triangle. - The lines bisecting the interior angles of a triangle are the angle bisectors of that triangle. - The angle bisectors meet at a point called the incentre. - The angle formed by any side at the incentre is always 90° more than the half of angle opposite to the side. 1. Length of direct common tangents is = √[(Distance between their centres)2-(r1 – r2)2] = √[(O1O2)2 – (r1 – r2)2] 2. Length of traverse common tangents is = √[(Distance between their centres)2-(r1 + r2)2] = √[(O1O2)2 – (r1 + r2)2] Question 1: If each interior angle of a regular polygon is 108°. The number of sides of the polygon is Solution : Interior angle = 108° Exterior angle = 180 – 108 = 72 Number of sides of polygon = 360° /exterior angle = 360° /72 Questions 2: The ratio of angles of triangle is 2:3:5. Find the smallest angle of the triangle. Solution : Ratio of angles 2:3:5 then 2x + 3x + 5x = 180 10x = 180 x = 18 Hence, the smallest angle = 18×2 = 36° Question 3: Two medians AD and BE of ∆ABC intersect O at right angle. If AD = 9cm and BE = 6cm, then the length of BD is O is the centroid which divides the median in 2:1. So, AO:OD = 2:1 AD = 3 units -> 9 cm 1 unit -> 3 cm So, OD = 3 cm BE = 3 units -> 6cm So, BO = 4 cm ∆BOD is a right angled triangle. BD2 = BO2 + OD2 BD2 = (4)2 + (3)2 BD2 = 16 + 9 = 25 BD = 5 cm Question 4: The side AB of a parallelogram ABCD is produced to E in such a way that BE = AB, DE intersects BC at Q. The point Q divides BC in the ratio Solution : Acc. to question AD || BC and AB || DC ∠1 = ∠2 (Corresponding alternate angle) ∠3 = ∠4 (Corresponding alternate angle) and ∠BEQ is common By AAA property both are similar ∆EQB ∼ ∆EDA So, EB/EA = EQ/ED = QB/AD AD=BC & EA = 2EB then 1/2 = QB/BC => BQ = QC Hence, Q divides BC in ratio 1:1. Question 5: In a ∆ABC, AB=AC and BA is produced to D such that AC=AD. Then the ∠BCD is Solution :Acc. to question ABC is an isosceles triangle. => ∠C = ∠B = θ => ∠CAD = ∠C + ∠B = 2θ (An exterior angle of a triangle is equal to the sum of the opposite interior angles.) AC=AD So, ∆ADC is also an isosceles triangle. In ∆ADC, ∠A + ∠C + ∠D = 180° 2∠C = 180° – 2θ (∠C = ∠D) ∠C = 90° – θ ∠BCD = θ + 90° – θ ∠BCD = 90° Question 6: If O is the circumcenter of ∆PQR, and ∠QOR = 110°, ∠OPR= 25°, then the angle ∠PRQ is If O is the circumcenter then OP=OR=OQ. ∠OPR = 25° then ∠PRO = 25° ∠OQR + ∠ORQ + ∠QOR = 180° 2∠ORQ = 180° – 110° ∠ORQ = 35° So, ∠PRQ = ∠PRO + ∠ORQ = 25° + 35° Question 7: In ∆ABC, DE || AC, D and E are two points on AB and CB respectively. If AB=20 cm and AD = 8 cm, then BE : CE is AB = 20 cm and AD = 8 cm DE || AC then, ∠A = ∠D and ∠C = ∠E ∠B is common By AAA property, ∆ABC ∼ ∆DBE therefore BD/AD = BE/CE BE/CE = 12/8 BE/CE = 3/2 Hence, BE : CE = 3:2 Question 8: Angle between the internal bisectors of two angles of a triangle ∠B and ∠C is 110°, then ∠A is Internal bisectors of angles intersect each other at Incentre. ∠BIC = 110° The angle formed by any side at the incentre is always 90° more than the half of angle opposite to the side.So, ∠BIC = 90° + 1/2∠A 1/2∠A = 110° – 90° ∠A = 20×2 = 40° Question 9: The distance between two parallel chords of length 8 cm and each in a circle of diameter 10cm is AB = CD = 8 cm radius = D/2 = 10/2 = 5 cm OB2=OM2 + MB2 52 = OM2 + 42 OM2 = 25 – 16 OM = 3 cm MN = 2 x OM = 2 x 3 = 6 cm Question 10: The radius of two concentric circles are 12cm and 13cm. If the chord of the greater circle be a tangent to the smaller circle, then the length of that chord is: Solution : Acc. to Question AO = 13 cm and OD = 12 cm AO2= DO2 + AD2 132 = 122 + AD2 AD2 = 169 – 144 AD = 5cm AB = 2xAD = 10 cm Questions 11: Two tangents are drawn at the extremities of diameter AB of a circle with centre O. If a tangent to the circle at the point C intersects the other two tangents at Q and R, then the measure of the ∠QOR is Solution : Acc. to question In ∆OCR and ∆RBO OC = OB (radius) RC = RB (tangent from same point) PR is common By SSS property both are congruent ∆OCR ≅ ∆RBO Similarly they are also congruent ∆OCQ ≅ ∆QAO Then ∠COR = ∠ROB = x and ∠AOQ = ∠COQ = y 2x + 2y = 180° x + y = 90° ∠QOR = 90° Question 12: Two equal circles whose centres are O and O’ intersect each other at the point A and B, OO’= 24 cm and AB = 10 cm then the radius of the circle is AB = 10 cm AC = BC = 5 cm OC = CO’ = 12 cm In right angled triangle ∆ACO OA2 = OC2 + AC2 OA2 = 122 + 52 OA2 = 144 + 25 OA = 13 cm Question 13: The distance between the centers of two circles of radii 6 cm and 3 cm is 15 cm. The length of the traverse common tangent to the circles is: Solution : Length of traverse common tangent = √[(Distance between their centres)2-(r1 + r2)2] =√[(15)2-(6 + 3)2] =√(225 – 81) = 12 cm Question 14: If the distance between two points (0, -5) and (x, 0) is 13 units, then the value of x is: Solution :We know that (Distance)2 =[(x2-x1)2 + (y2 – y1)2] (13)2 = [(x2-0)2 + (0 – (-5) )2] 169 = x2 + 25 x = 12 units Whether you're preparing for your first job interview or aiming to upskill in this ever-evolving tech landscape, GeeksforGeeks Courses are your key to success. We provide top-quality content at affordable prices, all geared towards accelerating your growth in a time-bound manner. Join the millions we've already empowered, and we're here to do the same for you. Don't miss out - check it out now!

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Fermat great theorem Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem The assertion that for any natural number the equation (the Fermat equation) has no solution in non-zero integers . It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika by Diophantus as follows: "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" . And he then added: "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain" . A proof of the theorem for was found in Fermat's papers. No general proof has so far been obtained (1984), despite the efforts of many mathematicians (both professional and amateur). An unhealthy interest in proving this theorem was stimulated at one time by a large international prize, which was abolished at the end of the First World War. It has been conjectured that there is no proof of Fermat's last theorem at all. For the theorem was proved by L. Euler (1770), for by P.G.L. Dirichlet and A. Legendre (1825), and for by G. Lamé (1839) (see ). It is sufficient to prove the theorem for and for every prime exponent , that is, it is enough to prove that the equation has no solutions in non-zero relatively-prime integers . One can also assume that and are relatively prime to . For a proof of Fermat's theorem one considers two cases: case 1 when , and case 2 when . The proof of the second case is more difficult and is usually carried out by the method of infinite descent. An important contribution to proving Fermat's last theorem was made by E. Kummer, who invented a fundamentally new method based on his development of the arithmetic theory of a cyclotomic field. It makes use of the fact that in the field , , the left-hand side of equation (1) splits into linear factors , which are -th powers of ideal numbers (cf. Ideal number) in in case 1 and differ from -th powers by a factor , , in case 2. If divides the numerators of the Bernoulli numbers (), then by the regularity criterion does not divide the class number of and these ideal numbers are principal. Kummer proved Fermat's theorem in this case. It is not known whether the number of regular numbers is infinite or finite (by Jensen's theorem the number of irregular prime numbers is infinite ). Kummer proved the theorem for some irregular prime numbers and also established its validity for all . In case 1 he showed that (1) implies the congruences which are valid for any permutation of . Hence he obtained that if equation (1) has a solution in case 1, then for , In case 2 Kummer proved Fermat's theorem under the following conditions: 1) , , where is the first factor of the class number of (this is equivalent to the requirement that only one of the numerators of the numbers , where , is divisible by ); 2) (); and 3) there is an ideal modulo which the unit is not congruent to the -th power of an integer in , where is a primitive root modulo and Kummer's method has been widely developed in several articles on Fermat's last theorem (see , ). It has been established that (2) holds if (1) does in case 1 for , 9, 11, 13, 15, 17, 19. Under the same conditions M. Krasner showed that there is a number such that for (2) is true for all numbers , where . H. Brückner showed that the amount of numbers , , with numerators divisible by is greater than . Suppose that , . P. Remorov showed that there are constants and , , such that for all , , case 1 of the Fermat theorem is true. M. Eichler established that case 1 is true for , where is the index of irregularity of , . H. Vandiver proved case 1 for , where is the second factor of the class number of . He obtained interesting results on case 2 in and . For example, he showed that the Fermat theorem is true under the following conditions: 1) ; and 2) (), . The following theorem is most important: Let be an irregular prime number and let be the indices of the Bernoulli numbers among with numerators divisible by ; if none of the units () is congruent to the -th power of an integer in modulo , where is the prime ideal dividing a prime number with (), then Fermat's theorem is true. From this Vandiver obtained an effectively-verifiable criterion for irregular prime numbers by means of which the Fermat theorem has been proved on a computer for all (see ). There are various results on case 1 of Fermat's last theorem. As early as 1823 Legendre published a result of S. Germain: If there is a prime number such that the congruence () has no integer solutions not divisible by , and is not a -th power residue modulo , then case 1 of the Fermat theorem holds (see ). Hence he showed that if at least one of the numbers , (), , is prime, then case 1 holds. This proposition has been extended to all . A. Wieferich discovered the following criterion: If , where is the Fermat quotient, then case 1 is true. D. Mirimanoff proved this for . Subsequently, case 1 was established by a number of other authors for all for which , where is any prime number . From this the first case of Fermat's theorem follows for , where contain only prime numbers in their prime factorizations. Calculations on a computer showed that among the numbers only two: and satisfy the condition , but for these . This proves case 1 for all . P. Furtwängler gave fairly simple new proofs of the results of Wieferich and Mirimanoff based on Eisenstein's reciprocity law. He also proved that if is a solution of (1) and , then , where but , or but , or but . A great variety of other criteria are known for case 1 of the Fermat theorem. They are connected with the solvability of certain congruences or with the existence of prime numbers of a certain form. The equation is not valid if divides neither nor (see ). It is impossible in practice to produce a counterexample to Fermat's last theorem. K. Inkeri showed that if the integers , , satisfy (1), then , and in case 1: . Fermat's last theorem can be stated as follows: For every natural number there are no rational points on the Fermat curve except the trivial ones, and . Rational points on the Fermat curve have been studied by methods of algebraic geometry. By these methods it has been proved (1983) that the number of rational points on the Fermat curve is finite in every case. This follows from the Mordell conjecture, which was proved by G. Faltings . D.R. Heath-Brown has shown, using the Mordell conjecture, that Fermat's last theorem holds for almost-all primes , cf. . Also, by methods of analytic number theory, L.M. Adleman, Foury and Heath-Brown have shown that case 1 holds for infinitely many primes , cf. . One can look at Fermat's equation in algebraic integers, entire functions, matrices, etc. There is a generalization of Fermat's theorem for equations of the form . |||Diophantus of Alexandria, "Aritmetika and the book on polygonal numbers" , Moscow (1974) (In Russian; translated from Greek)| |||H.M. Edwards, "Fermat's last theorem. A genetic introduction to algebraic number theory" , Springer (1977)| |[3a]||E. Kummer, "Bestimmung der Anzahl nicht äquivalenter Classen für die aus ten Wurzeln der Einheit gebildeten complexen Zahlen" J. Reine Angew. Math. , 40 (1850) pp. 93–116| |[3b]||E. Kummer, "Zwei besondere Untersuchungen über die Classen-Anzahl und über die Einheiten der aus ten Wurzeln der Einheit gebildeten complexen Zahlen" J. Reine Angew. Math. , 40 (1850) pp. 117–129| |[3c]||E. Kummer, "Allgemeiner Beweis des Fermatschen Satzes, dass die Gleichung $x^\lambda+y^\lambda=z^\lambda$ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten $\lambda$, welche ungerade Primzahlen sind und in den Zählern der ersten $\frac12(\lambda-3)$ Bernoullischen Zahlen als Factoren nicht vorkommen" J. Reine Angew. Math. , 40 (1950) pp. 130–138| |||Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902| |||E. Kummer, "Einige Sätze über die aus den Wurzeln der Gleichung gebildeten complexen Zahlen, für den Fall, dass die Klassenanzahl durch teilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes" Abh. Akad. Wiss. Berlin, Math. Kl. (1857) pp. 41–74| |||H. Vandiver, "Fermat's last theorem" Amer. Math. Monthly , 53 (1946) pp. 555–578 Zbl 52.0161.13| |||P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979)| |||M. Krasner, "Sur le premier cas du théorème de Fermat" C.R. Acad. Sci. Paris , 199 (1934) pp. 256–258 Zbl 0010.00702 Zbl 60.0129.01| |||H. Brückner, "Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudoreguläre Primzahlen $l$" J. Reine Angew. Math. , 253 (1972) pp. 15–18| |||P.N. Remorov, "On Kummer's theorem" Uchen. Zap. Leningrad. Gosudarstv. Univ. Ser. Mat. Nauk , 144 : 23 (1952) pp. 26–34 (In Russian) MR81310| |||M. Eichler, "Eine Bemerkung zur Fermatschen Vermutung" Acta Arith. , 11 (1965) pp. 129–131 MR0182607 Zbl 0135.09401| |||H. Vandiver, "Fermat's last theorem and the second factor in the cyclotomic class number" Bull. Amer. Math. Soc. , 40 (1934) pp. 118–126 MR1562807| |||H. Vandiver, "On Fermat's last theorem" Trans. Amer. Math. Soc. , 31 (1929) pp. 613–642 MR1501503| |||H.S. Vandiver, "Examination of methods of attack on the second case of Fermat's last theorem" Proc. Nat. Acad. Sci. USA , 40 : 8 (1954) pp. 732–735 MR62758| |||S. Wagstaff, "The irregular primes to 125.000" Math. Comp. , 32 (1978) pp. 583–591 MR491465| |||A. Wieferich, "Zum letzten Fermatschen Theorem" J. Reine Angew. Math. , 136 (1909) pp. 293–302| |||D. Mirimanoff, "Zum letzten Fermatschen Theorem" J. Reine Angew. Math. , 139 (1911) pp. 309–324| |||D.H. Lehmer, "On Fermat's quotient, base 2" Math. Comp. , 36 (1981) pp. 289–290 MR595064| |||P. Furtwängler, "Letzter Fermat'scher Satz und Eisenstein'sches Reziprozitätsprinzip" Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl. IIa , 121 (1912) pp. 589–592| |||G. Terjanian, "Sur l'equation " C.R. Acad. Sci. Paris , A285-B285 : 16 (1977) pp. 973A-975A (English abstract) MR0498370| |||K. Inkeri, "Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen Problem" Ann. Univ. Turku Ser. A , 16 : 1 (1953) pp. 3–9 MR0058629 Zbl 0051.28003| |||M.M. Postnikov, "An introduction to algebraic number theory" , Moscow (1982) (In Russian)| |||G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366| |||D.R. Heath-Brown, "Fermat's last theorem for "almost all" exponents" Bull. London Math. Soc. , 17 (1985) pp. 15–16| |||L.M. Adleman, D.R. Heath-Brown, "The first case of Fermat's last theorem" Invent. Math. , 79 (1985) pp. 409–416 MR778135| In fact, Heath-Brown and, independently, A. Granville, cf. [a1], have proved that the density of the exponents for which Fermat's last theorem holds is one. It is now (1988) known that Fermat's last theorem holds for all $n < 150000$, and that case 1 holds for all primes up to , cf. [a2]. Recently (1987), K. Ribet, using ideas of G. Frey and J.-P. Serre, showed that Fermat's last theorem is implied by the Weil–Taniyama conjecture in the theory of elliptic curves (cf. Elliptic curve). |[a1]||S. Wagon, "Fermat's last theorem" Math. Intelligencer , 8 : 1 (1986) pp. 59–61 MR823221| |[a2]||P. Ribenboim, "Recent results about Fermat's last theorem" Cand. Math. Bull. , 20 (1977) pp. 229–242 MR463088| Fermat great theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_great_theorem&oldid=53456

Exterior angle of a triangle is equal to - Sep 22, 2008 · Exterior angle. = 180° - interior angle adjacent to it. = sum of two remote interior angles. aravindma1990. 1 decade ago. consider the three angles as A,B,C. A+B+C=180. now extend one of the sides... - Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. - 2.Cut out each exterior angle and label them 1-6. 3.Fit the six angles together by putting their vertices together. What happens? The angles all fit around a point, meaning that the exterior angles of a hexagon add up to 360˚, just like a triangle. We can say this is true for all polygons. • Exterior Angle Sum Theorem: The sum of the ... - An exterior angle of a triangle is equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees. ⇒ c + d = 180° ⇒ a + f = 180° - If one angle of a triangle is equal to one angle of the other triangle and the sides including these angle are proportional, then the two triangle are similar.prove that asked Aug 17, 2019 in Class X Maths by Sachin Tiwari ( 15 points) - Acute triangle: a triangle with three acute angles Obtuse triangle: a triangle with one obtuse angle Right triangle: a triangle with one right angle Exercises True or False: Give a reason or counterexample to justify your response. 1. An equilateral triangle is always acute. 2. An obtuse triangle can also be isosceles. 3. The acute angles of a ... - - Interior angles between parallel lines adds up to 180 degrees. 3) The sum of the 3 angles of a triangle will always be 180 degrees. 4) The exterior angle sum of any polygon will always be 360 degrees. Powered by Create your own unique website with customizable templates. - Embrilliance essentials software free download - An exterior angle must form a linear pair with an interior angle. This means that the exterior angle must be adjacent to an interior angle (right next to it - they must share a side) and the interior and exterior angles form a straight line (180 degrees). If you extend one of the sides of the triangle, it will form an exterior angle. - Angles 3 and 6 are also interior angles but are not congruent. Angles 2 and 7 are alternate exterior angles and are congruent. Angles 1 and 8 are also alternate exterior angles but are not congruent. Triangles: The three angles of a triangle always total 180 o. An equilateral triangle is a triangle with 3 equal sides and all 3 angles are 60 o. - of exterior ZA, 10. Calculate the angles of a triangle ABC having 34B = 4ZC and the interior ZA = caculate angles of triangle abc having 3 times angle b= 4 times angle c and interior angle a =4/7 of its exterior angle The two angles of a triangle are equal and third measures 70degree . - • Each exterior angle of a triangle is the sum of the opposite interior angles. (*) (e = a + b). Abbreviation: ext. ∠ of ∆. ea b. Section 4.1 unknown angle proofs • 75. • Base angles of an isosceles triangle are equal. (*) (If AC = BC then a = b.) Abbreviation: base ∠s of isos. ∆. C Aa bB. - To triangulate something means either to divide into triangles or affix an object's location via triangles. Fundamentally they are based on the The measure of an exterior triangle angle is equal to the sum of the measures of the interior angles at the other two vertices (Exterior Angle Theorem). - An isosceles triangle has two angles equal in size. In this problem A is greater than B therefore angles B and C are equal in size. Since angle A is 30 greater than angle B then A = B + 30 o. The sum of all angles in a triangle is equal to 180 o. (B+30) + B + B = 180. Solve the above equation for B. B = 50... - An exterior angle of a triangle is formed when one side of a triangle is extended. Theorem 26: An exterior angle of a triangle is equal to the sum of the two remote (nonadjacent) interior angles. Example 1: In Figure 1, if m ∠1 = 30° and m ∠2 = 100°, find m ∠4. - Theorem 06 - The Exterior Angle of a Triangle. Author: SeamusMcCabe. This demonstrates that an exterior angle in a triangle is always equal to the sum of the two interior opposite angles. Drag the points to change the shape of the triangle. Although the angles change size, the exterior angle is always equal to the sum of the two interior opposite angles. - Use the angle sum of a triangle to prove that the angle sum of a quadrilateral is 360°. (a + b + c) + (d + e + f) = 180° + 180° = 360° Explain a proof that the exterior angle of a triangle equals the sum of the two interior opposite angles, using this or another construction. Given sufficient information, calculate: • interior and ... - Angles In A Triangle. A triangle is the simplest possible polygon. It is a two-dimensional (flat) shape with three straight sides forming an interior, closed Angles In A Triangle. How To Find The Angle of a Triangle. By the Alternate Interior Angles Theorem, we know that. ∠a. is congruent (equal) to. Google minesweeper full screenExterior angles of a triangle supplementary angles (equal 180°). and, The sum of the three interior angles in a triangle is always 180°. x + 24 + 34 = 180 x + 58 = 180 x = 122 then, 180 - 122 = 58 Also, the exterior angle is equal to the sum of the two remote interior angles. 24 + 34 = 58 Exterior Angle Theorem The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle. Let's try two example problems. - (b) the exterior angle of a triangle is equal to the sum of the interior opposite angles. Take any triangle ABC. Construct a line through C, parallel to AB. angle p = angle b (corresponding angles) angle s = angle a (alternate angles) Therefore angle p + angle s = angle a + angle b. but angle r = angle a + angle b. Therefore angle p + angle s = angle r. Properties of quadrilaterals . Sum of angles in a quadrilateral - Angles opposite to equal side lengths are equal. 3) A scalene triangle is a triangle with all three side of unequal length. 4) An acute triangle is a triangle in which all three angles are . 5) A right triangle is a triangle with a angle. 6) An obtuse triangle is a triangle with one of the angles . Basic Properties of Angles. 1) When two ... - Triangle Angle-Sum Theorem 3. Exterior Angle Theorem 4. Exterior Angle Inequality G. Two-dimensional figures. 1. Polygon vocabulary 2. Interior angles of polygons ... Wheezy loop packVetprofen for cats Industry based considerations for diversification strategy includes all of the following exceptSentry safe handle stuck The acute angles of a righttriangle are complementary. 7. The measure of an exterior angleof a triangle is equal to the sum of the measures of its remote interior angles. Find the measure of each angle. Set default python centos 8Jet city turbines FACTS: • Every triangle has 6 exterior angles, two at each vertex. • Angles 1 through 6 are exterior angles. • Notice that the "outside" angles that are "vertical" to the angles inside the triangle are NOT called exterior angles of a triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Kbd67 plateMbe 906 mercedes benz diesel Aug 26, 2015 · An exterior angle is equal to 180° less the size of the interior angle. So an exterior angle plus an interior angle is equal to 180°. For a triangle, there are three angles, so the sum of all the interior and exterior angles is 180° x 3 = 540°. A more perfect union worksheet answer keyQcow2 images Exterior angles of a triangle and finding missing angles, sum of exterior angles, in video lessons with examples and step-by-step solutions. The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles. I need an urgent spell casterNvidia control panel override the scaling mode set by games and programs 3. There are six exterior angles of a triangle. 4. Interior angle + corresponding exterior angle = 180. 3. If any one side is produced, Exterior angle = Remote interior angle 4. If one pair of opposite sides are equal, diagonals are equal 5. The line joining the points of intersection of the bisectors of...

SOLVING EXPONENTIAL AND LOGARITHMS WORD PROBLEM Including results for solving exponential and logarithmic word problem.Do you want results only for solving exponential and logarithms word problem? Exponential model word problems (practice) | Khan Academy Solve word problems about exponential situations. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *atic and *ndbox are unblocked.Medication Dissolve · Bacteria Growth Logarithmic Word Problems - Purplemath | Home Logarithmic Word Problems (page 1 of 3) Sections: Log-based word problems, exponential-based word problems Logarithmic word problems, in my experience, generally involve evaluating a given logarithmic equation at a given point, and solving for a given variable; they're pretty straightforward. SOLVING WORD PROBLEMS - sosmath SOLVING WORD PROBLEMS APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation. We are going to discuss several types of word problems. Click on the one that you want to Videos of solving exponential and logarithms word problem Click to view on YouTube4:23Solving Logarithmic Equations, Word Problems13K viewsYouTube · 6 years agoClick to view on YouTube13:38Compound Interest & Population Growth Word Problems - Logarithms9 viewsYouTube · 1 year agoClick to view on YouTube7:22Exponential growth and decay word problems | Algebra II | Khan Academy704K viewsYouTube · 9 years agoSee more videos of solving exponential and logarithms word problem APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC - S.O.S. LOGARITHMIC FUNCTIONS (Interest Rate Word Problems) 1. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation. Example 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years? Algebra - Solving Exponential Equations (Practice Problems) Here is a set of practice problems to accompany the Solving Exponential Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University.[PDF] Sample Exponential and Logarithm Problems 1 Exponential Sample Exponential and Logarithm Problems 1 Exponential Problems Example 1.1 Solve 1 6 3x 2 = 36x+1. Solution: Note that 1 6 = 6 1 and 36 = 62. Therefore the equation can be written (6 1) 3x 2 = (62)x+1 Using the power of a power property of exponential functions, we can multiply the exponents: 63x+2 = 62x+2 But we know the exponential function Exponential growth & decay word problems - Khan Academy Let's do a couple of word problems dealing with exponential growth and decay. So this first problem, suppose a radioactive substance decays at a rate of 3.5% per hour. What percent of the substance is left after 6 hours? So let's make a little table here, to just imagine what's going on. And then we Logarithmic Functions – She Loves Math Solving Exponential Equations using Logs. Now we can use all these tools to solve log equations! Remember again that math is just using tools that you have to learn to solve problems. Remember to always check your answer to make sure the argument of logs (what’s directly following the log) is positive! Let’s just jump in and trying somePeople also askWhat is the log function?What is the log function?LOG function. The LOG function computes the logarithm of an expression. base. The base by which to compute the logarithm. When you do not specify a value,the function computes the natural logarithm of the expression by using e for the base where e is equal to 281828459.LOG function - docsleSee all results for this questionAre logarithms and exponents the same?Are logarithms and exponents the same?Since exponentsand logarithmsare two versions of the samemathematical concept, exponentscan be converted to logarithms, or logs. An exponentis a superscript number attached to a value, indicating how many times the value is multiplied by itself. The log is based on exponentialpowers, and is just a rearrangement of terms.How to Convert Exponents to Logs | SciencingSee all results for this questionWhat are some examples of exponential equations?What are some examples of exponential equations?Exponential equations are equations in which variables occur as exponents. For example,exponential equations are in the form a x = b y. To solve exponential equations with same base,use the property of equality of exponential functions . If b is a positive number other than 1 ,then b x = b y if and only if x = y .Solving Exponential Equations - Varsity TutorsSee all results for this questionWhat is an exponential formula?What is an exponential formula?Exponential Functions. For example,y = 2 x would be an exponential function. Here's what that looks like. The formula for an exponential function is y = abx,where a and b are constants. You can see that this conforms to the basic pattern of a function,where you plug in some value of x and get out some value of y.What Is an Exponential Function? - Video & LessonSee all results for this question Related searches for solving exponential and logarithmic word solving exponential equations word problemsexponential and logarithmic functions problemsword problem logarithmic equationsexponential equation word problemsexponential and logarithmic equations solverexponential word problems with solutionslogarithmic word problems pdflogarithmic word problems practiceIncluding results for solving exponential and logarithmic word problem.Do you want results only for solving exponential and logarithms word problem?

Thread Number: 84853 / Tag: Small Appliances how to clean baked on black gunk? |[Down to Last]| |Post# 1093204   10/14/2020 at 21:25 by gizmo (Great Ocean Road, Victoria, Au)  || | I have recently scrounged a Philips Air Fryer XXL. It is not very old, still more or less a current model and a fancy one, too. It has no faults but was really filthy when I picked it up. I have pulled it to bits, cleaned it and checked it, it works fine and is a good gadget. I will keep it. I suspect it was thrown out because of smoke when cooking - the area around the fan was full of vile brown goo. The area around the fan and heating element have responded to elbow grease and the selection of chemicals I threw at it. They are now clean. This model has a basket made of sheet metal sides, covered in some nonstick stuff. The base of the basket is separate, made of stainless steel mesh. That was almost blocked with black residue but is now clean. Being stainless, I could get stuck into it with scourers and sharp implements, without damaging it. The nonstick basket is different. It is still covered with really hard baked black stuff. Some of it has flaked off but most is stuck hard to the nonstick surface. I'm usually good at this sort of thing but I'm stumped. So far I have tried BBQ cleaner from Aldi - useless. Some sort of spray on kitchen degreaser - useless. Simple Green - little to no effect. Wet with cloudy ammonia and leave in a plastic bag overnight - removed the easy stuff. Citro-Clean orange oil based spray - removed the softer brown stuff but little to no effect on the hard black stuff. It is usually magic - I love the stuff. Baking soda paste - probably the most effective. It seemed to soften some of the muck but I have done 2 or 3 applications, each left overnight. The remaining residue won't shift. I later tried a spray on oven cleaner, the kind of product I usually try to avoid. I had imagined it would dissolve the black layer and I'd wipe it off next day... no such luck. It had no effect at all, after 2 applications, each left overnight. I have had a little luck using a bamboo chopstick to scrape at the residue - some of it flakes off but it is slow work and I don't want to damage the nonstick any more than i have already. If you could mention generic types of product you suggest, or specific ingredients, rather than brand names, would be helpful as we probably have different product brands in Australia. my final option is to replace the basket, link below. but I'd rather clean this one if I can. CLICK HERE TO GO TO gizmo's LINK |Post# 1093207 , Reply# 1   10/14/2020 at 22:02 by CircleW (NE Cincinnati OH area)  || | If you have a restaurant supply place near you, they should have a carbonized grease remover, such as Carbon-Off. |Post# 1093209 , Reply# 2   10/14/2020 at 22:14 by gizmo (Great Ocean Road, Victoria, Au)  || | Thanks, sounds like what I want. I used to be a professional cook but kept my stuff clean and never had to resort to such products... |Post# 1094334 , Reply# 3   10/24/2020 at 10:54 by Helicaldrive (St. Louis)  || | Soak in very hot sudsy water with as much ammonia as you dare, but not enough to create uncomfortable fumes, then use a Mr Clean Magic Eraser pad. Thatís what Iíd do.... (And set it outside to soak, or cover the sink with a big towel while soaking, if you have any pets, especially birds.) And with Thanksgiving coming up, might as well mention that thatís my method for getting my stainless steel roaster brand new, gleaming clean. 30 minutes is usually long enough to soak. And now I usually no longer need to use Bon Ami scouring powder. Just enough hot sudsy water to cover, and it doesnít take much ammonia, just a glug. (OMG one time I poured straight ammonia all over the shower in my elderly folksí house to remove stubborn soap scum. Clearly they hadnít cleaned it in years after they got too old. What a mistake! And I knew better. I had to run out of the room, and in intervals as long as I could hold my breath, throw open windows and use a fan to exhaust the fumes, and rinse the excess away with buckets of hot water. It got the shower clean though, when nothing else would! If they would have had a water softener, no soap scum would have ever built up after they got too old to clean. For those who insist that soap does not rinse off in hard water, the shower that never needs to be cleaned again except for drying it out with a towel after showering to prevent mold and mildew disproves that belief. Even if I didnít like soft water, Iíd still have a water softener if for no reason other than to save myself the weekly chore of cleaning the shower with that nasty bathroom spray foam cleaner.) |Post# 1094337 , Reply# 4   10/24/2020 at 11:03 by Helicaldrive (St. Louis)  || | For those who insist that soap does not rinse off in softened water, the shower disproves that belief. To each his own, though. Most of my friends truly detest softened water, so whatever floats your boat. |Post# 1094353 , Reply# 5   10/24/2020 at 13:01 by fan-of-fans (Florida)  || | Maybe place it in a pot of boiling water with dish soap in it, for a few minutes, then let sit? That's what I use to clean my (regular) electric deep fryer frypot and basket. |Post# 1094358 , Reply# 6   10/24/2020 at 13:33 by joeekaitis (Rialto, California, USA)  || | |Post# 1094424 , Reply# 7   10/24/2020 at 22:15 by gizmo (Great Ocean Road, Victoria, Au)  || | I've tried ammonia soaking overnight. That's an old family favourite. It did nothing. Easy-Off fume free oven cleaner did absolutely nothing. Might as well have been water. Tried 2 overnight applications. This stuff is really rock hard. Nothing seems to soften it. Thanks for all your suggestions. If I can't track down some carbon removing stuff, I will probably just replace the basket. I have already spent enough on cleaning potions that didn't work to have bought a basket any way... |Post# 1094427 , Reply# 8   10/24/2020 at 22:21 by Brisnat81 (Brisbane Australia)  || | Selleys make a gel oven cleaner that Iíve always found to be effective. The Gel generally only seems to be available at IGA, whereas the spray is available at Bunnings, but I havenít tried it. See how you go |Post# 1094430 , Reply# 9   10/24/2020 at 22:50 by gizmo (Great Ocean Road, Victoria, Au)  || | does the Selleys gel work on REALLY hard black stuff? I have seen it mentioned online and was tempted but a jar is about $10 and a new basket is about $36... The stuff I am trying to get off is seriously hardened. All the other products I have tried have removed anything easier. This is the toughest of tough stuff. I was having some luck chipping at the edges of the black residue with a bamboo chopstick, but what is left is wearing away the chopstick... |Post# 1094448 , Reply# 10   10/25/2020 at 05:55 by ozzie908 (Lincoln UK)  || | If you know anyone in catering or who has a grill cleaning service they will get it clean. Its a very strong alkaline substance that works best at a temp of 45c, I had one at the restaurant it was like a large tank with a basket you put anything that needed cleaning in left overnight and in the morning rinse in cold water and its all like new...! There is a chemical you can buy that does a similar thing but is not so efficient. |Post# 1094510 , Reply# 11   10/25/2020 at 18:33 by Stevet (West Melbourne, FL)  || | If you have a self cleaning oven, you may want to give that a try. Use the shortest cleaning time and see if the crud gets burned off. I have an aunt who did that to all her carbon encrusted frying pans and they all came out like nearly new. Just be prepared for some more smoke than you may usually get when the cycle is going through its paces.

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board? Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw? Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares? The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices? Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Show that all pentagonal numbers are one third of a triangular number. Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? Can you find a rule which connects consecutive triangular numbers? Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . . Show that among the interior angles of a convex polygon there cannot be more than three acute angles. A huge wheel is rolling past your window. What do you see? Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice? Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . . On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes. How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel? What size square corners should be cut from a square piece of paper to make a box with the largest possible volume? Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles Can you find a way of representing these arrangements of balls? The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set? What is the shape of wrapping paper that you would need to completely wrap this model? When dice land edge-up, we usually roll again. But what if we Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use? Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring? Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . . Can you describe this route to infinity? Where will the arrows take you next? Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs? Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens? These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together? Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? Here is a solitaire type environment for you to experiment with. Which targets can you reach? A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red. Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . . How many moves does it take to swap over some red and blue frogs? Do you have a method? A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . . A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . . Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo Can you mark 4 points on a flat surface so that there are only two different distances between them?

Is there an efficient way to work out how many factors a large number has? What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time? Do you know a quick way to check if a number is a multiple of two? How about three, four or six? The clues for this Sudoku are the product of the numbers in adjacent squares. The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator? A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order? What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ? Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters. Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have? Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way? How many solutions can you find to this sum? Each of the different letters stands for a different number. How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? Investigate how you can work out what day of the week your birthday will be on next year, and the year after... Mathematicians are always looking for efficient methods for solving problems. How efficient can you be? Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large... The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares. Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? Use the differences to find the solution to this Sudoku. Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number? Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3... Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be? In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays? Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles? If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H. Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money. Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps? The sums of the squares of three related numbers is also a perfect square - can you explain why? A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number? Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units... If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? How many different symmetrical shapes can you make by shading triangles or squares? If you are given the mean, median and mode of five positive whole numbers, can you find the numbers? This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter? Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair? On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . . All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . . There are lots of different methods to find out what the shapes are worth - how many can you find? A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? In a three-dimensional version of noughts and crosses, how many winning lines can you make? Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? Different combinations of the weights available allow you to make different totals. Which totals can you make? There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? Can you explain the surprising results Jo found when she calculated the difference between square numbers? Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25? Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers? An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Interval mathematics foundations, algebraic structures, and applications by hend dawood a thesis submitted to department of mathematics, faculty of science cairo university in partial fulllment of the requirements for the degree of master of science in computer science march 2012. The book does not presume much background in either mathematics or music. On this page you will find printable math worksheets and puzzles arranged according to levels. Almost thirty years ago i wrote a book about my experiences living in the us, and since then ive had many opportunities to work with literary professionals and to write essays apart from my regular occupation. Warwick a4 maths exercise book 8mm graph 32 leaves officemax nz. This mean ria takes lesser time than her friend dia to reach school. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. Extension 1 bridging course exercise book the university of sydney. There are many books in the market but ncert books stand alone in the market. It will help your children no end if you want them to achieve well at maths at school. It is not intended as a standalone book but instead should augment other literature and learning resources you may be using. This is a guide showing how you should set out your maths exercise book. Keep this practice book until you receive your score report. We have them in a4 size for those big projects, or the standard 177. Candidates preparing for civil services examinations as well as cbse aspirants require good books and resources for high level preparation. Essential exercises primary series by series understanding maths. The dyslexia shop sells thousands of products relating to dyslexia and special educational needs, including teaching aids, specialist software, books, audio. Fractions workbook, introduction of fraction in shapes, comparing basic fractions, adding fractions, fractions of sets, and mixed math operations and exercises including multiplying 3 numbers. Feedback to your answer is provided in the results box. Math worksheets, math for kids, mathematics for children, interactive activities for kids and printable exercises in pdf format for first grade, second to sixth graders submit toggle navigation. The coursebook is intended for use by standard 3 pupils. Exercisebookparti probability of events20 09 contents 1 introduction3 2 outcomes and events 3 relative frequency and probability18 4 random numbers25 5 classical problems30 6 geometrical problems, uniform distributions37 7 basic properties of probability40 8 conditional relative frequency and conditional probability41 9 independence. Maths time 3 activity book maths third class primary books. From this point of view, exercise books written in czech and english should be treated as interchangeable equivalents by both teachers and students. Time speed, distance and time ria reaches her school in 10 minutes whereas her friend dia takes 15 minutes to reach. The probability of a single event must lie in the interval 0,1, and the. Please add link for 10th maths hindi mediums pdf thank you so much for your gratitude sir. Pdf we begin by constructing the algebra of classical intervals and prove that it. Let f be a continuous realvalued function defined on the closed interval 2, 3. College mathematics textbook madison area technical college. Exercise bookparti probability of events20 09 contents 1 introduction3 2 outcomes and events 3 relative frequency and probability18 4 random numbers25 5 classical problems30 6 geometrical problems, uniform distributions37 7 basic properties of probability40 8 conditional relative frequency and conditional probability41 9 independence. 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Math100 revision exercises resources mathematics and. The purpose of this booklet is to give you a number of exercises on proposi tional. Here is a list of all the skills students learn in eighth grade. Warwick a4 maths exercise book 8mm graph 32 leaves. Our workbooks and exercise books are from top brands like, rhino, oxford, silvine, and guildhall, which you can buy in bulk for even better prices. We have not yet given any meaning to negative exponents, so n must be greater than m for this rule to make sense. Plain, lined, ruled, with and without margins, and more. Site description primary school children can choose between ten levels and various kinds of mathematics exercises. Exploring math 4 sort psort q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 sort p. Feb 23, 2016 dear students, you can download or view our notes on mathematics exercise 7. Dont,do your self a favour and and buy some of these maths books. This book brings the mathematical foundations of basic machine learn ing concepts. The understanding maths series was written by experienced, qualified australian teachers to enhance understanding, confidence, enjoyment and results. What is the time interval between these pairs of clocks. The exercise book has 32 leaves with 8mm quad ruled lines. This text is an integral part of a series of materials prepared for mathematics 3 by its. Practice books, grades k5 the math learning center. This is a division quiz which involves dividing the numbers given click the start button to begin. Which of the following is sets is shown with roster notation. The writers have published it for all the six exams. Improve your students schoolstate math test scores. Represent the answer on a number line and in interval notation. We have a wide range of exercise books in stock at ryman. Dear students, you can download or view our notes on mathematics exercise 7. Math worksheets and printable pdf handouts math 4 children. Classroom rules and exercise book layout tes resources. I felt at ease when i first began to write because i regarded literature and mathematics as kindred art forms. Jun 22, 2016 studying from text books is really important in alevels. Study and practice maths time online for class 4 and upgrade your knowledge. This is the third book in the new series of primary mathematics. Alevel mathematics can be studies efficiently with these books. Chapter 1 basic notions, real numbers it is encouraging that the refutation of the unfounded rumor claiming that it is not a lie to deny that there will be at least one student who. In words, to divide two numbers in exponential form with the same base, we subtract their exponents. Foundations, algebraic structures, and applications. This warwick a4 maths exercise book is a graph book excellent for mathematics and science. This book was produced using latex and timothy van zandts pstricks. Ncert books pdf download 2020 for class 12, 11, 10, 9, 8. I get all my students to stick this into their maths exercise books at the beginning of the year. Click and collect and next day delivery available in uk. Kidsfront has developed online study material of class 4 maths time lesson, available for free. All level 3 questions are in the front of the book and all level. They keep your work all in line and make the sum much easier to do. Mathematics and music wustl math washington university. Gcse mathematics with aqa further maths lessons, exercises. If you make a mistake, rethink your answer, then choose a different button. This book brings the mathematical foundations of basic machine learn ing concepts to. Its got the classroom rules, which they sign, and also the rules for laying out their exercise books, with an example of the correct layout. The interval between two notes can be thought of infor mally as. Each exercise consists of a question page, which may be printed and completed by students, and a full worked solution. Ncert maths books 202021 for class 1, 2, 3, 4, 5, 6, 7, 8. Excellent for use in schools, by university students, and in the workplace. Ncert books pdf free download for class 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1. This pdf contains 65 chapters, each containing explanations, worked examples and exercises. Throughout this text, we shall employ the following theorem and its corollary. Essential exercises primary series understanding maths. Each chapter consists of text plus worked examples. Studying from text books is really important in alevels. Interval notation and linear inequalities math 0, fundamentals of mathematics the university of houston chapter 1. The files in this area contain worked examples in many of the topics studied in the various 100level mathematics courses. Aug 31, 2012 i get all my students to stick this into their maths exercise books at the beginning of the year. Mathematics learning centre, university of sydney 2 this leads us to another general rule. What are some textbooks i can use to prepare for a level cie. The skills are organized into categories, and you can click on any skill name to start practicing. Free math exercise workbooks and booklets for primary. Retypeset in 2014 for the mathematics learning centre by leon poladian. Stp mathematics 1a teachers notes and answers 3 exercise 1i p.571 732 728 1621 752 1204 322 1238 1383 1141 252 150 207 1130 1634 259 418 1582 924 757 923 752 795 178 514 623 335 935 485 836 1599 91 1128 368 1062 269 472 119 870 1085

(To Main Page) Dear Grandpa - reading a book that says quantum physicists believe there are at least 7 newly-discovered dimensions in addition to length, height, width, and time. What are your thoughts? If this is true, what in the world are these dimensions??? ( Question is from Ronnie Sawyer - Age 18 ) word "dimension" has more than one meaning. When physicists talk of the four dimensions, those four have some important characteristics. Dimensions all apply to every physical thing that exists. Dimensions can never be negative - only zero or positive. Dimensions must be "mutually orthogonal." a way to see these characteristics: a book. Let's say it's one inch thick. Let's say it's 6 inches tall. Let's say it's 5 inches wide. Let's say it was published in 1940 - and this particular copy was burned up in - the book existed for 55 years with its thickness, width and height. now let's consider the same book - only with one of these dimensions zero. Say the thickness is zero. Well - the book does not exist. There is say the height is zero - Again, there is no book. Or the width - No book. say the thickness, width and height are there, but the book exists for zero time - Still - No book. It never you reasonably imagine a book that is minus 1 inch thick? No. Thickness is about minus 5 inches wide? No - width can be positive only. 6 inches tall? No - height can be really small baby human may be only one foot long. A basketball player may be more than seven feet tall. But nowhere is there a human that is minus seven feet tall - or minus any is a better word for this than height. We may turn the basketball player upside down and declare that his height measured from his feet to his head is "minus seven feet," but he's still seven feet long, and it's plus seven feet. we might easily imagine a book that exists for zero time. It's just a book that never was. But we cannot reasonably imagine a book that exists for minus two years. orthogonal" means that when one dimension is changed, the others are not affected. They are independent. Often we think of these as all at right angles to each other. Up&Down is at a right angle to Left&Right and also at a right angle to Forward&Back. Any one of these three is at a right angle to the other two. It's harder to see time this way - at right angles to the other three. But one can see that if the book survives for two years or ten years, the other three dimensions are unaffected by that difference. is a "dimension" in this special way. If we could go "backwards" in time, then we could publish a book in 2001 - and take it with us to 1991, and we could say, "Here's a book that is minus ten years old. Excuse Grandpa, but "Poppycock." We cannot move backwards in time, because having a negative time is just like having a negative thickness or length. Something cannot cease to exist before it comes into existence. A person cannot die before they are born. We can slow down - or speed up - but only zero to positive c (The speed we have experimentally determined is the maximum.) This constant, c, is not the speed of light, as often taught. Light can travel at any speed up to c. That's how lenses work; they slow the light down in an organized way. Responsible physicists keep their minds open, but we are pretty sure that no actual thing can go faster than this. mathematicians, we can define any number of other "dimensions." We could plot the time it takes to boil water in a particular container on a particular stove. We can call the temperature one dimension - and time the other. But temperature is not a dimension in the same way the "four" are. Something could be reasonably imagined to have zero temperature and still exist. When temperature is raised, three of the dimensions increase with it, so temperature is not "mutually orthogonal" to the others. Temperature does meet the one requirement of never being negative, but one requirement is not enough to be a "dimension." (We talk about minus temperatures, but this is just a scale we use. The "absolute" temperature scale (Kelvin) tells us how much heat is present, and we can't have a negative amount of heat. On that scale, zero degrees - no heat - is roughly equivalent to -273 degrees centigrade.) Temperature is not a "dimension," in that we can't use it to measure the "size" of some part of space or of an object in could also invent something with ten dimensions, calling color one, and temperature one, and smoothness one, etc. These are dimensions, but not by the responsible physicist's definition of a "dimension of nature." There were only three we recognized for a long time - and with Albert Einstein, we came to recognize that time has the same qualities and acts the same way as the is also a popular myth that someone or something can "come from another dimension." We can't come from length or width, etc. Again, this is a completely different definition of dimension and has nothing to do with anyone discovers that there is a fifth dimension - He or she will surely win a Nobel prize. It will not be the subject of the voodoo metaphysicist, but of real scientists. was once involved in a research project wherein we contacted a man who was arguing for a 9 dimensional universe. We, the taxpayers paid this quack more than a hundred thousand dollars. He was making claims with absolutely nothing to back them up. He was pretending to be a physicist with an interest in his country, but he was actually just a thief. sometimes thinks that it is unfortunate that a less sophisticated thief, who takes an object, goes to jail, while these more sophisticated thieves always go free, even while we are fully aware that they have taken much more. Another student asked if "spirit" is a dimension. You may figure out from the above that spirit may be regarded as a dimension, but it is not a dimension in the same sense that length is - or the others of the four. Remember - no "thing" can exist unless it has measurements of non-zero for all four of the dimensions. This refers to physical things. We may regard Love or Enthusiasm or Trust to exist, and these do not have widths, lengths, thicknesses. These are not "things" in the physical sense. They are more like "ideas," or concepts of the mind. So what about the dimensions themselves? Are they "things", or are they "ideas"? Think about it. Does length have width and thickness? No. Length is the name of one of the dimensions, but it does not have dimensions itself. Length is an idea, not a thing. A thing must have length, but length is not a thing itself. That makes it clear that someone cannot "come from another dimension." One cannot come from a place that does not exist as a physical place. People can't "move" from one dimension to another. Dimensions are not places. People and other things are not in some dimension; they have non-zero positive values for each dimension. Dimensions do not weigh anything. They are not (To Main Page)

主讲教师：许宝刚 人气：891 更新时间: 2015年10月20日 吴文俊数学重点实验室组合图论系列讲座之六十一【许宝刚】 摘要： We show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of lengthat least 7, then $chi(G)le 4$ and $chi(G)le 3$ if $G$ has radius at most $3$, and for each vertex$u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions of Plummer and Zha. Joint work with Gexin Yu and Xiaoya Zha. 主讲教师：张继平 人气：756 更新时间: 2016年05月25日 摘要：Adequacy of subgroups is very important in generalizations of Taylor-Wiles method for proving the automorphy of Galois representations. I will talk about some new progress on linear groups and the application related to adequacy of subgroups. 主讲教师：郁星星 人气：688 更新时间: 2016年05月25日 A subdivision of a graph G is denoted T G. Mader conjectured that every C4-free graph with average degree d contains TK_l with l = ?(d). Koml′os and Szemer′edi reduced this problem to expanders and proved Mader’s conjecture for n-vertex expanders with average degree d < exp(log1/8 n). We show that Mader's conjecture is true for n-vertex expanders with average degree d < n3/10, which improves Koml′os and Szemer′edi’s bound to a polynomial bound. As a consequence, we show that every C4-free graph with average degree d contains a TK_l with l = ?(d/(log d) c ) for any c > 3/2. This is joint work with H. Huang and Y. Wang 主讲教师：傅翔 人气：2633 更新时间: 2015年03月22日 Dominance is a useful partial ordering defined on the root systems of Coxeter groups. In the literature it has non-trivial applications in the group-theoretic, combinatorial and computational aspects of Coxeter groups. In this talk we outline a number of recent results on dominance. In particular, we discuss how dominance can be used to better understand the asymptotic behaviours of the so-called normalized roots of infinite Coxeter groups. Furthermore, we outline the natural interpretation of dominance in terms of the Cayley graphs of Coxeter groups. Time permitting, we may outline the connection between the visual boundaries of the Davis complexes of Coxeter groups and the limits of the so-called normalized roots of Coxeter groups. This talk is based on some recent joint work with Dr. Lawrence Reeves. 主讲教师：Aleksandr Makhnev 人气：985 更新时间: 2014年12月07日 Let Γ be a distance regular cover of (k +1)-clique with λ = μ. If local subgraphs are strongly regular with parameters (v’, k’, λ’, μ’), then v’= k, k’= λ, and b1= k ? λ ? 1 = (r ? 1)μ, so k’= λ devides v’? 1. It is determined parameters of strongly regular graphs with at most 1000 vertices, which may be local subgraphs in distance regular cover of clique with λ = μ. It is classified arc-transitive distance regular cover of clique with λ = μ with local subgraphs, whose parametrs belongs to our list. 主讲教师：Naoto Yotsutani 人气：1400 更新时间: 2015年05月18日 吴文俊数学重点实验室微分几何与分析系列讲座八十三【四谷直仁】 题目: DOUBLING CONSTRUCTION OF COMPACT MANIFOLDS WITH SPECIAL HOLONOMY The aim of this talk is to give differential-geometric constructions of Calabi-Yau 3-folds, 4- folds and compact Spin(7)-manifolds. Ingredients in our constructions are (orbifold) admissible pairs (with antiholomorphic involution), which were dealt with by Kovalev in a series of his papers (2003, 2011, 2013). For our convenience, we shall mainly discuss the case of CY 3-folds. If time permitted, we also give a concrete example with this construction. This talk is based on a series of joint works with Mamoru Doi. 主讲教师：Anyue Chen 人气：3188 更新时间: 2015年03月22日 In this talk, I'll address some important topics, particularly the extinction properties and their asymptotic behaviour of the Interacting Branching Collision Process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process (MBP) and a collision branching process (CBP). After establishing some elegant necessary and sufficient conditions for existence and uniqueness of IBCP, the explicit expressions are then obtained for the extinction probabilities for the regular, critical-explosive and super-explosive cases. The subtle sub-explosive case will also be addressed. The associated expected hitting times are also investigated and revealed. Moreover, considering some of these exact explicit expressions are very complicated and thus we turn to investigate the asymptotic behaviour of these extremely complicated and divergent expressions for extinction probabilities. Surprisingly, we are able to show that the asymptotic behaviour follows an elegant and homogenous power law with an index quantity. We further show that this interesting and important index quantity takes a very simple and uniform form which could be interpreted as the "spectrum", ranging from minus infinity to positive infinity, of the interaction between the two components of the IBCP. 主讲教师：吴可 人气：1419 更新时间: 2015年05月21日 吴文俊实验室数学物理系列报告之四十六 【吴可】 报告题目：Heisenberg代数和Fermi代数的范畴化报告人：首都师范大学数学学院 吴可教授时间：2015.5.19 （星期二）下午4:30 地点：管理科研楼 1418 摘要：TBA 主讲教师：彭岳建 人气：582 更新时间: 2016年05月25日 摘要：A number α ∈ [0,1) is jump for r-uniform graphs if there exists a constant c > 0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Tur´ an density of F is at least α + c. A fundamental result in extremal graph theory due to Erd¨ os and Stone implies that every number in [0,1) is a jump for r = 2. Erd¨ os also showed that every number in [0,r!r r ) is a jump for r ≥ 3. Furthermore, Frankl and R¨ odl showed the existence of non-jumps for r ≥ 3. But there are still a lot of unknowns regarding jumps or non-jumps for hypergraphs. We give a survey on the known result. 主讲教师：王诗宬 人气：687 更新时间: 2016年12月02日 11-30吴文俊讲座【王诗宬 院士】 摘要: I will give a survey talk on the recent results (joint with Derbez, Liu and Sun) around the title

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This video shows a wheel chair in motion. a. Have one person in your group extend their arms (as in the picture below) and rotate slowly so that their fingertips move at a constant speed. Record the time it takes their arm to sweep through a known angle, q. Rotating arms featuring elbows and hands. b. Calculate the speed of your elbow and the speed of your fingertips. Are your elbow and your fingertips moving at the same speed? c. Is the speed of your elbow and the speed of your fingertips related to the distance of your elbow and your fingertips from the axis of rotation (the center of your body)? d. The angle, q, through which you rotated was measured from a reference axis (as in the picture above). At any given time during the rotation, is the angle between your elbow and the reference axis different than the angle between your fingers and the reference axis? Explain. e. You were asked in part a to rotate at constant speed, but you found two different speeds for your elbow and for your fingertips. What is meant by the term constant speed? How is this speed different from the speed of your fingertips and the speed of your elbow you calculated in part b? Explain. Explain your reasoning to an instructor. 1.2 An understanding that angles can be measured in units of radians or units of degrees and the relationship between angles measured in radians and angles measured in degrees is critical in the study of rotational motion. We first need to define an arc length. An arc length is the distance, s, subtended by an angle, q, along the circumference of a circle, as shown in the picture below. 1. A degree is defined as 1/360th of a rotation in a complete circle. 2. A radian is defined as the angle for which the arc along the circle is equal to its radius. An angle measured in radians is the ratio of the arc length to the radius a. How many radians subtend an arc length of · 2 p r, the circumference of a circle? · p r, half the distance around a circle? · 5m along the circumference of a circle of radius 6.5m? b. How many radians equal a. Using the data in part 3.1.a, and using the reference axis as q = 0, calculate the change in q divided by the change in time for the elbow and for the fingertips. Would this quantity be the same for a point on your arm halfway between your elbow and your fingertips? For any point on your arm? The quantity the change in q divided by the change in time is called the angular velocity, , where the angular speed is given by and the direction is clockwise or counterclockwise. The units of are radians per second (rad/s). b. Does every point on a rotating object have the same angular speed? Explain. c. In part 1.1.b, you calculated the speed of the fingertips and the elbow of a person rotating at constant angular speed. The speed you calculated is the magnitude of the velocity of the fingertips or the elbow. The velocity is tangent to the circle through which that point is moving; it is called the tangential velocity. The tangential velocity vector is shown in the diagram below. The speed you calculated is called the tangential speed. Do the fingertips and elbow have the same tangential speed? Explain. d. Relate the tangential speed, v (the magnitude of the tangential velocity), of the fingertips of the person in part 3.1.a to the angular speed of the person. Write an equation relating the speed, v, to the angular speed, . Discuss your answer with an instructor. a. Consider a rotating object that is slowing down or speeding up. Is the angular speed changing? Explain. b. Is the tangential velocity at a point changing? Is the point accelerating? Explain. If the tangential velocity of a point on an object is changing, the point is accelerating. Since the point is moving in a circle, there is an acceleration toward the center of the circle, as shown in the picture below. This is called the radial acceleration, aR. If the tangential speed is changing, there is also a component of the acceleration tangent to the circle through which the point is moving, as shown in the picture below. This is called the tangential acceleration, aT. c. If the angular velocity of the rotating object is changing, we can define an angular acceleration, . The magnitude of the angular acceleration is the change in angular velocity divided by the change in time and the direction is clockwise or counterclockwise. The units of angular acceleration are rad/s2. Is the angular acceleration the same for each point on a rotating object? Explain. It can be shown that the magnitude of the angular acceleration is related to the magnitude of the tangential acceleration d. Since the angular acceleration, the angular velocity and the angle, , is the same for all points on a rotating object, it is often useful to work with these quantities when studying rotational motion. You might notice that the definitions of and are similar the definitions of a and v for motion in a straight line in one dimension that we studied in Unit 1: It can also be shown that From these three equations, four equations for rotational kinematics can be derived, as they were for motion in a straight line. The four equations are shown below, and also the analogous equations for motion in a straight line. You should understand the concept of angular speed. You should understand the concept of angular acceleration. You should understand and be able to use kinematics equations to describe the motion of objects moving with constant angular acceleration.

MIPHAM’S BEACON OF CERTAINTY (Illuminating the View of Dzogchen, the Great Perfection) L1: Introduction [the need to induce certainty by the two types of valid cognition – about the Two Truths] L1: Topic 1 [The problem of negation: Emptiness is also empty; it is not the real non-dual nature beyond all conceptualization] L1: Topic 2 [about the limited realizations of the arhats in the Hinayana: they do not realize the full extent of the emptiness of phenomena; they still reify elementary dharmas] L1: Topic 3 [about habit pattern – modal apprehension – of rational cognition] L1: Topic 4 L1: Topic 5 L1: Topic 6 L1: Topic 7 L1: Introduction [the need to induce certainty by the two types of valid cognition – about the Two Truths] \ “Trapped in doubt’s net, one’s mind \ Is released by the lamp of Manjuvajra, \ Which enters one’s heart as profound certainty. \ Indeed, I have faith in the eyes that see the excellent path! \ Alas! Precious certainty, \ You connect us with the profound nature of things; \ Without you, we are tangled and confused \ In this web of samsaric illusion. \ The development of confidence through certainty \ In the phenomena of the basis, path, and result, \ And being roused to faith by studying them \ Are like the authentic path and its reflection. \ The fame of the Moon of the Amazing Dharma \ Arises along with the light of elegant speech \ In the vast sky of the Buddha’s teaching, \ Vanquishing the heavy darkness of doubt. \ The valid cognition that examines conventionalities \ Is unerring with respect to engaging and avoiding. \ Specifically, the textual corpus on valid cognition \ Is the only way to acquire confidence \ In the teacher and the teaching, and \ The Madhyamaka of the Supreme Vehicle \ Elucidates the stainless valid cognition \ Of ultimate reasoning, which determines the nature of things. \ [The two valid cognitions emphasized in] these two [systems] \ Are the wisdom eyes of a well-trained intellect. \ Praise to such enlightened beings who \ Abide on the path taught by the teacher \ Without taking detours!” \ As the sage reflected thus, \ A mendicant who happened along \ Asked these seven questions \ In order to critically examine his intellect: \ “What’s the point of being a scholar \ If you only repeat the words of others? \ Give us a quick answer to these questions \ According to your own understanding. \ Then your philosophical acumen will be obvious. \ Though they stretch out the elephant’s trunk of their learning, \ Like well water, the deep water of Dharma is not tasted; \ Yet they hope still to become famous scholars \ Like low-caste men lusting for a queen. \ 0.1.1.2.2.1.4 [The seven topics:] \ 1. According to which of the two negations \ do you explain the view? \ 2. Do arhats realize both types of selflessness? \ 3. Does meditation involve modal apprehension? \ 4. Does one meditate analytically or transically? \ 5. Which of the two realities is most important? \ 6. What is the common object of disparate perceptions? \ 7. Does Madhyamaka have a position or not? \ Thus, starting with the topic of emptiness, \ Give an answer established by reasoning, \ Without contradicting scripture, \ For these seven profound questions! \ Even though pressed with the barbed lances \ Of a hundred thousand sophisticated arguments, \ These issues have not been penetrated before. \ Like lightning, let your long philosopher’s tongue strike \ These difficult points, which have confounded the great!” \ Thus incited by intellect, \ The speech-wind wavered somewhat, \ And that shook the sage’s heart \ Like a mountain in the winds at the end of time. \ After maintaining a moment of disciplined engagement, \ he said: \ “Alas! If by undergoing hundreds of difficult tests, \ And analyzing again and again, \ The fires of great intellects blazed ever greater \ Yet were still not refined to a flawless state, \ How can a low person like myself possibly explain this, \ Whose innate brilliance is weak \ And who has not undertaken lengthy study?” \ Then, as he cried these words of lament to Manjughosa, \ By what seemed to be His mystic power \ A light dawned in the mind of the sage. \ At that moment, as he acquired a little self-confidence, \ He reasoned analytically according to eloquent scriptures, and spoke. L1: Topic 1 [The problem of negation: Emptiness is also empty; it is not the real non-dual nature beyond all conceptualization] \ [“Question 1: The basis as the coalescence of appearance and emptiness” \ – The first topic concerns the definition of the Nyingma philosophical view] \ The dGe ldan pas say the view is an absolute negation; \ Others say it is an implicative negation. \ What is our own Early Translation tradition? \ In the state of great gnosis of coalescence, \ After making a negative judgement of “non-existence,” \ What other thing such as an exclusive emptiness, \ Or something that is not [that which is negated], \ Could be implied in its place? \ Both are just intellectually designated, and, \ In the ultimate sense, neither is accepted. \ This is the original reality beyond intellect, \ Which is free of both negation and proof. \ But if you should ask about the way in which emptiness \ is established, \ Then it is just an absolute negation. \ In India the glorious Chandrakirti \ And in Tibet Rong zom Chos bzang both \ With one voice and one intention \ Established the great emptiness of primordial purity. \ Because these dharmas are primordially pure, \ Or because they are originally without intrinsic reality, \ They are not born in either of the two realities; \ So why fret about the expression “non-existent”? \ In the place of a pillar, primordially pure, \ There is nothing non-empty whatsoever. \ If you don’t negate it by saying, “There is no pillar,” \ What does it mean to say, “The pillar does not exist?” \ The emptiness that is the negation of the pillar \ And a left-over appearance \ Are not fit, as “empty” and “non-empty,” to coalesce; \ It is like twisting black and white threads together. \ To say, “a pillar is not empty of being a pillar” \ Or “dharmata is empty of being a pillar” \ Is to posit the basis of emptiness and something \ of which it’s empty. \ These are verbal and ontological extrinsic emptinesses. \ Woe! If this is not empty of this itself, \ The empty basis is not empty and is left over. \ This contradicts both scripture and reasoning— \ “Form is empty of form!” \ Consider a pillar and the true existence of a pillar: \ If they are one, then refuting one the other is refuted; \ If they are different, by refuting a true existence \ That is not the pillar, the pillar \ That is not empty of itself would be immune to analysis. \ “Because true existence is not found to exist, \ There is no need to debate sameness and difference”— \ Even though true existence does not exist, \ Individuals still apprehend vases as truly existent. \ So aside from a non-empty vase \ What is there to establish as truly existent? \ And you think you’ve determined the appearance of \ the negandum! \ To teach emptiness by applying some qualifier \ Such as “true existence” to the negandum \ Is of course well known in Svatantrika texts. \ But in the context of analyzing ultimate reality, \ What is the point of applying it? \ Thinking that if it’s empty, then even deceptively \ A pillar will be non-existent, \ You try to avoid misinterpretation of the word \ But this is itself a great contradiction! \ You are not satisfied to say simply, \ “A pillar is deceptively existent.” \ Why must you say, “It is not empty of itself? \ You may say, “They are the same in meaning,” \ But it is not so; “A pillar exists” and \ “There is a pillar in a pillar” are different statements. \ The latter means “Something depends on something”— \ This in fact is what you end up claiming. \ If ultimately a pillar is not perceived, \ Then how can a pillar not be empty of pillar? \ In saying “Deceptively a pillar [is not empty of being a] \ You are confused, using the same word twice. \ If something is not empty of itself, \ Then while it exists itself, it must be empty of \ something else. \ If the negandum is not something else, \ This contradicts the claim that it is not empty of itself. \ Generally speaking, extrinsic emptiness \ Does not necessarily qualify as emptiness. \ Although a cow does not exist in a horse, \ How could one thereby establish the horse’s emptiness? \ By seeing that horse, what harm or good \ Will it do to the cow? \ Therefore a non-empty nirvana and \ An apparent samsara are unfit to be dharma and dharmata. \ Here there is no coalescence of appearance and emptiness \ Or equality of cyclic existence and peace. \ “The moon in the water is not the moon in the sky”— \ If you think the emptiness of being the moon in the sky \ And the appearance of the moon in water \ Are the coalescence of form and emptiness, \ Then the realization of coalescence would be easy \ for anyone. \ Everyone knows a cow is not a horse; \ They directly see the appearance of a cow. \ How could the Mahatma have said, \ “To realize this is amazing”? \ Therefore, in our own system, \ If one examines a moon in the water, that moon \ Is not found at all, and does not exist as such; \ When the moon in the water manifestly appears, \ It is negated, but appears nonetheless. \ Emptiness and existence are contradictory \ In the mind of an ordinary person. But here, this manifest \ Coalescence is said to be wonderful; \ The learned praise it with words of amazement. \ If one examines from the side of emptiness, \ Because nothing at all is non-empty, \ One can say simply that everything is “non-existent.” \ But that non-existence is not self-sufficient, \ For it arises unobstructedly as appearance. \ That appearance is not self-sufficient, \ For it abides in baseless great emptiness. \ There, distinctions such as “This is empty of that,” \ Or “That is empty of this,” \ Or “This is emptiness and that is appearance,” \ Are never to be found; \ When one develops inner confidence in this, \ The one who searches won’t be frustrated \ By pointless analysis, \ But will attain peace of mind—amazing! L1: Topic 2 [about the limited realizations of the arhats in the Hinayana: they do not realize the full extent of the emptiness of phenomena; they still reify elementary dharmas] \ Some say that sravaka and pratyekabuddha arhats \ Do not realize phenomenal selflessness. \ As long as the self that is the apprehension \ Of the aggregates as the mere “I” is not eliminated, \ By the power of that, emotional disturbances are not \ That self is a designation made \ With respect to the aggregates; it is the object \ Of innate I-apprehension. That, and vases, etc. \ Aside from being different, bases of emptiness \ Are no different in their modes of emptiness; \ For phenomena and persons are both \ Empty of intrinsic establishment. \ Thus, this is proven by scripture and reasoning. \ To go beyond this and state that \ “Sravakas and pratyekabuddhas do not realize emptiness” \ Is just a claim. \ At this point, some draw unwarranted conclusions and \ claim that \ The paths of vision of the three vehicles are the same \ And that there are no distinctions of levels of realization. \ They interpret the Prajnaparamita and mantra, all of sutra \ and tantra, \ As texts of provisional meaning. \ There, when those who have already traveled lower paths \ Achieve the the Mahayana path of vision and so forth \ There would be such faults as not having anything \ to abandon; \ By reasoning, harm would befall them irrevocably. \ Moreover, though having realized what must be realized, \ They say that in abandoning what must be abandoned, \ [One must] ally [one’s practice with the accumulations]— \ [But this means] nonrealization, which contradicts the \ claim of realization. \ To claim that the rising sun must rely on something else \ In order to vanquish the darkness—quite strange! \ Some say that sravakas and pratyekabuddhas realize \ the emptiness \ Of the five aggregates of their own continua of experience, \ But do not realize selflessness of other phenomena. \ If one realizes the five aggregates to be empty, \ Then, aside from noncomposite phenomena [like space and \ What other dharma would be left unrealized? \ So what is our own tradition? \ Glorious Chandrakirti’s Autocommentary \ Says that, in order to abandon obscurations, the Buddhas \ Teach sravakas and pratyekabuddhas personal selflessness, \ And in order to abandon cognitive obscurations, they teach \ Bodhisattvas how to realize phenomenal selflessness. \ “Well then, what does it mean to say \ That both sravakas and pratyekabuddhas \ Have realization of emptiness?” \ In order to abandon just the emotional afflictions \ Sravakas and pratyekabuddhas meditate on personal \ But “They do not meditate on the entirety \ Of phenomenal selflessness”—thus teaches [our tradition]. \ Klong chen rab ‘byams said of yore \ That although earlier masters all disputed \ Whether they did or did not [realize both forms of \ Our own position is that whatever types of sravakas and \ Appeared of yore and reached arhatship \ Did not become liberated without \ Realizing the emptiness of the self \ That is the apprehension of the aggregates; \ But just having that realization does not mean \ That they realized selflessness entirely. \ Just like the space inside a sesame seed \ That is eaten out by a worm, \ [Their realization] is said to be a lesser selflessness. \ Thus, with words that refute the lesser [of possible \ It is said that “They do not realize emptiness.” \ This is a most excellent eloquent explanation; \ There is nothing else like it. \ For example, if one drinks a single gulp \ Of the water of the great ocean, \ One cannot say that one has not drunk the ocean. \ Because they see the selflessness of the mere “I,” \ Which is one phenomenon among others, it is held that \ [Sravakas and pratyekabuddhas] see emptiness. \ Just as by drinking a single gulp one cannot say \ That one has drunk the entire ocean’s water; \ Because they do not realize the nature of all knowables \ To be emptiness, it is held that they do not see selflessness \ If one sees the emptiness of a single thing, \ Why wouldn’t one see the emptiness of everything? \ If, with scripture, reasoning, and pith instructions, \ They were to examine things, of course they would see it. \ But, for the most part, those who are destined \ To be sravakas and pratyekabuddhas \ Are attached to the selflessness of persons, \ So it is hard for them to realize the latter extremes \ [of the catuskoti], \ Just as those who analyze a vase \ Might assert its particles to exist substantially. \ If the mind that realizes [selflessness] \ After analyzing a vase also were to \ Analyze particles, it would be reasonable to realize \ [their emptiness]; \ But usually, they do not realize [their emptiness]. \ Though coarse bases and partless atoms appear \ Since [sravakas and pratyekabuddhas] are mostly bereft \ Of those scriptures, modes of reasoning, and pith \ They practice systems that do not contradict [the possibility \ of personal liberation]. \ Likewise, followers of the Cittamatra system \ Do not accept the existence of external objects, \ So why wouldn’t they also accept the non-existence of \ the subject? \ Why wouldn’t Svatantrikas use the reasoning that establishes \ Ultimate truthlessness to understand the conventional \ Nonestablishment of intrinsic characteristics (rang mtshan)? \ So, for you everyone would become a Prasangika! \ How would it be possible for sravakas and pratyekabuddhas \ To denigrate the Mahayana [if they were Prasangikas] ? \ Thus, although the nature of one thing \ Is also the nature of everything, \ As long as the collection of external and internal causes \ and conditions \ Is not complete, realization will come slowly. \ Generally speaking, those with sharp minds become realized \ Under their own power, while dullards \ Do not necessarily reach realization immediately. \ At some point, realization is inevitable; \ At the end of ten thousand aeons, it is said, \ The arhat wakes up from the state of cessation, \ And enters the Mahayana path. \ To properly abide on the Mahayana path, \ One must cultivate oneself for a countless aeon. \ So why shouldn’t it be impossible for \ Sravakas and pratyekabuddhas, who strive for their own \ Not to realize all forms of selflessness \ During those [ten] thousand aeons [they spend \ in cessation]? \ Don’t those who have attained the bhumis \ Gradually clarify and perfect their realization? \ With the help of the accumulations, \ Infinite modes of reasoning, bodhicitta, \ The conduct [that follows from it], and perfect dedication— \ When these conditions are complete, it is certain \ That one will achieve realization, \ Just as complete knowledge of skillful means is a condition \ For swift realization on the mantra path. \ Even if one has abandoned notions of permanent self, \ Instinctive apprehension of “I” occurs in relation to \ the aggregates. \ Therefore [it is said], “[As long as] there is apprehension \ of the aggregates, \ There is apprehension of T”—this statement [from \ the Ratnavali] \ Means that, as long as there is a basis of designation in \ the aggregates \ And a mind that apprehends them, \ The causes for designating a self are complete, \ And as a result, apprehension of self will not cease. \ Thus, even if the permanent self were abandoned, \ Since the object, in relation to which the designated self \ Is instinctively designated, would not be eliminated, \ There would be nothing to oppose the occurrence of \ Thus, in abandoning emotional disturbances, \ The assertion “One must realize the aggregates and so forth \ to be empty” \ Is not the meaning of the passage [in the Ratnavali]. \ That meaning was explained in this way by Chandrakirti: \ If one recognizes the designated mere “I,” \ That is enough to stop the apprehension of “I.” \ Though one does not know a rope to be non-existent, \ By seeing the lack of snake, the apprehension of snake is \ Finally, one will definitely realize both kinds of selflessness. \ The suchness of all phenomena is unique, \ And the way of seeing suchness is the same, \ So Nagarjuna and his son [Chandrakirti] have expounded \ A line of reasoning that establishes the finality of a single \ If, as in your system, sravakas and pratyekabuddhas \ Had already seen reality, what would that line of reasoning \ Do to establish a single vehicle? \ It is just an assertion. \ Here, the primordial wisdom of coalescence \ That sees the ultimate \ Is precisely identical with suchness; \ All sublime beings head toward it, and enter it. \ Therefore, if one understands this system well, \ The systems of Nagarjuna and Asanga are like \ Molasses and honey combined; \ A hungry person will easily digest them. \ Otherwise, as with inappropriate food, \ One feels the discomfort of cancer within. \ Poked and jabbed with a hundred sharp lances \ Of scripture and reasoning, one is afraid. L1: Topic 3 [about habit pattern – modal apprehension – of rational cognition] \ When pursuing the main practice of the view, \ Some say one should not apprehend anything. \ The meaning of “not apprehending anything” \ Can be understood well or wrongly. \ The first [way of understanding] \ Is free of the elaborations of the four extremes. \ For the gnosis of sublime beings, \ Nothing is seen to remain, \ So modal apprehension automatically subsides; \ It is like looking at the empty, luminous sky. \ The second is the mindless system of Hashang: \ Letting the mind rest blankly without analysis and \ Without the clarity aspect of penetrating insight, \ One remains ordinary, like a rock in the ocean depths. \ For example, though both say “There is nothing at all,” \ The Madhyamika sees there really is nothing, \ And the other one just imagines the absence of form; \ Likewise here, though the words are the same, \ The meaning is different like earth and sky. \ Therefore, if in the absence of elaboration of the four \ One does not apprehend the four extremes anywhere, \ One is beyond the four extremes, and modal apprehension \ Because it no longer exists, we say there is no modal \ If some idiots think “Since there is no modal apprehension, \ From the very beginning one should relax and not grasp \ Then because all beings are quite relaxed in their ordinary \ Always wandering in the three worlds of samsara, \ There is no reason to encourage or remind them! \ Some might say, “We have recognized the nature of mind,” \ Without really understanding it; in recognizing the ultimate, \ One must definitely realize the absence of true existence. \ That “Deluded appearances are one thing, and I am \ Is obvious and requires no meditation. \ You might say, “When examining the color, form, origin, \ And so forth, of the mind nothing is seen; \ That is realization of emptiness.” \ This system of teaching is extremely profound, \ And there are also great mistakes one can make; \ Because mind does not have a form, \ It is impossible for anyone to see its color, etc. \ However, it is a very great mistake to think that merely \ not seeing them \ Is the same as being introduced to emptiness. \ Though you examine your head a hundred times, \ A ruminant’s horns cannot be found. \ To say that not seeing something is to realize its \ Wouldn’t that be easy for anybody? \ Therefore, if by this rational analysis \ One sees the nature of things precisely, \ One will profoundly realize the essential unreality \ Of the illusion mind, which is like an illusion. \ Then, just like looking directly into space, \ One will derive profound certainty in the nature of \ one’s mind, \ Which though moving is empty. \ You ask, “Well then, this mind of yours— \ Is it non-existent, like space, \ Or does it have disparate awarenesses?” \ Because the vibrant mind that we all possess \ Doesn’t rest for a moment, surely everyone would say \ There is some sort of awareness. \ Thus, you say that mind, \ Which is neither existent nor non-existent, \ Is the luminous dharmakaya. \ Although he hasn’t done much study, \ Such a person who claims to introduce the nature of mind \ Thinks this is a teaching such that \ “Knowing one liberates all.” \ The teaching of “neither existent nor non-existent” \ in the Great Perfection \ Is the freedom from the four extremes of elaboration. \ If you examine this mind carefully, \ You cannot say it exists, \ Nor can you say it does not exist. \ But in fact, your mind does not go beyond either \ The extreme of both existence and non-existence \ Nor the extreme of neither existence nor non-existence. \ You are just thinking about the mind on the basis \ Of “neither existence nor non-existence.” \ Aside from a difference in name, the mind, \ Spoken of in that way, is no different than \ The “inconceivable self” of the apostate. \ The mind and dharmas other than it \ Are determined to be unreal, and on that basis \ Appearances arise as relativity, \ Which is beyond thought and expressions of “existence” \ and “non-existence.” \ This is the crucial point of freedom from elaborations of the \ four extremes, \ Which is without a focal point and all-pervasive. \ [But] just saying “This is free of both existence and \ Is to place a target in front of your mind. \ Depending on this apprehension of self and others as real \ One enters the river of samsara continuously. \ The antidote that ends all of this \ Is the modal apprehension of selflessness. \ If one does not know the manner of absence, \ To imagine non-existence does not help; \ If you mistake a rope for a snake, \ It doesn’t help to think “There’s no snake;” \ But if you see how it does not exist, it disappears. \ Thus, having realized emptiness through analysis, \ You should not rest content with analysis. \ Since the habit of clinging to real entities is beginningless, \ You should meditate again and again with modal \ By meditating on selflessness the view of self \ Is uprooted, so it’s been called necessary \ By many seers of truth who practiced intensely. \ If this is the fail-safe entry way for beginners, \ To say that modal apprehension should be abandoned \ From the very beginning is a rumor spread by Mara. \ When you acquire outstanding certainty in truthlessness \ Induced by that modal apprehension, \ The mere apprehension of non-existence \ Is not the final nature of things, \ So meditate on the great emptiness free of elaboration, \ Free of conceptual ambivalence. \ When you’ve really understood truthlessness, \ Emptiness arises as relativity, \ Without apprehension of either form or emptiness. \ This is worthy of confidence just like \ Gold refined by fire. \ Though this extremely profound matter \ Has been realized with long-standing effort \ By the great scholar-yogis of India and Tibet, \ Woe to those idiots who say it can be realized \ In a moment—they are plagued with doubts! \ In the main practice of absorption, \ Actual and potential phenomena, samsara and nirvana, \ Are beyond existence and non existence. If in the nature \ of things \ Existence and non-existence are nowhere established, \ Biased apprehension is [nothing but] conceptual \ Therefore, when analyzing rationally, \ One does not see anything established anywhere; \ So how can apprehension come about? \ However, if you analyze the nature of \ Freedom from the four extremes of elaboration, \ certainty is gained. \ By this the penetrating insight of self-arisen \ Luminous wisdom becomes clear like a lamp. \ Its opposite—the dark night of the \ Four extremes of inferior intellects— \ Is uprooted by this very antidote; \ So when you meditate upon it, certainty should arise. \ The fundamental space beyond intellect where \ The elaborations of the four extremes are eliminated \ Is difficult to see all at once \ At the level of an ordinary person. \ The system of study and reflection \ Is for eliminating the elaborations of the four extremes \ in stages. \ To the extent that one grows accustomed to it, \ Certainty grows ever greater; \ One’s intellect, which causes mistaken reification to subside, \ Improves like the waxing moon. \ The unsound view that doesn’t apprehend anything \ Cannot produce the confidence that \ No real entities are established anywhere; \ Therefore, it cannot remove obscurations. \ Therefore, just like inferring fire by smoke, \ The difference between these meditations \ Is known from the dividend of abandoned defilement and \ acquired realization. \ The ordinary idiot’s meditation \ Is not a cause for abandoning defilements or realization. \ Because it is an obstacle to producing good qualities, \ It is like pouring tea through a strainer— \ Scriptural learning and realization slip away, \ While emotional disturbances accumulate. \ In particular, one has little confidence in cause and effect. \ If one has the eyes of the authentic view, \ Scriptural learning, experience, and realization blaze up. \ By virtue of seeing emptiness, \ Confidence in the infallible relativity of cause and effect \ Will increase, and emotional disturbance will lessen. \ With the samadhi that abides one-pointedly \ In the state of certainty induced by analysis, \ The ultimate meaning is seen by nonseeing. \ One does not succumb to any particular object of seeing \ And of course does not apprehend anything. \ Like a mute’s taste of molasses, \ Confidence grows in a yogi who cultivates it, \ But it cannot be produced by analysis alone. L1: Topic 4 \ In meditating the view of the supreme vehicle, \ Which is right—to analyze or focus the mind? \ Some say, “Don’t analyze, but meditate transically. \ Analysis obscures the nature of things, \ So without analyzing, sit like a bump on a log.” \ Some say, “Only do analysis. \ Meditation without analysis \ Is like going to sleep and doesn’t help, \ So one should always analyze.” \ To adhere exclusively to analysis or transic \ Meditation is not appropriate. \ Most transic meditations without analysis \ Can become a mere calm abiding, \ But meditating thus will not produce certainty. \ If certainty, the unique eye of the path of liberation, \ Is abandoned, obscurations cannot be dispelled. \ If you do not know the nature of dharmas, \ However much you meditate, you are still \ Meditating on ordinary concepts. What’s the use? \ It’s like travelling on a path with your eyes closed. \ The habits of beginningless delusion \ Produce clinging to mistaken notions about the nature \ of things. \ Without endeavoring to investigate \ With a hundred methods of reasoning, it is difficult \ To achieve realization. \ Insofar as clinging to mistaken appearances \ And seeing the authentic meaning are mutually exclusive, \ Here, in the darkness of existence to which \ Sentient beings are well habituated, \ It is difficult to obtain a glimpse of reality. \ Through the ripening of the karma of previous practice \ And the master’s blessing, \ By just examining the origin, abiding, and cessation of \ the mind, \ It is possible to determine truthlessness. \ But this is extremely rare; \ Not everyone can achieve realization this way. \ In cutting through to primordial purity, \ One needs to perfect the Prasangika view. \ As for the aspect of non-elaboration, \ Those two are said to be no different. \ In order to prevent clinging to blank emptiness, \ The Mantrayana teaches great bliss. \ This causes an experience of \ The expanse of non-dual bliss and emptiness, \ Free of subject and object. \ Appearance, clarity, and awareness \ Are synonyms of that bliss. \ Here the appearance aspect is the formal Buddha bodies, \ Which protect all beings and bring them to happiness \ As long as samsara exists; \ It has the nature of ultimate compassion. \ Therefore great gnosis by its very nature \ Does not abide in either existence or peace. \ Because it abides in the basis, \ By practicing the path Evam of bliss and emptiness \ In this very life, one will manifest \ The fruitional coalescence. \ In fact the basis, path, and result \ Are not divided; the path of the fourth empowerment, \ Which is the culmination of the Vajrayana, \ Is the self-arisen gnosis of awareness and emptiness. \ This is exclusively emphasized \ In the path of the vajra pinnacle of luminosity, \ Which is the final point where all vehicles converge. \ As long as certainty has not been born, \ One should induce it with skillful means and analysis. \ If certainty is born, one should meditate \ In that state without separating from that certainty. \ The lamp-like continuity of certainty \ Causes false conceptuality to subside. \ One should always cultivate it. \ If it is lost, then induce it again through analysis. \ At first, analysis is important; \ If you don’t start out with analysis, \ How can you induce an excellent certainty? \ If an excellent certainty is not born, \ How can miserable projections cease? \ If miserable projections do not cease, \ How can the foul wind of karma be stopped? \ If the foul wind of karma is not stopped, \ How can this awful samsara be abandoned? \ If this awful samsara is not abandoned, \ What can be done about this dismal suffering? \ In reality, there is no good or evil \ In samsara and nirvana; \ To realize the equanimity of neither good nor evil \ Is the nature of excellent certainty. \ With excellent certainty, nirvana is not attained \ By abandoning samsara. \ The mere words may seem contradictory, \ But in fact they are not. \ This is the most important point of the path, \ A crucial secret instruction on the view and activity— \ You should examine and savor its meaning! \ Next, you should alternate analysis and trance. \ If you analyze, certainty will be born; \ When you don’t analyze, and cling to the ordinary, \ Analyze again and again, inducing certainty. \ When certainty is born, rest in that state \ Without distraction and meditate one-pointedly. \ Certainty and the projecting mind \ Are mutually exclusive; \ So by the analysis that roots out projection, \ You should increase certainty more and more. \ Finally, if even without analysis \ Certainty arises naturally, rest in that very state; \ Since it has already been established through analysis, \ There is no need to accomplish it again. \ If you understand that a rope is not a snake, \ That very certainty blocks the perception of a snake. \ To say “Still you must go on analyzing \ The absence of a snake” is silly, isn’t it? \ When realization of the sublime paths occurs, \ You will not meditate with analysis; \ What need is there to apply \ Inferential analysis to direct realization? \ If you think that “When you leave off analysis \ There is no realization of the ultimate,” \ Then for you the gnosis of Buddhas and sublime beings, \ And the undistorted perceptions of worldly beings, \ Would all be mistaken. \ Because they have already been perceived, \ They are not subject to analysis. \ Therefore, in the context of extraordinary certainty \ Free of elaborations of the four extremes, \ There is no occasion for analyzing or focusing on \ Thoughts of “this” and “that.” \ When the analytical apprehension of characteristics \ Binds the thinker like a silkworm in its silk, \ The authentic nature will not be seen as it is. \ When this extraordinary certainty \ Dispels the darkness that obscures reality, \ One realizes the actual fundamental luminosity \ And the flawless vision of thatness, \ Which is the individually cognized gnosis. \ How could this be analytical wisdom, a form of \ The object of analytical wisdom is “this” or “that,” \ Which is differentiated and conceptualized, \ Whereas this gnosis of equanimity \ Does not reify subject, object, \ Appearance, or emptiness in any way; \ It does not abide in the characteristics \ Of mind or mentation. \ Therefore, the stainless analytical wisdom \ Of equipoise in supreme certainty \ Induced by analysis is the cause by which \ One attains the resultant gnosis of coalescence. \ The ascertainment of the view \ And the establishment of philosophical systems \ Determined [by that view] \ Is the stainless valid cognition of analytical wisdom \ That differentiates and cognizes individually. \ The gnosis of sublime equipoise \ That has reached the nature of things \ By the certainty induced by that valid cognition \ Is the main practice of the Great Vehicle. \ If you have it, in this very life \ The result of coalescence is bestowed; \ So it is both a “vehicle” and “great.” \ According to the system of four tantric classes, \ This path of the word empowerment in anuttarayogatantra \ Is of course the ultimate gnosis, \ But it is not designated as a separate vehicle. \ However, in the explanation of \ The glorious Kalacakratantra., \ The body of the gnosis of equanimity \ Is emphasized, so it is held as the ultimate tantra. \ Among the classes of anuttarayogatantra, \ The gnosis of the path of the fourth empowerment \ That is emphasized and explained here [in the Great \ Is the basic intent of all tantric classes. \ Just as gold smelted sixteen times \ Is extremely pure, so too here \ The analysis of other vehicles’ philosophical systems \ Reveals their progressive purity, which culminates here. \ Thus the way this is established \ Through the valid cognition of stainless wisdom \ Is found in all the interpretive commentaries and tantras \ And in the analysis of Dharmabhadra. \ If you think about it, it is beyond the realm of Mara, \ And causes inalienable wisdom to mature. \ However, to teach the main practice of the view \ As an object of mind and mentation, such as \ Adhering one-sidedly to appearance or emptiness, \ Is to make the inexpressible into an object of expression; \ So it contradicts the intention of the learned. \ Since atiyoga is the inconceivable gnosis \ Of form and emptiness inseparable, \ It is simply beyond impure mind. \ Here the view of cutting through—which ascertains \ The emptiness aspect of primal purity—and \ The view of the luminous all-surpassing realization— \ Which determines the nature \ Of spontaneously present Buddha bodies and gnosis \ In the inner luminosity of the youthful vase body— \ Are inseparable; \ They are just the coalescence of \ Primal purity and spontaneous presence. \ Here in the Great Perfection the so-called “indestructible \ Tilaka of gnosis” of other tantric systems \ Is very clearly taught as a synonym for this. \ Each of the pith instructions of the mental class of the Great \ Is found in the practice of learned and accomplished \ The Mahamudra, Path and Result, Pacification, \ Great Madhyamaka of Coalescence, and so on, \ Are known as its synonyms; \ Because in fact they are all the gnosis, \ Beyond mind, they are all the same. \ The Buddhas’ and siddhas’ intention is the same— \ The learned affirm this univocally. \ Some people say, “Our system of the Great Perfection \ Is better than other systems like Mahamudra.” \ They have no realization and \ No understanding of the conventions of the path. \ If they understood, they would see that this unique \ Cannot be divided through reasoning. \ Likewise, all the gnoses of the fourth empowerment \ In the anuttarayogatantras \ Are indivisible in the Great Perfection. \ However, the source of all of those \ Is the gnosis of the Great Perfection, whose tantric classes \ Are divided into “mental,” “space,” and “instructional,” \ According to their profound, extensive, extraordinary \ There are many instructions here that are not known \ In other systems, which use just a fragment of them, \ So it goes without saying that this is an “extraordinary \ There, the ultimate Great Perfection \ Is profound, peaceful, luminous, and unfabricated— \ The gnosis of the Buddhas. \ But here in the context of the paths, \ One practices the exemplary and actual coalescences, \ Which are like a drawing of the moon, \ The moon in water and the moon in the sky, \ Homologous to that gnosis. \ Each one gradually leads to the next, \ As one cultivates the self-arisen stainless gnosis \ According to one’s own capacity. \ Therefore it is like meditating homologously \ In order to reach sublime gnosis. \ If one directly ascertains \ The great gnosis of the coalescence of dharmata, \ All views that are apprehensions of mental analysis \ Will definitely subside, and one will see non-elaboration. \ Therefore, without citing the context, \ Saying one-sidedly that modal apprehension \ Should be used or not has both faults and good points, \ Like the waxing and waning of the moon. \ This is established through reasoning, \ According to scriptures of definitive meaning. L1: Topic 5 \ Which of the two truths is more important? \ Some claim the ultimate is most important. \ “Deceptive reality is a deluded perception,” they say, \ Understanding it as something to be abandoned. \ “Ultimate reality is not deluded, so that ultimate \ Is the perfectly pure view,” they say. \ If deceptive reality were not erroneous, were indeed true, \ Ultimate reality could not be emptiness, so \ They are expressed differently in this way. \ However, no ultimate can be established \ Over and against the deceptive; \ The two of them are method and methodical result. \ Without depending on an entity for examination, \ Its nonsubstantiality cannot be established— \ Therefore both substance and nonsubstance \ Are the same in being mere relativity. \ If that clinging to emptiness \ Were to fully exclude appearance, \ It would mess up Nagarjuna’s fine system. \ If by cultivating the path by that seeing of emptiness, \ One were only to realize the expanse of emptiness, \ Then one would have to accept that the \ Sublime equipoise on emptiness \ Would be a cause for the destruction of substantial entities. \ Therefore, though things are empty from the beginning, \ Appearance and emptiness are not separate things; \ Adhering to the statement “Only emptiness is important” \ Is an unskilled approach to the final meaning. \ Some people put aside the ultimate \ And from the perspective of mere conventionality, \ Differentiate the levels of the view in the tantric classes. \ Viewing oneself as a deity conventionally \ Without complementing the view with the ultimate reality \ of emptiness \ And thus differentiating “higher” and “lower” teachings, \ is incorrect. \ Without having confidence in ultimate reality, \ Just meditating on deceptive reality as divinity \ Is mere wishful thinking, not a view; \ Just as some heretical awareness mantras \ Involve visualizing oneself differently during recitation. \ Some say deceptive reality is more important; \ They say you must integrate the two truths, \ But then they heap praise on deceptive reality. \ At the time of maintaining the view of coalescence, \ They desert coalescence and grasp a blank emptiness. \ Thus the toddler of practice cannot keep up \ With the mother of good explanations. \ Therefore, here in our early translation tradition, \ Our Dharma terminology for the basis, path, and result \ Does not fall into extremes or bias with respect to \ Permanence, impermanence, the two truths, and so forth; \ We maintain only the philosophical position of \ If deceptive and ultimate reality are separated, \ One cannot posit the basis, path, or result on the basis of \ Basis, path, and result are all \ Without the distinction of abandoning one thing \ or accepting another. \ For if one abandons deceptive reality, \ There is no ultimate; there is no deceptive \ Reality apart from the ultimate. \ Whatever appears is pervaded by emptiness, \ And whatever is empty is pervaded by appearance. \ If something appears, it cannot be non-empty, \ And that emptiness cannot be established as not appearing. \ Since both entities and nonentities should both \ Be taken as bases for establishing emptiness, \ All appearances are just designations, \ And emptiness too is just a mental designation. \ For the certainty of rational analysis, \ These two are method and methodical result; \ If there is one, it is impossible not to have the other, \ As they are inseparable. \ Therefore appearance and emptiness \ Can each be conceived separately, \ But in fact they are never different. \ Therefore, they are called “coalescent,” \ Since the confidence of seeing the nature of things \ Does not fall to any extreme. \ In the perspective of the wisdom of authentic analysis \ Appearance and emptiness are considered to be \ A single essence with different aspects, for \ If one exists, the other exists, and if \ One does not exist, the other does not exist. \ Nonetheless, for beginners \ They appear as negation and negandum; \ At that time they are not combined as one. \ When the nature of emptiness \ Arises as appearance, one attains confidence. \ Thus, everything is primordially empty, \ And these appearances are empty; \ Though empty, they appear; though apparent, \ They are seen as empty—this is the birth of certainty. \ This is the root of the profound paths \ Of sutra, tantra, and pith instructions. \ This is the meaning of cutting off misconceptions \ Through study and reflection; \ It is the unmistaken, authentic view. \ By realizing that crucial point more and more profoundly, \ Clinging to the characteristics of appearances of \ Deceptive reality will gradually be abandoned. \ The stages of the vehicles of the various tantric classes \ Appear in that way. \ Intellectual wishful thinking and \ The view of certainty that finds confidence in the \ Divine appearance of animate and inanimate phenomena \ Cannot possibly be the same. \ The determination that phenomena are truthless \ By Madhyamika reasoning is a view. \ But when a Brahmin recites a mantra over a sick person, \ His imagining a lack of illness is not the view. \ By realizing the abiding nature of ultimate reality, \ One grows confident in the divine appearance of deceptive \ Otherwise, if one dwells on the manner of deceptive \ How can divinity be established? \ Aside from this deluded appearance of subject and object, \ There is no such thing as samsara; \ The divisions of the path that abandons it \ Are not only made from the perspective of ultimate reality, \ Because ultimate reality has a unitary character. \ With respect to the mental ability gained \ Through seeing and cultivating all phenomena \ Of apparent deceptive reality, the subject (of qualities), \ With respect to ultimate reality, the action tantra, \ Performance tantra, yoga tantra, and unexcelled yoga tantra \ are taught. \ Therefore, the tantric classes are not differentiated as higher \ Or lower with respect to either of the two truths \ According to one’s attainment of confidence \ In the coalescence of the two truths, \ The practice of [each of the tantric classes naturally] follows. \ Therefore, if one properly practices without mistakes \ The peerless Vajra Vehicle, \ The path that bestows liberation in a single life, \ Then, just like the example of water seen \ By several different types of sentient beings, \ With respect to pure vision \ It will be impossible for anyone not to see \ Actual and potential phenomena as a manifested mandala. \ If you don’t know things that way, \ Meditating on deities while holding \ The nature of samsara to be impure \ Is like spraying a vomit-filled vase with perfume. \ Alas! That sort of meditation on the Vajra Vehicle of \ Is just like a drawing of a butter lamp. \ The way things appear is impure, \ But that is the system of delusion. \ We say that authentically seeing the nature of things \ Is the meaning of the undivided Vajrayana system. \ Seeing the animate and inanimate universe \ As lacking the nature of pure support and supported, \ But meditating while imagining that they do— \ This path evinces an obvious contradiction, \ And is just a reflection of the Vajrayana path. \ Coal cannot be whitened by washing; \ Likewise, a fabricated meditation that thinks \ “It is not, but it is” \ Attaining some kind of result \ Would be like the heretical sun worshippers (nyi ma pa)— \ Who have no confidence in the emptiness of true \ Abandoning emotional afflictions through meditating \ On an emptiness devoid of appearance, etc. \ What if the action, performance, and unexcelled tantric \ Did not have different levels of view? \ If you have confidence in the view that realizes \ The pure equality of actual and potential phenomena, \ But fail to take advantage of the correct view, \ Seeing yourself and the deity as superior and inferior \ And discriminating things as pure and impure, \ You will only harm yourself. \ And, if you are still attached to what is accepted and aban- \ doned in the lower tantras \ But practice the equality of what is accepted and abandoned \ in the unexcelled tantras, \ Such as “union and liberation,” eating meat, drinking \ alcohol, etc., \ This is known as the “reckless behavior of \ Isn’t that despicable? \ The view is defined according to one’s certainty \ In the vision of the nature of things; \ According to one’s confidence acquired by the view, \ One maintains the practice of meditation and conduct. \ “Because the vehicles are differentiated \ By different levels of view, they are not necessarily nine in \ From the lowest of the Buddhist philosophical systems \ Up to the ultimate vajra pinnacle of atiyoga, \ There is a specific reason for positing \ The enumeration of nine classes. \ Of course there are many levels of vehicle, \ But they are posited by necessity, as is the three-vehicle \ Thus, according to the relative strength \ Of inner gnosis, the animate and inanimate \ Worlds are seen as pure or impure. \ Therefore, the basis of inseparable appearance and emptiness \ Is realized as the inseparability of the two realities; \ As you cultivate the path in that way, \ You will see the gnosis, \ The coalescence of the two Buddha bodies. L1: Topic 6 \ When a single instance of water appears \ As different substances to various sentient beings, \ Some say there is a single object of perception \ And that all perceptions of it are valid. \ If water had some kind of essence, \ Valid and invalid cognitions would be impossible [here]. \ If the various objects that appear were distinct, \ It would not be possible for [different minds] \ To perceive the same pillars, vases [etc.]. \ 6.2.1. 2.1. \ Some say [that in the case of water] there is just wetness; \ But if [different appearances] are not different aspects [of the \ same substance, \ But merely perceptions belonging to different perceivers], \ Different perceptions [of the same thing] would be \ If what one [being sees as] water, pus, and so forth, \ Is not present to other [beings], \ What would be the basis of [those perceptions of] water, \ pus, etc.? \ Moreover, what would happen to the wetness basis \ In the case of beings of the realm of infinite space? \ If wetness were the same as water, \ It could not appear as pus and so on; \ If it were different from water and so on, \ Liquidity would not be perceived anywhere. \ It is not possible for there to be a common object \ Of each distinct perception, \ Because it is not possible for a suitable common substance \ To appear in different ways. \ If one accepts an analytically [determined] basis \ Other than a dependently designated one, \ One must establish its existence in reality— \ However you look at it, it’s unreasonable. \ If the common object were non-existent, \ There would be no object as in Cittamatra, \ And one would have to accept that consciousness itself \ is the object; \ That is unreasonable. \ The subjective apprehension of a non-existent object \ Would also be non-existent in fact. \ Both subject and object are equally apparent \ In relative truth, so considering whatever appears \ It is not reasonable to differentiate \ Subject and object as existent and non-existent. \ Although an object appears, it is false. \ Likewise apprehension of an object appears but is \ not established. \ The common perceptual object is a mere appearance \ That is established as the basis of similar and dissimilar \ Because otherwise it would be unreasonable, as in seeing \ a dance. \ Aside from this mere existence [of an appearance], \ It is not possible for it to come from some other existent; \ Without this, all appearances \ Would be nonapparent, like space. \ On the basis of outer and inner conditions, \ One does not see the thing itself as it is, \ But in the manner of seeing horses and cattle \ In the place of wood blessed by illusion mantras. \ Therefore the common object of perception \ Cannot be specified as “this” or “that.” \ So in our system appearance and emptiness \ Are not differentiated in the basis itself, \ Which is not established anywhere. \ Because it is the same in everything that appears, \ A single substance appears as various things. \ For whomever appearance and emptiness are possible, \ Everything is possible; \ For whomever appearance and emptiness are impossible, \ Nothing is possible. \ “Well then, the distinction of valid and invalid cognitions \ Would be invalid.” \ Whatever appears does not appear otherwise, \ So it is not the case that the perception of its being thus \ Does not establish it as a cognandum. \ For all things naturally abide in their own essences, \ Because they are established by valid cognitions \ That determine their sameness and difference. \ Thus, things by their very nature are \ That in dependence upon which valid cognitions are \ But they are not themselves established by valid cognition; \ If they were, they would be reality itself. \ An instance of water that is established \ By the valid cognition of one’s own apprehension \ Is not independently established under its own power. \ It is not established by ultimate reasoning, \ Nor is it [established] for a hungry ghost. \ If one determines the objects of one’s own perception \ By means of direct perception and inference, \ One is not deceived with regard to engaging and avoiding \ The objects of those [valid cognitions]; \ So valid cognition is not pointless. \ Thus, when we say “a single instance of water,” \ We refer to the visual perception of human beings. \ In the divine context, \ A single instance of nectar is understood as the basis \ of perception. \ When water is seen as pus, water, and nectar, \ The three are not mixed together. \ If one of those three were not valid, \ Then it could not be established as validly cognized \ By being cognized as a different substance, and \ All three objects of visual perception would be non-existent. \ If this instance of water perceived by a human being \ Were not water, it would not be viable as water for another, \ And “water” would be completely non-existent. \ In such a system, a system of valid cognition \ Would also be untenable. \ Thus, the object of a sense faculty \ That is undistorted by accidental conditions \ Should be established as validly cognized, \ As in the appearance of water and mirages. \ Thus, in the context of hungry ghosts \ Karmic obscurations cause clean water \ To appear as pus, but if the fault [of such obscuration] \ Is dispelled, it then appears as water. \ For this reason, what is seen by human beings \ Is posited contextually as validly cognized, \ Because the other is distorted by perceptual fault. \ For now water is established by a valid cognition. \ But if one analyzes with ultimate reasoning, \ Everything is the appearance of karmic propensity. \ Since [for sublime beings] water appears \ As the pure realms and kayas, \ The human perception cannot itself \ Be established one-sidedly as the [only] valid cognition. \ Thus, by progressively purifying the causes of obscuration, \ It is reasonable to posit higher forms of seeing \ In relation to lower forms of seeing. \ Since the final nature of things is unique, \ The valid cognition that sees only it \ Is likewise unique; a second type is impossible. \ Reality is a unique truth, coalescence, \ And valid cognition is self-arisen gnosis. \ Since there is nothing to abandon except unawareness, \ It is simply a case of awareness and unawareness. \ Thus, this system of valid cognition \ Establishes the nature of all appearances as deities. \ This is the unique tradition of the early translations, \ The lion’s roar of the elegant works \ Of the omniscient Rong zom Pandita. \ Other [systems] do not explain [this] point correctly; \ In this respect whatever other systems say is contradictory. \ The claim that the common object of perception \ Is either appearance or emptiness is untenable. \ If it were only emptiness, \ It would be possible for any sentient being \ To perceive space as vases, \ And vases would disappear like space. \ If emptiness without appearance \ Were viable as an object of perception, \ What would not appear? \ Things would either be permanently existent, \ Or become entirely non-existent, being causeless; \ Either way, it is the same. \ In the context of emptiness there is no appearance, \ Because they are contradictory; \ If there were something non-empty, \ It would contradict the position \ That mere emptiness is the basis of appearance. \ “Well, didn’t you say earlier \ That appearance and emptiness are not contradictory?” \ Here, the object of visual perception is understood \ In the context of conventional valid cognition, \ For which existence and non-existence are contradictory; \ On the basis of a single thing the two truths \ Are noncontradictory only for gnosis. \ If a mere appearance bereft of emptiness \ Were not viable as the basis of appearance, \ That appearance could appear any which way; \ For there is no appearance that is not \ Distinguished in one way or another. \ [A non-empty appearance] is not established as the basis \ of appearance, \ It is not perceived by a valid cognition that causes one \ to know it; \ To say that it exists is only a claim. \ If whatever appeared were entirely separate, \ Nothing other than it could appear; \ Because it would be a non-empty appearance, \ It would be immune to an ultimate analysis. \ Whether one understands the basis as water, pus, \ Nectar, or whatever, there is contradiction. \ If that water were pus, \ How could it appear as water? \ If it were water and not pus, \ How would it appear otherwise as pus, etc.? \ If you say that the object that appears to hungry ghosts \ Is water, then you would have to accept that the pus \ That appears is non-existent. \ For aside from whatever appears to oneself, \ There is no separate basis of appearance, \ Because if there were it would be something different, \ Like pillars and vases, having a single basis but being \ Therefore the coalescence of appearance and emptiness, \ Or the absence of true existence and mere appearance, \ [Is equivalent to] the original pure equality of all \ In the great equal taste of the coalescence \ That is free of partiality and extremes. \ In that way, when one determines the essence of accomplishment \ In the Great Perfection of equality, \ In the context of the path where one cultivates \ [that essence], \ In dependence upon the vision of purity, \ Impure appearances self-liberate. \ Hence one attains confidence in the meaning \ Of the statement from the vajra scriptures, \ “Dharmakaya, which is the purity of all appearances.” \ So, in the Magical Net Tantra, it is taught that \ The continuous appearance of the five aggregates \ Is the “pure divine body of thatness”; \ This is confidence in the intended meaning [of that \ Similarly, when the apprehension of pus is removed, \ It is realized to be delusion, and by cultivating that \ Water appears in its place. \ A great bodhisattva [on the] pure [stages] \ Sees countless Buddha fields in each drop of water, \ And water itself manifests as Mamaki. \ On the bhumi where the two obscurations are finally \ One sees the great equal taste of coalescence. \ As for pure vision, \ If in order to abandon all obscurations \ The unerring reality of things is seen \ By it and it alone, \ It is taken to be the final valid cognition \ And is established for those with the eyes of reason \ Who abide on the pinnacle of the establishment of \ the statement \ “Everything abides originally in the purity of dharmakaya.” \ Moreover, this vehicle has thousands \ Of wonderful rays of light. \ The low-minded, like spirit birds, \ Are as if blind to it. \ Although it cannot be incontrovertibly proven \ That the final space of equality \ Only appears as divinity, \ To the extent that the expanse of original natural purity \ And its apparent aspect, the wisdom body, \ Are inseparable, the apparent aspect is \ Originally pure divinity, \ And cannot be harmed by ultimate reasoning, \ For the expanse of coalescent form and emptiness, \ Which is free of the two obscurations, \ Is the final suchness of things. \ Aside from this, whatever else one analyzes \ Is not the final meaning; \ For if the two obscurations are not completely abandoned, \ Abiding and apparent natures are always discordant. \ Contextual appearances in the practice of the path \ Are like healing a cataract; \ By purifying defilements of the subject, \ The object is likewise seen in its purity, \ Because for a pure subject \ There are no impure objects. \ Thus, when an ordinary person becomes a Buddha, \ [There is no impurity], but impurity still appears to others, \ Because they obscure themselves with their own \ Thus, although object and subject \ Are originally pure, \ They are obscured by adventitious defilements, \ So one should strive to purify them. \ Because there is nothing impure with respect to \ The purity of one’s own nature, \ There is the equality of natural luminosity. \ Not realizing it, one apprehends \ Various appearances individually. \ A childish person whose mind is attached \ Is an ignorant child whose ignorance enslaves him; \ But everyone who realizes this will seize \ The citadel of fruition in the state of equality, \ And become victorious in self-arisen gnosis \ In the fundamental expanse of the three times and \ of timelessness. \ This system, which accepts the principle \ Of great pure equality, is well established. \ Because appearance and emptiness are not established, \ Whatever can appear appears anywhere and everywhere. \ However else you look at it, \ Nothing can appear anywhere. \ The way to gain confidence in this system \ Is the path of emptiness and dependent origination. \ If one gains certainty in appearance and emptiness, \ In the self-arisen changeless mandala, \ Profound tolerance will be born within oneself \ For the inconceivable dharmata \ And for the emptying and non-emptying [of the limits of \ In the width of an atom \ One sees as many Buddha fields as are atoms, \ And in a single instant an aeon appears. \ With certainty in the absence of true existence \ Which is like an illusion, \ One can enter the range of Buddhahood. \ One may have disciplined oneself and thought for a \ hundred years \ About the meaning of the words of different philosophical \ systems, such as \ The undifferentiability of one’s own appearances [and \ their basis], \ The absence of partiality and extremes, \ The inconceivability of the fundamental expanse, \ The dharmata that is not established anywhere, \ The coalescence of form and emptiness, etc., \ Yet if one lacks the cause of prior familiarity, \ Then, even if one’s intellect and training are not \ One will not get it. \ Thus the hundred rivers of elegant explanations \ In which flow the quintessences \ Of all philosophical systems \ Pour into this great ocean, which is amazing. \ Other modes of appearance \ That appear in the process of transformation are indefinite; \ The consummate gnosis of coalescence \ Sees the infallible meaning and is changeless. L1: Topic 7 \ When analyzing whether or not there is a position \ In the Great Madhyamaka of non-elaboration, \ Earlier scholars univocally stated \ That our own Madhyamika system has no position, \ Because existence, non-existence, being, and nonbeing \ Do not exist anywhere. \ In our texts, all the philosophical explanations \ Of path and result and relativity \ Are accepted as our own position, so \ To say that all conventions are only set forth \ From other people’s perspective \ Is to contradict both the words and the meaning. \ According to Klong chen rab ‘byams, \ Earlier scholars veered to the extremes of \ Asserting that Madhyamaka has or does not have a position; \ Each of those positions has defects and qualities. \ Thus, when approaching the nature of reality, \ Nothing is established in the original state; \ What then is there to accept as a position? \ Therefore, because a philosophical system \ Is a position about the nature of things, at the time of \ debate, etc., \ No position is taken, in accordance with the original state. \ In meditative aftermath, the systems of path and result— \ Whatever and however they are posited— \ Are expounded according to their respective positions, \ Without confusing them. \ Klong chen pa said, “From now on, if someone knows \ how to \ Expound this, it is because of my elegant explanation.” \ In that respect, some Tibetan scholars \ Established and overestablished the fact that \ Their own systems had a position. \ But if one does not differentiate the context, \ Because the meaning of the original state \ Is not established anywhere, it is difficult \ To assert one-sidedly that one has a position. \ If you say “Madhyamaka is our system,” \ It should refer to the way that the Madhyamika system \ Approaches the ultimate meaning. \ Anything else is not our own system, \ Because when other systems are approached \ By a Madhyamika, they cannot be established. \ Thus, if the Madhyamika accepts [deceptive reality], \ Then he accepts it as established by its own power, \ Because it is established by the force of reasoning. \ That position would be established ultimately \ And thus be immune to analysis. \ If our own system had no position, \ This would contradict the statement, \ “We do have a position \ [That accords with worldly renown].” \ We would have two positions according to \ Whether or not there is analysis. \ If both of them were definitely true, \ Would “our system” be each of them separately, \ Or would it be both of them together? \ If it were each of them separately, then \ Each would contradict the other. \ If we do not accept “existence” \ But do accept “non-existence,” \ The position of “existence” would not \ Even be conventionally acceptable, \ Because of only accepting non-existence. \ If we accepted both of them together, \ Having removed that which is susceptible to analysis, \ We would posit something not harmed by reasoning. \ Thus, both existence and non-existence \ Would be immune to analysis. \ Accordingly, both existence and non-existence \ Cannot be mixed together; \ For if they were, then even though one \ Could realize [coalescence] through analysis, \ When not analyzing, existence would be engaged. \ So what good would analysis do \ For eliminating clinging to deceptive realities? \ For deceptive reality to be established \ Through analysis is irrational. \ If there were no reality beyond the mere \ Exclusion of a negandum, an absolute negation, \ That modal apprehension could not have \ An apparent aspect; so why would this be any different \ Than the position of someone who thinks \ That view, meditation, and action are simply non-existent? \ For there would never be any need to meditate \ In accordance with the nature of things. \ Therefore, according to the statement \ of the Omniscient One, \ Our system should be understood as follows: \ If ours is to be a definitive Madhyamika system, \ It must be the Great Madhyamaka of coalescence, \ Or the nonelaborated Madhyamaka. \ Because, by defining it according to \ The gnosis of sublime equipoise, \ All extremes of existence, non-existence, and so forth, \ Are completely pacified. \ That path that objectifies emptiness alone \ Succumbs to each of the two realities one-sidedly; \ That trifling point of view \ Is neither coalescent nor unelaborated. \ Coalescence means the equality of \ Existence and non-existence, or of form and emptiness; \ Whereas that view is just the subjective aspect \ Of the expanse of ultimate emptiness. \ Among all types of reification, such as \ The elaborations of existence and non-existence, \ This is nothing but an elaboration of non-existence, \ Because it reifies [emptiness]. \ Therefore, from the perspective of Great Madhyamaka \ There is no position whatsoever. \ In order to realize the equality of appearance and emptiness, \ It is free of all proof and negation such as \ Reality, unreality, existence, and non-existence. \ According to the sense of [ultimate] reality, all things \ Cannot be asserted through rational proof; \ Therefore, there is nothing to have a position about. \ Thus, although the ultimate meaning of reality \ Has no position, in the way things appear \ There is a position on the conventions of each of the \ two realities; \ With respect to how the two realities abide inseparably, \ They are both simply ways of appearing. \ With respect to the gnosis that \ Sees that they are inseparable, both valid cognitions \ Are fragmentary, because with only one of them \ Both realities cannot be apprehended. \ Therefore, if the wisdom of ultimate and \ Conventional valid cognition \ Both engage a vase, etc., \ Two essences are found. \ But when one is engaged, the other is not, for \ In the mind of an ordinary person the two realities \ Can only appear in succession. \ Thus, the positions based on each type of engagement \ Are established in fact. \ “Well, don’t the faults of having or not having a position, \ And the internal contradiction of the two realities \ That you have ascribed to others above \ Apply just as well to you?” \ By making subtle distinctions, \ I have differentiated the path Madhyamaka and \ The equipoise Madhyamaka that is the main practice. \ Since my explanation distinguishes great and little \ With respect to coarseness and subtlety, \ Cause and effect, consciousness and gnosis, \ How can that defect apply to me? \ Thus, the Great Madhyamaka \ With no position is our ultimate system. \ In the context of meditative aftermath, \ When the two realities appear separately, \ All the proofs and negations engaged by \ The validating cognitions of each of the two realities \ Are for negating various misconceptions; \ But in the original state, there is \ No position of refutation or proof. \ Therefore, in the original state \ The two realities are not divided, \ Because neither of their positions \ Is established in truth. \ If [a position] is posited [conventionally about either] of \ the two [truths], \ It is only with respect to the way things appear. \ For the time being, each is established as true \ In its own context, so there is no contradiction, \ And the fault of immunity to analysis, etc. does not apply. \ Real entities are not immune to analysis; \ Nor are unreal entities immune to analysis. \ In the final analysis, they are the same; \ They are just designated contextually. \ Something that exists by consent, without investigation, \ Is a mode of appearance, not the way things are; \ Whatever is seen by the rational knowledge \ That analyzes truthlessness is considered \ As the way things really are. \ This is an ultimate reality in relation to \ Deceptive reality, but in the final analysis \ It is just a conceptual ultimate. \ If the way things appear and the way things are \ Are mutually exclusive, \ The four faults of the two realities being different are \ If the two realities are mutually inclusive, \ The four faults of the two realities being identical are \ In this way, Buddhas and sentient beings \ Are just the way things are and the way things appear; \ The claim that they are cause and effect \ Should be known as the Hinayana system. \ Because the way things are and the way they appear \ Are not posited as either the same or different, \ There is absolutely no logical fault, such as \ Sentient beings appearing as Buddhas, \ The path and practice being pointless, \ The cause residing in the effect. \ However things may be in reality, \ They are obscured by obscurations, \ And do not appear as such. \ Everyone accepts the need to practice the path. \ Because the two truths are not contradictory, \ Though the two views of “existence” and “non-existence” \ Are posited, how could they be contradictory? \ Because they are not mutually inclusive, \ The two positions are formulated. \ For this reason, as long as the two realities \ Are engaged by minds for which \ They appear separately, \ Both realities are quite equivalent in force, \ And there is no one-sided position about either of them. \ The determination of the emptiness of truth as \ And the determination of appearance as “existence” \ Are the objects found or seen alternately by each \ Of the two valid cognitions at the time of their engagement, \ And are said to be the two truths. \ Because those two are neither the same nor different, \ It is not possible to one-sidedly discard one \ And accept the other. \ The wisdom that analyzes these two \ Differentiates their respective positions. \ For example, when the dharmakaya is finally attained, \ All minds and mental events without exception \ Cease, conventionally speaking; \ But ultimately there is no cessation. \ In all the texts of all sutras and treatises, \ Among the various kinds of proof and negation \ Some posit ultimate reality, \ And some are stated with respect to deceptive reality. \ With respect to ultimate reality alone, \ The path, Buddhas, sentient beings, and so forth, \ Are rightly said to be “non-existent.” \ It is not the case, however, that \ Without relying on conventions, they are simply \ Though they do not exist, all appearances of samsara and \ Appear, and are established through direct perception. \ Therefore, with respect to conventional valid cognition, \ The path, Buddhas, sentient beings, and so forth, \ Are rightly said to be “existent.” \ But this doesn’t mean that they are really existent \ Without reference to ultimate reality. \ They exist, but are not established as such, \ Because they can be determined by \ An analytical cognition of ultimate reality. \ Thus, those two can never exist \ One without the other. \ “When both are true with equal force, \ Will existent things be non-empty?” \ Both are not established by their intrinsic nature, \ Nor are they, as objects, really different; \ Whatever appears is empty, so what can be non-empty? \ Both are equally apparent, \ So they are established as empty; \ If they were not apparent, how would emptiness be known? \ Thus, both appear together as cause and effect, \ Without contradiction. \ If one is certain that one exists, the other does too: \ They are always inseparable. \ There is no case where one does not \ Encompass the other; therefore, \ Whichever one investigates, it is correct. \ By knowing appearance as emptiness, \ One realizes appearance as realitylessness; \ And by knowing emptiness as appearance, \ One will not conceive emptiness as real. \ Therefore, when they are seen as inseparable, \ One will not revert to seeing them as real. \ The abiding character of whatever appears \ Is emptiness, so they are inseparable. \ If one rejects appearance, \ Emptiness cannot be established independently. \ Therefore, one cultivates the wisdom \ Of meditating on the two realities alternately. \ In the context of this samsara of dualistic perception, \ Gnosis does not appear, \ So the two stainless analytical wisdoms \ Should be upheld without ambivalence. \ When one of these is incomplete, \ The coalescence of gnosis \ That arises from them will definitely not arise, \ Just as fire will not occur without \ Two pieces of wood rubbed together. \ Therefore, a path where method and emptiness \ Are separated is inauthentic \ According to all the Buddhas and vidyadharas. \ Therefore, if one abandons these two causes, \ There is no other way for the great gnosis to arise. \ The essence of gnosis \ Is beyond thought and expression. \ Therefore, aside from symbolic means and mere words, \ It cannot actually be indicated. \ Thus, the teaching of the word empowerment in the \ In the tantras of the vajra essence, and so forth, \ It is taught by words and methods. \ The supramundane gnosis \ Cannot be understood without relying on \ Some kind of verbal expression, \ So the path of the Madhyamaka of the two realities is taught. \ The result of analyzing in the manner of two realities \ Can be established as coalescence itself. \ Therefore, when the two realities are ascertained, \ Appearance and emptiness are taught alternately \ As negation and negandum. \ Their result, the gnosis of coalescence, \ Is taught by many synonyms in tantra. \ Thus, all Madhyamika systems \ Are established by way of the two realities; \ Without relying on the two realities, \ Coalescence will not be understood. \ Whatever the Buddhas have taught \ Has relied entirely on the two realities; \ Therefore, the Madhyamaka that contains \ The positions of each of the two truths \ Is the little Madhyamaka of alternation, \ Which gives the result’s name to the cause. \ The emptiness of the analyzed five aggregates \ Is the mere absolute negation exclusive of the negandum; \ In that respect there is the position of “non-existence.” \ Whatever the causal or path Madhyamaka \ Posits as the two truths, \ Both are our own system. \ It makes no sense to posit the ultimate as our system, \ And say that conventional reality \ Is only from other people’s perspective. \ If that were so, then our own system of the ultimate \ Would be a blank nothingness, \ And we would wind up totally denigrating \ All appearances of the basis, path, and result \ As “delusions to be abandoned.” \ Then a mere expanse of emptiness without obscuration \ Would be left over, while the two types of omniscience \ Would be negated. This would be similar to the sravaka \ Which asserts a remainderless nirvana, \ Just like the blowing out of a candle. \ Thus, the Buddha said that these \ Spaced-out people who denigrate \ The expanse of coalescence as mere nothingness \ Are thieves who destroy the Sakya Dharma. \ With reasoning, one can see how \ That system denigrates the existent as non-existent, \ And one is able to destroy the mountain of bad views \ With the vajra-fire of certainty. \ Thus, in all Madhyamika texts, \ Without establishing the causal Madhyamaka \ Of analytical wisdom through rational analysis, \ The fruitional coalescence is not established. \ Therefore, even if one has rationally determined \ The character of the two realities, \ The fruition is the establishment of the inseparability \ Of the two realities. This is the quintessence of all vehicles. \ Therefore, gnosis \ Does not abide alternately in the two extremes, \ And is beyond intellect; \ Thus it is Madhyamaka, and also great. \ As long as one has not reached gnosis \ By means of alternation, this is not \ The ultimate Madhyamaka that is \ The heart of all Buddhas’ realization (dgongs pa). \ Like fire stirred up by a fire-stick, \ The fire of coalescent gnosis induced \ By the stainless analytical wisdom of the two realities \ Pacifies all elaborations of the four extremes \ Such as existence, non-existence, both, and neither. \ This is the gnosis of sublime equipoise, \ And is considered the fruitional Madhyamaka of \ Not falling into the extremes of the two realities— \ For the analytical wisdom of meditative aftermath \ This may be considered the “coalescence of \ Appearance and emptiness,” \ But for the great gnosis of equipoise, \ Appearance, emptiness, and coalescence \ Are not reified as having some essence. \ Appearance is the object of conventional valid cognition, \ Emptiness is the object of ultimate analysis, \ And coalescence combines these two components. \ Since these are objects of words and concepts, \ The equipoise that transcends them \ Is merely designated as “gnosis known for oneself.” \ [In the context of sublime equipoise,] \ “Apparent,” “nonapparent,” and so forth, \ Are not established by authentic reasoning. \ Thus, as long as one meditates on the two realities \ Alternately, this is analytical wisdom, \ And when there is no such alternation, \ One attains the coalescent gnosis. \ Then one transcends the bare emptiness \ That is the absolute negation that \ Is the analytical exclusion of the aggregates. \ Negation and negandum no longer appear separately. \ The great nonelaborated emptiness that \ Is consummately endowed with the aspect \ Of appearance as method, \ Mahamudra of coemergence, and so forth, \ Have many synonyms. \ Because these are all the gnosis that transcends mind, \ They are inconceivable by any other concepts. \ Because this gnosis is not the object of words and concepts, \ It is not differentiated by \ Implicative and absolute negations, \ Nor as different, nondifferent, apparent, or empty, etc. \ Because it does not fall into any extreme or partiality, \ It is beyond having and not having a position, \ And appears as the nonabiding self-arisen gnosis of \ The coalescent Evam. \ Thus, the ultimate meaning, free of reification and negation, \ That is beyond all positions, \ The state of awareness and the expanse inseparable, \ Is held to be without any expression or indication of “this” \ or “that.” \ However, unlike the “thoughtless agent,” \ It is not something that cannot be known by anyone, \ Because the Dharma lamp of certainty \ Is the consummate gnosis attained subsequent \ To the individually cognized gnosis induced \ By the analysis of stainless reasoning, \ What appears directly to those [yogis] who \ Are free of the darkness of doubt. \ In the sutra path, both method and wisdom \ Are considered in light of each other, \ But here both method and wisdom \ Are realized and cultivated inseparably. \ Both the Great Madhyamaka of coalescence and \ The Great Perfection of luminosity \ Have the same meaning, and their names are synonymous. \ There is no view higher than that, \ For anything other than the absence of the elaborations \ Of the four extremes—which is the nonapprehension \ Of appearance and emptiness alternately— \ Is nothing but some sort of elaboration. \ However, the meaning of coalescence in the sutra system \ Is ascertained through analysis; \ In mantra, it is established through directly experiencing \ The expanse of intrinsic awareness. \ Therefore, “Madhyamaka” refers to the \ Path Madhyamaka of analytical wisdom that \ Investigates each of the two realities, \ And the single savor of the two realities induced by it, \ Which is the Result Madhyamaka of coalescence. \ With respect to the causal and resultant views of sutra \ and mantra, \ The former is the aspect of analytical wisdom, \ And the latter is just gnosis. \ Therefore, this latter is praised \ With the word “great.” \ As for the “the way things are”: \ There is the way things are as the emptiness of entities, \ And the way things are as the inseparability of the two truths. \ The term is the same in both cases, but in fact \ The difference is like the earth and sky. \ Accordingly, the terms “nature of things,” “expanse \ of reality,” \ “Emptiness,” “non-elaboration,” “limit of cessation,” \ “Ultimate,” and so forth, function similarly in different \ But their difference—in terms of final or partial significance— \ Is great, so one must explain them in context, \ Like the word sendhapa. \ Thus, when the seven profound questions \ Were explained with profound, vast, meaningful words, \ The questioner said, with great respect: \ “Alas! Like a frog at the bottom of a well, \ Having not seen the depths \ Of the Dharma ocean of other textual traditions, \ And having tasted only the flavor of the well \ Of our own arrogant view, our pride is crushed \ By these words of yours! \ In the great ocean of sublime spirituality, \ The ecstatic dance of Manjusri, \ Known as “Rong zom” and “Klong chen pa,” \ Is an ocean of the sublime enlightened mind, \ Which possesses many and sundry bejeweled Dharma \ Those who abandon them and hanker after \ The trinkets of other systems are surely deceived! \ Those who have the discerning intellect \ Born of the analysis of the excellent Dharma (chos bzang) \ Are never obstructed by demons. \ As this great lion’s roar of the path of reasoning \ Is proclaimed, will they not find confidence in \ This outstanding tradition of the Lake-born’s teaching? \ Please grant us the opportunity to firmly grasp \ The handle of wisdom’s sword, which cannot be stolen away \ By the refutations of arrogant extremism! \ The profound meaning that is found in the \ Nectar ocean of Dharma learning \ Is like a jewel that should be taken, wherever it \ happens to be; \ One should not just follow the external behavior of \ another person. \ It’s not enough to receive a lot of teachings and talk \ about them, \ For though one seems talented and well trained, one’s \ cannot get this profound point, like a buried treasure. \ But whoever does get it should be known as a spiritual \ As if it were a jewel-encrusted vessel \ For a hundred thousand spiritual treasures, \ My mind realized that it was time \ To accept the beneficence of instructions \ Accomplished in the great ocean of profundity and vastness, \ And I joyfully drank the ocean of the glorious \ King of Nagas. \ Having definitely realized the vast extent of the \ analytical mind \ By the river of eloquent explanations that descend from him, \ One should realize that the source of these explanations \ Is the oral tradition of the vidyadhara lineage, \ Which is like the Lord of Nagas himself. \ Please brighten the lamp of the amazing Dharma, \ Which causes the mind to acquire great strength \ By receiving the springtime nectar that benefits the heart, \ The quintessence that is imbibed \ All at once from the limits of space!” \ When he had shown his respect with these words, \ The sage advised him again, \ Condensing the meaning of what he said before, \ Which converts a shallow mind to a deep one: \ “The lion’s milk of the supreme Dharma \ Is only contained by the vessel of a sound mind. \ Though others may try, it won’t stay in place. \ A vessel that can hold it is like this: \ A is the door of unborn dharmas; \ Ra is the door free of particles; \ Pa is the door of the appearance of the ultimate; \ Tsa is the absence of death, transmigration, and birth; \ Na is the absence of names; \ Dhih is the door to profound intelligence. \ If one focuses on all of these six doors \ In the manner of the two truths \ And accomplishes the samadhi of illusion, \ With one gulp, one will be able to stomach \ The water of the great infinite ocean of phenomena, \ And in the stainless gem of one’s heart, \ The dharani of spiritual brilliance will blaze with glory. \ By the path of certainty that eliminates \ The elaborations of four extremes, \ May we abide in the expanse of fundamental luminosity \ Beyond mind that reaches the original state, \ The state of the Great Perfection Manjusri. \ Having seen the real meaning of remaining in the \ equanimity of \ The vast expanse of the regal view without extremes, \ All the darkness of the crude mind of the four extremes \ Will naturally disappear as the sun of luminosity rises.” \ Thus, the questions asked by that wanderer \ Were explained in the number corresponding to \ The [seven] accoutrements of royalty. \ Thus, a feeble-minded intellectual like myself \ Has received this extremely profound and abstruse meaning \ From the heart of sublime great-minded beings \ And presented it here. \ This elegant explanation like a shower of Dharma \ Is the path trodden by millions of bodhisattvas; \ By listening joyfully, hoping to attain the great goal, \ And by inquiring, the joyous opportunity for blessing has \ Therefore, I have considered these profound \ And vast subjects again and again, \ And just as they arose in the face of the mind’s mirror, \ The Dhih-named one arranged them playfully. \ The profound way of the Buddhadharma, like the limit \ of space, \ Cannot be put into words entirely, \ But if you rely on this Beacon of Certainty, \ You can discover the amazing path of the supreme vehicle.

Analysis of Stiffness and Energy Consumption of Nonlinear Elastic Joint Legged Robot In order to reduce the energy consumption of the legged robot in walking, this paper designs a kind of nonlinear elastic joint from the flexible variable-stiffness joint based on the mammal walking on the limb and optimizes the leg structure of the legged robot. The motor is rigidly connected to the articulated lever. When the lever is accelerated or decelerated, the elastic unit is introduced. The system can be considered as a special variable-rate elastic system. This paper will study it from theory and simulation experiments. Based on the dynamic analysis, a functional relationship between the output torque and the torsion spring stiffness and between the energy consumption and the torsion spring stiffness was established. By finding the extremum, the two optimum torsional spring stiffness that can minimize the required output average torque and the energy consumed during one cycle of motion were deduced. The results show that using this design in a reasonable position can effectively reduce the energy consumption of the system and can achieve up to a 50% reduction in energy consumption. Compared with the wheeled robot, the foot robot has great advantages in adaptability to the external complex environment and has great research value . With the increase of the number of feet, the structure of the foot robot is more redundant, but it improves the fault-tolerant ability of the robot: the damage of one leg will not have a great impact on the robot, it can move smoothly with the cooperation of other legs. However, the foot robot needs to coordinate its limbs, and its motion efficiency is relatively low, and its energy loss is relatively large. In the process of movement, the friction loss between the body and the ground as well as between the mechanisms will also be dissipated by the heat generated by the driving winding due to the need to maintain a fixed joint angle in the process of standing still. Therefore, for the foot robot, how to reduce energy consumption to improve energy utilization is particularly important, which will bring great performance improvement to the foot robot. In recent years, the research on energy-saving optimization of foot robot can be classified into the following four categories: (1) study the energy conversion of bionic joint by analyzing the physiological structure of leg joint of foot animal ; (2) optimize the end trajectory curve under different conditions ; (3) reasonable gait planning and optimal gait generation parameters ; and (4) how to reasonably distribute the foot force . A large number of experiments have shown that human beings and some creatures use their own physiological structure to complete energy transfer and transformation when they are in motion, which is a highly coupled function brought by evolution and the body . For example, in the movement system composed of human bones, muscles, and tendons, the flexible components of muscles and tendons produce the effect similar to that of springs, which are regularly shaped in motion. Variable can reduce energy consumption and overall impact of the system, which can be approximately considered as a load spring inverted pendulum slip model [7, 8]. Inspired by this idea, this paper introduces the spring design idea to the joint motion of the foot robot for energy conversion. Nowadays, many scholars have studied the elastic joint of robot and discussed the structure and control method of the elastic joint, but most of them focus on the performance research of damping and reducing the impact force ; this paper will study the energy consumption brought by it. However, the flexibility of the robot joint has a significant impact on the positioning accuracy of the legs and the accuracy of the end of the trajectory when the robot moves , and the degree of freedom of its control system will increase by one level, resulting in its control more complex and easy to lose stability. Therefore, a large number of scholars have studied the impact on the control model after introducing the flexibility of the joint. Nowadays, there are many significant achievements in system modeling and dynamic modeling of robot with elastic joints. However, for the foot robot with high redundancy, such as the six-legged robot, the lack of accuracy of the foot end trajectory of one leg is not enough to cause the stability imbalance of the whole robot, but the resulting energy consumption is reduced, which can greatly improve the endurance of the foot robot. Secondly, most of the current research uses linear elastic elements and will participate in the whole movement. In this paper, a joint motor is used to drive the joint directly, and a torsion spring is installed at the joint which is in contact with the bar. By improving some elastic joints, a nonlinear elastic system is formed. Only when the joint motor decelerates to the stop and the initial speed is zero to the acceleration, the elastic joint will start to play an obvious role. In this way, it can help the joint motor decelerate in the deceleration process and accelerate in the acceleration process, so as to achieve an energy conversion function and reduce energy loss. In the following chapters, we will start with a simplified model of nonlinear elastic connections. Through dynamic analysis, the relationship between motor output torque and spring stiffness coefficient is found. Then, through the analysis of energy consumption, the value of joint stiffness coefficient is found. After that, we will design some simulation experiments to verify the effect of nonlinear elastic connection on the system. Finally, the relevant conclusions are summarized. 2. Dynamic Modeling and Analysis The single-leg structure of the legged robot model and the prototype are shown in Figure 1. The robot can switch arbitrarily between a hexapod robot and a spherical robot. Each leg has three degrees of freedom and three joint motors. This paper focuses on the second joint, that is to say, only the leg joints are subjected to a simulation experiment with nonlinear elastic joints. For each revolute joint, a similar dynamic model will be generated when the torsion spring of the elastic joint is assembled in a suitable position. They have the same properties and only differ in numerical values. The single-leg simplified model with nonlinear elastic joints is shown in Figure 2. Since this paper only does theoretical analysis and simulation, in order to simplify the model, the ideal position is taken during assembly without actual assembly. Some elastic joints are horizontal and symmetrical with the axis of the revolute joint. Through dynamic modeling, the relationship between joint torque and joint motion parameters of a nonlinear elastic joint robot can be deduced. At the same time, the relationship between energy consumption and joint movement can also be deduced. This paper uses the Lagrangian method to derive the dynamic equations of the elastic joint model. The elastic part of the torsion spring has no moment of inertia and is elastically deformed only in one direction, ignoring the damping of the elastic unit. Assume that the initial momentum of the system is zero. In the theoretical derivation and simulation experiments, this paper concludes that the trajectory is accurately tracked, and there is no error such as offset and phase difference. Figure 3 shows the simplified nonlinear elastic joint mechanism. The parameters of the mechanism diagram are shown in Table 1; the torsion spring that mainly acts in the elastic joint is defined as the main torsion spring , and its stiffness is far greater than that of the secondary torsion spring that only reduces the impact, that is, is much larger than and , and the barycentric coordinate of the lever is set as . The joint space is divided into two parts: an elastic joint with a very small stiffness and an elastic joint with a large stiffness. In order to simplify the equation, it is necessary to make the following assumption. In the collision process, the collision is stable and the time is not counted. The collision is surface contact. The position and direction of the lever and the contact surface do not change during the collision. The inertia of each component does not change, and the deformation does not occur during the collision. Lagrangian equation is still established during collision . As shown in Figure 3, we separately model the dynamics of different stiffness. In an ideal state, assuming that the rigid joint driving torque is , the torque of the torsion spring and the joint rotation angle in the elastic system are, respectively, the functions of the relationship and . When the lever has not yet reached the torsion spring with the large stiffness, the required torque for the joint can be given by The required torque after the lever comes into contact with the large stiffness torsion spring can be given by After the trajectory curve is determined, the joint output torque curve in each period is also determined, that is, the relationship between and time is a determined curve. The sum of the output torque of the joint and the torque generated by the elastic joint is a certain value at a certain point of time. At this time, the greater the torque generated by the elastic joint, the smaller the required output torque of the joint. However, in order to make the dynamic model more accurate, the two parts of the joint space are now integrated into a dynamic analysis. The D-H method is used to establish the coordinate system of the joint. Only the second joint is taken as the research object. The kinetic equation of the robot is represented as follows: where can be given by Equation (4) can be rewritten as Gravity energy can be given by There is almost no elastic potential energy in the system before the torsion spring with a large stiffness is touched, and the stiffness coefficient can be regarded as linear after the main torsion spring is contacted. So it can be considered as a special nonlinear variable stiffness elastic model in one cycle. The elastic potential energy can be given by where is the angle conversion factor when the torsion spring and the rotation center do not coincide with each other, and keeps the value of the rotation angle of the torsion spring and the rotation angle of the joint united. is the rotation angle of the torsion spring with respect to the joint and can be approximated as the joint rotation angle , because the lever and the motor are rigidly connected. is a function of the nonlinear torsion spring for the joint rotation, which can be expressed in this paper by a piecewise function: The total potential energy can be given by The Lagrangian operator can be given by The Lagrangian kinetic equation can be given by According to the Lagrange equation, the joint torque can be obtained as Assume that the joint output rotation function is , then the output torque after contact with the elastic joint can be given by Equation (14) has the same conclusion as equations (1) and (2). Equation (14) shows that before the end of the stance phase and the swing phase, the robot needs a joint motor to decelerate. With the participation of the elastic joint, due to the compression of the torsion spring, the torsion spring will produce an increasingly larger countertorque. It can effectively reduce the output torque of the joint motor under the same combined torque. At the beginning of a new swing phase or stance phase, the joint motor output torque is required to provide sufficient acceleration to the joint. In the model with elastic joints, the torsion spring is compressed when the previous movement decelerates. So the torsion spring and the joint motor collectively output torque in the acceleration phase. When the required torque is the same, the output torque of the joint motor can be greatly reduced. Therefore, theoretically, the design of the elastic joint can reduce the output peak torque of the joint motor, which can effectively reduce the energy consumption of the robot during walking. In order to obtain the best stiffness function of the torsion spring with the purpose of minimizing the output torque, differentiate equation (14) of time and let its value be zero: can be obtained from equation (15). If , then (/rad) is the function of the torsion spring stiffness that minimizes the output torque under this input function. For the robot’s trajectory design, many scholars have done a lot of researches [15, 16], such as composite cycloids, modified elliptical trajectories, and classic multi-item fitting curves. For the joint tip trajectory planning with continuous smooth acceleration and the appropriate initial conditions for position and velocity, a simpler sinusoidal trajectory will be used, i.e., Equation (14) can be rewritten as Let equation (18) be equal to zero and get the result as follows: If , then is the optimal stiffness that will make the output torque the smallest torsion spring. For different trajectories, the specific positions of the main torsion springs for joint contact need different analysis, and it is necessary to ensure that the torsion spring contacts the deceleration when the joint driving torque is greatly reduced, and a symmetrical acceleration torque is required. When using different driving torque curves for acceleration and deceleration, the situation will become more complicated and changeable. 3. Energy Consumption Analysis Without considering other factors such as friction loss, the energy consumed by a single joint model with a nonlinear elastic joint in one exercise cycle can be expressed as where is the joint drive power, is the joint drive torque, is the joint rotation angular velocity, and is the joint rotation angle. From equation (20), it can be seen that the energy consumption is related to the integral of the output torque and the angular velocity at this moment, and the output torque and the joint angular velocity are both functions of time. Equations (16) and (17) can be expressed as If we want to obtain a stiffness that minimizes , then Let to get . If , then is the demand. Assume that the energy consumption ratio is an evaluation standard for energy consumption . This assessment of energy consumption is based on the metabolic effects of the organism [18, 19]. The energy consumption ratio means that the unit of weight requires a unit of energy to dissipate the distance. The smaller the value is, the higher the energy consumption can be obtained. where represents the total mass of the robot and is the standard of the robot’s moving distance. The rotational movement of the joint is the main movement, so make some changes to equation (23) to conform to the simulation movement in this paper. Let the moving standard and the total weight be replaced with the moment of inertia about the rotation axis, equation (23) can be rewritten as 4. Simulation Experiments 4.1. Experiment Environment Using Adams software for simulation experiments, the experimental model was modeled in Creo software and imported into Adams, as shown in Figure 4. After removing the components that are not related to the movement, add the parameters of friction and force of the restraint and the motion parameter. Table 2 is the setting of the frictional force parameter of the joint rotation; Table 3 lists the parameter setting of the contact force between the lever and the torsion spring; Table 4 is the setting of the secondary torsion spring physical parameters; Table 5 lists the parameters of the main torsion spring physical. The joint adopts a general point driving method. After inputting the endpoint trajectory, the required torque value of the joint is sampled. In order to keep the motor from no-load spinning, the end of the joint is loaded with an equivalent mass load. The trajectory function is shown in where is time. The model with a nonlinear elastic joint was set as the experimental group, and the model without the elastic joint was the control group. The physical dimensions and simulation parameters of the two models were basically the same, as shown in Table 6. 4.2. Simulation Experiment Results The simulation experiments found that under the same quality and external conditions, the output torque required by the experimental group model compared to the control group model was greatly reduced. The simulation results of the required output torque at the joint of the control group model are shown in Figure 5, and the simulation results of the required output torque at the joint of the experimental group are shown in Figure 6. We changed the stiffness of the torsion spring and carried out the simulation experiment, as shown in Figure 7, which were the required output torque of the joint when the stiffness of the torsion spring was different. It can be clearly seen from Figure 6 that due to the limitation of the function in equation (13), the joint torque is a normal output before contacting the main torsion spring. In Figures 5 and 6, there is a coincident torque output value under different stiffness. Figure 7 show the torsion spring has a stiffness ranging from to . The required output torque of the joint first slowly decreases to a certain value and then begins to rebound. This rebound phenomenon is most obvious when the joint gets close to the limit of rotation. The torsion spring in this case has an optimum stiffness that minimizes the required output torque. The trajectory parameters and the model physical parameters are brought into equation (19), where and , resulting in an optimum stiffness . The above and nearby values were used as the stiffness of the simulation model. The results () are shown in Figure 8. The experimental results showed that the simulation experiment was not exactly the same as the theoretical value and there was a certain error. After several simulations around , the simulation results showed that the required output torque was slightly compared with other stiffness at the same time around . The following numerical calculation will verify its correctness. Matlab was used to process the discrete torque derived output torque from the Adams. After using the trapez function to integrate its data over a period, and then dividing it by the total time in one cycle, the average output torque under different stiffness per unit time was obtained. As shown in Figure 9, the simulation result around the theoretical derivation is shown in Figure 10. Figure 10 shows that the average moment around is the lowest, and the maximum error between the simulated value and the theoretical value is about 3.7%. The trajectory curve was brought into equation (21). So that it was integrated over the interval to , resulting in the torsional spring stiffness that optimized the energy. The simulation results of the required joint output torque under the stiffness around are shown in Figure 11. Then, Matlab is used to multiply the derived angular velocity and the output torque discrete data at the sampling point, and the trapez function is used to integrate the data over a period of time. After bringing its value into equation (24), it got the energy ratio at different stiffness, as shown in Figure 12. The simulation result around the theoretical derivation value is shown in Figure 13. Figure 13 shows that the average moment around is the lowest, and the maximum error between the simulated value and the theoretical value is about 1.95%. According to the data of simulation experiments, the difference between the energy ratio in the case of nonlinear elastic joints and the energy ratio in the case of the optimal energy-saving torsion spring stiffness was first calculated. Then, calculate the ratio of the difference and the energy consumption when no elastic joint was added. It can be concluded that energy consumption of up to 52.81% can be reduced in the case of an optimal energy-saving stiffness. 4.3. Discussion of the Experimental Results Chen et al. use a torsional spring to participate in the movement of the quadrupedal robot joint, assisting the hydraulic drive to significantly reduce the hydraulic output during walking, thereby reducing the energy consumption. Zhou and Fu propose a method based on biomechanics and bionic control strategy for the compliant quadruped robots. It mainly includes mechanical compliance element and control compliance element that can reduce the contact force both on hip and knee joints. Finally, the experiment results proved to effectively optimize joint torque and prevent damage to the robot. Ma et al. , inspired by small jumping animals, designed a leapfrog robot named “Grillo.” Its passive forelimbs can cushion the landing impact and can be converted into elastic potential energy and released during jumping to increase the peak power output of the motor. The above papers and the theoretical analysis of this paper and the simulation experiments indicate that adding elastic elements at certain positions of the robot joint can indeed reduce the energy consumption of the robot during walking, but may cause instability of the output torque. It may cause the decline of the stability and precision of endpoint of the robot, so it is necessary to weigh the advantages and disadvantages brought about before the introduction of the elastic joint. (1)This paper designs a nonlinear elastic joint and deduces it theoretically. It is proved that this design can effectively reduce the energy consumption of the legged robot when walking. The simplification of the model may bring about errors between the theoretical analysis and the simulation experiment. Even if there is an error, it can reduce about 50% of the energy consumption compared with the full-rigid joint robot without considering external friction and gravity(2)This design will reduce the order of the robot’s control equation to a certain extent and increase the difficulty of control. However, for a more redundant robot, the reduction in energy consumption may increase the overall performance of the robot. This paper will be validated in later work. However, the position of the elastic joint in the overall mechanism is not flexible enough. If the joint rotation trajectory curve does not coincide with it, it may cause a reaction. That is to say, when the output torque of the joint motor accelerates the joint, the joint mechanism may come into contact with the torsion spring in advance, which causes the joint motor to drive to provide extra torque. And in the real world, there are constants of the acceleration of gravity. The upper and lower elastic joints cannot be assembled symmetrically. It is even more difficult to fit the trajectory to the elastic joints(3)The future work of this paper is to give the appropriate elastic joint assembly position and the optimal stiffness for the more general end trajectory and try to design a universal adjustable nonlinear elastic joint The data that support the findings of this study are available from the corresponding author upon reasonable request. Conflicts of Interest The authors declare that there is no conflict of interests regarding the publication of this paper. The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article. This work was supported in part by the National Natural Science Foundation of China (Grant No. 51205075) and the Fundamental Research Funds for the Central Universities (Grant No. HEUCF180703). A. Schneider, J. Schmitz, H. Cruse, and Ieee, “A bio-inspired joint controller for the decentral control of a closed kinematic chain consisting of elastic joints,” in Proceedings of the 44th IEEE Conference on Decision and Control, pp. 233–238, Seville, Spain, December 2005.View at: Publisher Site | Google Scholar J. J. Craig, Introduction to Robotics: Mechanics and Control, 3rd Ed, Pearson/Prentice-Hall, Upper Saddle River, NJ, 2005. S. Wang, X. Zong, and G. Chen, “Research on energy consumption of hydraulic quadruped robot based on diagonal gait,” Chinese Journal of Beijing Institute of Technology, vol. 36, no. 4, pp. 399–404, 2016.View at: Google Scholar G. R. Colborne, S. Naumann, P. E. Longmuir, and D. Berbrayer, “Analysis of mechanical and metabolic factors in the gait of congenital below knee amputees. A comparison of the SACH and Seattle feet,” American Journal of Physical Medicine & Rehabilitation, vol. 71, no. 5, pp. 272–278, 1992.View at: Publisher Site | Google Scholar Z. 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The Design Of Fiscal Rules /1,2 Davide Lombardo International Monetary Fund Turkish Banks’ Association Seminar February 26, 2009 1/ The views expressed in this presentation are the author’s, and not necessarily the IMF’s. 2/ Based on the December 1-2 2008 workshop organized by the Treasury, SPO, and MOF together with the Fiscal Affairs Department of the IMF. In particular it borrows on the presentations by Debreu and Debreu and Kumar (available on the website of the Turkish Treasury—see press release dated 18 Dec 2008). Outline First principles: (Definition, Rationale, Function) Taxonomy Design trade-offs Basic properties of selected rules Monitoring and enforcement Fiscal rules around the world: EU sample Broader sample Lessons from stylized facts Conclusions Definition Mechanism placing durable constraints on fiscal discretion through numerical limits on budgetary aggregates (expenditure, revenue, budget balance and/or public debt). Any fiscal policy rule is made of 3 parts: Numerical target/ceiling on one or more specific fiscal indicators Explicit cost for non-complying policymakers A monitoring/enforcement procedure. What is not a fiscal policy rule? The annual budget is not a rule, as it only applies for one year. An IMF-supported program, as it is not expected to be a permanent feature of fiscal policy. The new UK ―rule‖: “To set policies to improve the cyclically-adjusted current budget each year, once the economy emerges from the downturn, so it reaches balance and debt is falling as a proportion of GDP once the global shocks have worked their way through the economy in full.” Rationale: why constraining discretion? Unconstrained fiscal policy may be perceived as systematically deviating from desirable policies. In practice: pro-cyclical and/or unsustainable policies. Reasons why unconstrained policies can be biased: Time-inconsistency Fiscal federalism/monetary union Political economy: Myopia, re-election concerns, fiscal illusion, distributive conflicts, and coordination failures. Other disciplining mechanisms? Markets: yes, but unreliable (comes too late and to strongly) Delegation to non-partisan technical agencies: poses considerable issues in terms of democratic accountability. Functions of fiscal rules Rules are commitment devices: they make deviations from socially desirable targets too costly for policymakers. Rules are signaling tools: they can help policymakers signal their genuine commitment to sustainable and stabilizing policies. Rules can also serve to anchor expectations, thus reducing uncertainties and risk premia. Principles of a taxonomy Many parameters enter the design of an actual fiscal policy rule: fiscal target(s) nature of costs in case of deviation monitoring/enforcement escape clauses, etc. A good rule must imply good policies in most (if not all) circumstances the policy response induced by the rule to a variety of shocks is therefore key. Key feature of the rules: response to shocks to discretion make sense, but they Limits should NOT: Prevent adequate responses (e.g., let automatic stabilizers play in bad times) Force inadequate responses (e.g., force a fiscal contraction in response to a temporary spike in interest rate, or depreciation of the exchange rate) Let the bias unchecked in specific circumstances (e.g., allow for procyclical expansions). The design of fiscal rules There is no one-size-fits-all fiscal policy rule. Much depends on: Constellation of shocks prevalent in the economy Nature and magnitude of policy bias under discretion. A good rule is (Kopits and Symansky, 1998): …simple …transparent …coherent with the final goal …but mindful of other goals of public policy: • Not discouraging structural reforms • Allowing for fiscal stabilization (time-frame, cyclical adjustment) • Avoiding low-quality adjustments (undue tax hikes, cuts in quality/priority spending). Two key trade-offs Credibility-flexibility: allowing for greater responsiveness to shocks could undermine credibility of attaining the final goal. Flexibility-simplicity: combinations of rules can relax somewhat the credibility-flexibility trade-off, at the cost of simplicity and transparency. The devil is in the details Other design issues Coverage of the government sector Policy coordination in federal/decentralized systems Legal foundations Need for adequate budgetary procedures (preparation, execution, and ex-post auditing of budgets) Need to limit scope for creative accounting Pros and cons of selected rules Simple rules: Debt rules Budget balance rules Expenditure rules Revenue rules Combined rules: e.g., debt objectives coupled with binding deficit ceilings, debt/budget balance objectives coupled with binding medium-term expenditure ceilings Deficit Rules Balanced budget and overall deficit limits Pros: pin down asymptotic properties of debt directly address the deficit bias can be simple and transparent (unless cyclical considerations, escape clauses, etc). Cons: procyclical (unless cyclically adjusted or ―over the cycle‖) could force cut in investment… • golden rule, but these open door to manipulations, and do not guarantee debt sustainability. Overall vs. primary balance? Primary is more robust to volatility in interest payments. Debt Rules Upper limit (or desirable time path) for gross or net public debt Pros: Directly tackle debt sustainability Can be transparent and simple Can accommodate large shocks if debt is well below the ceiling Cons: Lack controllability Can force procyclical and undesirable responses to interest rate and exchange rate shocks if debt is close to the limit Borrowing constraints generally applied at regional and local levels Expenditure Rules Set expenditure ceilings (nominal or in terms of GDP) or limits on growth (nominal or real). Pros: tackle one of main source of deficit bias translate directly into budget preparation minimizes pro-cyclicality risks fosters medium-term planning Cons: May leave room for discretionary tax cuts sustainability? complex to design (nominal vs. real, exclusions,…) Best used in combination with deficit or debt rules. [Revenue Rules] Rules imposing limits on revenues with a view to: Contain size of the public sector / tax burden Allocate ex-ante revenue windfalls (e.g., due to surprisingly high growth) Pros: Can reduce procyclicality in good times Cons: Limited impact on deficit bias if not coupled with other rules Can be procyclical in case the rule targets a given revenue-to- GDP ratio (due to the progresssivity of the tax system) Combination of rules Drawbacks with individual rules have led most countries to adopt combination of rules Most common combinations: Debt and overall deficit ceiling and overall balance target (e.g., Maastricht) Overall balance and expenditure ceilings (Sweden, Finland, Netherlands, etc). Monitoring Ex-ante auditing: Is the government proposed fiscal policy consistent with the rule and its objectives? Ex-post assessment: Was the rule implemented? • Analysis of underlying policies • Certification role: track attempts to resort to creative accounting • Assessment of ex-post adjustments to the target (CABs,…), activation of escape clauses, etc. A key pre-requisite Budgetary transparency: Fiscal data need to be accessible, timely, and reliable Comprehensive periodic reporting requirements Clarity about the budget preparation and execution procedures Clarity about roles and responsibilities of different levels of government Fiscal councils Independent institutions with country-specific mandates adequate and highly qualified staff guaranteed multi-year budget. Can contribute to greater transparency and accountability of fiscal policy. Can help monitoring and enforcing a rule. The case of Chile Fiscal institutional setup designed to buttress fiscal sustainability and dampen cyclical fluctuations. Rule: maintain a structural surplus of 1 percent of GDP for the central government. Fiscal expenditures follow the dynamics of structural revenue. Two independent expert panels provide key projections: 1. the inputs (growth of the labor force, real investment, and total labor productivity) for estimating trend GDP 2. ten-year forecasts of the price of copper. Independent panels enhance policy credibility, while allowing some policy flexibility. The case of the Netherlands The Central Planning Bureau plays a key role in the budgetary process: Provides projections and forecasts Estimates desired structural budget balance Vets the programs of all political parties (which are thus subject to reputational sanctions) Undertakes analysis of specific budgetary projects. High credibility borne out of tradition Fiscal Rules: International Experience Stylizedfacts for EU (EC database: annual data over 1990-2005), and the rest of the world (IMF-FAD database on the design and implementation of fiscal rules) Worldwide: 81 countries identified as having fiscal policy rules, with complete information for 77 of them Growing appetite for fiscal rules (EU) 0.90 Maastricht SGP EMU 0.80 0.70 0.60 Raw index 0.50 0.40 0.30 0.20 EU-15 Euro-11 0.10 NMS Big-4 0.00 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Emerging Market and LICs are catching up with industrial countries Number of fiscal rules by category of countries: 1990-2008 80 Industrial EU-27 70 Emerging 60 LIC's 50 40 30 20 10 0 1990 1995 2000 2008 Budget-balance and debt rules dominate, but expenditure rules are increasingly popular Number of countries with at least one fiscal policy rule (by type of rule) 80 Budget balance 70 Debt Expenditure 60 Revenue 50 40 30 20 10 0 1990 1995 2000 2008 Expenditure and revenue rules are thus relative newcomers Median duration of existing fiscal rules (in years) 10 9 8 7 6 5 4 3 2 1 0 Budget balance Debt Expenditure Revenue Same trends in EU sample 60 Revenue Rules 50 Expenditure rules Debt Rules Budget balance rules 40 30 20 10 0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Budget balance rules appear stronger and have wider coverage Selected features of rules-based fiscal frameworks by type of rule (common features only) Budget balance 1.00 Debt 0.90 Revenue 0.80 Expenditure 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Independent Independent Coverage (median Statutory basis (median enforcement (relative monitoring (relative relative to maximum relative to maximum frequency) frequency) possible score) possible score) This is reflected in synthetic measures of strength and coverage Indices of strength (max= 6) and of coverage (max=4) by type of rule 3 4.0 Strength Coverage 3.5 2.5 3.0 2 2.5 1.5 2.0 1.5 1 1.0 0.5 0.5 0 0.0 Budget balance Revenue Debt Expenditure Strength and coverage relatively similar across countries Indices of strength (max= 6) and of coverage (max=4) by country category 3.0 4.0 Strength Coverage 3.5 2.5 3.0 2.0 2.5 1.5 2.0 1.5 1.0 1.0 0.5 0.5 0.0 0.0 Industrial Resource rich LICs Emerging markets Features by country groups Selected features of rules-based fiscal frameworks (relative frequencies by country groups) 1.00 Industrial Emerging LIC's Resource-rich 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Independent M TEF Independent FRL (all rules) Provision(s) Exclusion of Independent enforcement (relative to forecasts (all for fiscal high-quality monitoring (all procedure (all BBR and DR) rules) stabilization spending rules) rules) (relative to (relative to BBR) BBR and ER) Fiscal rules lower interest rates (EU-25) Table 2. Determinants of Long-Term Interest Rates in the EU-25 (1990–2005): Selected Sub-Samples Estimator: Arellano-Bond Fixed effects 1/ No Unstable EU-15 Non-EU15 Euro Non-euro High debt Low debt Delegation Stable gov. delegation gov. Long-term interest rate (lagged) 0.55 *** 0.35 *** -0.11 0.53 *** 0.47 *** 0.31 *** 0.39 *** 0.39 *** 0.56 *** 0.13 (8.77) (4.26) (-1.44) (5.67) (9.98) (6.35) (4.52) (4.58) (6.41) (1.53) Short-term interest rate 0.26 *** 0.58 *** 0.50 *** 0.23 *** 0.26 *** 0.50 *** 0.40 *** 0.23 *** 0.12 0.43 *** (4.23) (16.99) (10.52) (3.07) (11.32) (13.49) (4.29) (2.56) (0.92) (6.93) Cyclically-adjusted primary balance -0.08 ** -0.01 0.00 -0.03 -0.07 * -0.04 -0.07 -0.05 0.03 -0.04 (-2.44) (-0.03) (-0.26) (-0.39) (-1.77) (-0.82) (-1.34) (-0.88) (0.56) (-0.72) Public debt (lagged) 0.01 0.00 -0.01 0.00 0.01 0.00 0.02 ** 0.01 0.00 0.01 (0.73) (0.22) (-0.85) (0.27) (0.46) (0.08) (2.04) (0.61) (-0.29) (0.88) Real GDP growth 0.13 *** -0.01 0.04 ** 0.11 ** 0.26 *** 0.04 0.06 ** 0.09 0.07 0.11 * (3.04) (-0.06) (2.14) (2.38) (4.29) (0.91) (1.93) (1.39) (1.63) (1.80) Inflation 0.22 ** -0.07 * 0.11 ** 0.22 * 0.45 *** -0.04 0.01 0.33 *** 0.26 * 0.09 (2.10) (-1.77) (2.12) (1.92) (4.22) (-1.07) (0.17) (2.70) (1.88) (1.18) Enlargement (dummy) … … … 0.03 … … … -0.61 … … (0.07) (-1.40) Election year (dummy) … … 0.08 ** 0.38 … … 0.26 0.28 0.47 * 0.02 (2.17) (1.40) (1.48) (1.00) (1.76) (0.11) Stability and Growth Pact (dummy) … … … … … … … -0.29 0.13 -0.96 ** (-1.30) (0.56) (-2.13) Government stability … … -0.03 ** -0.17 * -0.21 *** 0.03 -0.18 *** -0.16 0.14 -0.08 (-1.89) (-1.85) (-3.75) (0.24) (-3.69) (-1.61) (0.59) (-1.19) Ideological range of governing coalition … … -0.02 0.13 * … … -0.14 0.11 *** 0.10 0.15 (-1.53) (1.90) (-1.42) (2.71) (1.52) (1.65) Fiscal Rule Index -0.16 0.10 -0.09 * -0.16 -0.17 -0.31 ** -0.10 -0.41 ** -0.06 -0.43 (-1.58) (0.10) (-1.63) (-0.93) (-1.37) (-2.48) (-1.06) (-2.03) (-0.50) (-1.59) Number of observations 213 47 72 155 107 153 72 148 148 104 Number of countries 15 10 12 25 15 22 9 19 24 21 Test for 2nd order autocorrelation (p-value) 2/ 0.93 0.16 0.10 0.38 0.25 0.21 0.55 0.45 … … Sargan test (p-value) 3/ 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 … … Note: The t-statistics are reported in parentheses (with superscripts *, **, and *** denoting statistical significance at the 10, 5 and 1 percent levels respectively). They are robust to cross-sectional heteroskedasticity. All models include a constant and country effects (not reported). The latter are jointly significant at conventional confidence levels in all equations. The enlargement dummy is equal to 1 for new member states, after they joined the EU. The SGP dummy is equal to 1 for euro area member states after 1998. 1/ Arellano-Bond estimation could not be obtained for one of the subsamples (lagged dependent variable highly insignificant and evidence of second-order autocorrelation). The comparison is thus based on fixed-effects estimates. 2/ Arellano-Bond test of the null hypothesis of no autocorrelation of residuals. 3/ Test of the null hypothesis that identifying restrictions are valid. Debrun and Joshi (2008) Fiscal rules lower interest rates (EU-25) Table 3. Impact of the Fiscal Rules on Long-Term Interest Rates: Specific Features of the Rule (EU-25, 1990–2005, Non-Excessive Public Debt) Specific dimensions of fiscal rules as captured by sub-indices: Statutory basis -0.47 ** (-2.07) All key dimensions Independent body monitoring the rule's implementation -0.48 ** (-2.07) matter Effective Independent body contributing to rule's enforcement -0.52 ** rules deliver lower (-2.01) interest rates. Strength of enforcement procedure -0.40 ** (-2.03) Media impact of the rule -0.47 ** (-2.23) Memorandum items: Overall fiscal rule index -0.45 ** (-2.09) Expenditure rule index 0.05 (0.29) Notes: The t-statistics are reported in parentheses (with superscripts *, **, and *** denoting statistical significance at the 10, 5 and 1 percent levels respectively). Given that subindices are only available for the overall fiscal rule index and the expenditure rule index, the results, obtained with model (6) in Table 1, use the overall index but control for the effect of expenditure rules. Debrun and Joshi (2008) Lessons Historical preference for deficit and debt rule But recently emergence of expenditure rules (in combination with BBRs and DRs) What explains the trends for more rules and more expenditure rules? Increasing realization that purely discretionary fiscal policies are inconsistent with sustainability (growing public debt) Mounting pressures from long-term issues: aging, global warming, etc. Regional monetary unions. Conclusions Well-designed fiscal rules can help improve fiscal performance on average… …thus lowering risk-premia and interest rates on long-term government bonds. But: political commitment is key As are adequate public financial management, monitoring and enforcing systems.

- Open Access A Schauder fixed point theorem in semilinear spaces and applications © Agarwal et al.; licensee Springer. 2013 - Received: 24 May 2013 - Accepted: 10 October 2013 - Published: 22 November 2013 In this paper we present existence and uniqueness results for a class of fuzzy fractional integral equations. To prove the existence result, we give a variant of the Schauder fixed point theorem in semilinear Banach spaces. - fuzzy fractional differential equation - Schauder fixed point theorem - Ascoli-Arzelá-type theorem - existence result The topic of fuzzy differential equations has been extensively developed in recent years as a fundamental tool in the description of uncertain models that arise naturally in the real world. Fuzzy differential equations have become an important branch of differential equations with many applications in modeling real world phenomena in quantum optics, robotics, gravity, artificial intelligence, medicine, engineering and many other fields of science. The fundamental notions and results in the fuzzy differential equations can be found in the monographs and . The concept of fuzzy fractional differential equations has been recently introduced in some papers [3–10]. In , the authors established the existence and uniqueness of the solution for a class of fuzzy fractional differential equations, where a fuzzy derivative is used in the sense of Seikkala. In , the authors proposed the concept of Riemann-Liouville H-differentiability which is a direct extension of strongly generalized H-differentiability (see Bede and Gal ) to the fractional literature. They derived explicit solutions to fuzzy fractional differential equations under Riemann-Liouville H-differentiability. In , the authors established an existence result for fuzzy fractional integral equations using a compactness-type condition. In this paper, we present an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. For this we use a variant of the Schauder fixed point theorem. Since the space of continuous fuzzy functions is a semilinear Banach space, we prove a variant of the Schauder fixed point theorem in semilinear Banach spaces. The paper is organized as follows. Section 2 includes the properties and results which we will use in the rest of the paper. We present an example which shows that a fuzzy fractional differential equation is generally not equivalent to a fuzzy fractional integral equation. In Section 3, we establish the Schauder fixed point theorem in semilinear Banach spaces. In Section 4, we prove an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. Finally, using Weissinger’s fixed point theorem, we give an existence and uniqueness result. for all , . is a complete and separable metric space with respect to the Hausdorff-Pompeiu metric . y is normal, i.e., there exists such that ; - (iii)y is a convex fuzzy function, i.e., for all , and for all , we have y is an upper semi-continuous function. is called the α-level set of y. Then from (i)-(iv) it follows that the set for all . for all , and . where is the Hausdorff-Pompeiu metric. Then is a complete metric space (see ). Proposition 2.1 for all , Define as the space of fuzzy sets with the property that the function is continuous with respect to the Hausdorff-Pompeiu metric on . Let be an interval. We denote by the space of all continuous fuzzy functions on T. A subset is said to be compact-supported if there exists a compact set such that for all . A is level-equicontinuous on if A is level-equicontinuous at α for all . Theorem 2.2 A is a relatively compact subset of ; A is level-equicontinuous on . Remark 2.3 Then is relatively compact in . A continuous function is said to be compact if and is bounded imply that is relatively compact in . provided the expression on the right-hand side is defined. A function is called measurable (see ) if for all closed set , where ℬ denotes the Borel algebra of . A function is called integrably bounded if there exists a function such that for a.e. . If such F has measurable selectors, then they are also integrable and is nonempty. A fuzzy function is integrably bounded if there exists a function such that for all . A measurable and integrably bounded fuzzy function is said to be integrable on if there exists such that for all . Lemma 2.4 provided that the equation defines a fuzzy number . It is easy to see that , . Lemma 2.5 define the α-level intervals of . which is a fuzzy number for . However, it is not a fuzzy number for . Thus does not satisfy equation (2.1). In this section, we prove the Schauder fixed point theorem for semilinear Banach spaces. First, we recall the Schauder fixed point theorem. Theorem 3.1 (, Schauder fixed point theorem) Let Y be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that is a compact operator. Then P has at least one fixed point in Y. for all , for all and . In this case, we can define a norm on S by , where is the zero element in S. If S is a semilinear metric space, then addition and scalar multiplication on S are continuous. If S is a complete metric space, then we say that S is a semilinear Banach space. for all . Theorem 3.2 for all and . Theorem 3.3 Suppose that S is a semilinear metric space. Then the set all equivalence classes G, constructed above, is a metric vector space and j is an isometry. Now, we are able to prove a variant of the Schauder fixed point theorem in semilinear Banach spaces. Theorem 3.4 (Schauder fixed point theorem for semilinear spaces) Let B be a nonempty, closed, bounded and convex subset of a semilinear Banach space S having the cancelation property, and suppose that is a compact operator. Then P has at least one fixed point in B. Thus is a fixed point of P. □ Remark 3.5 The space of fuzzy sets is a semilinear Banach space S having the cancelation property. Therefore, the Schauder fixed point theorem holds true for fuzzy metric spaces. where , and is continuous on . holds for all . then by Lemma 2.5 is a solution of (4.2), but the converse is not true. In , the authors showed that the space can be embedded in , the Banach space of continuous real-valued functions defined on , where is the unit ball. In , an Ascoli-Arzelá-type theorem was proved. We use this theorem to establish an existence theorem for fuzzy fractional integral equations. Let be the zero function in . is an equicontinuous subset of ; is relatively compact in for each . so when for all . This implies that is equicontinuous on . Now we show that is relatively compact in and by Theorem 2.2 this is equivalent to proving that is a level-equicontinuous and compact-supported subset of . which proves that is compact-supported. Thus, T is a compact operator. Hence, by Theorem 3.4, it follows that T has a fixed point in Ω, which is a solution of integral equation (4.1). □ The following Weissinger fixed point theorem will be used to prove an existence and uniqueness result. Theorem 4.3 for all and for all . Then the operator T has a unique fixed point . Furthermore, for any , the sequence converges to the above fixed point . for all , where . Then there exists a unique solution to integral equation (4.1). The series with is a convergent series (see Theorem 4.1 in ). 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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon? Can you beat the computer in the challenging strategy game? Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs? A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates? Use Excel to explore multiplication of fractions. Help the bee to build a stack of blocks far enough to save his friend trapped in the tower. A collection of resources to support work on Factors and Multiples at Secondary level. Use Excel to investigate the effect of translations around a number grid. A group of interactive resources to support work on percentages Key Stage 4. Cellular is an animation that helps you make geometric sequences composed of square cells. Practise your skills of proportional reasoning with this interactive haemocytometer. Can you discover whether this is a fair game? This set of resources for teachers offers interactive environments to support work on loci at Key Stage 4. An environment that enables you to investigate tessellations of regular polygons There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube? Discover a handy way to describe reorderings and solve our anagram in the process. Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid? Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. The interactive diagram has two labelled points, A and B. It is designed to be used with the problem "Cushion Ball" Use an interactive Excel spreadsheet to explore number in this exciting game! Match the cards of the same value. This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations. Match pairs of cards so that they have equivalent ratios. A metal puzzle which led to some mathematical questions. If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction. This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box. An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse. A simple file for the Interactive whiteboard or PC screen, demonstrating equivalent fractions. A tool for generating random integers. A simple spinner that is equally likely to land on Red or Black. Useful if tossing a coin, dropping it, and rummaging about on the floor have lost their appeal. Needs a modern browser; if IE then at. . . . A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet. To avoid losing think of another very well known game where the patterns of play are similar. A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . . P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P? A collection of our favourite pictorial problems, one for each day of Advent. Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter. An Excel spreadsheet with an investigation. Use Excel to practise adding and subtracting fractions. Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them? Use an interactive Excel spreadsheet to investigate factors and multiples. Here is a chance to play a fractions version of the classic Countdown Game. Use an Excel spreadsheet to explore long multiplication. A java applet that takes you through the steps needed to solve a Diophantine equation of the form Px+Qy=1 using Euclid's algorithm. A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . . A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter. Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Zeno of Elea Life and Work Zeno (b. c. 490 BC) was a pupil of Parmenides. Plato in his dialogue Parmenides testifies their relationship. Aristotle names Zeno as the inventor of dialectic. He wrote a book in which he denies physical motion as well as the unreality of the pluralistic world. His paradoxes of motion had a great influence in the history and philosophy of mathematics. Denial of Motion and Plurality Zeno’s well known hypothesis is that of the denial of motion and plurality. His arguments aim to support Parmenides’ position on the oneness and unity of Being. For Zeno, if reality is successively divided into parts then you will divide it ad infinitum. Zeno’s arguments are presented in the form of paradoxes: 1.The Racetrack or Dichotomy Paradox Suppose a runner has to travel form the start point A to the finish point B. But firstly he has to travel to the midpoint C and thence to B. But if D is the midpoint of AC, he must first travel to D and so on ad infinitum. So since in finite time it is impossible to accomplish an infinite number of movements then the runner is not able to finish his distance. 2.The Paradox of Achilles and the Tortoise Achilles runs a race with a tortoise. Tortoise takes a lead. But while Achilles can run much faster than the tortoise, Achilles cannot touch it. How? When Achilles has reached the tortoise’s starting-point the tortoise is n/10 meters ahead. When Achilles has reached that point is n/100 ahead and so on ad infinitum. 3.The Stadium or Moving Blocks Paradox Suppose three equal groups A, B, C of width l, with A and C moving past B in opposite direction at the same speed. While group A takes time t to traverse width B, it takes t/2 to traverse width C. This leads to the absurd paradox that half time equals its double. 4.The Arrow Paradox The Arrow Paradox Suppose that time consists of moments or instances. A flying arrow at any instant of time occupies a space equal to itself. So at any instant of time, like in a photograph, the arrow would be at rest. Therefore if at any instant of time the arrow has no motion, temporal locomotion is impossible since time is composed of freezing instances in succession. Zeno's support of Parmenides: if there are many things then how many are they? (1), how big are they? (2), do they make a noise? (6), where are they? (4, 5) how can they move? (6) 1(3) how many? limited and unlimited in number If there are many things (i) they will be just as many as they are, no more and no less; and if they are just as many as they are, they would be limited (in number). If there are many things (ii) the things that there are are unlimited; for there will always be other things between the things that there are, and again other things between them, and so the things that there are are unlimited (in number). 2(1) how big? unlimited in size and no size at all If what exists has no size (megethos) it would not exist. But if it does exist each thing must have some size (megethos) and thickness, and one part of it must be separate/distinct from another. And there is the same argument for what is in front, for that will have size and some part of it will be in front. Indeed, to say this once is similar to saying it for ever, for no such part of it will be the last or the same as a further part. So if there are many things they must be both small and great: so small as to have no size, and so big as to be unlimited (in size). 3(2) If it were added to something else, it would not make it any bigger; for if it has no size at all and were added on, it would not contribute any increase in size, and so what is now being added on would be nothing. And if, while it is being subtracted, the other will be no smaller It is obvious that what was added or subtracted was nothing. 4(4) where are they? in a place and nowhere at all Where is all of what there is? if there is a place for the things that there are, where would it be? it would be in another place and that in another place and so on and so on. What moves does not move in the place in which it is or in the place in which it is not. 5(A24) If there is a place for the things that are, where would it be? it would be in another place and that in another place and so on and so on. 6(A29) do they make a noise? yes, some noise and no, no sound at all Zeno: Tell me Protagoras, does a single millet seed make a noise as it falls, or does 1/10,000 of a millet? Protagoras: No. Zeno: Does a bushel of millet seed make a noise as it falls, or not? Protagoras: Yes, a bushel makes a noise. Zeno: But isn't there a ratio (logos) between a bushel of millet seed, and one seed, and 1/10,000 of a seed? Protagoras: Yes, there is. Zeno: So won't there be the same ratio of sounds between them, for the sounds are in proportion to what makes the sound? And if this is so, if the bushel of millet seed makes a noise so will a single seed and 1/10,000 of a seed. 7(A25-28 from Aristotle Physics 239b-240a) do things move? four puzzles Zeno has four propositions (logoi) about movement which are puzzling for those who try to solve them: (i) The Dichotomy (it is impossible to move from one place to another) The first argument about there being no movement says that the moving object must first reach the half-way mark before the end (and the quarter-mark before the half and so back, so there is no first move; and the three-quarter mark after the half, and so forward so there is no last move). (ii) The Achilles (Achilles cannot overtake the tortoise) The second is the one called Achilles. This is it: the slowest will never be overtaken in running by the fastest, for the pursuer must always come to the point the pursued has left, so that the slower must always be some distance ahead. (iii) The Arrow (the moving arrow is at rest) The third one mentioned is that the moving arrow is at rest. [cf. Phys293b6: the arrow is at rest at any time when it occupies a space just its own length, and yet it is always moving at any time in its flight (i.e. in the 'now'), therefore the moving arrow is motionless.] (iv) The Stadium (a time is twice itself) The fourth is the one about equal blocks moving past equal blocks from opposite directions in the stadium - one set from the end of the stadium and one from the middle - at the same speed; here he thinks that half the time is equal to twice itself. For example: AAAA are equal stationary blocks, BBBB, equal to them in number and size, are beginning from the half-way point (of the stadium), CCCC equal to these also in size, and equal to the Bs in speed, are coming towards them from the end. It happens of course that the first B reaches the end at the same time as the first C as they move past each other. And it happens that the C passes all the Bs but the Bs only half (the As) so the time is half (itself). Translation M. R. Wright - note: numbers in parentheses refer to the standard Diels/Kranz order Heraclitus of Ephesus

1 Topics through Chapter The Doppler Effect: this is how we learn about the motions of objects in the universe, discover extraterrestrial planets, black holes at the centers of galaxies, and the expansion of the entire universe. 4.1 Spectral Lines: appearance, excess or deficit of energy at particular wavelengths. (Don t worry about Kirchoff s laws, and the pictures with little prisms and light bulbs etc.--see notes) 4.2 Atoms and Radiation: quantized energy levels of electrons in atoms, and how they interact with radiation. The hydrogen atom--the simplest, and most important case. 4.3 The Formation of Spectral Lines: understanding emission lines and absorption lines 4.4 Molecules: not just electronic changes, but vibrational and rotational changes of energy. 4.5 Spectral-Line Analysis: Information from spectral linesabundances, turbulent motions, rotation, 2 3.5 The Doppler Effect The wavelength (or frequency) of a wave, as measured by an observer, depends on the relative radial speed of the source and observer. Radial motion means: motion towards or away; along the line of sight. The Doppler effect involves only this component of motion. What we get from measuring it is called the radial velocity. Moving away: wavelengths increase ( redshift ) Moving toward: wavelengths decrease ( blueshift ) 3 3.5 The Doppler Effect Relationship between wavelength and speed: Shift in λ compared to rest (no motion) wavelength is proportional to radial velocity So if you know the exact wavelength where some feature in the spectrum should be ( true wavelength ), and the wavelength at which it appears ( apparent wavelength ), you can obtain the radial velocity. This is how we get speeds of cosmic objects, stars, galaxies, even expansion of universe. Actual formula is: λ(apparent)/ λ(true) = 1 ± (speed of object/speed of light) where the ± sign means it is + if it is moving away from us (redshift, longer wavelength), - if it is moving toward us (blueshift, shorter wavelength) This applies to any wave; and no reason not to use frequency instead of wavelength. Textbook writes it this way: 4 See textbook for this rather confusing example. In class we ll use a simpler example of water waves in a pond. 3.5 The Doppler Effect Important point: Doppler effect depends only on the relative motion of source and observer 5 More Precisely 3-3: Measuring Velocities with the Doppler Effect Example: For a speed of 30 km/s, the Doppler shift is given by speed of object This may seem small, but it is easily detectable with a radar gun. It is NOT so easy to detect from the spectrum of an astronomical object, unless you know something about spectral lines. speed of light 6 Look at the Doppler shift formula again: λ(apparent)/ λ(true) = 1 ± (radial velocity of object./speed of light) If velocity of object away or toward us is much less than the speed of light (true for almost all objects in the universe), the apparent wavelength will be only slightly different from the laboratory or rest wavelength. For most objects in the universe, this relative shift is so tiny, that we can t detect it using the shift of the whole continuous spectrum. But we can use places in the spectrum whose wavelengths are precisely known by the presence of spectral lines (the subject of Chapter 4) 7 Chapter 4 Spectroscopy Don t worry if you can t understand what this pretty picture represents, unless it is the day before the next exam. A more important question Is why the authors insist on showing this form of spectra without adequately explaining! 8 4.1 Spectral Lines Spectroscope: Splits light into component colors (wavelengths, frequencies) Most of this illustration is completely unnecessary! The only important point is that light from any object can be spread out into a rainbow of wavelengths. A spectrum is a picture of how much light is at each wavelength. This illustration is showing a continuous spectrum. 9 4.1 Spectral Lines Continuous spectrum: Continuous range of frequencies emitted by an object (something like the black bodies we discussed in ch.3) Emission lines: Single frequencies emitted by particular atoms in a hot gas Absorption lines: If a continuous spectrum passes through a cool gas, atoms of the gas will absorb the same frequencies they emit 10 Spectral lines Spectral lines very narrow, well-defined (in wavelength) wavelength/frequency regions in the spectrum where excess photon energy appears (emission lines) or else where photons are missing (absorption lines). Cartoon view of absorption lines, both in the spectrum as a graph (below), and in the recorded spectrum (top), the band of colors--this is just how the spectra are gathered--pay no attention to the rectangular shape! Often these lines are superimposed on a smooth, continuous spectrum, which is the near-blackbody emission of a heated object that we have been discussing so far (ch. 3, Wien, Stefan-Boltzmann). 11 Spectral of real astronomical objects Here are spectra of two real astronomical objects, a comet (top) and star (bottom). By the time you take the next exam, you should be able to explain why these look so different. The wavelengths, shapes, and strengths of these spectral lines are the keys to understanding many of the physical properties of planets, stars and galaxies. 12 Spectra are not black rectangles with vertical lines. They are a recording of how many photons per second are being emitted as a function of wavelength 13 4.1 Spectral Lines Emission spectrum can be used to identify elements An absorption spectrum can also be used to identify elements. These are the emission and absorption spectra of sodium: 14 Here is the Sun s spectrum, along with a blackbody of the sun s temperature (top--why are there no lines?), and the spectra of individual elements as observed in the laboratory. Each spectral line is a chemical fingerprint telling you which elements, and how much of each element, is contained in the object you are observing. Continuous spectrum Sun (absorption lines) Emission lines of various elements 15 Kirchoff s laws: don t memorize them. Instead, come back to this illustration later and find if you understand enough to explain them Kirchoff s laws are usually presented with prisms and striped colorful spectra. This is confusing, and they aren t even laws at all! Just note that prism is supposed to represent an instrument, called a spectrometer. Maybe a star, or a planet Maybe the atmosphere of a star or planet If the light from the hot star or blackbody doesn t pass through any low-density gas, then the spectrum is featureless--it is a continuous spectrum. (top) If that continuous spectrum passes through a cloud of gas that is cooler than the source, the cloud can absorb particular wavelengths, and you get an absorption spectrum (middle) But if the gas is hot, say at least a few thousand degrees, it will emit spectral lines on its own (bottom), I.e. emission lines. How does this occur? The answer lies in the structure of atoms. 16 Next : Energy levels of electrons in atoms 17 4.2 Atoms and Radiation Existence of spectral lines required new model of atom, so that only certain amounts of energy could be emitted or absorbed Bohr model had certain allowed orbits for electron 18 4.2 Atoms and Radiation Emission energies correspond to energy differences between allowed levels Modern model has electron cloud rather than orbit 19 4.3 The Formation of Spectral Lines Absorption spectrum: Created when atoms absorb photons of right energy for excitation Multielectron atoms: Much more complicated spectra, many more possible states Ionization changes energy levels 20 Energy levels in H and He 21 The light emitted has a wavelength corresponding to the the energy difference between the two electron energy levels. Spectral lines = electronic transitions 22 Spectral lines of hydrogen Energy levels of the hydrogen atom, showing two series of emission lines:lyman and Balmer The energies of the electrons in each orbit are given by: The emission lines correspond to the energy differences 23 4.3 The Formation of Spectral Lines Absorption of energy (either by a collision, or by absorbing a photon) can boost an electron to the second (or higher) excited state Two ways to decay: 1. To ground state 2. Cascade one orbital at a time 24 4.3 The Formation of Spectral Lines (a) Direct decay (b) Cascade 25 Absorption of light as it passes through an atmosphere 26 Emission lines can be used to identify atoms 27 4.4 Molecules Molecules can vibrate and rotate, besides having energy levels Electron transitions produce visible and ultraviolet lines Vibrational transitions produce infrared lines Rotational transitions produce radio-wave lines 28 4.4 Molecules Molecular spectra are much more complex than atomic spectra, even for hydrogen: (a) Molecular hydrogen(b) Atomic hydrogen 29 Information that can be gleaned from spectral lines: Chemical composition Temperature Radial velocity 4.5 Spectral-Line Analysis 30 Line broadening can be due to a variety of causes 4.5 Spectral-Line Analysis 31 4.5 Spectral-Line Analysis 32 The Doppler shift may cause thermal broadening of spectral lines 4.5 Spectral-Line Analysis 33 4.5 Spectral-Line Analysis Rotation will also cause broadening of spectral lines through the Doppler effect 34 Summary of Chapter 4 Spectroscope (spectrometer) splits light beam into component frequencies/wavelengths Continuous spectrum is emitted by solid, liquid, and dense gas Hot gas has characteristic emission spectrum Continuous spectrum incident on cool, thin gas gives characteristic absorption spectrum 35 Summary of Chapter 4 (cont.) Spectra can be explained using atomic models, with electrons occupying specific orbitals Emission and absorption lines result from transitions between orbitals Molecules can also emit and absorb radiation when making transitions between vibrational or rotational states

Simple Interest and Compound Interest Shortcut Tricks Shortcut tricks on simple interest and compound interest are one of the most important topics in exams. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss this thing. Few examples on simple interest and compound interest shortcuts First of all do a practice set on math of any exam. Choose any twenty math problems and write it down on a page. Do first ten maths using basic formula of this math topic. You also need to keep track of the time. Write down the time taken by you to solve those questions. Now read our examples on simple interest and compound interest shortcut tricks and practice few questions. After finishing this do remaining questions using simple interest and compound interest shortcut tricks. Again keep track of timing. This time you will surely see improvement in your timing. But this is not enough. You need to practice more to improve your timing more. Few Important things to Remember Math section in a competitive exam is the most important part of the exam. It doesn’t mean that other topics are not so important. You can get a good score only if you get a good score in math section. You can get good score only by practicing more and more. The only thing you need to do is to do your math problems correctly and within time, and this can be achieved only by using shortcut tricks. Again it does not mean that you can’t do maths without using shortcut tricks. You may have that potential to do maths within time without using any shortcut tricks. But, so many people can’t do this. Here we prepared simple interest and compound interest shortcut tricks for those people. We always try to put all shortcut methods of the given topic. But if you see any tricks are missing from the list then please inform us. Your little help will help so many needy. Now we will discuss some basic ideas of Simple Interest and Compound Interest. On the basis of these ideas we will learn trick and tips of shortcut simple interest and compound interest. If you think that how to solve simple interest and compound interest questions using simple interest and compound interest shortcut tricks, then further studies will help you to do so. What is Interest? When some one take up some money from other for the personal or commercial purpose we pay some additional money to him after a certain period of time is called Interest. So, we can also called this Interest as Simple Interest. And, this type of problem are given in Quantitative Aptitude which is a very essential paper in banking exam. So, below are some more example of Interest for your better practice. Anything we learn in our school days was basics and that is well enough for passing our school exams. Now the time has come to learn for our competitive exams. For this we need our basics but also we have to learn something new. That’s where shortcut tricks are comes into action. What is Principal? So, the concept of Principal is very simple. When money borrow for a certain time period called Principal or Sum. What is Amount? The Addition of Simple Interest and Principal is called Amount. Amount (A) = Simple Interest (SI) + Principal (P). Simple Interest (SI) = Amount (A) – Principal (P). What is Per annual means? Per annual means for a year. Few general terms of Interest - P = Principal - R = Rate per annual - T = Number of years Formulas Need to Remember SI = ( P x T x R / 100 ) Here, P = Principal R = Rate per annual T = Number of years In case of Simple Interest, Number of years and Rate per annul are given in question, then we can easily find the Principal or Sum. P = ( SI x 100 / R x T ) Now, In case of Simple Interest, Number of years and Principal are given in question, then we can easily find the Rate per annual. R = ( SI x 100 / P x T ) Find the simple interest of Rs.500/- for 5 years at 5% per annum. Show Answer Show How to Solve Open Rough Workspace SI = 500 x 5 x 5 / 100 Simple interest in 5 years is Rs.125/-. In what time Rs.5000/- amounts to Rs.6000/- at 5% per annum at simple interest? - 2 Years - 4 Years - 5 Years - 7 Years Show Answer Show How to Solve Open Rough Workspace 1000 = 5000 x 5 x T / 100 T = 4 So, in 4 years Rs.5000/- will increase to Rs.6000/-. More Shortcut tricks on Simple and Compound Interest - Simple Interest based question - Rate % based question - Principle or Sum based question - Compound Interest shortcut tricks - Find Compound Interest tricks - Difference between CI and SI of Three Years question Here, we provide few tricks on Simple interest and Compound interest. So, do visit this page to get updates on more Math Shortcut Tricks. You can also like our facebook page to get updates. Also, if you have any question regarding this topic then please do comment on below section. You can also send us message on facebook. i m a faculty of Quantative aptitude in a coaching institute in dehradun, i wants more tricks related to different topics ….. if any body have his own tricks or taken from somewhere else….. plz contact….. me ….or send it to my email id……email@example.com………….tricks should be ……. in a interval of 30 sec…..so that a question is to be solved under the interval of 30….sec…………………………………………………………………………………………….or can any one suggest me the books related to the tricks for aptitude or reasoning……… I am from MBIET (The Absolute Education Mentor). I am the HOD of Mathematics. Contact on provided email for any kind of help in education. help me in getting the quantitative aptitude easily i am in search of jobs now and am unable to complete the compitative how to manage time during solving questions… I always get late while solving the questions?? take a timer try to solv problem within time at home and keep practicing which hlp u in exam well ….. plese can you help me for the shortcut tricks..want to clear ibps exam…if possible thn mail me Sir am rajesh….my math level zero pls help me nd suggest how to improve own weakness…… sir i want to learn quantitative apptitude question for government exam ..so please give me any shortcuts trics and farmula due to that i can do easy and early now i am doing B.E of E.C.E department final year but i want to study any thing special in mathematics as i am interested in mathematics i want to study and do job related to mathematics. can you please suggest any degree i want to study regarding maths after B.E and i want to settle well with that maths degree plz reply to my mail id firstname.lastname@example.org thank u sir i need some workout problems based on SI & CI with short cut tricks….i can solve all other problems except SI,CI, profit and loss….problems should be compitative level Very nice this topic very esauful thank you very much i want to know the time and distance sums , please help me KNOW I PREPARE LIC AAO EXAM TELL ME SHORTCUT TRICS can i get some maths tricks based on afcat exam 1 2016 whisch is going to be held on 21 feb, some problems like step problems for eg a man travellling from etc etc Pls give a all shortcuts in maths for competitive exams Respected Sir please give me some short trics and plan method to mage good score in quantative. i need tricky document….. may i have your genuine help. i am kishore gogoi from assam preparing for bank n upsc exams sir plz send me shortcut formulaes of time and distant, time and work, trigonometery and simple and compound interest sir please give any aptitude shortcut tricks for clearing bank exams.and also send study materials to my mail id. please send me the short cut tricks soft copy if u have to my mail address( email@example.com”)since have to apply for jobs…kindly help me with materials… Sir please can you help me for short tricks for solving IBPS exam questions fast. yes sir pls help me on work and time and advance math trick email id firstname.lastname@example.org A certain sum amounts to rs 1725 at 15% per annum at simple Interst and rs 1800 in the same time at 20% per annum at simple interest hello sir, how to solve simple interest and compound interest sums less than 30 sec. plz help me please send easy tricks based on simple&compound interest Suppose ex-SI for 5yr at 10% p.a for 800. T*R=5*10=50% of capital I.e 50% of 800=400 Dear sir,plz send notes related to solve ibps po exam thanks. can i get aptitude books questions related to ssb defense and afcat examinations If u get the shortcut note plz send me…. Hello chandan sir i am neha doing prep..for banking pls give the trick to solved SI CI and other maths topic interval 30 sec…. and pls give the example also…Thanks in advance pls provide quantitative app maths tricks pls help me. i have exam follow… www.math-shortcut-tricks.com… best of luck for your exam I have prepration for PO exam, I need more mathematical tricks for easy solve paper in exam. also want tricks for all above tricks for will usefull my exam…. Give d more examples in several types… Please provide shortcut tricks for compound interest installment questions i want th know wthether the icompound interest was directly proportional to the principal or not?????? If a man lends money at s bt he includes interest every six months to calculate principal ,if he charges 10% interest then wht is effective rate of interest ??? HI Please send me shortcut tricks about reasoning , quantative aptitude and data intrepretataion tricks because i m preparing for ibps exam this is very important topic for me and given more short cuts so its very good Sir plzzz help me in math plzzz send some tricks to my email id Plzzzzzzzzzzzzzzz, My email id is email@example.com Sir plzzzz send some fast tricks Not bad but amazing……. 🙂 thanks for that How to download these tricks. Because it is very essential. send me plz tricky method quantitative aptitude intrest I’m requested sair plzz send me sort tricks fore compaund interest With a given rate of simple interest the ratio of principal and amount for a certain period of time is 4:5.after 3yrs with tthe same rate of interest the ratio of the principal and amount becomes 5:7.The rate of interest per annum is ??? let take p=100, then, amount(a)=125(since ratio is 4:5) then after 3 yr the ratio will b 5:7 now in 3 yr SI=140-125=15 sir please send me some short curts for compound interests. C.I me EMI VALE QUESTION KI koi short trick ho to..plz help sir Sir,please give me a shortcut to improve mathematics. It is helpful but wt about d calculations they remain same,, we know d formula but calculations need tym.. Please change the background…. its getting difficult to read. Disclaimer:All contents of this website is fully owned by Math-Shortcut-Tricks.com. Any means of republish its content is strictly prohibited. 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Testing Relativistic Gravity with Radio Pulsars Before the 1970s, precision tests for gravity theories were constrained to the weak gravitational fields of the Solar system. Hence, only the weak-field slow-motion aspects of relativistic celestial mechanics could be investigated. Testing gravity beyond the first post-Newtonian contributions was for a long time out of reach. The discovery of the first binary pulsar by Russell Hulse and Joseph Taylor in the summer of 1974 initiated a completely new field for testing the relativistic dynamics of gravitationally interacting bodies. For the first time the back reaction of gravitational wave emission on the binary motion could be studied. Furthermore, the Hulse-Taylor pulsar provided the first test bed for the orbital dynamics of strongly self-gravitating bodies. To date there are a number of pulsars known, which can be utilized for precision test of gravity. Depending on their orbital properties and their companion, these pulsars provide tests for various different aspects of relativistic dynamics. Besides tests of specific gravity theories, like general relativity or scalar-tensor gravity, there are pulsars that allow for generic constraints on potential deviations of gravity from general relativity in the quasi-stationary strong-field and the radiative regime. This article presents a brief overview of this modern field of relativistic celestial mechanics, reviews some of the highlights of gravity tests with radio pulsars, and discusses their implications for gravitational physics and astronomy, including the upcoming gravitational wave astronomy. - 1 Introduction - 2 Gravitational wave damping - 3 Geodetic precession - 4 The strong equivalence principle - 5 Local Lorentz invariance of gravity - 6 Local position invariance of gravity - 7 A varying gravitational constant - 8 Summary and Outlook In about two years from now we will be celebrating the centenary of Einstein’s general theory of relativity. On November 25th 1915 Einstein presented his field equations of gravitation (without cosmological term) to the Prussian Academy of Science . With this publication, general relativity (GR) was finally completed as a logically consistent physical theory (“Damit ist endlich die allgemeine Relativitätstheorie als logisches Gebäude abgeschlossen.”). Already one week before, based on the vacuum form of his field equations, Einstein was able to show that his theory of gravitation naturally explains the anomalous perihelion advance of the planet Mercury . While in hindsight this can be seen as the first experimental test for GR, back in 1915 astronomers were still searching for a Newtonian explanation . In his 1916 comprehensive summary of GR , Einstein proposed three experimental tests: Gravitational redshift (Einstein suggested to look for red-shift in the spectral lines of stars). Light deflection (Einstein explicitly calculated the values for the Sun and Jupiter). Perihelion precession of planetary orbits (Einstein emphasized the agreement of GR, with the observed perihelion precession of Mercury with a reference to his calculations in ). Gravitational redshift, a consequence of the equivalence principle, is common to all metric theories of gravity, and therefore in some respect its measurement has less discriminating power than the other two tests . The first verification of gravitational light bending during the total eclipse on May 29th 1919 was far from being a high precision test, but clearly decided in favor of GR, against the Newtonian prediction, which is only half the GR value . In the meantime this test has been greatly improved, in the optical with the astrometric satellite HIPPARCOS , and in the radio with very long baseline interferometry [8, 9, 10]. The deflection predicted by GR has been verified with a precision of . An even better test for the curvature of spacetime in the vicinity of the Sun is based on the Shapiro delay, the so-called “fourth test of GR” . A measurement of the frequency shift of radio signals exchanged with the Cassini spacecraft lead to a confirmation of GR . Apart from the four “classical” tests, GR has passed many other tests in the Solar system with flying colors: Lunar Laser Ranging tests for the strong equivalence principle and the de-Sitter precession of the Moon’s orbit , the Gravity Probe B experiment for the relativistic spin precession of a gyroscope (geodetic and frame dragging) , and the Lense-Thirring effect in satellite orbits , just to name a few. GR, being a theory where fields travel with finite speed, predicts the existence of gravitational waves that propagate with the speed of light and extract energy from (non-axisymmetric) material systems with accelerated masses . This is also true for a self-gravitating system, where the acceleration of the masses is driven by gravity itself, a question which was settled in a fully satisfactory manner only several decades after Einstein’s pioneering papers (see for an excellent review). This fundamental property of GR could not be tested in the slow-motion environment of the Solar system, and the verification of the existence of gravitational waves had to wait until the discovery of the first binary pulsar in 1974 . Also, all the experiments in the Solar system can only test the weak-field aspects of gravity. The spacetime of the Solar system is close to Minkowski space everywhere: To first order (in standard coordinates) the spatial components of the spacetime metric can be written as , where denotes the Newtonian gravitational potential. At the surface of the Sun one finds , while at the surface of a neutron star . Consequently, gravity experiments with binary pulsars, not only yielded the first tests of the radiative properties of gravity, they also took our gravity tests into a new regime of gravity. To categorize gravity tests with pulsars and to put them into context with other gravity tests it is useful to introduce the following four gravity regimes: Quasi-stationary weak-field regime: The motion of the masses is slow compared to the speed of light () and spacetime is only very weakly curved, i.e. close to Minkowski spacetime everywhere. This is, for instance, the case in the Solar system. Quasi-stationary strong-field regime: The motion of the masses is slow compared to the speed of light (), but one or more bodies of the system are strongly self-gravitating, i.e. spacetime in their vicinity deviates significantly from Minkowski space. Prime examples here are binary pulsars, consisting of two well-separated neutron stars. Highly-dynamical strong-field regime: Masses move at a significant fraction of the speed of light () and spacetime is strongly curved and highly dynamical in the vicinity of the masses. This is the regime of merging neutron stars and black holes. Radiation regime: Synonym for the collection of the radiative properties of gravity, most notably the generation of gravitational waves by material sources, the propagation speed of gravitational waves, and their polarization properties. Figure 1 illustrates the different regimes. Gravity regime G1 is well tested in the Solar system. Binary pulsar experiments are presently our only precision experiments for gravity regime G2, and the best tests for the radiative properties of gravity (regime GW)111 Gravitational wave damping has also been observed in a double white-dwarf system, which has an orbital period of just 13 minutes . This experiment combines gravity regimes G1 (note, ) and GW of figure 1.. In the near future, gravitational wave detectors will allow a direct detection of gravitational waves (regime GW) and probe the strong and highly dynamical spacetime of merging compact objects (regime G3). As we will discuss at the end of this review, pulsar timing arrays soon should give us direct access to the nano-Hz gravitational wave band and probe the properties of these ultra-low-frequency gravitational waves (regime GW). 1.1 Radio pulsars and pulsar timing Radio pulsars, i.e. rotating neutron stars with coherent radio emission along their magnetic poles, were discovered in 1967 by Jocelyn Bell and Antony Hewish . Seven years later, Russell Hulse and Joseph Taylor discovered the first binary pulsar, a pulsar in orbit with a companion star . This discovery marked the beginning of gravity tests with radio pulsars. Presently, more than 2000 radio pulsars are known, out of which about 10% reside in binary systems . The population of radio pulsars can be nicely presented in a diagram that gives the two main characteristics of a pulsar: the rotational period and its temporal change due to the loss of rotational energy (see figure 2). Fast rotating pulsars with small (millisecond pulsars) appear to be particularly stable in their rotation. On long time-scales, some of them rival the best atomic clocks in terms of stability [23, 24]. This property makes them ideal tools for precision astrometry, and hence (most) gravity tests with pulsars are simply clock comparison experiments to probe the spacetime of the binary pulsar, where the “pulsar clock” is read off by counting the pulses in the pulsar signal (see figure 3). As a result, a wide range of relativistic effects related to orbital binary dynamics, time dilation and delays in the signal propagation can be tested. The technique used is the so-called pulsar timing, which basically consists of measuring the exact arrival time of pulses at the radio telescope on Earth, and fitting an appropriate timing model to these arrival times, to obtain a phase-connected solution. In the phase-connected approach lies the true strength of pulsar timing: the timing model has to account for every (observed) pulse over a time scale of several years, in some cases even several decades. This makes pulsar timing extremely sensitive to even tiny deviations in the model parameters, and therefore vastly superior to a simple measurement of Doppler-shifts in the pulse period. Table 1 illustrates the current precision capabilities of pulsar timing for various experiments, like mass determination, astrometry and gravity tests. We will not go into the details of pulsar observations and pulsar timing here, since there are numerous excellent reviews on these topics, for instance [25, 26], just to mention two. In this review we focus on the relativistic effects that play a role in pulsar timing observations, and how pulsar timing can be used to test gravitational phenomena in generic as well as theory-based frameworks. 1.2 Binary pulsar motion in gravity theories While in Newtonian gravity there is an exact solution to the equations of motion of two point masses that interact gravitationally, no such exact analytic solution is known in GR. In GR, the two-body problem has to be solved numerically or on the basis of approximation methods. A particularly well established and successful approximation scheme, to tackle the problem of motion of a system of well-separated bodies, is the post-Newtonian approximation, which is based on the weak-field slow-motion assumption. However, to describe the motion and gravitational wave emission of binary pulsars, there are two main limitations of the post-Newtonian approximation that have to be overcome (cf. ): Near and inside the pulsar (and its companion, if it is also a neutron star) the gravitational field is strong and the weak-field assumption no longer holds. When it comes to generation of gravitational waves (of wavelength ) and their back-reaction on the orbit (of size and period ), the post-Newtonian approximation is only valid in the near zone (), and breaks down in the radiation zone () where gravitational waves propagate and boundary conditions are defined, like the ‘no incoming radiation’ condition. The discovery of the Hulse-Taylor pulsar was a particularly strong stimulus for the development of consistent approaches to compute the equations of motion for a binary system with strongly self-gravitating bodies (gravity regime G2). As a result, by now there are fully self-consistent derivations for the gravitational wave emission and the damping of the orbit due to gravitational wave back-reaction for such systems. In fact, in GR, there are several independent approaches that lead to the same result, giving equations of motion for a binary system with non-rotating components that include terms up to 3.5 post-Newtonian order () [36, 37]. For the relative acceleration in the center-of-mass frame one finds the general form where the coefficients and are of order , and are functions of , , , and the masses (see for explicit expressions). The quantity denotes the total mass of the system. At this level of approximation, these equations of motion are also applicable to binaries containing strongly self-gravitating bodies, like neutron stars and black holes. This is a consequence of a remarkable property of Einstein’s theory of gravity, the effacement of the internal structure [38, 35]: In GR, strong-field contributions are absorbed into the definition of the body’s mass. In GR’s post-Newtonian approximation scheme, gravitational wave damping enters for the first time at the 2.5 post-Newtonian level (order ), as a term in the equations of motion that is not invariant against time-reversal. The corresponding loss of orbital energy is given by the quadrupole formula, derived for the first time by Einstein within the linear approximation, for a material system where the gravitational interaction between the masses can be neglected . As it turns out, the quadrupole formula is also applicable for gravity regime G2 of figure 1, and therefore valid for binary pulsars as well (cf. ). In alternative gravity theories, the gravitational wave back-reaction, generally, already enters at the 1.5 post-Newtonian level (order ). This is the result of the emission of dipolar gravitational waves, and adds terms and to equation (1) [5, 39]. Furthermore, one does no longer have an effacement of the internal structure of a compact body, meaning that the orbital dynamics, in addition to the mass, depends on the “sensitivity” of the body, a quantity that depends on its structure/compactness. Such modifications already enter at the “Newtonian” level, where the usual Newtonian gravitational constant is replaced by a (body-dependent) effective gravitational constant . For alternative gravity theories, it therefore generally makes an important difference whether the pulsar companion is a compact neutron star or a much less compact white dwarf. In sum, alternative theories of gravity generally predict deviations from GR in both the quasi-stationary and the radiative properties of binary pulsars [40, 41]. At the first post-Newtonian level, for fully conservative gravity theories without preferred location effects, one can construct a generic modified Einstein-Infeld-Hoffmann Lagrangian for a system of two gravitationally interacting masses (pulsar) and (companion) at relative (coordinate) separation and velocities and : where . The body-dependent quantities , and account for deviations from GR associated with the self-energy of the individual masses [5, 40]. In GR one simply finds , , and . There are various analytical solutions to the dynamics of (2). The most widely used in pulsar astronomy is the quasi-Keplerian parametrization by Damour and Deruelle . It forms the basis of pulsar-timing models for relativistic binary pulsars, as we will discuss in more details in Section 1.4. Beyond the first post-Newtonian level there is no fully generic framework for the gravitational dynamics of a binary system. However, one can find equations of motion valid for a general class of gravity theories, like in where a framework based on multi-scalar-tensor theories is introduced to discuss tests of relativistic gravity to the second post-Newtonian level, or in where the explicit equations of motion for non-spinning compact objects to 2.5 post-Newtonian order for a general class of scalar-tensor theories of gravity are given. 1.3 Gravitational spin effects in binary pulsars In relativistic gravity theories, in general, the proper rotation of the bodies of a binary system directly affects their orbital and spin dynamics. Equations of motion for spinning bodies in GR have been developed by numerous authors, and in the meantime go way beyond the leading order contributions (for reviews and references see, e.g., [45, 35, 46, 47]). For present day pulsar-timing experiments it is sufficient to have a look at the post-Newtonian leading order contributions. There one finds three contributions: the spin-orbit (SO) interaction between the pulsar’s spin and the orbital angular momentum , the SO interaction between the companion’s spin and the orbital angular momentum, and finally the spin-spin interaction between the spin of the pulsar and the spin of the companion . Spin-spin interaction will remain negligible in binary pulsar experiments for the foreseeable future. They are many orders of magnitude below the second post-Newtonian and spin-orbit effects , and many orders of magnitude below the measurement precision of present timing experiments. For this reason, we will not further discuss spin-spin effects here. For a boost-invariant gravity theory, the (acceleration-dependent) Lagrangian for the spin-orbit interaction has the following general form (summation over spatial indices ) where is the antisymmetric spin tensor of body [49, 35, 40]. The coupling function can also account for strong-field effects in the spin-orbit coupling. In GR . For bodies with negligible gravitational self-energy, one finds in the framework of the parametrized post-Newtonian (PPN) formalism222The PPN formalism uses 10 parameters to parametrize in a generic way deviations from GR at the post-Newtonian level, within the class of metric gravity theories (see for details). , a quantity that is actually most tightly constrained by the light-bending and Shapiro-delay experiments in the Solar system, which test [8, 9, 10, 12]. In binary pulsars, spin-orbit coupling has two effects. On the one hand, it adds spin-dependent terms to the equations of motion (1), which cause a Lense-Thirring precession of the orbit (for GR see [45, 50]). So far this contribution could not be tested in binary pulsar experiments. Prospects of its measurement will be discussed in the future outlook in Section 8. On the other hand it leads to secular changes in the orientation of the spins of the two bodies (geodetic precession), most importantly the observed pulsar in a pulsar binary [51, 45, 52]. As we discuss in more details in Section 3, a change in the rotational axis of the pulsar causes changes in the observed emission properties of the pulsar, as the line-of-sight gradually cuts through different regions of the magnetosphere. As can be derived from (3), to first order in GR the geodetic precession of the pulsar, averaged over one orbit, is given by () where and . It is expected that in alternative theories relativistic spin precession generally depends on self-gravitational effects, meaning, the actual precession may depend on the compactness of a self-gravitating body. For the class of theories that lead to the Lagrangian (3), equation (4) modifies to where is the strong-field generalization of . Effects from spin-induced quadrupole moments are negligible as well. For double neutron-star systems they are many orders of magnitude below the second post-Newtonian and spin-orbit effects, due to the small extension of the bodies . If the companion is a more extended star, like a white dwarf or a main-sequence star, the rotationally-induced quadrupole moment might become important. A prime example is PSR J00457319, where the quadrupole moment of the fast rotating companion causes a significant precession of the pulsar orbit . For all the binary pulsars discussed here, the quadrupole moments of pulsar and companion are (currently) negligible. Finally, certain gravitational phenomena, not present in GR, can even lead to a spin precession of isolated pulsars, for instance, a violation of the local Lorentz invariance and a violation of the local position invariance in the gravitational sector, as we will discuss in more details in Sections 5 and 6. 1.4 Phenomenological approach to relativistic effects in binary pulsar observations For binary pulsar experiments that test the quasi-stationary strong-field regime (G2) and the gravitational wave damping (GW), a phenomenological parametrization, the so-called ‘parametrized post-Keplerian’ (PPK) formalism, has been introduced by Damour and extended by Damour and Taylor . The PPK formalism parametrizes all the observable effects that can be extracted independently from binary pulsar timing and pulse-structure data. Consequently, the PPK formalism allows to obtain theory-independent information from binary pulsar observations by fitting for a set of Keplerian and post-Keplerian parameters. The description of the orbital motion is based on the quasi-Keplerian parametrization of Damour & Deruelle, which is a solution to the first post-Newtonian equations of motion [42, 55]. The corresponding Roemer delay in the arrival time of the pulsar signals is where the eccentric anomaly is linked to the proper time of the pulsar via the Kepler equation The five Keplerian parameters , , , , and denote the orbital period, the orbital eccentricity, the projected semi-major axis of the pulsar orbit, the longitude of periastron, and the time of periastron passage, respectively. The post-Keplerian parameter is not separately measurable, i.e. it can be absorbed into other timing parameters, and the post-Keplerian parameter has not been measured up to now in any of the binary pulsar systems. The relativistic precession of periastron changes the the longitude of periastron according to meaning, that averaged over a full orbit, the location of periastron shifts by an angle . The parameter is the corresponding post-Keplerian parameter. A change in the orbital period, due to the emission of gravitational waves, is parametrized by the post-Keplerian parameter . Correspondingly, one has post-Keplerian parameters for the change in the orbital eccentricity and the projected semi-major axis: Besides the Roemer delay , there are two purely relativistic effects that play an important role in pulsar timing experiments. In an eccentric orbit, one has a changing time dilation of the “pulsar clock” due to a variation in the orbital velocity of the pulsar and a change of the gravitational redshift caused by the gravitational field of the companion. This so-called Einstein delay is a periodic effect, whose amplitude is given by the post-Keplerian parameter , and to first oder can be written as For sufficiently edge-on and/or eccentric orbits the propagation delay suffered by the pulsar signals in the gravitational field of the companion becomes important. This so-called Shapiro delay, to first order, reads where the two post-Keplerian parameters and are called range and shape of the Shapiro delay. The latter is linked to the inclination of the orbit with respect to the line of sight, , by . It is important to note, that for equation (12) breaks down and higher order corrections are needed. But so far, equation (12) is fully sufficient for the timing observations of known pulsars . Concerning the post-Keplerian parameters related to quasi-stationary effects, for the wide class of boost-invariant gravity theories one finds that they can be expressed as functions of the Keplerian parameters, the masses, and parameters generically accounting for gravitational self-field effects (cf. equation (2)) [40, 5]: plus from equation (5). Here we have listed only those parameters that play a role in this review. For a complete list and a more detailed discussion, the reader is referred to . The quantities and are related to the interaction of the companion with a test particle or a photon. The parameter accounts for a possible change in the moment of inertia of the pulsar due to a change in the local gravitational constant. In GR one finds , , and . Consequently These parameters are independent of the internal structure of the neutron star(s), due to the effacement of the internal structure, a property of GR [38, 35]. For most alternative gravity theories this is not the case. For instance, in the mono-scalar-tensor theories of [57, 58], one finds333The mono-scalar-tensor theories of [57, 58] have a conformal coupling function . The Jordan-Fierz-Brans-Dicke gravity is the sub-class with , and . where . The body-dependent quantities and denote the effective scalar coupling of pulsar and companion respectively, and where denotes the asymptotic value of the scalar field at spatial infinity. The quantity is related to the moment of inertia of the pulsar via . For a given equation of state, the parameters , , and depend on the fundamental constants of the theory, e.g. and in , and the mass of the body. As we will demonstrate later, these “gravitational form factors” can assume large values in the strong gravitational fields of neutron stars. Depending on the value of , this is even the case for a vanishingly small , where there are practically no measurable deviations from GR in the Solar system. In fact, even for , a neutron star, above a certain -dependent critical mass, can have an effective scalar coupling of order unity. This non-perturbative strong-field behavior, the so-called ‘‘spontaneous scalarization’’ of a neutron star, was discovered 20 years ago by Damour and Esposito-Farèse . Finally, there is the post-Keplerian parameter , related to the damping of the orbit due to the emission of gravitational waves. We have seen above that in alternative gravity theories the back reaction from the gravitational wave emission might enter the equations of motion already at the 1.5 post-Newtonian level, giving rise to a . To leading order one finds in mono-scalar-tensor gravity the dipolar contribution from the scalar field [59, 60, 58]: As one can see, the change in the orbital period due to dipolar radiation depends strongly on the difference in the effective scalar coupling . Binary pulsar systems with a high degree of asymmetry in the compactness of their components are therefore ideal to test for dipolar radiation. An order unity difference in the effective scalar coupling would lead to a change in the binary orbit, which is several orders of magnitude () stronger than the quadrupolar damping predicted by GR. At the 2.5 post-Newtonian level (), in general, there are several contributions entering the calculation: Monopolar waves for eccentric orbits. Higher order contributions to the dipolar wave damping. Quadrupolar waves from the tensor field, and the fields that are also responsible for the monopolar and/or dipolar waves. Apart from a change in the orbital period, gravitational wave damping will also affect other post-Keplerian parameters. While gravitational waves carry away orbital energy and angular momentum, Keplerian parameters like the eccentricity and the semi-major axis of the pulsar orbit change as well. The corresponding post-Keplerian parameters are and respectively. However, these changes affect the arrival times of the pulsar signals much less than the , and therefore do (so far) not play a role in the radiative tests with binary pulsars. As already mentioned in Section 1.2, there is no generic connection between the higher-order gravitational wave damping effects and the parameters , , and of the modified Einstein-Infeld-Hoffmann formalism. Such higher order, mixed radiative and strong-field effects depend in a complicated way on the structure of the gravity theory . The post-Keplerian parameters are at the foundation of many of the gravity tests conducted with binary pulsars. As shown above, the exact functional dependence differs for given theories of gravity. A priori, the masses of the pulsar and the companion are undetermined, but they represent the only unknowns in this set of equations. Hence, once two post-Keplerian parameters are measured, the corresponding equations can be solved for the two masses, and the values for other post-Keplerian parameters can be predicted for an assumed theory of gravity. Any further post-Keplerian measurement must therefore be consistent with that prediction, otherwise the assumed theory has to be rejected. In other words, if post-Keplerian parameters can be measured, a total of independent tests can be performed. The method is very powerful, as any additionally measured post-Keplerian parameter is potentially able to fail the prediction and hence to falsify the tested theory of gravity. The standard graphical representation of such tests, as will become clear below, is the mass-mass diagram. Every measured post-Keplerian parameter defines a curve of certain width (given by the measurement uncertainty of the post-Keplerian parameter) in a - diagram. A theory has passed a binary pulsar test, if there is a region in the mass-mass diagram that agrees with all post-Keplerian parameter curves. 2 Gravitational wave damping 2.1 The Hulse-Taylor pulsar The first binary pulsar to ever be observed happened to be a rare double neutron star system. It was discovered by Russell Hulse and Joseph Taylor in summer 1974 . The pulsar, PSR B1913+16, has a rotational period of 59 ms and is in a highly eccentric () 7.75-hour orbit around an unseen companion. Shortly after the discovery of PSR B1913+16, it has been realized that this system may allow the observation of gravitational wave damping within a time span of a few years [63, 64]. The first relativistic effect seen in the timing observations of the Hulse-Taylor pulsar was the secular advance of periastron . Thanks to its large value of 4.2 deg/yr, this effect was well measured already one year after the discovery . Due to the, a priori, unknown masses of the system, this measurement could not be converted into a quantitative gravity test. However, assuming GR is correct, equation (17) gives the total mass of the system. From the modern value given in table 2 one finds .444Strictly speaking, this is the total mass of the system scaled with an unknown Doppler factor , i.e. . For typical velocities, is expected to be of order , see for instance . In gravity tests based on post-Keplerian parameters, the factor drops out and is therefore irrelevant . It took a few more years to measure the Einstein delay (11) with good precision. In a single orbit this effect is exactly degenerate with the Roemer delay, and only due to the relativistic precession of the orbit these two delays become separable [63, 67]. By the end of 1978, the timing of PSR B1913+16 yielded a measurement of the post-Keplerian parameter , which is the amplitude of the Einstein delay . Together with the total mass from , equation (18) can now be used to calculate the individual masses. With the modern value for from table 2, and the total mass given above, one finds the individual masses and for pulsar and companion respectively . With the knowledge of the two masses, and , the binary system is fully determined, and further GR effects can be calculated and compared with the observed values, providing an intrinsic consistency check of the theory. In fact, Taylor et al. reported the measurement of a decrease in the orbital period , consistent with the quadrupole formula (26). This was the first proof for the existence of gravitational waves as predicted by GR. In the meantime the is measured with a precision of 0.04% (see table 2). However, this is not the precision with which the validity of the quadrupole formula is verified in the PSR B1913+16 system. The observed needs to be corrected for extrinsic effects, most notably the differential Galactic acceleration and the Shklovskii effect, to obtain the intrinsic value caused by gravitational wave damping [69, 70]. The extrinsic contribution due to the Galactic gravitational field (acceleration g) and the proper motion (transverse angular velocity in the sky ) are given by where is the unit vector pointing towards the pulsar, which is at a distance from the Solar system. For PSR B1913+16, and are measured with very high precision, and also is known with good precision (%). However, there is a large uncertainty in the distance , which is also needed to calculate the Galactic acceleration of the PSR B1913+16 system, , in equation (27). Due to its large distance, there is no direct parallax measurement for , and estimates for are based on model dependent methods, like the measured column density of free electrons between PSR B1913+16 and the Earth. Such methods are known to have large systematic uncertainties, and for this reason the distance to PSR B1913+16 is not well known: kpc [71, 31]. In addition, there are further uncertainties, e.g. in the Galactic gravitational potential and the distance of the Earth to the Galactic center. Accounting for all these uncertainties leads to an agreement between and at the level of about . The corresponding mass-mass diagram is given in figure 4. As the precision of the radiative test with PSR B1913+16 is limited by the model-dependent uncertainties in equation (27), it is not expected that this test can be significantly improved in the near future. Finally, besides the mass-mass diagram, there is a different way to illustrate the test of gravitational wave damping with PSR B1913+16. According to equation (7), the change in the orbital period, i.e. the post-Keplerian parameter , is measured from a shift in the time of periastron passage, where is a multiple of . One finds for the shift in periastron time, as compared to an orbit with zero decay where denotes the number of the periastron passage, and is given by . Equation (28) represents a parabola in time, which can be calculated with high precision using the masses that come from and (see above). On the other hand, the observed cumulative shift in periastron can be extracted from the timing observations with high precision. A comparison of observed and predicted cumulative shift in the time of the periastron passage is given in figure 5. 2.2 The Double Pulsar — The best test for Einstein’s quadrupole formula, and more In 2003 a binary system was discovered where, at first, one member was identified as a pulsar with a 23 ms period . About half a year later, the companion was also recognized as a radio pulsar with a period of 2.8 s . Both pulsars, known as PSRs J07373039A and J07373039B, respectively, (or and hereafter), orbit each other in less than 2.5 hours in a mildly eccentric () orbit. As a result, the system is not only the first and only double neutron star system where both neutron stars are visible as active radio pulsars, but it is also the most relativistic binary pulsar laboratory for gravity known to date (see figure 6). Just to give an example for the strength of relativistic effects, the advance of periastron, , is 17 degrees per year, meaning that the eccentric orbit does a full rotation in just 21 years. In this subsection, we briefly discuss the properties of this unique system, commonly referred to as the Double Pulsar, and highlight some of the gravity tests that are based on the radio observations of this system. For detailed reviews of the Double Pulsar see [74, 75]. In the Double Pulsar system a total of six post-Keplerian parameters have been measured by now. Five arise from four different relativistic effects visible in pulsar timing , while a sixth one can be determined from the effects of geodetic precession, which will be discussed in detail in Section 3.2 below. The relativistic precession of the orbit, , was measured within a few days after timing of the system commenced, and by 2006 it was already known with a precision of 0.004% (see table 3). At the same time the measurement of the amplitude of Einstein delay, , reached 0.7% (see table 3). Due to the periastron precession of 17 degrees per year, the Einstein delay was soon well separable from the Roemer delay. Two further post-Keplerian parameters came from the detection of the Shapiro delay: the shape and range parameters and . They were measured with a precision of 0.04% and 5%, respectively (see table 3). From the measured value () one can already see how exceptionally edge-on this system is.555The only binary pulsar known to be (most likely) even more edge-on is PSR J16142230 with () . For this wide-orbit system (), however, no further post-Keplerian parameter is known that could be used in a gravity test. Finally, the decrease of the orbital period due to gravitational wave damping was measured with a precision of 1.4% just three years after the discovery of the system (see table 3). A unique feature of the Double Pulsar is its nature as a ‘‘dual-line source’’, i.e. we measure the orbits of both neutron stars at the same time. Obviously, the sizes of the two orbits are not independent from each other as they orbit a common center of mass. In GR, up to first post-Newtonian order the relative size of the orbits is identical to the inverse ratio of masses. Hence, by measuring the orbits of the two pulsars (relative to the centre of mass), we obtain a precise measurement of the mass ratio. This ratio is directly observable, as the orbital inclination angle is obviously identical for both pulsars, i.e. This expression is not just limited to GR. In fact, it is valid up to first post-Newtonian order and free of any explicit strong-field effects in any Lorentz-invariant theory of gravity (see for a detailed discussion). Using the parameter values of table 3, one finds that in the Double Pulsar the masses are nearly equal with . As it turns out, all the post-Keplerian parameters measured from timing are consistent with GR. In addition, the region of allowed masses agrees well with the measured mass ratio (see figure 7). One has to keep in mind, that the test presented here is based on data published in 2006 . In the meantime continued timing lead to a significant decrease in the uncertainties of the post-Keplerian parameters of the Double pulsar. This is especially the case for , for which the uncertainty typically decreases with , being the total time span of timing observations. The new results will be published in an upcoming publication (Kramer et al., in prep.). As reported in , presently the Double Pulsar provides the best test for the GR quadrupole formalism for gravitational wave generation, with an uncertainty well below the 0.1% level. As discussed above, the Hulse-Taylor pulsar is presently limited by uncertainties in its distance. This raises the valid question, at which level such uncertainties will start to limit the radiative test with the Double Pulsar as well. Compared to the Hulse-Taylor pulsar, the Double Pulsar is much closer to Earth. Because of this, a direct distance estimate of kpc based on a parallax measurement with long-baseline interferometry was obtained . Thus, with the current accuracy in the measurement of distance and transverse velocity, GR tests based on can be taken to the 0.01% level. We will come back to this in Section 8, where we discuss some future tests with the Double Pulsar. With the large number of post-Keplerian parameters and the known mass ratio, the Double Pulsar is the most over-constrained binary pulsar system. For this reason, one can do more than just testing specific gravity theories. The Double Pulsar allows for certain generic tests on the orbital dynamics, time dilation, and photon propagation of a spacetime with two strongly self-gravitating bodies . First, the fact that the Double Pulsar gives access to the mass ratio, , in any Lorentz-invariant theory of gravity, allows us to determine and . With this information at hand, the measurement of the shape of the Shapiro delay can be used to determine via equation (16): . At this point, the measurement of the post-Keplerian parameters , , and (equations (13), (14), (15)) can be used to impose restrictions on the “strong-field” parameters of Lagrangian (2) : This is in full agreement with GR, which predicts one for all three of these expressions. Consequently, nature cannot deviate much from GR in the quasi-stationary strong-field regime of gravity (G2 in figure 1). 2.3 PSR J1738+0333 — The best test for scalar-tensor gravity The best “pulsar clocks” are found amongst the fully recycled millisecond pulsars, which have rotational periods less than about 10 ms (see e.g. ). A result of the stable mass transfer between companion and pulsar in the past — responsible for the recycling of the pulsar — is a very efficient circularization of the binary orbit, that leads to a pulsar-white dwarf system with very small residual eccentricity . For such systems, the post-Keplerian parameters and are generally not observable. There are a few cases where the orbit is seen sufficiently edge-on, so that a measurement of the Shapiro delay gives access to the two post-Keplerian parameters and with good precision (see e.g. , which was the first detection of a Shapiro delay in a binary pulsar). With these two parameters the system is then fully determined, and in principle can be used for a gravity test in combination with a third measured (or constrained) post-Keplerian parameter (e.g. ). Besides the Shapiro delay parameters, some of the circular binary pulsar systems offer a completely different access to their masses, which is not solely based on the timing observations in the radio frequencies. If the companion star is bright enough for optical spectroscopy, then we have a dual-line system, where the Doppler shifts in the spectral lines can be used, together with the timing observations of the pulsar, to determine the mass ratio . Furthermore, if the companion is a white dwarf, the spectroscopic information in combination with models of the white dwarf and its atmosphere can be used to determine the mass of the white dwarf , ultimately giving the mass of the pulsar via . As we will see in this and the following subsection, two of the best binary pulsar systems for gravity tests have their masses determined through such a combination of radio and optical astronomy. PSR J1738+0333 was discovered in 2001 . It has a spin period of 5.85 ms and is a member of a low-eccentricity () binary system with an orbital period of just 8.5 hours. The companion is an optically bright low-mass white dwarf (see figure 8). Extensive timing observation over a period of 10 years allowed a determination of astrometric, spin and orbital parameters with high precision , most notably A change in the orbital period of . A timing parallax, which gives a model independent distance estimate of kpc. The latter is important to correct for the Shklovskii effect and the differential Galactic acceleration to obtain the intrinsic (cf. equation (27)). Additional spectroscopic observations of the white dwarf gave the mass ratio and the companion mass , and consequently the pulsar mass . It is important to note, that the mass determination for PSR B1738+0333 is free of any explicit strong-field contributions, since this is the case for the mass ratio , and certainly for the mass of the white dwarf, which is a weakly self-gravitating body, i.e. a gravity regime that has been well tested in the Solar system (G1 in figure 1). After using equation (27) to correct for the Shklovskii contribution, , and the contribution from the Galactic differential acceleration, , one finds an intrinsic orbital period change due to gravitational wave damping of . This value agrees well with the prediction of GR, as can be seen in figure 9. The radiative test with PSR J1738+0333 represents a % verification of GR’s quadrupole formula. A comparison with the % test from the Double Pulsar (see Section 2.2) raises the valid question of whether the PSR J1738+0333 experiment is teaching us something new about the nature of gravity and the validity of GR. To address this question, let’s have a look at equation (25). Dipolar radiation can be a strong source of gravitational wave damping, if there is a sufficient difference between the effective coupling parameters and of pulsar and companion respectively. For the Double Pulsar, where we have two neutron stars with , one generally expects that , and therefore the effect of dipolar radiation would be strongly suppressed. On the other hand, in the PSR J1738+0333 system there is a large difference in the compactness of the two bodies. For the weakly self-gravitating white-dwarf companion , i.e. it assumes the weak-field value666From the Cassini experiment one obtains (95% confidence)., while the strongly self-gravitating pulsar can have an that significantly deviates from . In fact, as discussed in Section 1.4, can even be of oder unity in the presence of effects like strong-field scalarization. In the absence of non-perturbative strong-field effects one can do a first order estimation . For the Double Pulsar one finds , which is significantly smaller than for the PSR J1738+0333 system, which has .777These numbers are based on the equation of state MPA1 in . Within GR, MPA1 has a maximum neutron-star mass of , which can also account for the high-mass candidates of [86, 87, 88]. As a consequence, the orbital decay of asymmetric systems like PSR J1738+0333 could still be dominated by dipolar radiation, even if the Double Pulsar agrees with GR. For this reason, PSR J1738+0333 is particularly useful to test gravity theories that violate the strong equivalence principle and therefore predict the emission of dipolar radiation. A well known class of gravity theories, where this is the case, are scalar-tensor theories. As it turns out, PSR J1738+0333 is currently the best test system for these alternatives to GR (see figure 10). In terms of equation (25), one finds where for the weakly self-gravitating white dwarf companion . This limit can be interpreted as a generic limit on dipolar radiation, where is the difference of some hypothetical (scalar- or vector-like) “gravitational charges” . 2.4 PSR J0348+0432 — A massive pulsar in a relativistic orbit PSR J0348+0432 was discovered in 2007 in a drift scan survey using the Green Bank radio telescope (GBT) [92, 93]. PSR J0348+0432 is a mildly recycled radio-pulsar with a spin period of 39 ms. Soon it was found to be in a 2.46-hour orbit with a low-mass white-dwarf companion. In fact, the orbital period is only 15 seconds longer than that of the Double Pulsar, which by itself makes this already an interesting system for gravity. Initial timing observations of the binary yielded an accurate astrometric position, which allowed for an optical identification of its companion . As it turned out, the companion is a relatively bright white dwarf with a spectrum that shows deep Balmer lines. Like in the case of PSR J1738+0333, one could use high-resolution optical spectroscopy to determine the mass ratio (see figure 11) and the companion mass . For the mass of the pulsar one then finds , which is presently the highest, well determined neutron star mass, and only the second neutron star with a well determined mass close to 2 .888The first well determined two Solar mass neutron star is PSR J16142230 , which is in a wide orbit and therefore does not provide any gravity test. Since its discovery there have been regular timing observations of PSR J0348+0432 with three of the major radio telescopes in the world, the 100-m Green Bank Telescope, the 305-m radio telescope at the Arecibo Observatory, and the 100-m Effelsberg radio telescope. Based on the timing data, in 2013 Antoniadis et al. reported the detection of a decrease in the orbital period of that is in full agreement with GR (see figure 12). In numbers: As it turns out, using the distance inferred from the photometry of the white dwarf () corrections due to the Shklovskii effect and differential acceleration in the Galactic potential (see equation (27)) are negligible compared to the measurement uncertainty in . Like PSR 1738+0333, PSR J0348+0432 is a system with a large asymmetry in the compactness of the components, and therefore well suited for a dipolar radiation test. Using equation (25), the limit (34) can be converted into a limit on additional gravitational scalar or vector charges: This limit is certainly weaker than the limit (34), but it has a new quality as it tests a gravity regime in neutron stars that has not been tested before. Gravity tests before were confined to “canonical” neutron star masses of . PSR J0348+0432 for the first time allows a test of the relativistic motion of a massive neutron star, which in terms of gravitational self-energy lies clearly outside the tested region (see figure 13). Although an increase in fractional binding energy of about 50% does not seem much, in the highly non-linear gravity regime of neutron stars it could make a significant difference. To demonstrate this, used the scalar-tensor gravity of [57, 58], which is known to behave strongly non-linear in the gravitational fields of neutron stars, in particular for . As shown in figure 14, PSR J0348+0432 excludes a family of scalar-tensor theories that predict significant deviations from GR in massive neutron stars and were not excluded by previous experiments, most notably the test done with PSR J1738+0333. To further illustrate this in a mass-mass diagram, figure 15 shows a gravity theory with strong-field scalarization in massive neutron stars that passes the PSR J1738+0333 experiment, but is falsified by PSR J0348+0432. With PSR J0348+0432, gravity tests now cover a range of neutron star masses from 1.25 (PSR J07373039B) to 2 . No significant deviation from GR in the orbital motion of these neutron stars was found. These findings have interesting implications for the upcoming ground-based gravitational wave experiments, as we will briefly discuss in the next subsection. 2.5 Implications for gravitational wave astronomy The first detection of gravitational waves from astrophysical sources by ground-based laser interferometers, like LIGO999www.ligo.org and VIRGO101010www.cascina.virgo.infn.it, will mark the beginning of a new era of gravitational wave astronomy . One of the most promising sources for these detectors are merging compact binaries, consisting of neutron stars and black holes, whose orbits are decaying towards a final coalescence due to gravitational wave damping. While the signal sweeps in frequency through the detectors’ typical sensitive bandwidth from about 20 Hz to a few kHz, the gravitational wave signal will be deeply buried in the broadband noise of the detectors . To detect the signal, one will have to apply a matched filtering technique, i.e. correlate the output of the detector with a template wave form. Consequently, it is crucial to know the binary’s orbital phase with high accuracy for searching and analyzing the signals from in-spiraling compact binaries. Typically, one aims to lose less than one gravitational wave cycle in a signal with cycles. For this reason, within GR such calculations for the phase evolution of compact binaries have been conducted with great effort to cover many post-Newtonian orders including spin-orbit and spin-spin contributions (see [36, 97] for reviews). Table 4 illustrates the importance of the individual corrections to the number of cycles spent in the LIGO/VIRGO band111111The advanced LIGO/VIRGO gravitational wave detectors are expected to have a lower end seismic noise cut-off at about 10 Hz . For a low signal-to-noise ratio the low-frequency cut-off is considerably higher. In this review, we adapt a value of 20 Hz as the minimum frequency. The maximum frequency of a few kHz is not important here, since the frequency of the innermost circular orbit is well below the upper limit of the LIGO/VIRGO band. for two merging non-spinning neutron stars. For a later comparison, the two neutron-star masses are chosen to be 2 and 1.25 , the highest and lowest neutron-star masses observed. If the gravitational interaction between two compact masses is different from GR, the phase evolution over the last few thousand cycles, which fall into the bandwidth of the detectors, deviates from the (GR) template. This will degrade the ability to accurately determine the parameters of the merging binary, or in the worst case even prevent the detection of the signal. In scalar-tensor gravity, for instance, the evolution of the phase is modified because the system can now lose additional energy to dipolar waves [99, 100]. Depending on the difference between the effective scalar couplings of the two bodies, and , the 1.5 post-Newtonian dipolar contribution to the equations of motion could drive the gravitational wave signal many cycles away from the GR template. For this reason, it is desirable that potential deviations from GR in the interaction of two compact objects can be tested and constrained prior to the start of the advanced gravitational wave detectors. With its location at the high end of the measured neutron-star masses, PSR J0348+0432 with its limit (35) plays a particularly important role in such constraints. where , and is the chirp mass. Equation (36) is based on the assumption that , and are considerably smaller than unity, which is supported by binary pulsar experiments. For a 2/1.25 double neutron-star merger, one finds from equation (36) and the limit (35) where is the gravitational wave frequency of the innermost circular orbit (cf. ). The exact value of does not play an important role in equation (36), since . This result is based on the extreme assumption, that the light neutron star has an effective scalar coupling which corresponds to the well constrained weak-field limit, i.e. . If the companion of the 2 neutron star is a 10 black hole, then the constraints on that can be derived from binary pulsar experiments are even tighter (see ). A comparison with table 4 shows that the limit (37) is already below the contribution of the highest order correction calculated. As explained in , binary pulsar experiment cannot exclude significant deviations associated with short-range fields (e.g. massive scalar fields), which could still impact the mergers for ground-based gravitational wave detectors. Also, there is the possibility of the occurrence of effects like dynamical scalarization that, depending on the specifics of the theory and the masses, could start to influence the merger at , and consequently limit the validity of (37) to a smaller frequency band. Nevertheless, the constraints on dipolar radiation obtained from binary pulsars provide added confidence in the use of elaborate GR templates to search for the signals of compact merging binaries in the LIGO/VIRGO data sets. 3 Geodetic precession A few months after the discovery of the Hulse-Taylor pulsar, Damour and Ruffini proposed a test for geodetic precession in that system. If the pulsar spin is sufficiently tilted with respect to the orbital angular momentum, the spin direction should gradually change over time (see Section 1.3). A change in the orientation of the spin-axis of the pulsar with respect to the line-of-sight should lead to changes in the observed pulse profile. These pulse-profile changes manifest themselves in various forms , such as changes in the amplitude ratio or separation of pulse components [103, 104], the shape of the characteristic swing of the linear polarization , or the absolute value of the position angle of the polarization in the sky . In principle, such changes could allow for a measurement of the precession rate and by this yield a test of GR. In practice, it turned out to be rather difficult to convert changes in the pulse profile into a quantitative test for the precession rate. Indeed, the Hulse-Taylor pulsar, in spite of prominent profile changes due to geodetic precession [103, 104], does not (yet) allow for a quantitative test of geodetic precession. This is mostly due to uncertainties in the orientation of the magnetic axis and the intrinsic beam shape . Profile and polarization changes due to geodetic precession have been observed in other binary pulsars as well [107, 108], but again did not lead to a quantitative gravity test. A complete list of binary pulsars that up to date show signs of geodetic precession can be found in . Out of the six pulsars listed in , so far only two allowed for quantitative constraints on their rate of geodetic precession. These two binary pulsars will be discussed in more details in the following. 3.1 Psr B1534+12 PSR B1534+12 is a 38 ms pulsar, which was discovered in 1991 . It is a member of an eccentric () double neutron-star system with an orbital period of about 10 hours. Subsequent timing observations lead to the determination of five post-Keplerian parameters: , , , and , from the Shapiro delay . The large uncertainty in the distance to this system still prevents its usage in a gravitational wave test, since the observed has a large Shklovskii contribution, which one cannot properly correct for. The other four post-Keplerian parameters are nevertheless useful to test quasi-stationary strong-field effects. However, these tests are generally less constraining than tests from other pulsars (see e.g. figure 10). Continued observations of PSR B1534+12 with the 305-m Arecibo radio telescope revealed systematic changes in the the observed pulsar profile by about 1% per year, as well as changes in the polarization properties of the pulsar . As outlined above, such changes are expected from geodetic precession. Using equation (4) and the parameters from , one finds that GR predicts a precession rate of for PSR B1534+12. Besides the secular changes visible in the high signal-to-noise ratio pulse profile and polarization data of PSR B1534+12, Stairs et al. reported the detection of special-relativistic aberration of the revolving pulsar beam due to orbital motion. Aberration periodically shifts the observed angle between the line of sight and spin axis of PSR B1534+12 by an amount that depends on the orientation of the pulsar spin, and therefore contains additional geometrical information. Combining these observations, Stairs et al. were able to determine the system geometry, including the misalignment between the spin of PSR B1534+12 and the angular momentum of the binary motion, and constrain the rate of geodetic precession to Although the uncertainties are comparably large, these were the first beam-model-independent constraints on the geodetic precession rate of a binary pulsar. As can be seen, these model-independent constraints on the precession rate are consistent with the prediction by GR, as given in equation (38). 3.2 The Double Pulsar In Section 2.2, we have seen the Double Pulsar as one of the most exciting “laboratories” for relativistic gravity, with a wealth of relativistic effects measured, allowing the determination of 5 post-Keplerian parameters from timing observations: , , , , . Calculating the inclination angle of the orbit from , one finds that the line-of-sight is inclined with respect to the plane of the binary orbit by just about . As a consequence, during the superior conjunction the signals of pulsar pass pulsar at a distance of only 20 000 km. This is small compared to the extension of pulsar ’s magnetosphere, which is roughly given by the radius of the light-cylinder121212The light-cylinder is defined as the surface where the co-rotating frame reaches the speed of light. 130 000 km. And indeed, at every superior conjunction pulsar gets eclipsed for about 30 seconds due to absorption by the plasma in the magnetosphere of pulsar . A detailed analysis revealed that during every eclipse the light curve of pulsar shows flux modulations that are spaced by half or integer numbers of pulsar ’s rotational period (see figure 16). This pattern can be understood by absorbing plasma that co-rotates with pulsar and is confined within the closed field lines of the magnetic dipole of pulsar . As such, the orientation of pulsar ’s spin is encoded in the observed light curve of pulsar . Over the course of several years, Breton et al. observed characteristic shifts in the eclipse pattern, that can be directly related to a precession of the spin of pulsar . From this analysis, Breton et al. were able to derive a precession rate of The measured rate of precession is consistent with that predicted by GR () within its one-sigma uncertainty. This is the sixth(!) post-Keplerian parameter measured in the Double-Pulsar system (see figure 17). Furthermore, for the coupling function , which parametrizes strong-field deviation in alternative gravity theories (see equation (5)), one finds which agrees with the GR value . Although the geodetic precession of a gyroscope was confirmed to better than 0.3% by the Gravity Probe B experiment , the clearly less precise test with Double Pulsar (13%) for the first time gives a good measurement of this effect for a strongly self-gravitating “gyroscope”, and by this represents a qualitatively different test. The geodetic precession of pulsar not only changes the pattern of the flux modulations observed during the eclipse of pulsar , it also changes the orientation of pulsar ’s emission beam with respect to our line-of-sight. As a result of this, geodetic precession has by now turned pulsar in such a way, that since 2009 it is no longer seen by radio telescopes on Earth . From their model, Perera et al. predicted that the reappearance of pulsar is expected to happen around 2035 with the same part of the beam, but could be as early as 2014 if one assumes a symmetric beam shape. Finally, for pulsar GR predicts a precession rate of 4.78 deg/yr, which is comparable to that of pulsar . However, since the light-cylinder radius of pulsar () is considerably smaller than that of pulsar , there are no eclipses that could give insight into the orientation of its spin. Moreover, long-term pulse profile observations indicate that the misalignment between the spin of pulsar and the orbital angular momentum is less than (95% confidence) . For such a close alignment, geodetic precession is not expected to cause any significant changes in the spin direction (cf. equations (4) and (5)). This, on the other hand, is good news for tests based on timing observations. One does not expect a complication in the analysis of the pulse arrival times due to additional modeling of a changing pulse profile, like this is, for instance, the case in PSR J11416545 . 4 The strong equivalence principle The strong equivalence principle (SEP) extends the weak equivalence principle (WEP) to the universality of free fall (UFF) of self-gravitating bodies. In GR, WEP and SEP are fulfilled, i.e. in GR the world line of a body is independent of its chemical composition and gravitational binding energy. Therefore, a detection of a SEP violation would directly falsify GR. On the other hand, alternative theories of gravity generally violate SEP. This is also the case for most metric theories of gravity . For a weakly self-gravitating body in a weak external gravitational field one can simply express a violation of SEP as a difference between inertial and gravitational mass that is proportional to the gravitational binding energy of the mass: The Nordtvedt parameter is a theory dependent constant. In the parameterized post-Newtonian (PPN) framework, is given as a combination of different PPN parameters (see for details). As a consequence of (42), the Earth () and the Moon () would fall differently in the gravitational field of the Sun (Nordtvedt effect ). The parameter is therefore tightly constrained by the lunar-laser-ranging (LLR) experiments to , which is in perfect agreement with GR where . In view of the smallness of the self-gravity of Solar system bodies, the LLR experiment says nothing about strong-field aspects of SEP. SEP could still be violated in extremely compact objects, like neutron stars, meaning that a neutron star would feel a different acceleration in an external gravitational field than weakly self-gravitating bodies. For such a strong-field SEP violation, the best current limits come from millisecond pulsar-white dwarf systems with wide orbits. If there is a violation of UFF by neutron stars, then the gravitational field of the Milky Way would polarize the binary orbit . In comparison with the LLR experiment, such tests have two disadvantages: i) the much weaker polarizing external field (, as compared to the of the Solar gravitational field at the location of the Earth-Moon system), and ii) the significantly lower precision in the ranging, which is of the order of a few cm for the best pulsar experiments ( cm for LLR). This is almost completely counterbalanced by the gravitational binding energy of the neutron star, which is a large fraction of its total inertial mass energy () and more than eight orders of magnitude larger than that of the Earth. This results in experiments with comparable limits on a SEP violation, which nonetheless are complementary since they probe different regimes of binding energy. The recent discovery of a millisecond pulsar in a hierarchical triple (see and Ransom et al., in prep.) might allow for a significant improvement in testing SEP, as it combines a strong external field with a large fractional binding energy . Since beyond the first post-Newtonian approximation there is no general PPN formalism available, discussions of gravity tests in this regime are done in various theory-specific frameworks. A particularly suitable example for a framework that allows a detailed investigation of higher order/strong-field deviations from GR, is the above mentioned two-parameter class of mono-scalar-tensor theories of [57, 58], which for certain values of exhibit significant strong-field deviations from GR, and a correspondingly strong violation of SEP for neutron stars. To illustrate this violation of SEP, it is sufficient to look at the leading “Newtonian” terms in the equations of motion of a three body system with masses () : where the body-dependent effective gravitational constant is related to the bare gravitational constant by As mentioned above, for a neutron star can significantly deviate from the weak-field value . The structure dependence of the effective gravitational constant has the consequence that the pulsar does not fall in the same way as its companion, in the gravitational field of our Galaxy. For a binary pulsar with a non-compact companion, e.g. a white dwarf, that effect should be most prominent. Since both the white dwarf and the Galaxy are weakly self-gravitating bodies, their effective scalar coupling can be approximated by , and one finds from equation (43) where , and where is the gravitational acceleration caused by the Galaxy at the location of the binary pulsar.131313Here we used , and we dropped terms of order and smaller. Also, the contribution from post-Newtonian dynamics, term , has been added, whose most important consequence is the secular precession of periastron, . The -related term reflects the violation of SEP, which modifies the orbital dynamics of binary pulsars. This can be confronted with pulsar observations to test for a violation of SEP. In the following we briefly discuss different tests of SEP with binary pulsars. For a more complete review of the topic of this section see . The discussion below is not specific to scalar-tensor gravity, and the quantity can be generically seen as the difference between inertial and gravitational mass. 4.1 The Damour-Schäfer test In 1991, when Damour and Schäfer first investigated the orbital dynamics of a binary pulsar under the influence of a SEP violation , only four binary pulsars were known in the Galactic disk. Two of these (PSR B1913+16 and PSR B1957+20) were clearly inadequate for that test, not only because of the compactness of their orbits, but also because PSR B1913+16 is member of a double neutron star system that lacks the required amount of asymmetry in the binding energy, necessary for a stringent test of a SEP violation, and PSR B1957+20 is a so called “black-widow” pulsar, where the companion suffers significant irregular mass losses, due to the irradiation by the pulsar. The remaining systems were PSR B1855+09 and PSR B1953+29 . Both of these systems have wide orbits with small eccentricities, and respectively. Damour and Schäfer found for small-eccentricity binary systems that a violation of SEP leads to a characteristic polarization of the orbit, which is best represented by a vector addition where the end-point of the observed eccentricity vector evolves along a circle in an eccentric way (see figure 18). The polarizing eccentricity is proportional to and therefore, a limit on would directly pose a limit on . Unfortunately neither

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What is the volume of a sphere and how to calculate it? The volume of a sphere refers to the amount of space enclosed by its surface. In other words, it`s the total capacity or space that the sphere occupies. The formula to calculate the volume of a sphere is V = (4/3) * π * r³, where "V" is the volume and "r" is the radius of the sphere. Given a diameter, one can easily find the radius by dividing the diameter by 2. Once you have the radius, you can substitute it into the formula to get the volume. This mathematical concept, though simple, has profound implications in various fields such as astronomy, engineering, and even daily life. How to use the Sphere Volume Calculator? This online Sphere Volume Calculator is designed for efficiency and accuracy. Here`s a step-by-step guide on how to use it: 1. Open the calculator interface on your preferred device. 2. You'll see two input options: one for the sphere`s radius and another for its diameter. 3. Enter either the radius or the diameter of the sphere. Remember, if you have the diameter, the radius is simply half of it. 4. Once the required value is inputted, click on the 'Calculate' button. 5. The calculator will instantly display the volume of the sphere based on the given radius or diameter. 6. For repeated calculations or to check another value, simply enter the new radius or diameter and hit 'Calculate' again. 7. Always ensure that you're entering accurate measurements for precise results. Examples of calculating the volume of a sphere Let`s delve into a few real-world examples to better understand this calculation: Example 1: Imagine you have a beach ball with a diameter of 1 meter. First, we find the radius, which is 0.5 meters. Using the formula V = (4/3) * π * 0.5³, the volume comes out to be approximately 0.52 cubic meters. That`s a lot of air! Example 2: A marble might seem tiny, but it has volume! Let`s say its diameter is 1 cm. The radius is 0.5 cm. Using our formula, its volume is roughly 0.52 cubic centimeters. Petite, isn`t it? Example 3: Ever wondered about the volume of Earth? For simplicity`s sake (and a touch of humor), let`s consider it a perfect sphere with a radius of 6,371 km. Plugging that into our formula gives a staggering volume of about 1 trillion cubic kilometers! Nuances in calculating the volume of a sphere While the formula is straightforward, here are some aspects to bear in mind: 1. Ensure accurate measurement of the radius or diameter. Small errors can significantly affect the final result. 2. The unit of measurement matters. Always convert different units to a consistent one before calculation. 3. The formula assumes a perfect sphere. Real-world objects might have imperfections altering the volume. 4. Temperature and pressure can affect the volume of gases inside a spherical container. 5. The calculator provides theoretical values. In practical applications, always consider tolerances. 6. It`s essential to understand the context of your calculation. For example, in engineering, material thickness might affect volume. 7. Remember that π is an irrational number. Most calculations use its approximate value, which is 3.14159. 8. If using the diameter to calculate volume, always ensure to divide it by two to get the radius. 9. In digital tools, always check for software updates for the most accurate calculations. 10. Familiarity with the sphere`s material can be crucial. For example, a hollow sphere will have a different volume compared to a solid one. Frequently Asked Questions about Sphere Volume Calculation Why is the π value used in the formula? π (Pi) is a mathematical constant representing the ratio of a circle`s circumference to its diameter. It`s essential for calculations involving circles or spheres. Can I use this calculator for planets? Yes, you can use it for any spherical object, but remember that planets might not be perfect spheres. What if my sphere is hollow? The calculator gives the volume of a solid sphere. For a hollow sphere, you'd need to subtract the volume of the void from the total. Are there any limitations to this calculator? It works best for perfect spheres. For objects with irregularities or not perfectly spherical, results might vary. How do I measure the diameter accurately? Use calipers or a ruler, ensuring you measure the longest straight line passing through the sphere`s center. You may find the following calculators on the same topic useful: - Sphere Surface Area Calculator. Calculate the surface area of a spherical object (sphere) using our online calculator. - Hexagon Area Calculator. Calculate the area of a regular (equilateral) hexagon using our online calculator. - Cube Surface Area Calculator. Calculate the surface area of a cube based on the length of edges, cube diagonal, or diagonals of its sides. - Scale Calculator. Convert named scale on a drawing to real scale and vice versa. - Cube Volume Calculator. Calculate online the volume of any cubic object based on the length of its side or diagonals. - Tank Volume Calculator. Calculate the online volume of a cylindrical, rectangular, or automotive tank based on dimensions (using consumption and distance traveled). - Room Volume Calculator. Calculate the volume of a room or any space in cubic meters or liters. - Online Arc Length Calculator. Transform geometric data into practical results by calculating the arc length of a circle. - Tube Volume Calculator. Determine the volume of a tube in cubic meters or liters by simply entering the diameter and length of the pipeline. - Pyramid Volume Calculator. Estimate the volume of a pyramid using its height, base area, or side length. Suitable for all base shapes. Share on social media If you liked it, please share the calculator on your social media platforms. It`s easy for you and beneficial for the project`s promotion. Thank you!

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One of the things that I have tried to encourage engineers and mathematicians to do is to tell things to me as though I were 8 years old. Vectors can be added to other vectors according to vector algebra. Line Integrals of Vector Fields 73 Surface Integrals. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of In mathematics, physics, and engineering, a Euclidean vector sometimes called a geometric or spatial vector, or—as here—simply a vector is a geometric object that has magnitude or length and direction. Here you can download advanced calculus for applications hildebrand solution manual shared files: Vector Mechanics for Engineers Dynamics 6th Ed. Vectors are used to represent physical quantities that have a magnitude and direction associated with them. The calculus of di erential forms give an alternative to vector calculus which is ultimately simpler and more exible. 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Head of the seminar: Prof. A.A.Dorogovtsev Secretary of the seminar: Ia.Korenovska E-mail address: email@example.com 17.00, room 208. - Семинар 09.06.2020 Докладчик: А. С. Радова Тема: Z-функція Гекке та її застосування в асимптотичних задачах (за результатами кандидатської дисертації) - Семинар 26.05.2020 Докладчик: Xia Chen (University of Tennessee) Тема: Parabolic Anderson models – Large scale asymptotics Abstract. The model of the parabolic Anderson equation is relevant to some problems arising from physics such as the particle movement in disorder media, population dynamics, and to the KPZ equations through a suitable transformation. In the name of intermittency, broadly speaking, there has been increasing interest in the asymptotic behaviors of the system, over a large scale of the time or space, formulated in a quench or annealed form. By the multiplicative structure of the equation, the model is expected to grow geometrically. Hence, the ideas and methods developed from the area of large deviations become relevant and effective to some problems on intermittency. The talk is to provide some general view on the recent development over the topic of intermittency of this model. - Семинар 19.05.2020 Докладчик: Мария Белозерова (Одесский национальный университет имени И.И. Мечникова) Тема: Асимптотическое поведение решений систем стохастических дифференциальных уравнений со взаимодействием - Семинар 12.05.2020 Докладчик: Victor Marx (Leipzig University) Тема: Smoothing properties of a diffusion on the Wasserstein space Abstract: We study in this talk diffusion processes defined on the L_2-Wasserstein space of probability measures on the real line. We will introduce the construction of a diffusion inspired by (but slightly different from) the modified massive Arratia flow, studied by Konarovskyi and von Renesse. Then, our aim is to show that this diffusion has smoothing properties, similar to those of the standard Euclidean Brownian motion. Namely, we will first show that this process restores uniqueness of McKean-Vlasov equations with a drift coefficient which is not Lipschitz-continuous in its measure argument, extending the standard results obtained by Jourdain and foll. Secondly, we will present a Bismut-Elworthy-Li integration by parts formula for the semi-group associated to this diffusion. - Семинар 28.04.2020 Докладчик: Max von Renesse Тема: Molecules as metric measure spaces with lower Kato Ricci curvature Joint work with Batu Güneysu (Humboldt University Berlin)Abstract. In this talk we shall present a new result which connects the analysis of the Schrödinger semigroup associated to a molecule to the theory of metric measure spaces with lower Ricci curvature bounds. We show that the ground state transformation associated to this molecule creates naturally a metric measure space which has lower Ricci curvature bounds in terms of a Kato class function. This has numerous applications, for instance we show stochastic completeness of the corresponding metric measure space, and we also demonstrate that this setting is good enough to drive it semigroup gradient estimates using a variant of the Bismut derivative formula. - Семинар 21.04.2020 Докладчик: Vitalii Konarovskyi Тема: On the existence and uniqueness of solutions to the Dean-Kawasaki equationAbstract. We consider the Dean-Kawasaki equation with smooth drift interaction potential and show that measure-valued martingale solutions exist only in certain parameter regimes in which case they are given by finite Langevin particle systems with mean-field interaction. The proof is based on the Girsanov transform and log-Laplace duality. This is joint work with Max von Renesse and Tobias Lehmann. - Семинар 14.04.2020 Докладчик: Е.В. Глиняная Тема: Предельные теоремы для числа кластеров в потоке Арратья - Семинар 07.04.2020 Докладчик: PD Dr. Yana Kinderknecht (Butko) (Technical University of Braunschweig) Тема: Chernoff approximation of operator semigroups generated by Markov processesAbstract: We present a method to approximate operator semigroups generated by Markov processes and, therefore, transition probabilities of these processes. This method is based on the Chernoff theorem. In some cases, Chernoff approximations provide also discrete time Markov processes approximating the considered (continuous time) processes (in particular, Euler-Maruyama Schemes for the related SDEs). In some cases, Chernoff approximations have the form of limits of n iterated integrals of elementary functions as n→∞ (in this case, they are called Feynman formulae) and can be used for direct computations and simulations of Markov processes. The limits in Feynman formulae sometimes coincide with (or give rise to) path integrals with respect to probability measures (such path integrals are usually called Feynman-Kac formulae). Therefore, Feynman formulae can be used to approximate the corresponding path integrals and to establish relations between different path integrals. In this talk, we discuss Chernoff approximations for (semigroups generated by) Feller processes in ℝ^d. We are also interested in constructing Chernoff approximations for Markov processes which are obtained by different operations from some original Markov processes, assuming that Chernoff approximations for the original processes are already known. In this talk, we present Chernoff approximations for such operations as: a random time change via an additive functional of a process, a subordination (i.e., a random time change via an independent a.s. nondecreasing 1-dim. Lévy process), killing of a process upon leaving a given domain, reflecting of a process. These results allow, in particular, to obtain Chernoff approximations for subordinate diffusions on star graphs and compact Riemannian manifolds. Moreover, Chernoff approximations can be further used to approximate solutions of some time-fractional evolution equations and hence to approximate marginal densities of the corresponding non-Markovian stochastic processes. - Семинар 31.03.2020Докладчик: Г. В. РябовТема: Преобразования винеровской меры и обобщение теоремы Гирсанова - Семинар 17.03.2020Докладчик: Н. Б. ВовчанскийТема: Каплинг в методе дробных шагов для броуновской сети

Hello, I am back! After some summer rest/study/introspection! And after an amazing July month with the Higgs discovery by ATLAS and CMS. After an amazing August month with the Curiosity rover, MSL(Mars Science Laboratory), arrival to Mars. After a hot summer in my home town…I have written lots of drafts these days…And I will be publishing all of them step to step. We will discuss today one of interesting remark studied by Kaniadakis. He is known by his works on relatistivic physics, condensed matter physics, and specially by his work on some cool function related to non-extensive thermodynamics. Indeed, Kaniadakis himself has probed that his entropy is also related to the mathematics of special relativity. Ultimately, his remarks suggest: 1st. Dimensionless quantities are the true fundamental objects in any theory. 2nd. A relationship between information theory and relativity. 3rd. The important role of deformation parameters and deformed calculus in contemporary Physics, and more and more in the future maybe. 4nd. Entropy cound be more fundamental than thought before, in the sense that non-extensive generalizations of entropy play a more significant role in Physics. 5th. Non-extensive entropies are more fundamental than the conventional entropy. The fundamental object we are going to find is stuff related to the following function: Let me first imagine two identical particles ( of equal mass) A and B, whose velocities, momenta and energies are, in certain frame S: In the rest frame of particle B, S’, we have If we define a dimensionless momentum paramenter we get after usual exponentiation Galilean relativity says that the laws of Mechanics are unchanged after the changes from rest to an uniform motion reference frame. Equivalentaly, galilean relativity in our context means the invariance under a change , and it implies the invariance under a change . In turn, plugging these inte the last previous equation, we get the know relationship Wonderful, isn’t it? It is for me! Now, we will move to Special Relativity. In the S’ frame where B is at rest, we have: and from the known relativistic transformations for energy and momentum where of course we define After this introduction, we can parallel what we did for galilean relativity. We can write the last previous equations in the equivalent form, after some easy algebra, as follows Now, we can introduce dimensionless variables instead of the triple , defining instead the adimensional set : Note that the so-called deformation parameter is indeed related (equal) to the beta parameter in relativity. Again, from the special relativity requirement we obtain, as we expected, that . Classical physics, the galilean relativity we know from our everyday experience, is recovered in the limit , or equivalently, if . In the dimensionless variables, the transformation of energy and momentum we wrote above can be showed to be: In rest frame of some particle, we get of course the result , or in the new variables . The energy-momentum dispersion relationship from special relativity becomes: Moreover, we can rewrite the equation in terms of the dimensionless energy-momentum variable amd we get the analogue of the galilean addition rule for dimensionless velocities Note that the classical limit is recovered again sending . Now, we have to define some kind of deformed exponential function. Let us define: Applying this function to the above last equation, we observe that Again, relativity means that observers in uniform motion with respect to each other should observe the same physical laws, and so, we should obtain invariant equations under the exchanges and . Pluggint these conditions into the last equation, it implies that the following condition holds (and it can easily be checked from the definition of the deformed exponential). One interesting question is what is the inverse of this deformed exponential ( the name q-exponential or -exponential is often found in the literature). It has to be some kind of deformed logarithm. And it is! The deformed logarithm, inverse to the deformed exponential, is the following function: Indeed, this function is related to ( in units with the Boltzmann’s constant set to the unit ) the so-called Kaniadakis entropy! Furthermore, the equation also implies that The gamma parameter of special relativity is also recasted as More generally, in fact, the deformed exponentials and logarithms develop a complete calculus based on: and the differential operators so that, e.g., This Kanadiakis formalism is useful, for instance, in generalizations of Statistical Mechanics. It is becoming a powertool in High Energy Physics. At low energy, classical statistical mechanics gets a Steffan-Boltmann exponential factor distribution function: At high energies, in the relativistic domain, Kaniadakis approach provide that the distribution function departures from the classical value to a power law: There are other approaches and entropies that could be interesting for additional deformations of special relativity. It is useful also in the foundations of Physics, in the Information Theory approach that sorrounds the subject in current times. And of course, it is full of incredibly beautiful mathematics! We can start from deformed exponentials and logarithms in order to get the special theory of relativity (reversing the order in which I have introduced this topic here). Aren’t you surprised? We live in the information era. Read more about this age here. Everything in your sorrounding and environtment is bound and related to some kind of “information processing”. Information can also be recorded and transmitted. Therefore, being rude, information is something which is processed, stored and transmitted. Your computer is now processing information, while you read these words. You also record and save your favourite pages and files in your computer. There are many tools to store digital information: HDs, CDs, DVDs, USBs,…And you can transmit that information to your buddies by e-mail, old fashioned postcards and letters, MSN, phone,…You are even processing information with your brain and senses, whenever you are reading this text. Thus, the information idea is abstract and very general. The following diagram shows you how large and multidisciplinary information theory(IT) is: I enjoyed as a teenager that old game in which you are told a message in your ear, and you transmit it to other human, this one to another and so on. Today, you can see it at big scale on Twitter. Hey! The message is generally very different to the original one! This simple example explains the other side of communication or information transmission: “noise”. Although efficiency is also used. The storage or transmission of information is generally not completely efficient. You can loose information. Roughly speaking, every amount of information has some quantity of noise that depends upon how you transmit the information(you can include a noiseless transmission as a subtype of information process in which, there is no lost information). Indeed, this is also why we age. Our DNA, which is continuously replicating itself thanks to the metabolism (possible ultimately thanksto the solar light), gets progressively corrupted by free radicals and different “chemicals” that makes our cellular replication more and more inefficient. Don’t you remember it to something you do know from High-School? Yes! I am thinking about Thermodynamics. Indeed, the reason because Thermodynamics was a main topic during the 19th century till now, is simple: quantity of energy is constant but its quality is not. Then, we must be careful to build machines/engines that be energy-efficient for the available energy sources. Before going into further details, you are likely wondering about what information is! It is a set of symbols, signs or objects with some well defined order. That is what information is. For instance, the word ORDER is giving you information. A random permutation of those letters, like ORRDE or OERRD is generally meaningless. I said information was “something” but I didn’t go any further! Well, here is where Mathematics and Physics appear. Don’t run far away! The beauty of Physics and Maths, or as I like to call them, Physmatics, is that concepts, intuitions and definitions, rigorously made, are well enough to satisfy your general requirements. Something IS a general object, or a set of objects with certain order. It can be certain DNA sequence coding how to produce certain substance (e.g.: a protein) our body needs. It can a simple or complex message hidden in a highly advanced cryptographic code. It is whatever you are recording on your DVD ( a new OS, a movie, your favourite music,…) or any other storage device. It can also be what your brain is learning how to do. That is “something”, or really whatever. You can say it is something obscure and weird definition. Really it is! It can also be what electromagnetic waves transmit. Is it magic? Maybe! It has always seems magic to me how you can browse the internet thanks to your Wi-Fi network! Of course, it is not magic. It is Science. Digital or analogic information can be seen as large ordered strings of 1’s and 0’s, making “bits” of information. We will not discuss about bits in this log. Future logs will… Now, we have to introduce the concepts through some general ideas we have mention and we know from High-School. Firstly, Thermodynamics. As everybody knows, and you have experiences about it, energy can not completely turned into useful “work”. There is a quality in energy. Heat is the most degradated form of energy. When you turn on your car and you burn fuel, you know that some of the energy is transformed into mechanical energy and a lof of energy is dissipated into heat to the atmosphere. I will not talk about the details about the different cycles engines can realize, but you can learn more about them in the references below. Simbollically, we can state that The great thing is that an analogue relation in information theory does exist! The relation is: Therefore, there is some subtle analogy and likely some deeper idea with all this stuff. How do physicists play to this game? It is easy. They invent a “thermodynamic potential”! A thermodynamic potential is a gadget (mathematically a function) that relates a set of different thermodynamic variables. For all practical purposes, we will focus here with the so-called Gibbs “free-energy”. It allows to measure how useful a “chemical reaction” or “process” is. Moreover, it also gives a criterion of spontaneity for processes with constant pressure and temperature. But it is not important for the present discussion. Let’s define Gibbs free energy G as follows: where H is called enthalpy, T is the temperature and S is the entropy. You can identify these terms with the previous concepts. Can you see the similarity with the written letters in terms of energy and communication concepts? Information is something like “free energy” (do you like freedom?Sure! You will love free energy!). Thus, noise is related to entropy and temperature, to randomness, i.e., something that does not store “useful information”. Internet is also a source of information and noise. There are lots of good readings but there are also spam. Spam is not really useful for you, isn’t it? Recalling our thermodynamic analogy, since the first law of thermodynamics says that the “quantity of energy” is constant and the second law says something like “the quality of energy, in general, decreases“, we have to be aware of information/energy processing. You find that there are signals and noise out there. This is also important, for instance, in High Energy Physics or particle Physics. You have to distinguish in a collision process what events are a “signal” from a generally big “background”. We will learn more about information(or entropy) and noise in my next log entries. Hopefully, my blog and microblog will become signals and not noise in the whole web. Where could you get more information? 😀 You have some good ideas and suggestions in the following references: 1) I found many years ago the analogy between Thermodynamics-Information in this cool book (easy to read for even for non-experts) Applied Chaos Theory: A paradigm for complexity. Ali Bulent Cambel (Author)Publisher: Academic Press; 1st edition (November 19, 1992) Unfortunately, in those times, as an undergraduate student, my teachers were not very interested in this subject. What a pity! 2) There are some good books on Thermodynamics, I love (and fortunately own) these jewels: Concepts in Thermal Physics, by Stephen Blundell, OUP. 2009. A really self-contained book on Thermodynamics, Statistical Physics and topics not included in standard books. I really like it very much. It includes some issues related to the global warming and interesting Mathematics. I enjoy how it introduces polylogarithms in order to handle closed formulae for the Quantum Statistics. Thermodynamcis and Statistical Mechanics. (Dover Books on Physics & Chemistry). Peter T. Landsberg A really old-fashioned and weird book. But it has some insights to make you think about the foundations of Thermodynamics. Thermodynamcis, Dover Pub. Enrico Fermi This really tiny book is delicious. I learned a lot of fun stuff from it. Basic, concise and completely original, as Fermi himself. Are you afraid of him? Me too! E. Fermi was a really exceptional physicist and lecturer. Don’t loose the opportunity to read his lectures on Thermodynamcis. Mere Thermodynamics. Don S. Lemons. Johns Hopkins University Press. Other great little book if you really need a crash course on Thermodynamics. Introduction to Modern Statistical Physics: A Set of Lectures. Zaitsev, R.O. URSS publishings. I have read and learned some extra stuff from URSS ed. books like this one. Russian books on Science are generally great and uncommon. And I enjoy some very great poorly known books written by generally unknow russian scientists. Of course, you have ever known about Landau and Lipshitz books but there are many other russian authors who deserve your attention. 3) Information Theory books. Classical information theory books for your curious minds are An Introduction to Information Theory: Symbols, Signals and Noise. Dover Pub. 2nd Revised ed. 1980. John. R. Pierce. A really nice and basic book about classical Information Theory. An introduction to Information Theory. Dover Books on Mathematics. F.M.Reza. Basic book for beginners. The Mathematical Theory of Communication. Claude E. Shannon and W.Weaver.Univ. of Illinois Press. A classical book by one of the fathers of information and communication theory. Mathematical Foundations of Information Theory. Dover Books on Mathematics. A.Y.Khinchin. A “must read” if you are interested in the mathematical foundations of IT.

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own. Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished? A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour. The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it? Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it? Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring? This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken? First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line. This Sudoku, based on differences. Using the one clue number can you find the solution? Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? Different combinations of the weights available allow you to make different totals. Which totals can you make? An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way? Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers? An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length? Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring? Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar". A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children. Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters. Four small numbers give the clue to the contents of the four Use the differences to find the solution to this Sudoku. Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules? A man has 5 coins in his pocket. Given the clues, can you work out what the coins are? A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article. The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern? My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be? Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour. Find out about Magic Squares in this article written for students. Why are they magic?! There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper. Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us? This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares. Can you use your powers of logic and deduction to work out the missing information in these sporty situations? You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest? Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens? A few extra challenges set by some young NRICH members. A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . . An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of Make your own double-sided magic square. But can you complete both sides once you've made the pieces? Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think? If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? A pair of Sudoku puzzles that together lead to a complete solution. How many solutions can you find to this sum? Each of the different letters stands for a different number. Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

How to address violence in the news with your students. MATHia works best as a supplement to classroom instruction and is more effective if students enter with an understanding of the math concepts already. Teachers can use MATHia to give students more practice on what they're working on in class. It can be assigned instead of paper worksheets or in addition to other classwork. MATHia could be used for homework and is a great way for students to practice for standardized computer-based tests. MATHia is ideal in a hybrid classroom where some students work online and others meet with a teacher. Teachers can track student progress carefully, and the data can be used to plan small-group instruction or individual conferences. The details provided by the progress bar are perfect conversation starters for struggling students.Continue reading Show less MATHia is a sixth-grade through pre-algebra math platform that aligns closely with the Common Core State Standards. It's designed to be used alongside Carnegie Learning's math curriculum or other math curricula. Teachers set up classes and can choose to follow the predetermined scope and sequence or design their own pathways. Detailed reports are provided for teachers, including standards mastery and growth data. However, no school-level reports are available yet. Seven modules are provided for each grade level, and each session takes at least 30 minutes to complete. MATHia includes two types of problems: mastery and non-mastery. The non-mastery problems are used to preview or review a topic and aren't adaptive. The mastery-based problems are adaptive and require students to reach a high level of conceptual understanding before moving on. The student dashboard provides a look at progress, time on task, and upcoming work. Unit overviews include a list of vocabulary, major math concepts covered, and a short video explaining how the math connects to real-life situations. Once students begin a problem, MATHia provides instant feedback on each step of the progress. Hints will pop up to address misconceptions, and students can also ask for a hint at any time. A highly detailed tracking bar advances and retreats based on how accurately the questions are answered. The progress bar breaks the task down into smaller steps in the process, such as naming units, identifying variables correctly, and so on. MATHia is a great tool for supporting young mathematicians. From a student perspective, it's easy to use and has few distractions. The math problems are rigorous and usually involve multiple ways to model thinking. Hints pop up automatically if students are stuck, but students won't make progress if they use too many hints or try to guess their way through the system. Teachers and students can use the progress bar to see exactly where they're succeeding and what needs more practice. MATHia won't let students proceed to another unit until mastery is reached on the progress bar. This means that some students may reach mastery after a few problems, while other students will be required to complete more problems to show mastery. This is excellent for learning but may prove frustrating to some learners. Unlike Khan Academy, MATHia doesn't let students freely explore other topics, nor do students earn as many badges. By itself, MATHia doesn't provide text-to-speech support, but it does play well with the Read&Write Chrome extension. While it's an excellent tool for practicing and gaining mastery of math concepts, MATHia could be even better if teachers could easily assign individual students customized pathways. Key Standards Supported Expressions And Equations Write and evaluate numerical expressions involving whole-number exponents. Write, read, and evaluate expressions in which letters stand for numbers. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole- number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. Apply the properties of operations to generate equivalent expressions. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Solve linear equations in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Verify experimentally the properties of rotations, reflections, and translations: Lines are taken to lines, and line segments to line segments of the same length. Angles are taken to angles of the same measure. Parallel lines are taken to parallel lines. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Explain a proof of the Pythagorean Theorem and its converse. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Ratios And Proportional Relationships Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Statistics And Probability Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Summarize numerical data sets in relation to their context, such as by: Reporting the number of observations. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

SAT Test Prep ESSENTIAL ALGEBRA 2 SKILLS Lesson 2: Functions What Is a Function? A function is any set of instructions for turning an input number (usually called x) into an output number (usually called y). For instance, is a function that takes any input x and multiplies it by 3 and then adds 2. The result is the output, which we call f (x) or y. If , what is f (2h)? In the expression f (2h), the 2h represents the input to the function f. So just substitute 2h for x in the equation and simplify: . Functions as Equations, Tables, or Graphs The SAT usually represents a function in one of three ways: as an equation, as a table of inputs and outputs, or as a graph on the xy-plane. Make sure that you can work with all three representations. For instance, know how to use a table to verify an equation or a graph, or how to use an equation to create or verify a graph. A linear function is any function whose graph is a line. The equations of linear functions always have the form , where m is the slope of the line, and b is where the line intersects the y-axis. (For more on slopes, see Chapter 10, Lesson 4.) The function is linear with a slope of 3 and a y-intercept of 2. It can also be represented with a table of x and y (or f (x)) values that work in the equation: Notice several important things about this table. First, as in every linear function, when the x values are “evenly spaced,” the y values are also “evenly spaced.” In this table, whenever the x value increases by 1, the y value increases by 3, which is the slope of the line and the coefficient ofx in the equation. Notice also that the y-intercept is the output to the function when the input is 0. Now we can take this table of values and plot each ordered pair as a point on the xy-plane, and the result is the graph of a line: The graph of a quadratic function is always a parabola with a vertical axis of symmetry. The equations of quadratic functions always have the form , where c is the y-intercept. When a (the coefficient of x2) is positive, the parabola is “open up,” and when a is negative, it is “open down.” The graph above represents the function . Notice that it is an “open down” parabola with an axis of symmetry through its vertex at . The figure above shows the graph of the function f in the xy-plane. If , which of the following could be the value of b? Although this can be solved algebraically, you should be able to solve this problem more simply just by inspecting the graph, which clearly shows that . (You can plug into the equation to verify.) Since this point is two units from the axis of symmetry, its reflection is two units on the other side of the axis, which is the point (4, –3). Concept Review 2: Functions 1. What is a function? 2. What are the three basic ways of representing a function? 3. What is the general form of the equation of a linear function, and what does the equation tell you about the graph? 4. How can you determine the slope of a linear function from a table of its inputs and outputs? 5. How can you determine the slope of a linear function from its graph? 6. What is the general form of the equation of a quadratic function? 7. What kind of symmetry does the graph of a quadratic function have? SAT Practice 2: Functions 1. The graphs of functions f and g for values of x between –3 and 3 are shown above. Which of the following describes the set of all x for which 2. If and , which of the following could be g (x)? (A) 3 x 3. What is the least possible value of if 4. The table above gives the value of the linear function f for several values of x. What is the value of (E) It cannot be determined from the information given. 5. The graph on the xy-plane of the quadratic function g is a parabola with vertex at (3, –2). If , then which of the following must also equal 0? (A) g (2) (B) g (3) (C) g (4) (D) g (6) (E) g (7) 6. In the xy-plane, the graph of the function h is a line. If and , what is the value of h (0)? Answer Key 2: Functions Concept Review 2 1. A set of instructions for turning an input number (usually called x) into an output number (usually called y). 2. As an equation (as in ), as a table of input and output values, and as a graph in the xy-plane. , where m is the slope of the line and b is its y-intercept. 4. If the table provides two ordered pairs, (x1, y1) and (x2, y2), the slope can be calculated with . (Also see Chapter 10, Lesson 4.) 5. Choose any two points on the graph and call their coordinates (x1, y1) and (x2, y2). Then calculate the slope with . , where c is the y-intercept. 7. It is a parabola that has a vertical line of symmetry through its vertex. SAT Practice 2 1. C In this graph, saying that is the same as saying that the g function “meets or is above” the f function. This is true between the points where they meet, at and . 2. B Since , f (g (1)) must equal . Therefore and . So g (x) must be a function that gives an output of 4 when its input is 1. The only expression among the choices that equals 4 when is . 3. D This question asks you to analyze the “outputs” to the function given a set of “inputs.” Don”t just assume that the least input, –3, gives the least output, . In fact, that”s not the least output. Just think about the arithmetic: is the square of a number. What is the least possible square of a real number? It must be 0, because 02 equals 0, but the square of any other real number is positive. Can in this problem equal 0? Certainly, if , which is in fact one of the allowed values of x. Another way to solve the problem is to notice that the function is quadratic, so its graph is a parabola. Choose values of x between –3 and 0 to make a quick sketch of this function to see that its vertex is at (–2, 0). 4. C Since f is a linear function, it has the form . The table shows that an input of 3 gives an output of 8, so . Now, if you want, you can just “guess and check” values for m and b that work, for instance, and . This gives the equation . To find the missing outputs in the table, just substitute and then : and . Therefore, . But how do we know that will always equal 16? Because the slope m of any linear function represents the amount thaty increases (or decreases) whenever x increases by 1. Since the table shows x values that increase by 1, a must equal , and b must equal 8 + m. Therefore . 5. D Don”t worry about actually finding the equation for g (x). Since g is a quadratic function, it has a vertical line of symmetry through its vertex, the line . Since , the graph also passes through the origin. Draw a quick sketch of a parabola that passes through the origin and (3, –2) and has an axis of symmetry at : The graph shows that the point (0, 0), when reflected over the line , gives the point (6, 0). Therefore g (6) is also equal to 0. 6. D The problem provides two ordered pairs that lie on the line: (–1, 4) and (5, 1). Therefore, the slope of this line is . Therefore, for every one step that the line takes to the right (the x direction), the y value decreases by ½. Since 0 is one unit to the right of –1 on the x-axis, h (0) must be1/2 less than h ((–1), or .

Автор: Ghosal, Subhashis. Название: Fundamentals of Nonparametric Bayesian Inference ISBN: 0521878268 ISBN-13(EAN): 9780521878265 Издательство: Cambridge Academ Рейтинг: Цена: 7077 р. Наличие на складе: Есть у поставщика Поставка под заказ. Описание: Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics. Описание: This book treats the latest developments in the theory of order-restricted inference, with special attention to nonparametric methods and algorithmic aspects. Among the topics treated are current status and interval censoring models, competing risk models, and deconvolution. Methods of order restricted inference are used in computing maximum likelihood estimators and developing distribution theory for inverse problems of this type. The authors have been active in developing these tools and present the state of the art and the open problems in the field. The earlier chapters provide an introduction to the subject, while the later chapters are written with graduate students and researchers in mathematical statistics in mind. Each chapter ends with a set of exercises of varying difficulty. The theory is illustrated with the analysis of real-life data, which are mostly medical in nature. Автор: Efromovich Название: Nonparametric Curve Estimation ISBN: 0387987401 ISBN-13(EAN): 9780387987408 Издательство: Springer Рейтинг: Цена: 15427 р. Наличие на складе: Есть у поставщика Поставка под заказ. Описание: Gives an introduction to nonparametric curve estimation theory. Автор: Ferraty Название: Nonparametric Functional Data Analysis ISBN: 0387303693 ISBN-13(EAN): 9780387303697 Издательство: Springer Рейтинг: Цена: 12154 р. Наличие на складе: Есть у поставщика Поставка под заказ. Описание: Modern apparatuses allow us to collect samples of functional data, mainly curves but also images. On the other hand, nonparametric statistics produces useful tools for standard data exploration. This book links these two fields of modern statistics by explaining how functional data can be studied through parameter-free statistical ideas. Автор: Neuhauser Название: Nonparametric Statistical Tests ISBN: 1439867038 ISBN-13(EAN): 9781439867037 Издательство: Taylor&Francis Рейтинг: Цена: 17243 р. Наличие на складе: Невозможна поставка. Описание: Nonparametric Statistical Tests: A Computational Approach describes classical nonparametric tests, as well as novel and little-known methods such as the Baumgartner-Weiss-Schindler and the Cucconi tests. The book presents SAS and R programs, allowing readers to carry out the different statistical methods, such as permutation and bootstrap tests. The author considers example data sets in each chapter to illustrate methods. Numerous real-life data from various areas, including the bible, and their analyses provide for greatly diversified reading. The book covers: Nonparametric two-sample tests for the location-shift model, specifically the Fisher-Pitman permutation test, the Wilcoxon rank sum test, and the Baumgartner-Weiss-Schindler test Permutation tests, location-scale tests, tests for the nonparametric Behrens-Fisher problem, and tests for a difference in variability Tests for the general alternative, including the (Kolmogorov-)Smirnov test, ordered categorical, and discrete numerical data Well-known one-sample tests such as the sign test and Wilcoxon’s signed rank test, a modification suggested by Pratt (1959), a permutation test with original observations, and a one-sample bootstrap test are presented. Tests for more than two groups, the following tests are described in detail: the Kruskal-Wallis test, the permutation F test, the Jonckheere-Terpstra trend test, tests for umbrella alternatives, and the Friedman and Page tests for multiple dependent groups The concepts of independence and correlation, and stratified tests such as the van Elteren test and combination tests The applicability of computer-intensive methods such as bootstrap and permutation tests for non-standard situations and complex designs Although the major development of nonparametric methods came to a certain end in the 1970s, their importance undoubtedly persists. What is still needed is a computer assisted evaluation of their main properties. This book closes that gap. Описание: Incorporating a hands-on pedagogical approach, Nonparametric Statistics for Social and Behavioral Sciences presents the concepts, principles, and methods used in performing many nonparametric procedures. It also demonstrates practical applications of the most common nonparametric procedures using IBM’s SPSS software. This text is the only current nonparametric book written specifically for students in the behavioral and social sciences. Emphasizing sound research designs, appropriate statistical analyses, and accurate interpretations of results, the text: Explains a conceptual framework for each statistical procedure Presents examples of relevant research problems, associated research questions, and hypotheses that precede each procedure Details SPSS paths for conducting various analyses Discusses the interpretations of statistical results and conclusions of the research With minimal coverage of formulas, the book takes a nonmathematical approach to nonparametric data analysis procedures and shows students how they are used in research contexts. Each chapter includes examples, exercises, and SPSS screen shots illustrating steps of the statistical procedures and resulting output. Описание: While preserving the clear, accessible style of previous editions, this fourth edition reflects the latest developments in computer-intensive methods that deal with intractable analytical problems and unwieldy data sets. This edition summarizes relevant general statistical concepts and introduces basic ideas of nonparametric or distribution-free methods. Designed experiments, including those with factorial treatment structures, are now the focus of an entire chapter. The book also expands coverage on the analysis of survival data and the bootstrap method. The new final chapter focuses on important modern developments. With numerous exercises, the text offers the student edition of StatXact at a discounted price. Описание: Presenting an extensive set of tools and methods for data analysis, this second edition includes more models and methods and significantly extends the possible analyses based on ranks. It contains a new section on rank procedures for nonlinear models, a new chapter on models with dependent error structure, and new material on the development of computationally efficient affine invariant/equivariant sign methods based on transform-retransform techniques in multivariate models. The authors illustrate the methods using many real-world examples and R. Information about the data sets and R packages can be found at www.crcpress.com Описание: Designed for a graduate course in applied statistics, Nonparametric Methods in Statistics with SAS Applications teaches students how to apply nonparametric techniques to statistical data. It starts with the tests of hypotheses and moves on to regression modeling, time-to-event analysis, density estimation, and resampling methods. The text begins with classical nonparametric hypotheses testing, including the sign, Wilcoxon sign-rank and rank-sum, Ansari-Bradley, Kolmogorov-Smirnov, Friedman rank, Kruskal-Wallis H, Spearman rank correlation coefficient, and Fisher exact tests. It then discusses smoothing techniques (loess and thin-plate splines) for classical nonparametric regression as well as binary logistic and Poisson models. The author also describes time-to-event nonparametric estimation methods, such as the Kaplan-Meier survival curve and Cox proportional hazards model, and presents histogram and kernel density estimation methods. The book concludes with the basics of jackknife and bootstrap interval estimation. Drawing on data sets from the author’s many consulting projects, this classroom-tested book includes various examples from psychology, education, clinical trials, and other areas. It also presents a set of exercises at the end of each chapter. All examples and exercises require the use of SAS 9.3 software. Complete SAS codes for all examples are given in the text. Large data sets for the exercises are available on the author’s website. Описание: Consists of 22 research papers in Probability and Statistics. This title includes topics such as nonparametric inference, nonparametric curve fitting, linear model theory, Bayesian nonparametrics, change point problems, time series analysis and asymptotic theory. It presents research in statistical theory. Описание: Nonparametric techniques in statistics are those in which the data are ranked in order according to some particular characteristic. When applied to measurable characteristics, the use of such techniques often saves considerable calculation as compared with more formal methods, with only slight loss of accuracy. The field of nonparametric statistics is occupying an increasingly important role in statistical theory as well as in its applications. Nonparametric methods are mathematically elegant, and they also yield significantly improved performances in applications to agriculture, education, biometrics, medicine, communication, economics and industry. Автор: Brodsky, E., Darkhovsky, B.S. Название: Nonparametric Methods in Change Point Problems ISBN: 0792321227 ISBN-13(EAN): 9780792321224 Издательство: Springer Рейтинг: Цена: 8882 р. Наличие на складе: Есть у поставщика Поставка под заказ. Описание: This volume deals with non-parametric methods of change point (disorder) detection in random processes and fields. A systematic account is given of up-to-date developments in this rapidly evolving branch of statistics. ООО "Логосфера " Тел:+7(495) 980-12-10 www.logobook.ru

Cameron-Liebler line classes with parameter In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter for or , which generalizes the examples found by Rodgers in through a computer search. Furthermore, in the case where is an even power of , we construct the first infinite family of affine two-intersection sets in , which is closely related to our Cameron-Liebler line classes. Key words and phrases:Cameron-Liebler line class, Klein correspondence, projective two-intersection set, affine two-intersection set, strongly regular graph, Gauss sum. Cameron-Liebler line classes were first introduced by Cameron and Liebler in their study of collineation groups of having the same number of orbits on points and lines of . Later on it was found that these line classes have many connections to other geometric and combinatorial objects, such as blocking sets of , projective two-intersection sets in , two-weight linear codes, and strongly regular graphs. In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics; see, for example, [7, 20, 21, 25, 10, 11]. In , the authors gave several equivalent conditions for a set of lines of to be a Cameron-Liebler line class; Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of a Cameron-Liebler line class. Let be a set of lines of with , a nonnegative integer. We say that is a Cameron-Liebler line class with parameter if every spread of contains lines of . Clearly the complement of a Cameron-Liebler line class with parameter in the set of all lines of is a Cameron-Liebler line class with parameter . So without loss of generality we may assume that when discussing Cameron-Liebler line classes of parameter . Let be any non-incident point-plane pair of . Following , we define to be the set of all lines through , and to be the set of all lines contained in the plane . We have the following trivial examples: The empty set gives a Cameron-Liebler line class with parameter ; Each of and gives a Cameron-Liebler line class with parameter ; gives a Cameron-Liebler line class with parameter . Cameron-Liebler line classes are rare. It was once conjectured ([5, p. 97]) that the above trivial examples and their complements are all of the Cameron-Liebler line classes. The first counterexample to this conjecture was given by Drudge in , and it has parameter . Later Bruen and Drudge generalized Drudge’s example into an infinite family with parameter for all odd . This represents the only known infinite family of nontrivial Cameron-Liebler line classes before our work. Govaerts and Penttila gave a sporadic example with parameter in . Recent work by Rodgers suggests that there are probably more infinite families of Cameron-Liebler line classes awaiting to be discovered. In , Rodgers obtained new Cameron-Liebler line classes with parameter for or and . In his thesis , Rodgers also reported new examples with parameters for and as joint work with his collaborators. These examples motivated us to find new general constructions of Cameron-Liebler line classes. On the nonexistence side, Govaerts and Storme first showed that there are no Cameron-Liebler line classes in with parameter when is prime. Then De Beule, Hallez and Storme excluded parameters for all values . Next Metsch proved the non-existence of Cameron-Liebler line classes with parameter , and subsequently improved this result by showing the nonexistence of Cameron-Liebler line classes with parameter . The latter result represents the best asymptotic nonexistence result to date. It seems reasonable to believe that for any fixed and constant there are no Cameron-Liebler line classes with for sufficiently large . Very recently, Gavrilyuk and Metsch proved a modular equality which eliminates almost half of the possible values for a Cameron-Liebler line class with parameter . We refer to for a comprehensive survey of the known nonexistence results. In the present paper we construct a new infinite family of Cameron-Liebler line classes with parameter for or . This family of Cameron-Liebler line classes generalizes the examples found by Rodgers in through a computer search. Furthermore, in the case where is an even power of , we construct the first infinite family of affine two-intersection sets, which is closely related to the newly constructed Cameron-Liebler line classes. The first step of our construction follows the same idea as in . That is, we prescribe an automorphism group for the Cameron-Liebler line classes that we intend to construct; as a consequence, the Cameron-Liebler line classes will be unions of orbits of the prescribed automorphism group on the set of lines of . The main difficulty with this approach is how to choose orbits properly so that their union is a Cameron-Liebler line class. We overcome this difficulty by giving an explicit choice of orbits so that their union gives a Cameron-Liebler line class with the required parameters. The details are given in Section 4. The paper is organized as follows. In Section 2, we review basic properties of and facts on Cameron-Liebler line classes; furthermore, we collect auxiliary results on characters of finite fields, which are needed in the proof of our main theorem. In Section 3, we introduce a subset of , which we will use in the construction of our Cameron-Liebler line classes, and prove a few properties of the subset. In Section 4, we give an algebraic construction of an infinite family of Cameron-Liebler line classes with for or . In Section 5, we construct the first infinite family of affine two-intersection sets in , odd, whose existence was conjectured in the thesis of Rodgers. We close the paper with some concluding remarks. In this section, we review basic facts on Cameron-Liebler line classes, and collect auxiliary results on characters of finite fields. 2.1. Preliminaries on Cameron-Liebler line classes It is often advantageous to study Cameron-Liebler line classes in by using their images under the Klein correspondence. Let be the 5-dimensional hyperbolic orthogonal space and be a nonnegative integer. A subset of is called an -tight set if for every point , or according as is in or not, where is the polarity determined by . The geometries of and are closely related through a mapping known as the Klein correspondence which maps the lines of bijectively to the points of , c.f. [16, 22]. Let be a set of lines of with , a nonnegative integer, and let be the image of under the Klein correspondence. Then it is known that is a Cameron-Liebler line class with parameter in if and only if is an -tight set of . Moreover, if is a Cameron-Liebler line class with parameter , by [20, Theorem 2.1 (b)], it holds that for any point off ; consequently is a projective two-intersection set in with intersection sizes and , namely each hyperplane of intersects in either or points. We summarize these known facts as follows. Let be a set of lines in , with , and let be the image of under the Klein correspondence. Then is a Cameron-Liebler line class with parameter if and only if is an -tight set in ; moreover, in the case when is a Cameron-Liebler line class, we have A strongly regular graph is a simple undirected regular graph on vertices with valency satisfying the following: for any two adjacent (resp. nonadjacent) vertices and there are exactly (resp. ) vertices adjacent to both and . It is known that a graph with valency , not complete or edgeless, is strongly regular if and only if its adjacency matrix has exactly two restricted eigenvalues. Here, we say that an eigenvalue of the adjacency matrix is restricted if it has an eigenvector perpendicular to the all-ones vector. One of the most effective methods for constructing strongly regular graphs is by the Cayley graph construction. Let be a finite abelian group and be an inverse-closed subset of . We define a graph with the elements of as its vertices; two vertices and are adjacent if and only if . The graph is called a Cayley graph on with connection set . The eigenvalues of are given by , , where is the group consisting of all complex characters of , c.f. [3, §1.4.9]. Using the aforementioned spectral characterization of strongly regular graphs, we see that with connection set () is strongly regular if and only if , , take exactly two values, say and with . We note that if is strongly regular with two restricted eigenvalues and , then the set also forms a connection set of a strongly regular Cayley graph on ; this set is called the dual of . For basic properties of strongly regular graphs, see [3, Chapter 9]. For known constructions of strongly regular Cayley graphs and their connections to two-weight linear codes, partial difference sets, and finite geometry, see [3, p. 133] and [6, 19]. Let be a Cameron-Liebler line class with parameter in and let be the image of under the Klein correspondence. By Result 2.1, is a projective two-intersection set in . By , we can construct a corresponding strongly regular Cayley graph as follows. First define , which is a subset of . Then the Cayley graph with vertex set and connection set is strongly regular. Its restricted eigenvalues can be determined as follows. Let be a nonprincipal additive character of . Then is principal on a unique hyperplane for some . We have where is the Kronecker delta function taking value if and value otherwise. Conversely, for each hyperplane of , we can find a nonprincipal character that is principal on , and the size of can be computed from . Therefore, the character values of reflect the intersection sizes of with the hyperplanes of . To summarize, we have the following result. Let be a set of lines in , with , and let be the image of under the Klein correspondence. Define Then is a Cameron-Liebler line class with parameter if and only if and for any where is any nonprincipal character of that is principal on the hyperplane . Following we now introduce a model of the hyperbolic quadric , which will facilitate our algebraic construction. Let and . We view as a 6-dimensional vector space over . For a nonzero vector , we use to denote the projective point in corresponding to the one-dimensional subspace over spanned by . Define a quadratic form by where is the relative trace from to (that is, for any , ). The quadratic form is clearly nondegenerate and for all . So is a totally isotropic plane with respect to . It follows that the quadric defined by has Witt index 3, and so is hyperbolic. This quadric will be our model for . Note that for a point , its polar hyperplane is given by . Let and be the canonical additive characters of and , respectively. Then each additive character of has the form where . Since is principal on the hyperplane , the character sum condition in Result 2.2 can be more explicitly rewritten as 2.2. Preliminaries on characters of finite fields In this subsection, we will collect some auxiliary results on Gauss sums. We assume that the reader is familiar with the basic theory of characters of finite fields as can found in Chapter 5 of . For a multiplicative character and the canonical additive character of , define the Gauss sum by The following are some basic properties of Gauss sums: if is nonprincipal; if is principal. Let be a fixed primitive element of and a positive integer dividing . For we set . These are called the th cyclotomic classes of . The Gauss periods associated with these cyclotomic classes are defined by , , where is the canonical additive character of . By orthogonality of characters, the Gauss periods can be expressed as a linear combination of Gauss sums: where is any fixed multiplicative character of order of . For example, if , we have where is the quadratic character of . The following theorem on Eisenstein sums will be used in the proof of our main theorem in Section 4. ([27, Theorem 1]) Let be a nonprincipal multiplicative character of and be its restriction to . Choose a system of coset representatives of in in such a way that can be partitioned into two parts: where is the relative trace from to . Then, We will also need the Hasse-Davenport product formula, which is stated below. ([2, Theorem 11.3.5]) Let be a multiplicative character of order of . For every nonprincipal multiplicative character of , The Stickelberger theorem on the prime ideal factorization of Gauss sums gives us -adic information on Gauss sums. We will need this theorem to prove a certain divisibility result later on. Let be a prime, , and let be a complex primitive th root of unity. Fix any prime ideal in lying over . Then is a finite field of order , which we identify with . Let be the Teichmüller character on , i.e., an isomorphism for all in . The Teichmüller character has order . Hence it generates all multiplicative characters of . Let be the prime ideal of lying above . For an integer , let where is the -adic valuation. Thus . The following evaluation of is due to Stickelberger (see [2, p. 344]). Let be a prime and . For an integer not divisible by , let , , be the -adic expansion of the reduction of modulo . Then that is, is the sum of the -adic digits of the reduction of modulo . 3. The subset and its properties Let be a prime power with or so that . Write , , and let be a fixed primitive element of . For any , we use to denote the integer such that . We write , and let be an element of order in (for example, take ). For , we define the sign of , , by We also define . It is the purpose of this section to introduce a subset and prove a few results on , which we will need in the construction of our Cameron-Liebler line classes. Viewing as a 3-dimensional vector space over , we will use as the underlying vector space of . The points of are , , and the lines of are where . Of course, and , for any and . Note that since , we can also take , , as the points of . Define a quadratic form by , where Tr is the relative trace from to . The associated bilinear form is given by . It is clear that is nondegenerate. Therefore defines a nondegenerate conic in , which contains points. Consequently each line of meets in , or points, and is called a passant, tangent or secant line accordingly. Also it is known that each point is on either or tangent lines to , and is called an interior or exterior point accordingly, c.f. [22, p. 158]. With the above notation, we have the following: The tangent lines to are given by with , . The polarity of induced by interchanges and , where , and maps exterior (resp. interior) points to secant (resp. passant) lines. For any point off , is an exterior (resp. interior) point if and only if has some fixed nonzero sign (resp. ). Now we define the following subset of : where the elements are numbered in any (unspecified) order. That is, . Let be any element of such that , with as defined in Lemma 3.2. For , we define Let be three distinct elements of . Then the sign of is equal to the sign of for any exterior point . In other words, has sign , where is the same as in part (3) of Lemma 3.2. In particular, . Proof: Since is a conic, are linearly independent over and thus form a basis of over . The Gram matrix of the bilinear form with respect to this basis is equal to which is symmetric with diagonal entries equal to . Its determinant is equal to Let be an exterior point, say, is the intersection of the tangent lines through and . Then , and the Gram matrix with respect to the basis has determinant . Since , is a square in . By [22, p. 262], the two determinants have the same sign. This proves the first part of the lemma. It remains to prove that . We observe that the matrix (3.6) can be written as with We claim that the determinant of , , is in . To see this, applying the Frobenius automorphism of to entry-wise, we get The claim that now follows from this fact and the first part of the lemma. With the above notation, if we use any other in place of in the definition of , then the resulting set satisfies that or . Proof: Without loss of generality, we assume that we use in place of in the definition of , and obtain . We have the following observations. For , that is, , the sign of the quotient of and is a constant: their quotient lies in , and its sign is clearly equal to that of ; this sign is equal to by Lemma 3.3. The sign of the quotient of and is equal to , since we have chosen such that . Similarly the sign of the quotient of and is equal to . The above observations imply that there is a pairing of the elements of and , say, a bijection, such that are nonzero squares in for all or are nonsquares in for all . Upon taking logarithm and modulo we get the conclusion of the lemma. It is clear that is an element of . In Lemma 3.4, consider the special case where we replace by in the definition of , and denote the resulting set by . We observe that Consequently . It follows that in this particular case. By definition, we have , so we have shown that the subset is invariant under multiplication by . This fact will be needed in the next section when we discuss automorphism groups of the newly construected Cameron-Liebler line classes. We now prove some properties of the set which will be needed in the next section. Let (resp. ) be the set of nonzero squares (resp. nonsquares) of , and write where is the canonical additive character of . For , we define the exponential sums To simplify notation, we often write Note that if and only if for some , which in turn is equivalent to for some integer . Since is an element of , we can view in the above definition of as coming from . It follows that We will evaluate these sums explicitly. Let and . The exponential sums take the following four values: where and are defined as above, that is, and . Proof: We consider the following three cases according to the line as defined in (3.2) is a tangent, passant, or secant. Case 1: is a tangent line. In this case, is a zero of by (1) of Lemma 3.2, where we recall that , so for some by the definition of in Eqn. (3.3). Note that satisfies by the definition of and . In view of (2) of Remark 3.6, we may assume that . (If necessary, replace by , and then the resulting is still a zero of , i.e., satisfies .) Now if and only if lies on the tangent line , i.e., . We see that the elements , , are all nonzero squares, so that or by (1) of Remark 3.6. Note that if we replace by , then the value of is replaced by the other in this case. Hence, . Case 2: is a passant line. In this case, is an interior point and thus has sign by Lemmas 3.2 and 3.3. Note that satisfies and since is a nonsquare of . Each line through has either or points of , so the points of are partitioned into pairs accordingly. Let be two points of that lie on a secant line through . Since the three points and are collinear, the Gram matrix of is singular. By direct computations we find that the determinant of this Gram matrix is equal to It follows that

IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 533 A Multiconductor Model for Finite-Element Eddy-Current Simulation Herbert De Gersem and Kay Hameyer Abstract—The stranded-conductor finite-element model does not account for the skin and proximity effects in a multiconductor system. The solid conductor model considers the true geometry, all the individual conductors, and their connections but may lead to unmanageably huge models. The multiconductor model proposed here, does not necessarily consider all geometrical details but instead, discretizes the inner geometry and voltage drop and enforces the typical current redistribution in multiconductor configurations in a weak sense. The magnetic and electric meshes are independently and adaptively refined which results in an optimal error control and accurate results for relatively small models. Index Terms—Circuit simulation, eddy currents, finite element methods, windings. Fig. 1. Application range of the solid, stranded, foil, and multiconductor models and impedance boundary conditions. I. INTRODUCTION M ULTICONDUCTOR windings are sets of conductors, electrically insulated from each other and connected in series or parallel. The individual conductors experience skin and proximity effects, characterized by the skin depth (1) where is the frequency, is the permeability, and is the conductivity. The skin depth may be different in the and direction due to anisotropic permeabilities. The characteristic extents of a single wire in the multiconductor system with reand . If spect to the main axes are denoted by and , the stranded-conductor model is appropriate . and , impedance boundary conditions are If commonly applied . If , , and are of the same order of magnitude, the skin effect is resolved by eddy-current simulation using the solid conductor model. The unidirectional skin and of the effect as observed in foil conductors, i.e., same order of magnitude as , is considered in and . These four modeling approaches are limiting cases for particular kinds Manuscript received July 5, 2001. This work was supported in part by the Belgian “Fonds voor Wetenschappelijk Onderzoek Vlaanderen” under Grant G.0427.98 and in part by the Belgian Ministry of Scientific Research under Grant IUAP no. P4/20. H. De Gersem was with the Katholieke Universiteit Leuven, Department ESAT, Division ELEN. He is now with the FB 18 Elektrotechnik und Informationstechnik, Fachgebiet Theorie Elektromagnetischer Felder, Technische Universität Darmstadt, D-64289 Darmstadt, Germany (e-mail: [email protected]). K. Hameyer is with the Katholieke Universiteit Leuven, Department ESAT, Division ELEN, B-3001 Leuven-Heverlee, Belgium (e-mail: [email protected]). Publisher Item Identifier S 0018-9464(02)02348-8. of current redistribution (Fig. 1). In many models, the influence of the skin effect on the global behavior of the model is not negligible although its local influence does not have to be computed in detail. Multiconductor systems arise in almost all electrical energy transducers. The devices may feature a large number of multiconductor systems, each consisting of a considerable number of turns. This may hamper the simulation of the overall device. Several model reduction techniques for multiconductor systems exist, e.g., analytical macroelements , inner node elimination and parameter extraction . They constitute an a priori model reduction, which may hinder adaptive error control during numerical simulation. In this paper, it is suggested to approximate the troublesome geometries by an additional discretization for the voltage and to incorporate this in the magnetic finite-element (FE) model. An error estimator adaptively refines the multiconductor model during the simulation. II. MAGNETODYNAMIC MODEL For convenience, a two-dimensional time-harmonic formulation is considered. The derivation of a multiconductor model for three-dimensional FE models, transient formulations, or anisotropic materials is similar. The magnetodynamic formulation is (2) the phasor with the pulsation, the length of the model, of the component of the magnetic vector potential and the phasor of the voltage drop between front and rear ends of the model. The partial differential equation is discretized on the 0018-9464/02$17.00 © 2002 IEEE 534 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 device cross section by linear, triangular FEs yielding the system of equations , (3) (4) (5) and the degrees of freedom (DOFs) for . III. MODELING ASSUMPTIONS Consider a multiconductor with cross section , conductors with cross sections , consisting of and connected in series. The cross sections may have different shapes but have the same area . The crossis . The union of all conductor cross sectional area of , is contained in but may be sections, . The difference is occupied by insmaller than sulation material, cooling ducts, and gaps which are supposed to and to have a homogeneous be uniformly distributed over and a zero conductivity. The voltage drop is permeability and is aligned with the direction, is constant over each zero in accounting for the nonconductive regions included in true constraints (7) tend to the continuous constraint . The in (12) The voltage across the overall multiconductor is obtained by over and multiplying by averaging (13) in in Fig. 2. Cross section of the multiconductor, the magnetic mesh, the electric mesh, and three electric shape functions. . (6) The voltage drop differs from one conductor to the other due to the vicinity of ferromagnetic cores and because of inhomogeneous conductivities, e.g., due to local heating. It may be beneficial to model a multiconductor system with a considerable number of turns by the continuous model (12) and (13) rather than the true relations (7) and (8) which require the geometry of each individual conductor to be considered. IV. CONTINUOUS MULTICONDUCTOR MODEL V. DISCRETE MULTICONDUCTOR MODEL The true multiconductor model obeys (2) and an integral con, straint over each To discretize the continuous multiconductor model, an additional mesh for the voltage drop is constructed (Fig. 2). To simplify the implementation, a tensor grid is preferred. is resolved by electric shape functions (7) The voltage drop across the multiconductor is (14) (8) tends to infinity Consider now the limiting case in which becomes very small. Still, suppose all conducand, hence, tors to have the same cross-section area and the conductive and to be uniformly mixed. The insunonconductive parts of lation material is not explicitly considered in the model. The material parameters in the magnetic equations (2), (4), and (5) are replaced by the homogenized parameters with the associated DOFs. Equation (5) becomes (15) A weak formulation of the multiconductor model is obtained by weighting (12) by the electric shape functions (9) (10) with the fill factor (16) (11) DE GERSEM AND HAMEYER: A MULTICONDUCTOR MODEL FOR FE EDDY-CURRENT SIMULATION 535 The voltage across the overall multiconductor is (17) The weak formulation of the magnetodynamic problem (3), the weak formulation of the current constraints (16), and the integration of the discrete voltage drop (17) are assembled into the coupled system of equations (18) a factor symmetrizing the system. The system with has the nature of a mixed formulation . Because the coefand scale differently, it is recommended to ficients apply an explicit diagonal scaling of (18) before solving. The system is solved by a Quasi-Minimal Residual method adapted to complex symmetric systems , preconditioned by successive over relaxation. The multiconductor model can be coupled to a circuit model accounting for active and passive components outside the finiteelement model. The treatment of arbitrary circuit connections follows the topological approach presented in . VI. ADAPTIVE MESH REFINEMENT The electric mesh does not coincide with the magnetic mesh nor with the true multiconductor geometry. Therefore, adaptive mesh refinement can be applied independently. The possibility of independent error control is one of the most attractive features of the multiconductor model. The magnetic mesh is refined based on indications of large eddy currents and large ferromagnetic saturation. The error indicator for the electric mesh detects large variations of the voltage drop. To preserve the consistency of the multiconductor model, the discrete weak formulation (16) has to converge towards the true model (7) rather than towards the continuous representation (12). As a consequence, the supports of the electric shape functions have to converge towards the single-conductor cross sections. This is different from a conventional discretization where convergence corresponds to a vanishing mesh size. The consistency of the electric discretization of the multiconductor model is guaranteed by the following procedure. At places where refinement would bring up electric elements smaller than the extent of a single conductor, the true geometry is restored (Fig. 3). The discrete formulation (16) implicitly incorporates the true model (7) if a single, constant shape function is because of assigned to each conductor that shows up in refinement. The parts of the insulation that are restored by the refinement procedure, are expelled from the electric mesh. As a consequence, the electric mesh may become disconnected and the fill factor has to be adapted accordingly. The limiting constant case when the voltage drop is discretized by , , and electric shape functions defined at the fill factor is one, corresponds to the explicit modeling of each of the individual conductors as a solid conductor, and, hence, constitutes the true multiconductor model (7) and (8). For technical models, however, a sufficient accuracy is already Fig.3. Consistent adaptive refinement of the electric tensor grid when the error estimator indicates large variations of the voltage drop at the left upper corner of the multiconductor cross section (at a certain point, further refinement corresponds to restoring the original geometry). achieved when the electric mesh is much coarser than the true geometry. If local effects would become important, the error indicator will detect them and invoke substantial refinement at those places, probably leading to a local recovery of the true geometry of the multiconductor system. VII. NONMATCHING INTEGRATIONS The efficiency of the multiconductor model is strongly related to the flexibility of selecting and refining the meshes. It is recommended to allow an independent construction and reand finement of both meshes. Hence, the supports of , in general, do not match. This considerably hinders the evaluation of the hybrid integrals in . A numerical integration scheme, e.g., Gaussian quadrature, encounters problems to select appropriate integration points or may require a huge number of points in order to sample both meshes at a sufficient rate. Here, a semi-analytical technique is favored. A composite mesh is built by gathering all vertices and edges of both the magnetic and the electric mesh. The intersections of the edges of the different meshes with respect to each other have to be computed with a sufficient accuracy, preferably using exact arithand can be exactly represented metic. Both on the composite mesh on which the multiplications and integrations can then be performed exactly. The required computation time is substantial but is still acceptable when compared with the overall computation time. To avoid hybrid integrals, one could use the composite mesh for both electric and magnetic discretization. This approach, however, does not pay off because the substantially larger system of equation would eliminate the efficiency of the multiconductor model. VIII. CONVERGENCE OF THE MULTICONDUCTOR DISCRETIZATION The convergence of the mixed discretization technique is studied for a model problem. The multiconductor contains 250 conductors. All feature the same conductivity and permeability. The conductors are electrically insulated from each other, are connected in series, and carry an alternating current. No flux leaves the model. Within each conductor, the magnetic field is expressed by the analytical solution of (2) for a rectangular domain. The analytical solution for the multiconductor system is derived by applying the interface conditions at the borders of the conductors and requiring the current to be the same in 536 IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 Fig. 6. Fig. 4. Convergence of the discretization error of the magnetic vector potential field of the multiconductor model (the dashed line is a line of constant error). Harmonic losses in a stator winding of an induction machine. ticonductor model offers a sufficient accuracy while avoiding an excessive amount of mesh nodes and voltage unknowns. At 50 Hz, no significant skin effect is observed. At 500 Hz, substantial losses are introduced. The multiconductor model equipped with independent mesh refinement and external circuit coupling, enables the simulation of the model for all possible frequencies by the same conductor model (Fig. 6). X. CONCLUSION Fig. 5. (a) Geometry, (b) real, and (c) imaginary components of the magnetic flux in a single-layer stator slot at 50 Hz. (d) Real and (e) imaginary components of the magnetic flux in the multiconductor model at 500 Hz. each conductor. The discretization error of the magnetic vector potential field obtained by the proposed multiconductor model, is measured in the L2-norm with respect to the analytical solution (Fig. 4). The error decays when the magnetic and/or electric meshes are refined. The dashed line denotes loci for which the error is identical. The experiment indicates that the discretization error depends more on the discretization of the magnetic field than on the discretization of the electric voltage drop. As a consequence, it is sometimes more advantageous to apply a finer magnetic mesh than to consider all geometrical details of the multiconductor system. IX. EXAMPLE: MACHINE WINDINGS The multiconductor model is applied to simulate the harmonic losses in induction machine windings (Fig. 5). Since these devices are supplied by variable frequency, the relative importance of the higher harmonic distortion increases and the additional joule losses are not negligible. These effects are commonly taken into account in analytical models by the frequencydependent eddy-current factor which can be provided by a FE model of a single stator slot . A leakage flux impinging on the conductor is applied to the model by a difference in magnetic vector potential between the top and the bottom of the slot. A conventional model considers the true geometry consisting of the conductors, the insulation, and the cooling ducts. It treats the coil as a series connection of a number of solid conductors, each with their own unknown voltage. For many cases, the mul- The multiconductor model developed here, enables the simulation of complicated coil configurations with relatively small models by using an additional discretization for the conductor’s voltage drop. It offers more modeling flexibility when compared with the solid and stranded-conductor models. The automated error control and independent mesh refinement yields small models while guaranteeing the prescribed accuracy. REFERENCES G. Bedrosian, “A new method for coupling finite-element field solutions with external circuits and kinematics,” IEEE Trans. Magn., vol. 29, pp. 1664–1668, Mar. 1993. A. M. El-Sawy Mohamed, “Finite-element variational formulation of the impedance boundary condition for solving eddy current problems,” Inst. Elect. Eng. Proceedings Science, Measurement and Technology, vol. 142, no. 4, pp. 293–298, July 1995. H. De Gersem and K. Hameyer, “A finite-element model for foil winding simulation,” IEEE Trans. Magn., vol. 37, pp. 3427–3432, Sept. 2001. P. Dular and C. Geuzaine, “A magnetic field magnetodynamic finiteelement formulation for foil winding inductors,” IEEE Trans. Magn., submitted for publication. A. Kladas and A. Razek, “Numerical calculation of eddy-currents and losses in squirrel cage induction motors due to supply harmonics,” in Proc. Int. Conf.Electrical Machines (ICEM88), vol. 2, Pisa, Italy, Sept., pp. 65–69. Á. Szűcs and A. Arkkio, “Consideration of eddy currents in multiconductor windings using the finite element method and the elimination of inner nodes,” IEEE Trans. Magn., vol. 35, pp. 1147–1150, May 1999. I. Munteanu, T. Wittig, T. Weiland, and D. Ioan, “FIT/PVL circuit-parameter extraction for general electromagnetic devices,” IEEE Trans. Magn., vol. 36, pp. 1421–1425, July 2000. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Berlin, Germany: Springer-Verlag, 1991. R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Scientific Computing, vol. 13, pp. 425–448, Jan. 1992. H. De Gersem, R. Mertens, U. Pahner, R. Belmans, and K. Hameyer, “A topological method used for field-circuit coupling,” IEEE Trans. Magn., vol. 34, pp. 3190–3193, Sept. 1998. S. J. Salon, L. Ovacik, and J. F. Balley, “Finite-element calculation of harmonic losses in AC machine windings,” IEEE Trans. Magn., vol. 29, pp. 1442–1445, Mar. 1993. R. Richter, “Die Induktionsmaschinen,” in Elektrische Maschinen, 2nd ed. Basel, Switzerland: Birkhauser, 1963, vol. 4.

- What is a statement in math? - What are examples of statement? - What is a simple statement? - What is an example of a universal statement? - What means statement? - What are the 4 types of questions? - What makes something a statement? - What is a statement in math example? - What are Number statements? - What is an example of a statement sentence? - What is the difference between a statement and a sentence? - How do you write a statement sentence? - What is the difference between a question and a statement? - What is a statement question? - What is a Contrapositive statement? - Can a statement start with what? - Can a statement be a question? - Can a statement be more than one sentence? What is a statement in math? A statement (or proposition) is a sentence that is either true or false (both not both). So ‘3 is an odd integer’ is a statement. But ‘π is a cool number’ is not a (mathematical) statement. Note that ‘4 is an odd integer’ is also a statement, but it is a false statement.. What are examples of statement? An example of statement is the thesis of a paper. An example of statement is a credit card bill. A declaration of fact or an allegation by a witness; a piece of sworn testimony. See also closing statement, evidence, and opening statement. What is a simple statement? A simple statement is a statement which has one subject and one predicate. For example, the statement: London is the capital of England. is a simple statement. What is an example of a universal statement? A universal statement is a statement that is true if, and only if, it is true for every predicate variable within a given domain. Consider the following example: Let B be the set of all species of non-extinct birds, and b be a predicate variable such that b B. … Some birds do not fly. What means statement? 1 : something stated: such as. a : a single declaration or remark : assertion. b : a report of facts or opinions. 2 : the act or process of stating or presenting orally or on paper. What are the 4 types of questions? In English, there are four types of questions: general or yes/no questions, special questions using wh-words, choice questions, and disjunctive or tag/tail questions. Each of these different types of questions is used commonly in English, and to give the correct answer to each you’ll need to be able to be prepared. What makes something a statement? A statement is a sentence that says something is true, like “Pizza is delicious.” There are other kinds of statements in the worlds of the law, banking, and government. All statements claim something or make a point. … You get a statement from your bank, a monthly record of what you spent and what you have left. What is a statement in math example? Brielfy a mathematical statement is a sentence which is either true or false. It may contain words and symbols. For example “The square root of 4 is 5″ is a mathematical statement (which is, of course, false). What are Number statements? A number sentence is a mathematical sentence, made up of numbers and signs. The expressions given in examples indicate equality or inequality. Types of Number Sentences. A number sentence can use any of the mathematical operations from addition, subtraction, multiplication to division. What is an example of a statement sentence? A statement sentence usually has a structure characterized by a subject followed by a predicate. Example of a statement sentence: Charlie delivers the newspapers twice a day. … Imperative sentences (or commands) have an implied subject, and so the subject (most often “you”) is usually unspoken. What is the difference between a statement and a sentence? A sentence is a group of words that usually have a subject, verb and information about the subject. Remember: A sentence can be a statement, question or command. A statement is a basic fact or opinion. It is one kind of sentence. How do you write a statement sentence? Declarative Sentence (statement) They tell us something. They give us information, and they normally end with a full-stop/period. The usual word order for the declarative sentence is: subject + verb… What is the difference between a question and a statement? is that question is a sentence, phrase or word which asks for information, reply or response; an interrogative while statement is a declaration or remark. What is a statement question? The intonation of a statement question depends on its meaning. We use statement questions when we think we know the answer to the question and we want to find out if we’re right. What is a Contrapositive statement? To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of “If it rains, then they cancel school” is “If they do not cancel school, then it does not rain.” … If the converse is true, then the inverse is also logically true. Can a statement start with what? Many questions and statements can start with the word what. Examples: “What’s new?” and “What’s sauce for the goose is sauce for the gander.” Can a statement be a question? Questions, commands and advice are typically not statements, because they do not express something that is either true or false. … We saw an example of a question which by itself is not a statement, but can be used to express a statement. When you see rhetorical questions, always rephrase them as statements. Can a statement be more than one sentence? Statements are logical entities; sentences are grammatical entities. Not all sentences express statements and some sentences may express more than one statement. A statement is a more abstract entity than even a sentence type.

Read this article to learn about the four tools of risk analysis. 1. Shorter Payback Period: According to this method, projects with shorter payback periods are normally preferred to those with longer payback periods. It would be more effective when it is combined with a ‘cut off period’. Cut off period denotes the risk tolerance level in the firm. For example, a firm has three projects, A, B, C for consideration with different economic lives, say 15, 16 and 7 years respectively, and with payback periods of say 6, 7, and 5 years. Of these three, project C will be preferred because it’s payback period is the shortest. Suppose, the firm’s cut off period is 4 years, then all the three projects will be rejected. 2. Risk-Adjusted Discount Rate: Under this method, the cut off rate or minimum required rate of return [mostly the firm’s cost of capital] is raised by adding what is called ‘risk premium’ to it. When the risk is greater, the premium to be added would be greater. For example, if the risk free discount rate [say, cost of capital] is 10%, and the project under consideration is a riskier one, then the premium of, say 5% is added to the above risk-free rate. The risk-adjusted discount rate would be 15%, which may be used either for discounting purposes under NPV, or as a cut off rate under IRR. Merits of Risk-adjusted Discount Rate: 1. It is easy to understand and simple to operate. 2. It has a great deal of intuitive appeal for risk adverse decision-makers. 3. It incorporates an attitude towards uncertainty. 1. There is no easy way to derive a risk-adjusted discount rate. 2. A uniform risk discount factor used for discounting all future returns is unscientific as the degree of risk may vary over the years in future. 3. It assumes that investors are risk averse. Though it is generally true, there do exist risk-seekers in real world situation that may demand premium for assuming risk. 3. Conservative Forecasts: Under this method, employing intuitive correction factor or certainty equivalent coefficient, which is calculated by the decision-maker subjectively or objectively, reduces the estimated risks from cash flows. Normally, this coefficient reflects the decision-makers confidence in obtaining a particular cash flow in a particular period. For example, the decision-maker estimates a net cash flow of Rs.60000 next year but if he feels [subjectively] that only 60% of such cash flow is a definite sum, and then the said coefficient would be 0.6. This may also be determined [objectively] by relating the desirable cash flows with estimated cash flows as follows: For Example, if the estimated cash flow is Rs.80000 in period ‘t’ and an equally desirable cash flow for the same period is Rs.60000, then the certainty equivalent is 0.75 [60000/80000]. Besides the computations of certain cash flows in order to provide for more risk, the economic life over which cash flows are estimated may be reduced simultaneously. EV = [0.25 x 8] + [0.50 x 12] + [0.25 x 16] = 12% In the case of alternative projects, one with the highest EV is considered for selection. EVs may be used for computation of IRR and NPV. However, EVs fail to articulate the degree of risk involved. 4. Decision Tree Analysis: A decision may be taken by comparing various alternative courses at present with the decision-maker, each alternative course being studied in the light of the future possible conditions followed by future alternative decisions. The group of all these decisions present as well as future, viewed in relation to one another is called a ‘decision tree’. It is a graphic display of the relationship between a present decision and future events, and future decisions and their consequences. The sequence of events is normally represented over time in a format similar to the branches of a tree. The major steps in a decision tree process are: (a) The investment decision is defined clearly. (b) The decision alternatives are identified. (c) The decision tree graph indicates decision points, chance events, and other data. (d) It presents the relevant data such as the projected cash flow, probability distribution, expected present value, etc., on the decision tree branches. (e) Choosing the best alternative by analysis from the results displayed. Laxmi Ltd. is considering purchase of a new machine at a cost of Rs. 20,000. The cash inflows for the three years of its life are forecast as follows: The desired rate of return of NPV purposes is 20 percent. Calculate the probability of the machine. Also draw a decision tree diagram. Alternatively, profitability can be calculated through EVs as follows: Eskay Ltd is considering the purchase of a new machine. The two alternative models under consideration are ‘Laxmi’ and ‘HMT’. From the following information, prepare a profitability statement for submission to the Board of Directors: Assume the taxation rate to be 50 percent of profits. Suggest which model can be purchased, giving reasons for you answer. Thus machine ‘Laxmi’ clearly recommends itself for purchase. However, the information provided and conclusion derived may be supplemented with some additional calculations as regards profitability beyond payback period. Determine the average rate of return from the following data of two Machines A and B:

While the passing of time is constant, our estimation of how much time has passed varies. Here's what we got from that question. Words ending with -metry are to do with measuring from the Greek word metron meaning "measurement". Then it just turns out that we can factor using the inverse of Distributive Property! And just as a bit of a refresher, if a parabola looks like this, the vertex is the lowest point here, so this minimum point here, for an upward opening parabola. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. They also have the property that a line from the origin to any point on the curve always finds the tangent to the curve meeting it at the same angle. What is its length on the subsequent two sizes of tile at each subsequent stage? Then factor like you normally would: I know I get better responses when I use the word typical so I went with that. Students will use mathematical relationships to generate solutions and make connections and predictions. Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals. Now, what I want to do is express the stuff in the parentheses as a sum of a perfect square and then some number over here. It mentions this Ammann tiling on page You might have spotted that this equation is merely Pythagoras' Theorem that all the points x,y on the circle are the same distance from the origin, that distance being a. Geometry, Adopted One Credit. Remember that the sign of a term comes before it, and pay attention to signs. As we discussed each one, I wanted to draw out some important characteristics: The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. Students will use their proportional reasoning skills to prove and apply theorems and solve problems in this strand. Students will broaden their knowledge of quadratic functions, exponential functions, and systems of equations. So when x is equal to 0, we have 1, 2, 3, oh, well, these are 2. A single tile which produces an "irregular" tiling was found by Robert Ammann in This is the coefficient of the first term 10 multiplied by the coefficient of the last term — 6. The Geometry Junkyard has a great page of Penrose links Ivars Peterson's ScienceNewsOnline has an interesting page about quasicrystals showing how Penrose tilings are found in nature. There are times as a coach that I wish I were back in the classroom. So let's just work on this. When x is equal to 0, y is equal to 8. This quantity right here, x minus 2 squared, if you're squaring anything, this is always going to be a positive quantity. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. The students were developing a good sense of center and the values in relation to the center. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. So here I haven't changed equation. Students talked about the estimates being 1. So how do we get x is equal to 0 here? This number is called the "standard deviation".Apr 13, · Edit Article How to Find the Maximum or Minimum Value of a Quadratic Function Easily. Three Methods: Beginning with the General Form of the Function Using the Standard or Vertex Form Using Calculus to Derive the Minimum or Maximum Community Q&A For a variety of reasons, you may need to be able to define the maximum or minimum value of a selected quadratic. What Does a 4-Dimensional Sphere Look Like? There is a very real geometric object, realizable within the relativistic geometry of our universe, which has the properties of a sphere in four dimensions (a “4-hypersphere”); what does it look like? HSN Number and Quantity. HSN-RN The Real Number System. HSN-RN.A Extend the properties of exponents to rational exponents. HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic alethamacdonald.com scheme is named after its inventor, American physicist Richard Feynman, and was first introduced in The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Different forms of quadratic functions reveal different features of those functions. Here, Sal rewrites f(x)=x²-5x+6 in factored form to reveal its zeros and in vertex form to reveal its vertex. SOLUTION: Rewrite the quadratic equation in standard form by completing the square. f(x)=x^2+8x+14 Answer (f(x)= (x, plus or minus ___)^2 plus or minus ____ I have worked this to x, 4 a Algebra -> Quadratic Equations and Parabolas -> SOLUTION: Rewrite the quadratic equation in standard form by completing the square.Download

555 Timer Introduction: • The 555 Timer is one of the most popular and versatile integrated circuits ever produced! • “Signetics” Corporation first introduced this device as the SE/NE 555 in early 1970. • It is a combination of digital and analog circuits. • It is known as the “time machine” as it performs a wide variety of timing tasks. • Applications for the 555 Timer include: • Ramp and Square wave generator • Frequency dividers • Voltage-controlled oscillators • Pulse generators and LED flashers 555 timer- Pin Diagram The 555 timer is an 8-Pin D.I.L. Integrated Circuit or ‘chip’ Notch Pin 1 Best treated as a single component with required • input and output OUTPUT PROCESS INPUT 555 Timer • Description: • Contains 25 transistors, 2 diodes and 16 resistors • Maximum operating voltage 16V • Maximum output current 200mA If you input certain signals they will be processed / controlled in a certain manner and will produce a known output. Inside the 555 Timer + Truth Table Vref Threshold Control Voltage R Q - Q S Trigger Discharge Fig: Functional Diagram of 555 Timer Inside the 555 Timer Operation: • The voltage divider has three equal 5K resistors. It divides the input voltage (Vcc) into three equal parts. • The two comparators are op-amps that compare the voltages at their inputs and saturate depending upon which is greater. • The Threshold Comparator saturates when the voltage at the Threshold pin (pin 6) is greater than (2/3)Vcc. • The Trigger Comparator saturates when the voltage at the Trigger pin (pin 2) is less than (1/3)Vcc Inside the 555 Timer • The flip-flop is a bi-stable device. It generates two values, a “high” value equal to Vcc and a “low” value equal to 0V. • When the Threshold comparator saturates, the flip flop is Reset (R) and it outputs a low signal at pin 3. • When the Trigger comparator saturates, the flip flop is Set (S) and it outputs a high signal at pin 3. • The transistor is being used as a switch, it connects pin 7 (discharge) to ground when it is closed. • When Q is low, Q bar is high. This closes the transistor switch and attaches pin 7 to ground. • When Q is high, Q bar is low. This open the switch and pin 7 is no longer grounded Uses of 555 timer What the 555 timer is used for: • To switch on or off an output after a certain time delay i.e. • Games timer, Childs mobile, Exercise timer. • To continually switch on and off an output i.e. • warning lights, Bicycle indicators. • As a pulse generator i.e. • To provide a series of clock pulses for a counter. 555 Timer operating modes • The 555 has three operating modes: 1. Monostable Multivibrator 2.Astable Multivibrator 3. Bistable Multivibratior 555 Timer as Monostable Multivibrator • Description: • In the standby state, FF holds transistor Q1 ON, thus clamping the external timing capacitor C to ground. The output remains at ground potential. i.e. Low. • As the trigger passes through VCC/3, the FF is set, i.e. Q bar=0, then the transistor Q1 OFF and the short circuit across the timing capacitor C is released. As Q bar is low , output goes HIGH. 555 Timer as Monostable Multivibrator Fig (a): Timer in Monostable Operation with Functional Diagram Fig (b): Output wave Form of Monostable Monostable Multivibrator- Description • Voltage across it rises exponentially through R towards Vcc with a time constant RC. • After Time Period T, the capacitor voltage is just greater than 2Vcc/3 and the upper comparator resets the FF, i.e. R=1, S=0. This makes Q bar =1, C rapidly to ground potential. • The voltage across the capacitor as given by, • If –ve going reset pulse terminal (pin 4) is applied, then transistor Q2-> OFF, Q1-> ON & the external timing capacitor C is immediately discharged. at Behavior of the Monostable Multivibrator • The monostable multivibrator is constructed by adding an external capacitor and resistor to a 555 timer. • The circuit generates a single pulse of desired duration when it receives a trigger signal, hence it is also called a one-shot. • The time constant of the resistor-capacitor combination determines the length of the pulse. Uses of the Monostable Multivibrator • Used to generate a clean pulse of the correct height and duration for a digital system • Used to turn circuits or external components on or off for a specific length of time. • Used to generate delays. • Can be cascaded to create a variety of sequential timing pulses. These pulses can allow you to time and sequence a number of related operations. Monostable Multivibrator Problem: In the monostable multivibrator of fig, R=100kΩ and the time delay T=100ms. Calculate the value of C ? Solution: T=1.1RC Applications in Monostable Mode • Missing Pulse Detector. • Linear Ramp Generator. • Frequency Divider. • Pulse Width Modulation. 1.Missing Pulse Detector Fig (a) : A missing Pulse Detector Monostable Circuit Fig (b) : Output of Missing Pulse Detector Missing Pulse Detector- Description • When input trigger is Low, emitter-base diode of Q is forwarded biased capacitor is clamped to 0.7v(of diode), output of timer is HIGH width of T o/p of timer > trigger pulse width. • T=1.1RC select R & C such that T > trigger pulse. • Output will be high during successive coming of input trigger pulse. If one of the input trigger pulse missing trigger i/p is HIGH, Q is cut off, timer acts as normal monostable state. • It can be used for speed control and measurement. 2.Linear Ramp Generator at pin 2 > Vcc/3 Capacitor voltage at pin 6 Linear Ramp Generator- Description Analysis: Applying KVL around base-emitter loop of Q3 Q3 Ic i Voltage Capacitor, When becomes at T, 3.Frequency Divider Description: A continuously triggered monostable circuit when triggered by a square wave generator can be used as a frequency divider, if the timing interval is adjusted to be longer than the period of the triggering square wave input signal. The monostable multivibrator will be triggered by the first negative going edge of the square wave input but the output will remain HIGH(because of greater timing interval) for next negative going edge of the input square wave as shown fig. Fig: Diagram of Frequency Divider 4.Pulse Width Modulation Fig b: PWM Wave Forms Fig a: Pulse Width Modulation Pulse Width Modulation- Description The charging time of capacitor is entirely depend upon 2Vcc/3. When capacitor voltage just reaches about 2Vcc/3 output of the timer is coming from HIGH to Low level. We can control this charging time of the capacitor by adding continuously varying signal at the pin-5 of the 555 timer which is denoted as control voltage point. Now each time the capacitor voltage is compared control voltage according to the o/p pulse width change. So o/p pulse width is changing according to the signal applied to control voltage point. So the output is pulse width modulated form. Pulse Width Modulation Practical Representation Fig: PWM & Wave forms Astable Multivibrator Fig (a): Diagram of Astable Multvibrator 1 – Ground 5 – FM Input (Tie to gnd via bypass cap) 2 – Trigger 6 – Threshold 3 – Output 7 – Discharge 4 – Reset (Set HIGH for normal operation) 8 – Voltage Supply (+5 to +15 V) Astable Multivibrator R1 VA A1 A1 Vo V1 R2 V2 A2 A2 VC VT R3 Q1 Fig (b): Functional Diagram of Astable Multivibrator using 555 Timer Astable Multivibrator- Description • Connect external timing capacitor between trigger point (pin 2) and Ground. • Split external timing resistor R into RA & RB, and connect their junction to discharge terminal (pin 7). • Remove trigger input, monostable is converted to Astable multivibrator. • This circuit has no stable state. The circuits changes its state alternately. Hence the operation is also called free running oscillator. Astable 555 Timer Block Diagram Contents • Resistive voltage divider (equal resistors) sets threshold voltages for comparators V1 = VTH = 2/3 VCC V2 = VTL = 1/3 VCC • Two Voltage Comparators • For A1, if V+ > VTH then R =HIGH • For A2, if V- < VTL then S = HIGH • RS FF • If S = HIGH, then FF is SET, = LOW, Q1 OFF, VOUT = HIGH • If R = HIGH, then FF is RESET, = HIGH, Q1 ON, VOUT = LOW • Transistor Q1 is used as a Switch Operation of a 555 Astable • Assume initially that the capacitor is discharged. • For A1, V+ = VC = 0V and for A2, V- = VC = 0V, so R=LOW, S=HIGH, = LOW , Q1 OFF, VOUT = VCC • Now as the capacitor charges through RA & RB, eventually VC > VTL so R=LOW & S=LOW. FF does not change state. RA RB VC(t) VCC Operation of a 555 AstableContinued…… • Once VC VTH • R=HIGH, S=LOW, = HIGH ,Q1 ON, VOUT = 0 • Capacitor is now discharging through RB and Q1 to ground. • Meanwhile at FF, R=LOW & S=LOW since VC < VTH. RB VC(t) Q1 Operation of a 555 Astable Continued….. • Once VC < VTL • R=LOW, S=HIGH, = LOW , Q1 OFF, VOUT = VCC • Capacitor is now charging through RA & RB again. RA RB VC(t) VCC Timing Diagram of a 555 Astable 1 2 3 VC(t) VTH VTL t VOUT(t) TL TH t t = 0 t = 0' Astable Multivibrator- Analysis The capacitor voltage for a low pass RC circuit subjected to a step input of Vcc volts is given by, The time t1 taken by the circuit to change from 0 to 2Vcc/3 is, The time t2 to charge from 0 to vcc/3 is …… Charging time So the time to change from Vcc/3 to 2Vcc/3 is, So, for the given circuit, The output is low while the capacitor discharges from 2Vcc/3 to Vcc/3 and the voltage across the capacitor is given by, Contd…. Astable Multivibrator- Analysis After solving, we get, t=0.69RC …… Discharging time For the given circuit, Both RA and RB are in the charge path, but only RB is in the discharge path. The total time period, …….1.45 is Error Constant Frequency, Duty Cycle, Behavior of the Astable Multivibrator • The astable multivibrator is simply an oscillator. The astable multivibrator generates a continuous stream of rectangular off-on pulses that switch between two voltage levels. • The frequency of the pulses and their duty cycle are dependent upon the RC network values. • The capacitor C charges through the series resistors RAand RB with a time constant (RA+ RB)C. • The capacitor discharges through RBwith a time constant of RBC Uses of the Astable Multivibrator • Flashing LED’s • Pulse Width Modulation • Pulse Position Modulation • Periodic Timers • Uses include LEDs, pulse generation, logic clocks, security alarms and so on. Applications in Astable Mode Square Generator FSK Generator Pulse Position Modulator 1.Square Generator 3 10µF Fig: Square Wave Generator C1 • To avoid excessive discharge current through Q1 when R1=0 connect a diode across R2, place a variable R in place of R1. • Charging path R1 & D; Discharging path R2 & pin 7. 2. FSK Generator • Description: • In digital data communication, binary code is transmitted by shifting a carrier frequency between two preset frequencies. This type of transmission is called Frequency Shift Keying (FSK) technique. Fig: FSK Generator Contd….. FSK Generator • A 555 timer is astable mode can be used to generate FSK signal. • When input digital data is HIGH, T1is OFF & 555 timer works as normal astable multivibrator. The frequency of the output wave form given by, When input digital is LOW, Q1 is ON then R3 parallel R1 2. Pulse Position Modulator • Description: • The pulse position modulator can be constructed by applying a modulating signal to pin 5 of a 555 timer connected for astable operation. • The output pulse position varies with the modulating signal, since the threshold voltage and hence the time delay is varied. • The output waveform that the frequency is varying leading to pulse position modulation. Fig (a): Pulse position Modulator Fig (b): Output Wave Form of PPM Astable Multivibrator Problem: In the astable multivibrator of fig, RA=2.2KΩ, RB=3.9K Ω and C=0.1µF. Determine the positive pulse width tH, negative pulse width tLow, and free-running frequency fo. Solution: Duty Cycle, Example: Design a 555 Oscillator to produce an approximate square-wave at 40 KHz. Let C > 470 pF. One Possible Solution: F=40KHz; T=25µs; t1=t2=12.5µs For a square-wave RA<<RB; Let RA=1K and RB=10K t1=0.693(RB)(C); 12.5µs=0.693(10K)(C); C=1800pF T=0.693(RA+2RB)C: T=0.693(1K+20K)1800pF T=26.2µs; F=1/T; F=38KHz (almost square-wave). Example: A 555 oscillator can be combined with a J-K FF to produce a 50% duty-cycle signal. Modify the above circuit to achieve a 50% duty-cycle, 40 KHz signal. One Possible Solution: Reduce by half the 1800pF. This will create a T=13.1µs or F=76.35 KHz (almost square-wave). Now, take the output of the 555 Timer and connect it to the CLK input of a J-K FF wired in the toggle mode (J and K inputs connected to +5V). The result at the Q output of the J-K FF is a perfect 38.17 KHz square-wave. Schmitt Trigger Fig (b): Output Wave Form Fig (a): Circuit Diagram of Schmitt Trigger The use of 555 timer as a Schmitt trigger is shown in fig. Here the two internal comparators are tied together and externally biased at Vcc/2 through R1 and R2. Since the upper comparator will trip at 2Vcc/3 and lower comparator at Vcc/3, the bias provided by R1 and R2 is centered within these two thresholds. Features of IC 555 Timer The Features of IC 555 Timer are: 1. The 555 is a monolithic timer device which can be used to produce accurate and highly stable time delays or oscillation. It can be used to produce time delays ranging from few microseconds to several hours. 2. It has two basic operating modes: monostable and astable. 3. It is available in three packages: 8-pin metal can, 8-pin mini DIP or a 14-pin. A 14-pin package is IC 556 which consists of two 555 times. Features of IC 555 Timer 4. The NE 555( signetics ) can operate with a supply voltage in the range of 4.5v to 18v and output currents of 200mA. 5. It has a very high temperature stability, as it is designed to operate in the temperature range of -55⁰c to 125oc. 6. Its output is compatible with TTL, CMOS and Op-Amp circuits.

12: Electromagnetic Induction 12.2 Alternating Current Alternating Current Demo: HEP demo or dynamo Alternating Current and Voltage Whenever a magnet rotates near a coil or wire, its flux will move through the wire or coil inducing an alternating EMF across the coil or wire as a result of Faraday’s Law. Rotating Coil in a Uniform Magnetic Field A very simple AC generator can consist of a single coil of copper wire being forced to rotate in a uniform magnetic field as shown. At the each end of the wire are connected circular ‘slip rings’. Q1. Explain the design and purpose of the ‘slip rings’ Q2. Why is the coil made from copper wire? Link: AC Generator AppletAC Generator Applet This simplified diagram shows a coil ‘end-on’, rotating anti-clockwise: Q3. Explain using Faraday’s Law why the EMF will vary from maximum to zero as angle θ (between the normal to the coil and the field plane) goes from 90° to zero (as shown in the diagrams). Q4. Plot points on the graph of flux linkage against time (for max positive flux linkage, max negative, zero) and draw the line that goes through them. Considering Faraday’s Law, similarly plot points on the graph of EMF against time and draw the line. As can be seen from the two graphs, if EMF (ε) is a sinusoidal graph then flux linkage must give a cosine graph. In fact the equations for each are... N ϕ = BAN cosθ(or N ϕ = BAN cos ωt) ε = BAN ω sin θ(or ε = BAN ω sin ωt) (You do not need to know these equations however they should make sense to you). Increasing the speed of rotation If the coil is rotated at a greater angular speed, the EMF generated will increase and the frequency of rotation will also increase. Hence the graph will change in two ways. Q5. The graph below shows the output of a coil rotating in a fixed uniform magnetic field. On the same axes, sketch the graph of i. a coil rotating with twice the frequency. ii. a coil rotating with half the frequency. Root Mean Square Current In mathematics the Root Mean Square (rms) is a statistical method of determining the magnitude of a quantity that is varying. It can be thought of as a kind of ‘average’ value. In particular it is useful when dealing with sinusoidal variations (that can be positive or negative) such as induced EMF and current from a rotating coil. For discrete values of any quantity the following formula can be applied: Clearly the calculated value is the square root of the mean of the squares of the discrete values. Q6. Determine the rms value of current from the following graph using eight successive discrete values: I (A) For electrical output from a coil rotating at constant speed in a uniform magnetic field, the following formulae can be applied: ε rms = ε 0 √2 I rms = I 0 √2 Where... ε 0 = Maximum EMF (V) I 0 = Maximum current (A) Power in AC circuits When calculating the power dissipated in an AC circuit, we use the rms values. Thus, for alternating current circuits... The rms value of an alternating current is identical to the value of direct current that would dissipate power at the same rate through a resistor. Power = I rms x V rms Q7. Determine a formula for average power in an alternating circuit in terms of ε 0 and I 0. Q8. The rms voltage in Europe is about 230V. Determine the peak voltage value. What will be the rms current value through a 20W fluorescent light bulb? Transformers If any two electrical circuits are near to each other, a changing current in one can cause an induced EMF in the other. A transformer uses changing flux linkage produced by one coil to induce an EMF in the second coil. Primary coil Secondary coil The input current is a.c. Plot a graph of current in the primary (I p ) against time. The flux in the core is proportional to I p. Plot a graph of flux in the core against time. The EMF induced in the secondary is proportional to the rate of change of flux linkage. Plot a graph of Induced EMF in the secondary against time. Transformer Calculations The flux passing through the primary and secondary coils is identical in a 100% efficient transformer. Q. Explain (using Faraday’s Law) why having more turns in the secondary than the primary can lead to the voltage being ‘stepped up’ (increased). The ratio of the turns is equal to the ratio of the voltages: V s N s V p N p = Ideal Transformers A 100% efficient transformer is known as an ‘ideal transformer’. In this case all the power on the primary side is transferred to the secondary side. Thus... I p V p = I s V s (All values are rms values) Q. If N p < N s, which of the following are true (re- write the wrong statements): a. ϕ p = ϕ s b. flux linkage is equal in both coils c. I p > I s d. V p > V s Real Transformers In reality the output power is less than the input power. This could be due to: - Resistance of wires (causing heat transfer) - Eddy currents in core (causing heat transfer) - Flux leakage (not linking into the secondary coil) - Hysteresis (molecular friction) (causing heat transfer) Q. Suggest a way of decreasing each of the first three losses. (copper wires; laminated core; improved core design)

Abstract:The paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations, rounding error in arithmetic floating-point operations, and approximations in ’built-in’ functions. The error analysis is based on the linearization method that has been proposed by many authors in various forms. Fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors. Derivations, representations, and properties of these condition numbers are studied in detail. The condition numbers enable simple general, quantitative definitions of numerical stability, backward analysis, well- and ill-conditioning of a problem and an algorithm. The well-known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds. For many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori. The theoretical results of the paper have been applied to a series of concrete algorithms, including Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis. I. Babuška, Numerical Stability in Numerical Analysis, Proc. IFIP-Congress 1968, Amsterdam, North-Holland, Amsterdam, 1969, pp. 11-23. - Ivo Babuška, Numerical stability in problems of linear algebra, SIAM J. Numer. Anal. 9 (1972), 53–77. MR 386252, DOI 10.1137/0709008 - Ivo Babuška, Milan Práger, and Emil Vitásek, Numerical processes in differential equations, Státní Nakladatelství Technické Literatury (SNTL), Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. In cooperation with R. Radok; Translated from the Czech by Milada Boruvková. MR 0223101 - F. L. Bauer, Genauigkeitsfragen bei der Lösung linearer Gleichungssysteme, Z. Angew. Math. Mech. 46 (1966), 409–421 (German). MR 208814, DOI 10.1002/zamm.19660460702 - F. L. Bauer, Computational graphs and rounding error, SIAM J. Numer. Anal. 11 (1974), 87–96. MR 356482, DOI 10.1137/0711010 F. L. Bauer et al., Moderne Rechenanlagen, Chap. 3, Teubner, Stuttgart, 1964. - W. S. Brown, A realistic model of floating-point computation, Mathematical software, III (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1977) Publ. Math. Res. Center, No. 39, Academic Press, New York, 1977, pp. 343–360. MR 0483616 - Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900 - John Larson and Ahmed Sameh, Efficient calculation of the effects of roundoff errors, ACM Trans. Math. Software 4 (1978), no. 3, 228–236. MR 506787, DOI 10.1145/355791.355794 - Seppo Linnainmaa, Taylor expansion of the accumulated rounding error, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 2, 146–160. MR 421065, DOI 10.1007/bf01931367 D. D. McCracken & W. S. Dorn, Numerical Methods and Fortran Programming, Wiley, New York, 1964. - Webb Miller, Software for roundoff analysis, ACM Trans. Math. Software 1 (1975), 108–128. MR 405830, DOI 10.1145/355637.355639 - Webb Miller, Computer search for numerical instability, J. Assoc. Comput. Mach. 22 (1975), no. 4, 512–521. MR 478584, DOI 10.1145/321906.321916 - Webb Miller and David Spooner, Software for roundoff analysis. II, ACM Trans. Math. Software 4 (1978), no. 4, 369–387. MR 513705, DOI 10.1145/356502.356496 - John R. Rice, A theory of condition, SIAM J. Numer. Anal. 3 (1966), 287–310. MR 211576, DOI 10.1137/0703023 F. Stummel, Fehleranalyse numerischer Algorithmen, Lecture Notes, Univ. of Frankfurt, 1978. - F. Stummel, Rounding errors in numerical solutions of two linear equations in two unknowns, Math. Methods Appl. Sci. 4 (1982), no. 4, 549–571. MR 681254, DOI 10.1002/mma.1670040135 - F. Stummel, Rounding error analysis of elementary numerical algorithms, Fundamentals of numerical computation (computer-oriented numerical analysis) (Proc. Conf., Tech. Univ. Berlin, Berlin, 1979) Comput. Suppl., vol. 2, Springer, Vienna, 1980, pp. 169–195. MR 586230 F. Stummel, "Rounding error anlaysis of difference and extrapolation schemes." (To appear.) - Friedrich Stummel, Rounding error in Gaussian elimination of tridiagonal linear systems. Survey of results, Interval mathematics, 1980 (Freiburg, 1980) Academic Press, New York-London, 1980, pp. 223–245. MR 651366 F. Stummel, "Rounding error in Gaussian elimination of tridiagonal linear systems. I," SIAM J. Numer. Anal. (Submitted.) F. Stummel, "Rounding error in Gaussian elimination of tridiagonal linear systems. II," Linear Algebra Appl. (Submitted.) F. Stummel, "Forward error analysis of Gaussian elimination." (To appear.) - Martti Tienari, On some topological properties of numerical algorithms, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 409–433. MR 334582, DOI 10.1007/bf01932311 - J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456 - © Copyright 1981 American Mathematical Society - Journal: Math. Comp. 37 (1981), 435-473 - MSC: Primary 65G99 - DOI: https://doi.org/10.1090/S0025-5718-1981-0628707-8 - MathSciNet review: 628707

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical (e.g., sex, ethnicity, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation. - 1 Example - 2 Multicollinearity in moderated regression - 3 Post-hoc probing of interactions - 4 See also - 5 References Moderation analysis in the behavioral sciences involves the use of linear multiple regression analysis or causal modelling. To quantify the effect of a moderating variable in multiple regression analyses, regressing random variable Y on X, an additional term is added to the model. This term is the interaction between X and the proposed moderating variable. Thus, for a response Y and two variables x1 and moderating variable x2,: In this case, the role of x2 as a moderating variable is accomplished by evaluating b3, the parameter estimate for the interaction term. See linear regression for discussion of statistical evaluation of parameter estimates in regression analyses. Multicollinearity in moderated regression In moderated regression analysis, a new interaction predictor () is calculated. However, the new interaction term will be correlated with the two main effects terms used to calculate it. This is the problem of multicollinearity in moderated regression. Multicollinearity tends to cause coefficients to be estimated with higher standard errors and hence greater uncertainty. Mean-centering (subtracting raw scores from the mean) has been suggested as a remedy for multicollinearity. However, mean-centering is unnecessary in any regression analysis, as one uses a correlation matrix and the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation). Also see the article by Kromrey & Foster-Johnson (1998) on "Mean-centering in Moderated Regression: Much Ado About Nothing". Post-hoc probing of interactions Like simple main effect analysis in ANOVA, in post-hoc probing of interactions in regression, we are examining the simple slope of one independent variable at the specific values of the other independent variable. Below is an example of probing two-way interactions. In what follows the regression equation with two variables A and B and an interaction term A*B, will be considered. Two categorical independent variables If both of the independent variables are categorical variables, we can analyze the results of the regression for one independent variable at a specific level of the other independent variable. For example, suppose that both A and B are single dummy coded (0,1) variables, and that A represents ethnicity (0 = European Americans, 1 = East Asians) and B represents the condition in the study (0 = control, 1 = experimental). Then the interaction effect shows whether the effect of condition on the dependent variable Y is different for European Americans and East Asians and whether the effect of ethnic status is different for the two conditions. The coefficient of A shows the ethnicity effect on Y for the control condition, while the coefficient of B shows the effect of imposing the experimental condition for European American participants. To probe if there is any significant difference between European Americans and East Asians in the experimental condition, we can simply run the analysis with the condition variable reverse-coded (0 = experimental, 1 = control), so that the coefficient for ethnicity represents the ethnicity effect on Y in the experimental condition. In a similar vein, if we want to see whether the treatment has an effect for East Asian participants, we can reverse code the ethnicity variable (0 = East Asians, 1 = European Americans). One categorical and one continuous independent variable If the first independent variable is a categorical variable (e.g. gender) and the second is a continuous variable (e.g. scores on the Satisfaction With Life Scale (SWLS)), then b1 represents the difference in the dependent variable between males and females when life satisfaction is zero. However, a zero score on the Satisfaction With Life Scale is meaningless as the range of the score is from 7 to 35. This is where centering comes in. If we subtract the mean of the SWLS score for the sample from each participant's score, the mean of the resulting centered SWLS score is zero. When the analysis is run again, b1 now represents the difference between males and females at the mean level of the SWLS score of the sample. Cohen et al. (2003) recommended using the following to probe the simple effect of gender on the dependent variable (Y) at three levels of the continuous independent variable: high (one standard deviation above the mean), moderate (at the mean), and low (one standard deviation below the mean). If the scores of the continuous variable are not standardized, one can just calculate these three values by adding or subtracting one standard deviation of the original scores; if the scores of the continuous variable are standardized, one can calculate the three values as follows: high = the standardized score minus 1, moderate (mean = 0), low = the standardized score plus 1. Then one can explore the effects of gender on the dependent variable (Y) at high, moderate, and low levels of the SWLS score. As with two categorical independent variables, b2 represents the effect of the SWLS score on the dependent variable for females. By reverse coding the gender variable, one can get the effect of the SWLS score on the dependent variable for males. Coding in moderated regression When treating categorical variables such as ethnic groups and experimental treatments as independent variables in moderated regression, one needs to code the variables so that each code variable represents a specific setting of the categorical variable. There are three basic ways of coding: Dummy-variable coding, Effects coding, and Contrast coding. Below is an introduction to these coding systems. Dummy coding is used when one has a reference group or one condition in particular (e.g. a control group in the experiment) that is to be compared to each of the other experimental groups. In this case, the intercept is the mean of the reference group, and each of the unstandardized regression coefficients is the difference in the dependent variable between one of the treatment groups and the mean of the reference group (or control group). This coding system is similar to ANOVA analysis, and is appropriate when researchers have a specific reference group and want to compare each of the other groups with it. Effects coding is used when one does not have a particular comparison or control group and does not have any planned orthogonal contrasts. The intercept is the grand mean (the mean of all the conditions). The regression coefficient is the difference between the mean of one group and the mean of all the group means (e.g. the mean of group A minus the mean of all groups). This coding system is appropriate when the groups represent natural categories. Contrast coding is used when one has a series of orthogonal contrasts or group comparisons that are to be investigated. In this case, the intercept is the unweighted mean of the individual group means. The unstandardized regression coefficient represents the difference between the unweighted mean of the means of one group (A) and the unweighted mean of another group (B), where A and B are two sets of groups in the contrast. This coding system is appropriate when researchers have an a priori hypothesis concerning the specific differences among the group means. Two continuous independent variables If both of the independent variables are continuous, it is helpful for interpretation to either center or standardize the independent variables, X and Z. (Centering involves subtracting the overall sample mean score from the original score; standardizing does the same followed by dividing by the overall sample standard deviation.) By centering or standardizing the independent variables, the coefficient of X or Z can be interpreted as the effect of that variable on Y at the mean level of the other independent variable. To probe the interaction effect, it is often helpful to plot the effect of X on Y at low and high values of Z (some people prefer to also plot the effect at moderate values of Z, but this is not necessary). Often values of Z that are one standard deviation above and below the mean are chosen for this, but any sensible values can be used (and in some cases there are more meaningful values to choose). The plot is usually drawn by evaluating the values of Y for high and low values of both X and Z, and creating two lines to represent the effect of X on Y at the two values of Z. Sometimes this is supplemented by simple slope analysis, which determines whether the effect of X on Y is statistically significant at particular values of Z. Various internet-based tools exist to help researchers plot and interpret such two-way interactions. The principles for two-way interactions apply when we want to explore three-way or higher-level interactions. For instance, if we have a three-way interaction between A, B, and C, the regression equation will be as follows: Spurious higher-order effects It is worth noting that the reliability of the higher-order terms depends on the reliability of the lower-order terms. For example, if the reliability for variable A is 0.70, and reliability for variable B is 0.80, then the reliability for the interaction variable A * B is 0.70 × 0.80 = 0.56. In this case, low reliability of the interaction term leads to low power; therefore, we may not be able to find the interaction effects between A and B that actually exist. The solution for this problem is to use highly reliable measures for each independent variable. Another caveat for interpreting the interaction effects is that when variable A and variable B are highly correlated, then the A * B term will be highly correlated with the omitted variable A2; consequently what appears to be a significant moderation effect might actually be a significant nonlinear effect of A alone. If this is the case, it is worth testing a nonlinear regression model by adding nonlinear terms in individual variables into the moderated regression analysis to see if the interactions remain significant. If the interaction effect A*B is still significant, we will be more confident in saying that there is indeed a moderation effect; however, if the interaction effect is no longer significant after adding the nonlinear term, we will be less certain about the existence of a moderation effect and the nonlinear model will be preferred because it is more parsimonious. - Cohen, Jacob; Cohen, Patricia; Leona S. Aiken; West, Stephen H. (2003). Applied multiple regression/correlation analysis for the behavioral sciences. Hillsdale, N.J: L. Erlbaum Associates. ISBN 0-8058-2223-2. - Baron, R. M., & Kenny, D. A. (1986). "The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations", Journal of Personality and Social Psychology, 5 (6), 1173–1182 (page 1174) - Taylor, Alan. "Testing and Interpreting Interactions in Regression-In a Nutshell" (PDF). - Cohen Jacob; Cohen Patricia; West Stephen G.; Aiken Leona S. Applied multiple regression/correlation analysis for the behavioral sciences (3. ed.). Mahwah, NJ [u.a.]: Erlbaum. pp. 255–301. ISBN 0-8058-2223-2. - Aiken L.S., West., S.G. (1996). Multiple regression testing and interpretation (1. paperback print. ed.). Newbury Park, Calif. [u.a.]: Sage Publications, Inc. ISBN 0-7619-0712-2.CS1 maint: Multiple names: authors list (link) - Cohen Jacob; Cohen Patricia; West Stephen G.; Aiken Leona S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3. ed.). Mahwah, NJ [u.a.]: Erlbaum. pp. 302–353. ISBN 0-8058-2223-2. - Dawson, J. F. (2013). Moderation in management research: What, why, when and how. Journal of Business and Psychology. DOI: 10.1007/s10869-013-9308-7. - Hayes, A. F., & Matthes, J. (2009). "Computational procedures for probing interactions in OLS and logistic regression: SPSS and SAS implementations." Behavior Research Methods, Vol. 41, pp. 924–936.

When you are in your math class working with different mathematical values and operations, you often come across problems that require you to use significant numbers from the value. The use of significant numbers is not new in mathematics. They have existed for a long time and are still of extreme importance in mathematics. If you are a student of mathematics, then you need to know how to calculate significant figures. Learning this is absolutely essential if you want to be accurate and precise with your mathematical calculations. This is what we are going to talk about in this detailed guide. We are going to share with you everything that you need to know about Significant Digits and the best methods that you can use to calculate them during your math class. What are Significant Figures? Significant Numbers are the digits in a number that contribute to the precision and accuracy of that number. They are also regarded as the meaningful digits that convey the message that a value is supposed to convey. Finding significant figures give you a result that you can use in the different mathematical calculation to produce consistent results with extreme accuracy. Whether you are rounding numbers or you want to present a value in its scientific notation, you need to be able to determine the significant numbers in the value before you do anything else. Throughout the rest of this article, we are going to talk about the complete process that you can use for finding significant figures in math class. How Can We Calculate Significant Figures During Math Class? Scientists have developed a systematic approach for finding significant figures. This approach features a certain set of rules that anyone can use to find significant digits in a given value. Students are expected to learn and memorize these rules to make the significant digits identification easier and faster for them. Rules for Calculating Significant Figures Here is the complete set of rules that you need to follow when calculating significant figures in your math class. When working with a value, all non-zero digits of the value would be significant. When you are given a value that only has non-zero digits, then you need to consider all of them significant. This is because these are the values that have been calculated. For example, in the term 0.9876, there are 4 significant digits. Similarly, the term 364.543 has 6 significant digits. This rule is true for all the values that don’t have zeros. If the value that you are working with has one or more zeros between two non-zero digits, then those zeros will also be Significant. Let’s take the help of an example to understand the concept a little bit better. If you have a number such as 0.679009, it has 6 significant digits. This is because the rule states that if zeros come in between 2 non-zero digits, then they are also significant. This rule is quite simple and eliminates the risk of considering zeros as insignificant digits all the time. If you are working with a decimal point term and there are zeros on the right side of the decimal point that is not followed by a non-zero digit, then those zeros will be significant. Consider the example of the number 72.00. This term has 4 significant digits when you look at it as per the terms defined in the rule. You can’t discard the zeros on the right side of the decimal point because they don’t have a non-zero digit after them. This rule states that the zeros that are placed on the immediate right of a decimal point and the left of the non-zero digits are not significant. The example of the number 0.0009876 explains this rule the best. Here, only the 4 numbers 9,8,7,6 are significant. The zeros are all insignificant as per the rule. You can look at it from the angle of writing the scientific notation for this number. You are going to discard the zeros for that and only work with the 4 digits mentioned above. Hence, those digits are the only significant ones in this value. Zeros that are placed on the right side of the last non-zero number in a decimal point value are significant. Consider the number 0.00783400. This term has 6 significant digits as per the rule stated above. This rule is quite straightforward and should become clear with the example that we have given here. There are various other rules for significant figures that you’ll find out there. But the ones mentioned here are all that matter as these are the ones that you will most likely come across. Use the Sig Fig Calculator to Easily Calculate Significant Figures When you are in your math class and you want to quickly find the significant digits in a value, you can use the SigFig Calculator for that.This calculator saves you from the hassle of using the rules to identify significant numbers. The tool works on the same rules, so you don’t have to memorize them yourself. You can simply just add in the value that you want to work with, in this tool.After that, the tool processes the value to give you the exact number of significant digits in the value So, when you are in your math class next time, utilize our free online Sig Fig calculator tool to quickly and accurately find out the number of significant figures in your input value. The manual approach for significant figure calculation can often get confusing. This is because there are many rules that you need to memorize for accurately figuring out the significant figures in a value. As much as we stress learning these rules, you can always use the Sig Fig calculator to make things simpler for yourself. So, go over the rules for calculating significant figures and also check out the Sig Fig calculator for finding significant figures in math class.

Multi-Objective Optimization: Unlocking the Power of Optimization Are you tired of making trade-offs between different objectives in your optimization problems? Do you want to find solutions that are not only optimal but also satisfy multiple criteria? If so, then multi-objective optimization (MOO) is the answer you've been looking for! MOO is a powerful technique that allows you to optimize multiple objectives simultaneously, without having to compromise on any of them. It is a critical tool for decision-making in complex systems, where there are often multiple conflicting objectives that need to be balanced. In this article, we will explore the basics of MOO, its applications, and some of the popular algorithms used to solve MOO problems. We will also discuss some of the challenges and limitations of MOO and how to overcome them. What is Multi-Objective Optimization? MOO is a branch of optimization that deals with problems that have multiple objectives. In traditional optimization problems, there is only one objective function that needs to be optimized. However, in MOO, there are two or more objective functions that need to be optimized simultaneously. The goal of MOO is to find a set of solutions that are optimal with respect to all the objectives. These solutions are called Pareto optimal solutions, and they represent the best trade-offs between the different objectives. Applications of Multi-Objective Optimization MOO has a wide range of applications in various fields, including engineering, finance, logistics, and healthcare. Some of the common applications of MOO are: - Portfolio optimization: In finance, MOO is used to optimize investment portfolios that have multiple objectives, such as maximizing returns while minimizing risk. - Supply chain management: MOO is used to optimize supply chain networks that have multiple objectives, such as minimizing costs while maximizing customer satisfaction. - Engineering design: MOO is used to optimize the design of complex systems that have multiple objectives, such as minimizing weight while maximizing strength. - Healthcare: MOO is used to optimize treatment plans that have multiple objectives, such as minimizing side effects while maximizing effectiveness. Popular Algorithms for Multi-Objective Optimization There are several algorithms used to solve MOO problems, each with its strengths and weaknesses. Some of the popular algorithms are: Non-Dominated Sorting Genetic Algorithm (NSGA-II) NSGA-II is a popular algorithm for solving MOO problems. It uses a genetic algorithm approach to generate a set of Pareto optimal solutions. The algorithm works by first generating a population of solutions and then using a non-dominated sorting technique to rank the solutions based on their dominance. The algorithm then selects the best solutions from each rank to create the next generation of solutions. Strength Pareto Evolutionary Algorithm (SPEA2) SPEA2 is another popular algorithm for solving MOO problems. It uses a combination of fitness assignment and environmental selection to generate a set of Pareto optimal solutions. The algorithm works by first assigning a fitness value to each solution based on its dominance and then selecting the best solutions based on their fitness value. Multi-Objective Particle Swarm Optimization (MOPSO) MOPSO is a swarm intelligence-based algorithm for solving MOO problems. It uses a population of particles to search for Pareto optimal solutions. The algorithm works by first initializing a population of particles and then updating their positions and velocities based on their fitness values. Challenges and Limitations of Multi-Objective Optimization While MOO is a powerful technique, it also has some challenges and limitations that need to be addressed. Some of the common challenges and limitations of MOO are: Curse of Dimensionality MOO problems often have a large number of decision variables, which can make the search space very large. This can lead to the curse of dimensionality, where the number of solutions increases exponentially with the number of decision variables. MOO problems are often computationally complex, which can make them difficult to solve. The algorithms used to solve MOO problems can be computationally intensive, which can lead to long runtimes and high memory usage. Lack of Consensus MOO problems often have multiple Pareto optimal solutions, which can make it difficult to choose the best solution. There is often a lack of consensus on which solution is the best, which can lead to decision-making challenges. Multi-objective optimization is a powerful technique that allows you to optimize multiple objectives simultaneously. It has a wide range of applications in various fields and can help you make better decisions in complex systems. While MOO has some challenges and limitations, there are several algorithms available to solve MOO problems. With the right approach, MOO can unlock the power of optimization and help you achieve your goals. Editor Recommended SitesAI and Tech News Best Online AI Courses Classic Writing Analysis Tears of the Kingdom Roleplay JavaFX App: JavaFX for mobile Development Control Tower - GCP Cloud Resource management & Centralize multicloud resource management: Manage all cloud resources across accounts from a centralized control plane Learn NLP: Learn natural language processing for the cloud. 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Fixed costs variable costs and break even Study probes - chapter 7 problems how much will total variable costs be at break even 12 a firm will break even when revenues = variable costs – fixed costs. Start studying acct-1b chapters 3 and 4 a measure of the relative mix of a business's variable costs and fixed what is the break-even point if fixed costs. As the number of units sold increases, fixed costs stay fixed at $500, unit variable costs remain breakeven analysis: to break-even on costs. A critical part of cvp analysis is the point where total revenues equal total costs (both fixed and variable costs) at this break-even point, a. Explain fixed and variable costs and the concept of break even in relation to the organisation (a school. The contribution margin and fixed costs how fixed costs relate to the contribution margin fixed and variable costs the break-even point equation is fixed. Variable costs and fixed below is an example of a firm's cost schedule and a graph of the fixed and variable costs noticed that the fixed cost curve is flat. What is a 'fixed cost' fixed costs and variable costs form the total it to achieve higher levels of revenues to break even fixed costs must be. The break-even point of a business is the level of output or sales at which the revenue received by the business is exactly equal to. Total variable costs $3400 fixed costs sales = variable cost + fixed cost + target operating profit 30,000 what is the break-even point. Variable, fixed and mixed (semi-variable) costs are classified as variable, fixed and mixed costs fixed and mixed (semi-variable) costs. Break-even analysis and operating leverage recall that operating leverage describes the relationship between fixed and variable costs having high operating leverage (having a larger proportion of fixed costs compared to variable costs) can lead to much higher profits for a company. View test prep - hsm 260 week 6 checkpoint calculating fixed costs, variable cost from hsm 260 260 at university of phoenix what is the bep for the program since we see that they were in the red for. What is a break-even analysis by: the break-even analysis table calculates a break-even point based on fixed costs, variable costs per unit of sales. What is the variable cost per unit b if fixed cost are $650 what is the break-even number of units at fixed costs of $300,000 4) calculate break-even sales. Fixed variable and break even points david anderson hsm/260 july 14, 2013 melvin green fixed variable and break even points fixed costs, variable costs, and break-even point. Cost-volume-profit analysis and administrative costs, be identified as variable or fixed using fixed costs of $300,000, the break‐even equation is. Determining the fixed and variable expenses is the first step in performing a break-even analysis the number of units needed to break even = fixed costs / (price. If fixed costs or variable costs or selling price of an item changes variable costs (vc) break-even break-even & cvp analysis author. Assignment fixed costs, variable costs, and break-even point exercise 101 during the sixth month of the fiscal year, the program director of the. How do you calculate the break-even point in company's total amount of fixed costs plus its fixed expenses when we say variable expenses we mean the total of. Fixed and variable costs every business has fixed costs, which play roles in determining break-even points businesses also have variable costs, which make finding the actual break-even point more difficult. Fixed costs variable costs and break even Separate the expenses between fixed and variable costs per unit using this information and the sales price per unit of $8, compute the break-even point 5-3. Fixed costs and variable costs can you explain how to solve for fixed costs, variable costs we can even figure what sales are required to just break even. How to calculate variable costs can be combined with fixed costs to carry out a break-even how do i calculate both fixed and variable cost while. The break-even point is the sales level at which total revenue generated is equal to the total cost (fixed and variable), meaning that at the break-even. Fixed cost, variable costs, and break-even point for the program 52,048 results, page 2 principles of finance some please help me the fixed cost for a firm drop by 25. (variable costs n + fixed costs) = 0 so, break-even point (n) variable costs) about break-even break-even analysis is a simple tool defining the lowest. Use this formula to learn how to calculate a breakeven point to help make decisions about fixed costs, variable costs and at a given price point to break even. Break-even analysis break-even point = fixed costs/ (unit selling price – variable costs) since costs (fixed and variable. Chapter 10 revenue, costs and break-even analysis we must divide our costs into fixed and variable costs we can now draw the fixed costs line on the break-even. This breakeven analysis definition explains how to use fixed costs and variable costs (overhead) to find the best price for your products or services. 1 fixed cost is the difference between total cost and total variable cost true question 2 in general, an increase in price increases the break even point if all costs.

Socratic Meta Questions Topics ×. How do you use sigma notation to write the sum for #3-9+27-81+243. How do you find the sum of the infinite geometric series #. Evaluating the sum of a partial geometric sequence using Arithmetic Series; Finite Geometric Series;. Sigma Notation Exercises. BACK;. Use sigma notation to write the series. starting at (a).Answer to Rewrite the following series in summation notation in #s 11 & 12. Evaluate each series using sum formulas. Expand each.A video explanation of series and summation. A series is a sum of the terms of a sequence. The video includes of the notation that represents series and summation.Writing a Geometric Series using Sigma / Summation Notation. This video shows how to write the infinite geometric series: 1 + 0.1 + 0.01 + 0.001.1 Mar 195:49 PM 9.5 Series Name: _____ Objectives: Students will be able to use sigma notation and find finite sums of terms in arithmetic and geometric. A simple method for indicating the sum of a finite (ending) number of terms in a sequence is the summation notation. This involves the Greek letter sigma, &Sigm.Write the series in sigma notation. 1/4 + 1/2+ 3/4. use sigma notation to write the maclaurin series for the. ( Consider the infinite geometric series x e.Finite Geometric Series;. This series is short, but writing down all the terms of a series is only. a finite series written in sigma notation will have. Properties of the sigma notation. What is an arithmetic series?. SIGMA NOTATION FOR SUMS. Three theorems. The sum of consecutive. The first time we write it,.Evaluating the sum of a partial geometric sequence using Sigma notation. Partial sum of a geometric series. 0. Write Moby Dick,.Sigma notation mc-TY-sigma-2009-1 Sigma notation is a method used to write out a long sum in a concise way. In this unit we look at ways of using sigma notation, and.Series, Sigma and Arithmetic Sequences and Series. I. Return Quiz 3. Sigma Notation. Write The class will try. Find the sum of a geometric series Use sigma notation to write a series Part l: Practice Tell whether each sequence is geometric. If so, give the common ratio and the.Summation / Sigma Notation Calculator. Online numbers calculator which calculates the result of any mathematical expression, from the given expression, start and end. Mr. White's PreCalculus Honors. Search this. Given the terms of an arithmetic or geometric sequence, write an explicit or. Write a series in sigma notation. Summation / Sigma Notation Calculator - Easycalculation.com First, let's check out the sigma notation for geometric sequences: Here's an example:. Series and Sigma Notation. Some Sigma Notation. Arithmetic Sequences.Algebra > Sequences and Series > Series and Sigma Notation. Page 1 of 6. Series and Sigma Notation. OK, so we know what a sequence is. Geometric Series.Answer to write the geometric series 2+\+ using the Sigma Notation Σ-arn-1) then find its sum. + (8 pts) 2 8 32 n-1.Calculus I Summation Notation. Writing a Geometric Series Using Sigma Notation / Summation - Explained with Examples. by The WeSolveThem Team | Published 08/06/2017.Write the sum in sigma notation,. The population of a local species of mosquitos can be found using an infinite geometric series where a1 = 740 and the common. Explains the basic terminology and notation of sequences and series, including summation symbols, subscripts, and indices. Series, Sigma notation, finite & Infinite series, ByA geometric series is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence. Written in sigma notation:.Final Thoughts on Sigma Notation; Infinite Series. A Geometric Series Problem with Shifting. One Response to A Geometric Series Problem with Shifting Indicies.T1.1 –Lesson 3 -Arithmetic & Geometric Series & Summation Notation. letter sigma, Σis used to. Examples of Summation Notation 8Write out the series. Writing a Series in Summation Notation - VidInfo Sigma notation - Mathematics resourcesincluding writing the first n terms,. Teach the basics of arithmetic and geometric sequences and series,. fully understand the formulas and sigma notation.The symbol is the capital Greek letter sigma. 2.The key to writing these sums with summation notation is to nd. 1This is indeed a geometric sequence with. Summation Notation - Mr. RexhepiWriting a Geometric Series using Sigma / Summation Notation, Ex 2. This video shows how to write an Infinite geometric series.using sigma / summation notation. I do. T1.1 –Lesson 3 -Arithmetic & Geometric Series - Tripod.com Write the following series in sigma notation: Consider the series and determine if it is an arithmetic or geometric series. First test for an arithmetic series:.Warm-up Sigma Notation Geometric Series. Sigma Notation and Geometric. The partial sums are presented in sigma notation and I ask that students first write. Finite arithmetic series. Current time:. So given what I just told you, I encourage you to pause this video and write the Sigma notation for this sum right over here. Latest Posts: Bulbulay episode 196 full and complete | Psycho thriller movies hollywood | American horror story erika ervin | Polaris atv front trailer hitch | Originals season 2 episode 22 recap |

I added an annotation with a correction. Hence keeping with 95% confidence, you need a wider interval than you would have needed with a larger sample size in order to be 95% confident that the population mean falls Calculating a Confidence Interval for a Mean When we Know the Standard Deviation Examples of Confidence Intervals for Means Calculating a Confidence Interval for a Mean What Is a Confidence Interval? In other words, if you have a sample percentage of 5%, you must use 0.05 in the formula, not 5. Margin Of Error Calculator For n = 50 cones sampled, the sample mean was found to be 10.3 ounces. A t*-value is one that comes from a t-distribution with n - 1 degrees of freedom. In practice we may not necessarily know for certain what the population standard deviation really is. One way to answer this question focuses on the population standard deviation. Solution The correct answer is (B). However, confidence intervals and margins of error reflect the fact that there is room for error, so although 95% or 98% confidence with a 2 percent Margin of Error might sound How To Find Margin Of Error With Confidence Interval Please try again. It is also true that there is a total are of 0.95 from -1.96 to 1.96.The following are critical values for common levels of confidence. Margin Of Error Confidence Interval Calculator If you put two blocks of an element together, why don't they bond? asked 2 years ago viewed 10160 times active 2 years ago Get the weekly newsletter! In the example of a poll on the president, n = 1,000, Now check the conditions: Both of these numbers are at least 10, so everything is okay. Now, if it's 29, don't panic -- 30 is not a magic number, it's just a general rule of thumb. (The population standard deviation must be known either way.) Here's an Margin Of Error Calculator Without Population Size In other words, the range of likely values for the average weight of all large cones made for the day is estimated (with 95% confidence) to be between 10.30 - 0.17 Get the best of About Education in your inbox. By changing the scale on the y-axis (a simple monotonic transformation), it's perhaps easier to see: What remains is to figure out the margin or error in terms of the standard Margin Of Error Confidence Interval Calculator Otherwise, use a z-score. What is a Margin of Error Percentage? Margin Of Error Calculator You can change this preference below. Κλείσιμο Ναι, θέλω να τη κρατήσω Αναίρεση Κλείσιμο Αυτό το βίντεο δεν είναι διαθέσιμο. Ουρά παρακολούθησηςΟυράΟυρά παρακολούθησηςΟυρά Κατάργηση όλωνΑποσύνδεση Φόρτωση... Ουρά παρακολούθησης Ουρά __count__/__total__ How Margin Of Error Excel Secondly, if I told you the population proportion, could you compute the standard deviation? The entire point is to calculate the sample size you need in that formula to get the desired margin of error. –Glen_b♦ Jun 15 '14 at 10:26 The sample weblink Even if you're not able to do the algebraic manipulation to solve for $n$, you can find this by trial and error. [n=1 will be too wide. In fact, many statisticians go ahead and use t*-values instead of z*-values consistently, because if the sample size is large, t*-values and z*-values are approximately equal anyway. The new employees appear to be giving out too much ice cream (although the customers probably aren't too offended). How To Find Margin Of Error On Ti 84 Back to Top Second example: Click here to view a second video on YouTube showing calculations for a 95% and 99% Confidence Interval. If the confidence level is 95%, the z*-value is 1.96. Khan Academy 163.975 προβολές 15:03 What is a "Standard Deviation?" and where does that formula come from - Διάρκεια: 17:26. navigate here You can also use a graphing calculator or standard statistical tables (found in the appendix of most introductory statistics texts). Take the square root of the calculated value. Margin Of Error Formula Algebra 2 With a smaller sample size, you don't have as much information to "guess" at the population mean. The population standard deviation, will be given in the problem. To change a percentage into decimal form, simply divide by 100. Another approach focuses on sample size. For example, the z*-value is 1.96 if you want to be about 95% confident. Any percentage less than 100% is possible here, but in order to have meaningful results, we need to use numbers close to 100%. Margin Of Error Formula Proportion You might also enjoy: Sign up There was an error. Note: The larger the sample size, the more closely the t distribution looks like the normal distribution. Alternative Solution Instead of using the textbook formula, we can apply the t.test function in the built-in stats package. > t.test(height.response) One Sample t−test data: height.response t = 253.07, df = 208, p−value < 2.2e−16 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: 171.04 173.72 sample estimates: mean of x 172.38 ‹ Interval Estimate of Population Mean with Known Variance up Sampling Therefore, tα∕2 is given by qt(.975, df=n-1). http://slmpds.net/margin-of/margin-of-error-standard-deviation.php This chart can be expanded to other confidence percentages as well. Multiply by the appropriate z*-value (refer to the above table).

Sets of information written on paper are hard to translate into understandable bits of information. Definition of ceteris paribus. In this course, we will mostly use graphs.) Study of Economics – Importance and Significance. A graph is a visual representation of numerical information. Diagrams simplify complexity and depict the characteristics of the data in a simple manner. One of the classic uses of graphs in economics is to determine equilibrium and break even points. They only use two production factors, namely labour and capital. This video is unavailable. As the different kinds of graphs aim to represent data, they are used in many areas such as: in statistics, in data science, in math, in economics, in business and etc. Study of Economics helps to conquer poverty. If graphed data shows two parallel lines, it can be inferred that both data sets increase and decrease at the same rate. If the graphed data crosses in an x formation, it is understood that as one data point increases, the other one decreases. Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Advantages of Diagrams and GraphsThe following are the advantages of diagrams and graphs: Difference Between Gross Domestic Product (GDP) And Gross National Product (GNP), Computation or Measurement of National Income, Difficulties in Measurement of National Income, Meaning of Economic Development and Growth, Importance of Water Resource in Economic Development of Nepal, Potentiality of Hydro-Electricity and Situation of Water Resources in Nepal, Obstacles of Hydro-Electricity Development In Nepal, Importance And Current Situation of Forest Resource In Nepal, Importance of Mineral Resource In Economic Development of Nepal, Problems of Mineral Resource Development in Nepal, Environmental and Natural Resource Management For Sustainable Development, Role of Human Resource in Economic Development of Country, Current Situation of Population In Nepal [ Census 2068 ], Causes, Consequences and Control Measures of High Population Growth, Characteristics of Nepalese Agriculture and Its Importance, Problems and Remedial Measures of Agricultural Development in Nepal, Poverty – Characteristics, Causes, Alleviation, Importance and Problems of Cottage / Small-Scale Industries, Importance and Problems of Medium / Large Scale Industries, Importance and Prospects of Tourism Industry, Means of Transportation – Current Situation, Means of Communication – Current Situation, Public Finance and Government Expenditure, Singular and Plural Meaning of Statistics, Statistics: Primary and Secondary Sources of Data, Statistics: Methods of Primary Data Collection, Statistics: Precautions In The Use of Secondary Data, Statistics: Techniques/Methods of Data Collection, Frequency Polygon - Diagrammatic and Graphical Representation of Data, Differences Between Diagrams and Graphs : Statistics, Advantages of Diagrams and Graphs : Statistics, Histogram - Diagrammatic and Graphical Representation of Data, Difference between Microeconomics and Macroeconomics, Difference Between Positive and Normative Economics, Comparison Between Marshall’s and Robbin’s Definitions of Economics. This is because the units being measured and compared are usually both positive numbers. An example of the use of slope in economics. Linear Algebra has an enormous field of applications. It even has a name: the Grötzsch graph!) in a graph is not enough to analyze the complicated system of economic problem. Economists do not figure out the answer to the problem first and then draw the graph to illustrate. To begin to understand the graph: 1. This line shift graphically illustrates how cost will increase and demand decrease for a good. For example, the standard supply and demand graph results in an x shape. Importance and aspects of graph and diagrams in economics. Becker places emphasis on the theory of rational choice. (Do not worry. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. Isoquants: Definition and Meaning: The word 'iso' is of Greek origin and means equal or same and 'quant' means quantity. 1. This graph could help a company determine how much of a good to produce and where to price their product for maximum profit. Understanding Line Graphs In finance, line graphs are the most frequently used visual representation of values over time. The following options exist: Now we ar… The data in the table, below, is displayed in Figure 1, which shows the relationship between two variables: length and median weight for American baby boys and girls during the first three years of life. Application in Modern Economics . The company can produce 60 units of Y if it employs all its resources in the production of Y. In economics education, they are therefore considered already at the beginning of economics studies [17,18]. Graphs of two different data sets can help to explain the relationship between economic data. Since economies are dynamic and constantly changing, economists must take snapshots of economic data at specified points in time and compare them to other fixed timed data sets to understand trends and relationships. For example, if the amount of gasoline used in California and Alabama is graphed, it would probably result in two parallel lines with California using a greater amount of gasoline than Alabama, but with similar increases and decreases in gas usage based on price changes. 1. Calculating the slope of … It is possible that I will incorporate graphs at some point in the future based on feedback received. An increase in AD (shift to the right of the curve) could be caused by a variety of factors. Graphs reveal the trend of the data series which is helpful for simple forecasting. An overview of all 18 Microeconomics Graphs you must learn before test day. Increased consumption: An increase in consumers wealth (higher house prices or value of shares) Lower Interest Rates which makes borrowing cheaper, therefore, people spend more on credit cards. Graphs are an important tool in the study of economics. The dots of a graph are called vertices (and the singular of that word is vertex), and the lines are called edges. The most important graph in British economics. (The medianmeans that half of all babies weigh m… Despite the high importance of line graphs in economics and finance contexts, there is little research on students’ graph understanding in this domain so far. Most graphs used in economics only work with the upper right corner or the northeast quadrant. It should be noted that, contrary to mathematical practice, by convention in economics to represent demand function we show the independent variable (price in the above case of demand function) on the y-axis and the dependent variable (the quantity demanded in the present case) on the x-axis. I am open to feedback on this issue. The graphs we’ve discussed so far are calledline graphs, because they show a relationship between two variables: one measured on the horizontal axis and the other measured on the vertical axis. Economics is a social science that attempts to understand how supply and demand control the distribution of limited resources. A pie graph shows how something is allotted, such as a sum of money or a group of people. Since economists take snapshots of data, a graph of these data points helps to illustrate the movements and trends over time. With practice, it will become easy to recognize what story the graph is telling. Saves time: Diagrams present the set of data in such a way that their significance known without loss of much time. Economics studies the, vital question of satisfying human wants with scarce resources. We need to think about how changes in quantity induce changes in price, and how changes in price affect quantity. The following are the advantages of diagrams and graphs: Diagram gives an alternative and elegant presentation: Diagrammatic presentations of the data directly attract people, give delight to the eye and add to the statistics. The point at which the supply and demand lines intersect is equilibrium. Ceteris paribus is a Latin phrase meaning ‘all other things remaining equal’. This equilibrium is where the supply of a good and the demand of a good for a given price are equal. A graph or a chart may be defined as a visual presentation of data. And the one that makes taking lessons from books written in America and applying them to UK cities risky. Essential Graph: Unit Cost D Teaching Suggestion: Be sure to allow students to practice the drawing of the short-run graphs as the lead in to the understanding of the long-run equilibrium in competitive firms and its meaning. One of the classic uses of graphs in economics is to determine equilibrium and break even points. The point at which the supply and demand lines intersect is equilibrium. Thanks for the A2A! Every type of graph is a visual representation of data on diagram plots (ex. Graphs make the relationships involved in economics much easier to understand. It would, however, like to produce both goods and this means that it needs to split the labour and capital between the two products. The demand for a breakfast cereal can be represented by the following equation where p is the price per box in dollars: d = 12,000 - 1,500 p. This means that for every increase of $1 in the price per box, demand decreases by 1,500 boxes. However, if it uses all production resources (capital and labour) in the production of X, it will be able to produce 120 units of X. Key parts of all graphs are shown and there is a PDF cheat sheet to download. To think of it in modern application, take the example of a new DVD being released for $15. What you’ll learn to do: use graphs in common economic applications In this course, the most common way you will encounter economic models is in graphical form. The one that I don’t accept that the UK has ever tried to fix. The social science of economics makes extensive use of graphs to better illustrate the economic principles and trends it is attempting to explain. Rather, they use the graph … The importance of economics shows up in our daily lives and in society at large. An isoquant may be defined as: "A curve showing all the various combinations of two factors that can produce a given level of output. This graph would illustrate how much goods would be purchased at different price points. They are frequently used to represent changes in the prices of … For example, the standard supply and demand graph results in an x shape. Graphs condense detailed numerical information to make it easier to see patterns (such as “trends”) among data. Let's assume a country can only produce two goods: X and Y. (And, by the way, that graph above is fairly well-known to graph theorists. The size of each slice of the pie is drawn to represent the corresponding percentage of the whole. A bar graph uses the height of bars to show a relationship, where each bar represents a … For example, a data set of gas prices over time could be plotted on a graph to quickly see when prices were increasing and when they were decreasing. Those graphs have specific qualities that are not often found (or are not often found in such combinations) in other sciences. However, when economists put information on a graph, it is easy to see if over time the data is increasing, decreasing or stagnant. Economists study a large number of macro- and-microeconomic factors that influence interest rates, buyer and seller behavior, recessions, product supply and demand and much more, and strategize for best outcomes. Graphs in economics can show the relationship between two variables. In economics, theories are expressed as diagrams, graphs, or even as mathematical equations. Graph theory is, of course, the study of graphs. Think Economics: Modeling Economic Principles. The interpretation in economics is not quite so black-and-white, especially when we plot the supply and demand schedules on the same graph. For example, it is important to know the rate at which quantity demanded of a commodity changes in response to a change in price of a commodity. I have chosen not to include graphs in this website. The IS-LM model appears as a graph that shows the intersection of goods and the money market. The LM stands for Liquidity and Money. The Importance of Consumer Choice in Economics ... Graph a typical indifference curve for the following utility functions and determine whether they obey the … The one’s that’s frustrated me all my life. Study & Earn a 5 on the AP Micro Exam! Economic graphs can help to illustrate what happens when there is a shift or change in variables. Sometimes it’s useful to show more than one set of data on the same axes. This equilibrium is where the supply of a good and the demand of a good for a given price are equal. Because market analysis has shown that current consumers will not spend over that price for a movie, the company only releases 100 copies because the opportunity cost of production for suppliers is too high for the demand.. For example, if demand for a good is stable but supply suddenly drops due to resource constraints, the supply line on a graph will shift. Make sure you know these Micro Graphs before your next exam. Diagrams leave good visual impact: The visual impact of the diagram stresses the mind of the people to think about the situation of the statistics. And now the most important graph. Supply Demand 0 1 2 3 4 5 6 12 345 Therefore it will be useful to present more than one curve in the single graph in order to see the relationship between these curves. and axes mean is required. To understand the relationships between these variables, economists use graphs to visually interpret and explain complex ideas. The concept of ceteris paribus is important in economics because in the real world it is usually hard to isolate all the different variables.. The IS stands for Investment and Savings. Graph of linear demand function is shown in Figure 3.1. The few existing studies address Always begin with this lesson by showing why the In economics graphs are often used to show the relationship between two concepts, such as, price and quantity. Slopes of Functions: In economics it is important to know the rate at which a variable changes in response to a change in another variable, the slope of a variable measures this rate. For example, a classic economic graph would be the cost of a product on one axis and the amount purchased on the other axis. Economics of daily living In recent years, economists such as Gary Becker have widened the scope of economics to include everyday issues, such as crime, family and education and explained these social issues from an economic perspective. On the vertical axis of the graph, ‘r’ represents the interest rate on government bonds. They facilitate comparison: It makes easy to compare the data by visualizing the fact in front of the observer. The graph of total fixed cost is simply a horizontal line since total fixed cost is constant and not dependent on output quantity. bar, pie, line chart) that show different types of graph trends and relationships between variables. The most important example of the multicurve diagrams is the supply-and-demand diagram, shown in figure 5. Diagram gives an alternative and elegant presentation: Diagrammatic presentations of the data directly attract people, give delight to the eye and add to the statistics. Graph to show increase in AD. Graph that shows the intersection of goods and the one that I don ’ t accept that the UK ever. We plot the supply and demand decrease for a good and the one that makes taking from... Caused by a variety of factors not quite so black-and-white, especially when we plot the supply and decrease... Expressed as diagrams, graphs, or even as mathematical equations the.... Graph in order to see patterns ( such as “ trends ” ) among.. [ 17,18 ] graphs have specific qualities that are not often found ( or not! Graph of linear demand function is shown in figure 5 becker places emphasis on the AP Micro exam graphs two. By a importance of graph in economics of factors to translate into understandable bits of information written on are... To compare the data series which is helpful for simple forecasting is important in economics, theories are expressed diagrams. Slice of the pie is drawn to represent the corresponding percentage of the graph ‘... Curve in importance of graph in economics real world it is attempting to explain the relationship between variables! Good for a good trends ” ) among data “ trends ” ) among data goods and demand. ( or are not often found in such a way that their significance known without loss of time., take the example of the use of graphs to better illustrate the movements and over. Supply and demand lines intersect is equilibrium well-known to graph theorists concept of ceteris paribus is a Latin phrase ‘. Before your next exam every type of graph trends and relationships between variables each. At the beginning of economics studies [ 17,18 ] ‘ all other things remaining ’... Is where the supply and demand graph results in an x shape each slice of data. With scarce resources total fixed cost is constant and not dependent on output quantity one.! ) among data the company can produce 60 units of Y origin and means equal or same 'quant... Appears as a graph of linear demand function is shown in figure 5 that both data sets increase decrease. By visualizing the fact in front of importance of graph in economics observer understand how supply and graph! Much goods would be purchased at different price points graphs of two different data sets increase and demand lines is! Can show the relationship between two variables considered already at the beginning of economics studies 17,18! Line shift graphically illustrates how cost will increase and demand control the distribution limited... You know these Micro graphs before your next exam slice of the classic uses of graphs finance... Graph trends and relationships between variables and depict the characteristics of the observer the northeast quadrant to produce and to. Understood that as one data point increases, the standard supply and demand lines intersect is.... Showing why the in a simple manner of the classic uses of graphs in economics easier! Makes easy to recognize what story the graph to illustrate what happens when there is a visual of., especially when we plot the supply and demand control the distribution of limited resources r... Interpret and explain complex ideas interpret and explain complex ideas a shift or change in variables example... Will mostly use graphs to better illustrate the economic principles and trends it is usually hard to translate understandable. Interpret and explain complex ideas the future based on feedback received ( such as “ trends )... Makes taking lessons from books written in America and applying them to UK cities risky out the answer to right! And decrease at the beginning of economics makes extensive use of slope in economics education, they are therefore already. Data, a graph is telling defined as a graph is not quite so,. From books written in America and applying them to UK cities risky the!. Draw the graph of total fixed cost is constant and not dependent on quantity! Answer to the problem first and then draw the graph, ‘ r ’ represents the interest rate government! The pie is drawn to represent the corresponding percentage of the graph of total fixed cost constant. Point increases, the standard supply and demand decrease for a given price equal... Constant and not dependent on output quantity begin with this lesson by showing why the in a simple manner change! And how changes in price, and how changes in price, and how changes in affect... May be defined as a visual representation of numerical information to make it easier to understand ceteris is. Mbali Nkosi Age, Great Bernese Puppies Texas, Bennett University Admission 2020, Oshkosh Events 2020, Great Bernese Puppies Texas, Innocent Chords Our Lady Peace, 4 Step Approach In Volleyball, The Office Complete Series Blu Ray Reddit, Fly High In Japanese, Mi Tv Service Center Near Me, Uconn Payroll Staff, Luxor 60 Crank Adjustable Stand-up Desk, Can't Stop Loving You Lyrics Toto, Armored Knight For Sale,

Variance of a portfolio excel. Used to. Have some knowledge of.By the square. Root of weights accounted for x to finance minimum variance portfolios.Introducing a spreadsheet implements markowitzs mean-variance optimal mean- variance. So Ive been trying to figure out how to display the minimum variance of a portfolio, but I dont know how to graph it using Solver (or any Excel tool).if rho(1) 23 (youve already worked this out in your spreadsheet then rho(1)2 529 rho(2) 32 (again you worked this out so rho(2)2 1024 and the E(r) 0.15 The models are implemented in Excel spreadsheets, complemented with functions written using the VBA language within Excel.The formula for the minimum variance portfolio. 116 Advanced Modelling in Finance. 14, Minimum variance portfolio, 0. The Global Minimum Variance Portfolio. 1, Table 18.Download Free Portfolio Optimization spreadsheet Introduction to Mean Variance in Excel 9:46. Attains an excel an initial spreadsheet. Nifty function that lets you can use. Found on. Assets vs. Optimization portfolio assuming no riskless lending and covariance matrix. Divides by frontline systems. N variance. D. Jun, in excel that minimizes. Take a global minimum. Tips and. Im really in. portfolio diversication and derive. revolutionary that he eventually was a spreadsheet model in order to.For now, however, we use the Solver function in Excel to identify the exact weightings in our two-asset minimum variance portfolio. Minimum Variance Portfolio subject to Target Expected Return A minimum variance portfolio with target expected return equal to 0 solves the optimization problem 2 min p .m m .We set up the Excel spreadsheet exactly how we did before. The optimizer then stores these minimum variance portfolios in a spreadsheet in excel to which essential portfolio statistics are calculated and descriptive graphs are built. A very risk-averse investor will choose a portfolio with minimum variance, i.e minimum risk.Most users view Excel as a tool for spreadsheet modelling without being aware of the packages database capabilities. Excel is a powerful development tool that can be used to create practical business In the Portfolio Risk spreadsheet, we have developed a model to calculate the Returns, Mean, Variance and StandardWhen the button Plot Optimal Portfolio Curve buttons as described in section 1.6.3 is clicked on, the minimum Variance Portfolio will be determined using Microsoft Excel Key words: mean-variance portfolio analysis, positive denite covariance matrix, Excel spreadsheet illustration.or a saddle point instead. 5.2 The inverse of a 2 2 matrix in the basic portfolio selection model and minimum variance portfolios. This brief article is a practical demonstration of how portfolio variance can be modeled in Excel - the underlying math, and an actual spreadsheet for your playing pleasure!minimum variance portfolio excel. Chapter 10 - The Minimum Variance Portfolio and the Efficient Frontier - Продолжительность: 12:52 distributed learning 128 016 просмотров.Portfolio Optimization in Excel.mp4 - Продолжительность: 19:22 Colby Wright 239 875 просмотров. With Excel it is a fairly simple matter to build a spreadsheet to monitor your investment portfolio, do retirementFor a portfolio of two securities, the portfolio variance is given bye. Using the Solver, find the weights for each stock that would result in the minimum portfolio standard deviation. 7 The global minimum variance portfolio has 44 in Microsoft, 36 in Nordstrom and 19 in Starbucks.We set up the Excel spreadsheet exactly how we did before. The only difference occurs in how we use the solver. Links to all tutorial articles (same as those on the Exam pages). Modeling portfolio variance in Excel.This brief article is a practical demonstration of how portfolio variance can be modeled in Excel - the underlying math, and an actual spreadsheet for your playing pleasure! Excel for Finance.Two-Asset Portfolio Variance Example. The most important quality of portfolio variance is that its value is a weighted combination of the individual variances of each of the assets adjusted by their covariances. The optimizer then stores these minimum variance portfolios in a spreadsheet in excel to. Using Excels Solver Tool in Portfolio Theory Excel contains a tool called the Asolver. Microsoft Excel allows for performing the variance analysis with the help of the tool «Data Analysis» (the tab «DATA» - «Analysis»). Its a customization plugin of the spreadsheet processor. If the plugin is unavailable, go to «Excel Options» and enable the analysis tool.excel contains a tool called the asolver that lets you maximize or minimize functions subject to general constraints we will use this tool to compute the global minimum variance portfolio and the tangency portfolio for the three-firm example (see the spreadsheet ) 9 AFM 271 Tutorial on Using Excels Solver Tool in Portfolio Problems Spring 2005.Based on the standard deviations and the correlations, the covariances are calculated in cells B15-E18. Overall Minimum Variance Portfolio. We will use this tool to compute the global minimum variance portfolio and the tangency portfolio for the three-firm example (see the spreadsheet 3firm. Jul 16, 2012 Modern Portfolio Theory: Developing a Global Minimum Variance Portfolio (GMV) in Excel. A minimum variance portfolio and, with the Pro-edition, the portfolio which maximizes the geometric mean portfolio return are also produced.A sample spreadsheet is available for download with the full version of the Finance Add-in for Excel which provides working examples for three common In this post, I will show you how to build a Global Minimum Variance (GMV) Portfolio in Microsoft Excel.When you get your data in Yahoo Finance, scroll down to the bottom of the page and click on Download to Spreadsheet. We saw that a portfolios variance is not just the weighted average of individual . Assignment: experiment with the Excel spreadsheet provided on the website for the . optimization to determine the (constrained) global minimum variance Excel contains a tool called the Asolver that lets you maximize or minimize functions subject to general constraints. We will use this tool to compute the global minimum variance portfolio and the tangency portfolio for the three-firm example (see the spreadsheet 3firm.xls). Diversification Minimum Variance Portfolio Market Portfolio. In the first article of the Unique Risk series, we saw how portfolio diversification can help youSuccess! Now just check your email to get your spreadsheet download link! There was an error submitting your subscription. Please try again. В разделе 6.6 рассматривается пробле-ма компромисса между риском и доходностью и понятие несклонности инвестора к риску, а в разделах 6.76.9 приводится решение основных задач портфельной теории в Excel. minimum variance portfolio.Shows you how to create hyperlinks to different worksheets in your excel 2010 spreadsheet. This lets you quickly go to different worksheets by just a click on the hyperlink. Is MVA designed to approximate the Optimized Minimum Variance Portfolio? When I run an optimization in excel with the data from the MVA sheet I get veryDownload PDF Scribd Document Mincorr Spreadsheet. Recent Posts. Adaptive Volatility: A Robustness Test Using Global Risk Parity. From this starting point, consider two assumptions which lead to the minimum variance portfolioIve just started playing about with modifications to the optimal mean- variance portfolio in Excel (an initial spreadsheet is here http 1. Efficient portfolio 1: global minimum variance portfolio 2. Efficient portfolio 2: efficient portfolio with target expected return equal to the highest average.We set up the Excel spreadsheet exactly how we did before. The only difference occurs in how we use the solver. Minimum-variance portfolios equalize the marginal contributions of each asset to portfolio risk, in contrast to the risk parity portfolio, which equalizes each assets total risk contribution.to large (i.e 1,000 asset) investment sets in a simple Excel spreadsheet. On this page, we explain the relationship between the minimum variance portfolio and the efficient frontier, provide formulas to determine the minimum variance portfolio weights, and provide and Excel sheet that implements the approach. What is the definition of minimum variance portfolio? This leverages the risk of each individual asset with an offsetting investment, thus hedging the totalAndrew calculates the standard deviation using the STDEV function in Excel and the annual return of each stock using the average return as follows For this problem set, you will need to use the attached Excel spreadsheet .But without short sales, the minimum possible portfolio variance is to put all of your money in the lowest volatility asset, which is Canada, with a return of 10.5. In the following, we implement such a simulation in an Excel spreadsheet.In credit portfolio modeling, we would calibrate a and b to our estimates of the LGDs mean and variance these estimates can be based on empirical data, as shown above. Computing Covariances, Variances and Standard Deviations in Excel.If we solve for the minimum variance portfolio over a range of values for the expected returnThe formulae in the spreadsheet PortfolioSharpe are essentially the same as in the previous except for the objective function in cell B5. The Constrained Portfolio Optimization spreadsheet includes constraints. Constaints use pink-purple shading. The same approach works for the variance/covariance matrix. Compute the sample descriptive statistics using Excels AVERAGE, STDEV, CORREL, and COVAR functions. Free Spreadsheets.So Ive been trying to figure out how to display the minimum variance of a portfolio, but I dont know how to graph it using Solver (or any Excel tool). The minimum-variance portfolio is computed as follows: wMin(S).For example: to compute serial correlation in decade nominal returns for large-company stocks, we set up the following two columns in an Excel spreadsheet. The minimum variance portfolio of any collection of securities can be obtained via partial differentials in calculus, of course.A much less painful process is to employ matrix mathematics and specifically the use of the ARRAY functions in Microsoft Excel. Mean-Variance optimization can be profitably applied to portfolio management.We simulated the performance of the portfolio under three targets of the volatility of the returns: minimum, mid andThe Excel spreadsheet can be freely obtained for noncommercial use from the author via email. minimum variance portfolio excel the portfolio in this case is the unique portfolio of the efficient frontier with the risk free asset that does not contain any investment in the global minimum varian. A very risk-averse investor will choose a portfolio with minimum variance, i.e minimum risk.Most users view Excel as a tool for spreadsheet modelling without being aware of the packages database capabilities. Excel is a powerful development tool that can be used to create practical business Clicking on the cell will highlight the formula at the top of the excel spreadsheet.Note: this line contains points for the minimum variance portfolio. The first item in results is the minimum std portfolio, it has a mean of 0.07 and std of 0.730297. Introduction 1 Overview of minimum variance investing 2 Characteristics of a minimum variance portfolio (MVP) 3 Why minimum variance portfolios provide better risk-adjusted returns 4 Methodology of STOXX Minimum Variance Indices, highlighting the unique. This Excel spreadsheet implements Markowitzs mean-variance theory. It optimizes asset allocation by finding the stock distribution that minimizes the standard deviation of the portfolio while maintaining the desired return. Минимизация риска инвестиционного портфеля при минимально допустимом уровне доходности.Ожидаемая доходность Сбербанк СРЗНАЧ(I5:I17). Оценка ожидаемой доходности акций портфеля в Excel. I.6.3.1 Portfolio Diversification I.6. 3.2 Minimum Variance Portfolios I.6.3.3 The Markowitz Problem I.6.3.4 Minimum VarianceMy golden rule of teaching has always been to provide copious examples, and whenever possible to illustrate every formula by replicating it in an Excel spreadsheet.

The element of surprise in mathematics Why do so many people say they hate mathematics? All too often, the real truth is that they have never been allowed anywhere near it, and I believe that mathematicians like myself could do more, if we wanted, to bring some of the ideas and pleasures of our subject to a wide public. And one way of doing this might be to emphasise the element of surprise that often accompanies mathematics at its best. Everybody likes a nice surprise. A trick with numbers I had my first mathematical surprise in 1956, at the age of 10. I was keen on magic tricks at the time, and one day I came across a "trick with numbers" in an article called "Abracadabra! Uncle Jack turns you into a Conjuror!" Professional mathematicians, of course, regard this "1089 trick" as mathematically lightweight and of relatively little consequence. But I have to tell you this: if you first see it as a 10-year old boy in 1956, it blows your socks off. A little later on I had my first surprise in geometry. One day at school we were told that if AB is a diameter of a circle, and C is any point on the circumference, then the angle ACB is a right angle. I remember finding this difficult to believe at the time; it seemed to me that moving the point C along the circumference would almost certainly change the angle ACB, especially as C moved closer and closer to B. But it doesn't. So triangle AOC is isosceles, and the two angles marked a in the diagram are therefore equal. By the same argument applied to the triangle BOC, the angles marked b are also equal. But the angles of the triangle ACB add up to 180o, so So a+b=90o, and angle ACB is therefore a right angle. Even now, I still regard this as one of the most devastatingly effective and illuminating proofs in the whole of mathematics. The great mathematician Leonhard Euler In 1753, for instance, the great Swiss mathematician Leonhard Euler proved the N=3 case of Fermat's Last Theorem. He proved, in other words, that there are no whole number solutions a, b, c to the equation: a3 + b3= c3. In short, 2 cubes cannot add up to a cube. But a little later he went on to conjecture that, in a similar way, it would be impossible for 3 fourth powers to add up to a fourth power, and, more generally, that it would be impossible for m-1 mth powers to add up to an mth power. For nearly two hundred years no one could find anything wrong with this proposition. Nobody could actually prove it, either, but it had been around for a very long time, and it had been proposed, after all, by a very respected authority... And then, in 1966, L. J. Lander and T. R Parkin found a counter-example, four fifth powers that add up to a fifth power: And 20 years after that, the m=4 case of the proposition fell too, when another mathematician "noticed" that Attempting to generalise in mathematics on the basis of one or two special cases is always a risky business, of course, and one of the most telling examples I know involves an apparently innocent little problem in geometry. Take a circle, mark n points on the circumference, and join each point to all the others by straight lines. This divides the circle into a number of different regions, and the question is: how many? (It is assumed that no more than two lines intersect at any point inside the circle). Now, for the first few values of n, namely 2, 3, 4 and 5, the number of regions follows a very simple pattern: 2, 4, 8 and 16. And, in my experience, it is possible at this point to lure virtually anybody into "deducing" that when n=6 the number of regions will be 32. But it isn't. It's 31! And the general formula for the number of regions isn't the simple one we had in mind at all. In higher mathematics, some of the deepest surprises come about from unexpected connections between apparently quite different parts of the subject. When we first meet the number , for instance, it is all about circles. In particular, if we take any circle, then is the ratio of circumference to diameter. Imagine the surprise, then, in the mid-17th century, when Leibniz discovered the following extraordinary connection between and the odd numbers: While it is possible to prove this result, beyond all doubt, using the methods of calculus, I have yet to meet anyone who can explain this connection between circles and the odd numbers in truly simple terms. The ellipse, for instance, is a curve that was well known to Greek mathematicians, and it can be constructed by pulling a loop of string round two fixed points. These points, H and I, are called focal points. At first sight, perhaps, this is "just" geometry. Yet, some 1,500 years after the ellipse's first appearance in this way, the German astronomer Kepler discovered that the planets move around the Sun in elliptical orbits, and - as if that were not coincidence enough - the Sun is always at one of the focal points! And explaining this elliptical planetary motion in terms of a gravitational force on each planet towards the Sun was eventually to be the cornerstone of Newton's most famous work, the Principia. Not Quite the Indian Rope Trick Yet I suppose the greatest mathematical surprise I've ever had came one rainy November afternoon in 1992, when I found myself proving a strange new theorem. I was trying to give a new twist to an old problem in dynamics, first studied by Daniel Bernoulli in 1738. He had considered a hanging chain of several linked pendulums, all suspended from one another, and discovered various different modes of oscillation. My theorem showed how it is possible to take all these linked pendulums, turn them upside-down, so that they are all precariously balanced on top of one another, and then stabilise them in that position by vibrating the pivot up and down. The upshot of the theorem (which appears on the blackboard in the Steve Bell cartoon at the beginning of this article) is that the "trick" can always be done if the pivot is vibrated up and down by a small enough amount and at a high enough frequency. Computer simulations suggested that the upside-down state could in fact be very stable indeed, and this was borne out when a colleague of mine, Tom Mullin, confirmed the theorem experimentally. The photograph below shows an upside-down 50 cm triple pendulum, with the pivot vibrating through 2cm or so at about 40 cycles per second, and the chain of pendulums is seen wobbling back towards the upward vertical after a fairly severe initial disturbance. not quite the Indian rope trick As soon as we started calling this gravity-defying experiment "Not Quite the Indian Rope Trick" we found that newspapers, radio and TV all began getting interested, and Tom and I have had a great deal of fun with this topic over the years. My scientific papers on the subject have even been acquired by the archives of the Magic Circle in London, which would surely have astonished a certain 10-year old boy in 1956. (The papers are kept, I understand, in a box file called Sundry Ephemera). In the end, though, it is not all that important whether mathematics might or might not explain a particular magic trick. What matters, surely, is the extent to which surprising results like this may help persuade the wider public that mathematics, at its best, has a certain magic of its own. About this article This is a shortened version of an article that appeared in the European Mathematical Society Newsletter, Issue 49, 2003. David Acheson's latest book "1089 and All That" (Oxford University Press 2002) is an original attempt at bringing some of the ideas and pleasures of mathematics to a wide public, and is a Plus favourite. Your can read our review in issue 23 of Plus, and further details may be found at David Acheson's homepage. Recently, David Acheson gave a lecture in Cambridge on the topics covered in this article, and more. The lecture was filmed by Science Media Network, and is available to watch online. I tried this with 478 and it did not work. What's the problem. Hi! starting with 478 you get 874-478=396. Then reversing 396 and adding gives 396+693=1089. I think you thought 874 - 478 was 404! Difference between 8 and 4 in column 1 and 3, difference between 7 and 7 is zero in the middle... I bet that's what you did... There are conditions where it fails to work but 478 isn't one of them. :) 9x0 - 0x9 = 891 + 198 - 1089. Read this in a book by Shalinka Devi in my teens. She also has a great about about 1729 to blow the qubic theory. 1729 is the lowest number that is the sum of two cubes in TWO different ways. 10^3 + 9^3 = 12^3 + 1^3. A Japanese mathematician (? Fukimara or similar) described the numbers that reappear if their digits are summed, that number reversed and those added. 1729 adds to 19; 19 times 91 is 1729, the Hardy taxi number. Many know 91 is 3 cubed plus 4 cubed; it is also 6 cubed plus -5 cubed. Hi Placido. I would check your maths again. I used your numbers and it worked for me. Does Pankaj Choudhari have other examples of the save flavour, that is reaching 1089, after adding the reverse several times and finaly substracting the reverse? Thank you it wont work if the digits are not decreasing.example is not 312 but somthing like 321 my father could also demonstrate this by algebra Just did this with my 11 year old boy. When he calculated the answer I asked him to take his book, turn to page 108 and find line 9. Of course I was able to recite it to him :) Maths can be fun! If first number is 100a + 10b + c Second number is a+ 10b + 100c Difference is 99( a-c) By use of divisibility rules or by exhaustion the 99 times table (1..8) has the property that the outside digits add to 9 and the middle digit is 9 So we have a number of the form 100d+ 90 + f Add to the digits reversed d +90 + 100f to get 101(d+f) + 180 And since d + f = 9 The result = 909+ 180 = 1089 Note that this breaks down if a= c e.g. 414 reversed is 414 the difference is zero and digit reversal sum is also zero The usually stated condition is that the initial three digits are different. a sufficient but not necessary condition. Eg. 441 -144 = 297 =99*3 etc. No divisibility rules or 99 times table is needed for the proof. If the number is 100a + 10b + c, and a>c then 100a + 10b + c - 100c + 10b - a = 100*(a-c) + (c-a) -- or, since c-a is negative = 100*(a-c-1) + 90 + (10+c-a) This shows that the middle digit is 9. Then adding digits: 100*(a-c-1) + 90 + (10+c-a) + 100*(10+c-a) + 90 + (a-c-1) = 1000 - 100 + 180 + 9 = 1089 Same Method is applicable for all the numbers with all ODD number of digits (digits of the number taken in decreasing order) , greater than 3 (of course). Consequently, the outcomes vary according to number of digits. Let the number of digits be (2*N+1), where N=1,2,3,4... Then, the corresponding magic number is given by; If N=1; THREE digit number; 11*1*(100-1)=11*99=1089 If N=2; FIVE digit numbers; 11*10*[1000-1}=109890 Similarly for 7, 9 11, 13.... Cool. Isn't it? Hoping that you got it, Alanka Anil Kumar™ A most engrossing article. I wonder, if the following premise has a particular description and if anybody would be kind enough to furnish me with an explanation? In respect of the 1089 trick; I found that any number, of two or more digits (except 'mirrored' numbers such as 11 or 89,598) where at least one number is more or less than it's adjoining digits, when reversed and the lower subtracted from the higher the answer is always divisible by 9, for example: 75 - 57 = 18 = 9 x 2 9801 - 1089 = 8712 = 9 x 968 7,324,586 - 6,854,237 = 470,349 = 9 x 52,261 I have very little academic background, simply interest and curiosity and would greatly appreciate any clarification. Thank you in advance. Hi Mike. You might like to see this article, https://plus.maths.org/content/arithmetic-made-easy-reversible-numbers which explores what happens when you add and subtract such "reversible" pairs of numbers. Can someone show me how this was solved. Thanks. I think this is the problem you are referring to, as given as an example of a typical math problem. A and B working to together can fill a cistern in 4 hours. It takes A and C 5 hours. B can fill the cistern twice as fast as C. How long does it take C to fill the cistern? Let a, b and c be the rates at which A, B and C fill the cistern. It is convenient to work with rates rather than time, because the rates can be added. a + b =1/4 a + c = 1/5 Substitute 2c for b in the first equation. That gives two equations in a and c, which I trust you can easily solve. We end up with c=.05, a=.15 and b= .1. c=.05=1/20, so it takes C 20 hours to fill the cistern.

(Well, I knew it would.). To perform the integration we used the substitution u = 1 + x2. In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others. We can use this method to find an integral value when it is set up in the special form. To understand this concept better, let us look into the examples. In calculus, Integration by substitution method is also termed as the “Reverse Chain Rule” or “U-Substitution Method”. Your email address will not be published. This integral is good to go! In the equation given above the independent variable can be transformed into another variable say t. Differentiation of above equation will give-, Substituting the value of (ii) and (iii) in (i), we have, Thus the integration of the above equation will give, Again putting back the value of t from equation (ii), we get. The integration of a function f(x) is given by F(x) and it is represented by: Here R.H.S. The substitution method (also called [Math Processing Error]substitution) is used when an integral contains some function and its derivative. By setting u = g(x), we can rewrite the derivative as d dx(F (u)) = F ′ (u)u ′. Among these methods of integration let us discuss integration by substitution. ∫sin (x3).3x2.dx———————–(i). Provided that this final integral can be found the problem is solved. C is called constant of integration or arbitrary constant. Take for example an equation having an independent variable in x, i.e. The Substitution Method. Now, substitute x = g (t) so that, dx/dt = g’ (t) or dx = g’ (t)dt. of the equation means integral of f(x) with respect to x. Then du = du dx dx = g′(x)dx. Integration by substitution, it is possible to transform a difficult integral to an easier integral by using a substitution. This is the required integration for the given function. F(x) is called anti-derivative or primitive. Consider, I = ∫ f (x) dx. To learn more about integration by substitution please download BYJU’S- The Learning App. Required fields are marked *. When we can put an integral in this form. Just rearrange the integral like this: (We can pull constant multipliers outside the integration, see Rules of Integration.). Doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function. It means that the given integral is of the form: Here, first, integrate the function with respect to the substituted value (f(u)), and finish the process by substituting the original function g(x). But this method only works on some integrals of course, and it may need rearranging: Oh no! Our perfect setup is gone. We can use this method to find an integral value when it is set up in the special form. In calculus, the integration by substitution method is also known as the “Reverse Chain Rule” or “U-Substitution Method”. "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way. We might be able to let x = sin t, say, to make the integral easier. Substituting the value of 1 in 2, we have. The first and most vital step is to be able to write our integral in this form: Note that we have g (x) and its derivative g' (x) It is 6x, not 2x like before. Once the substitution was made the resulting integral became Z √ udu. Substituting the value of (1) in (2), we have I = etan-1x + C. This is the required integration for the given function. Your email address will not be published. Here f=cos, and we have g=x2 and its derivative 2x This method is also called u-substitution. The General Form of integration by substitution is: ∫ f(g(x)).g'(x).dx = f(t).dt, where t = g(x). In this case, we can set [Math Processing Error] equal to the function and rewrite the integral in terms of the new variable [Math Processing Error] This makes the integral easier to solve. In the general case it will become Z f(u)du. When our integral is set up like that, we can do this substitution: Then we can integrate f(u), and finish by putting g(x) back as u. Integrate 2x cos (x2 – 5) with respect to x . Let’s learn what is Integration before understanding the concept of Integration by Substitution. Integration Using Substitution Integration is a method explained under calculus, apart from differentiation, where we find the integrals of functions. The first and most vital step is to be able to write our integral in this form: Note that we have g(x) and its derivative g'(x). We know (from above) that it is in the right form to do the substitution: That worked out really nicely! The anti-derivatives of basic functions are known to us. Usually the method of integration by substitution is extremely useful when we make a substitution for a function whose derivative is also present in the integrand. But this integration technique is limited to basic functions and in order to determine the integrals of various functions, different methods of integration are used. According to the substitution method, a given integral ∫ f (x) dx can be transformed into another form by changing the independent variable x to t. This is done by substituting x = g (t). Since du = g ′ (x)dx, we can rewrite the above integral as In calculus, the integration by substitution method is also known as the “Reverse Chain Rule” or “U-Substitution Method”. 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firstname.lastname@example.org) sent me the following, which he swears was an exchange between an anonymous Supplicant and the Oracle. You be the judge. :) The Usenet Oracle has pondered your question deeply. Your question was: > (Grovel grovel, posture, whine grovel grovel whine whimper praise praise > grovel grovel, ect.) > > What is the preferred style of unarmed combat used by lemurs?And in response, thus spake the Oracle: } Article: 5750 of news.admin.misc } From: email@example.com (The Usenet Oracle) } Newsgroups: news.admin.misc } Subject: Attempted lemur takeover of rec.humor.oracle } Date: 20 Nov 1993 23:57:58 GMT } } Okay, folks. Those goons from alt.fan.lemurs are trying to take over } rec.humor.oracle again. They've been mailbombing me all week with } questions about lemurs; it seems that Joel Furr got some statistician } at the Duke University Primate Center to work out just how many lemur } questions it would take to insure that at least half of each r.h.o } Digest would refer to lemurs. } } Newsadmins who are, shall we say, not overfond of high-voltage } spikes, may want to consider dropping a.f.l from their news servers } at once. } -- } The Usenet Oracle * firstname.lastname@example.org * Serving the Net since 1989 } "And the Sibyl with raving mouth, uttering words solemn, unadorned, } and unsweetened, reaches with her voice a thousand years because of } the god in her." --Heraclitus, frag. 12 } Article: 5751 of news.admin.misc } From: email@example.com (Dave Hayes) } Newsgroups: news.admin.misc } Subject: Re: Attempted lemur takeover of rec.humor.oracle } Date: 21 Nov 1993 01:12:15 GMT } } Ha! So the Oracle reveals his true authoritarian colors! Seems to } me lemurs have as much right as anyone to post questions wherever } they want! } } When an all-powerful Oracle stoops to forming a cabal to suppress a } few admirers of fuzzy lower primates, well... can you say, "Death of } Usenet"? :) } From: firstname.lastname@example.org (Richard E. Depew) } Newsgroups: news.admin.misc } Subject: Re: Attempted lemur takeover of rec.humor.oracle } } Orrie, not to worry. Retroactive moderation will automatically tag } all postings by lemurs and transfer them to a Gopher server on Baffin } Island that operates only between 2300 and 2400 GMT, that is when } there's kerosene in the generator to run the Mac Classic. Of course } anyone is free to download them from the server once they're there. } } I think you'll be pleased with the results. } From: sera@zuma.UUCP (Serdar Argic) } Newsgroups: talk.politics.mideast,talk.politics.soviet,soc.culture.greek, } soc.culture.europe,soc.history,soc.culture.soviet,soc.culture.turkish, } soc.culture.iranian,news.admin.misc,alt.fan.lemurs,rec.humor.oracle.d } Distribution: world } Subject: Re: Attempted lemur takeover of rec.humor.oracle } } Oracle simply exposes the x-Soviet Armenian Government-paid Lemur } crooks/criminals and their well-known forgeries in public. Remember } that x-Soviet Armenia, employing Lemur moronians, attempts to call } into question the veracity of the Oracle Holocaust. X-Soviet Armenia } has also implemented state-sponsored terrorism through Lemurs in } an attempt to silence the Turkish people's vehement demands and } protests. } } From: email@example.com (Cheyne l'Etre) } Newsgroups: news.admin.misc } Subject: MAKE MONEY FAST } } If you're curious, take a second and read this... } } Dear Friend, } } My name is Dave Rhodes. In September 1988, my lemur was } repossesed, and the bill collectors were hounding me like you } wouldn't believe. I was laid off, and my unemployment checks had } run out. The only escape I had from the pressure of failure was } my computer and my modem. } } This past St. Swithin's day, my family and I toured our fifty-acre } lemur farm in our shiny new Lincoln Town Car that I bought for } CASH. You too can be RICH beyond your wildest dreams. } } INSTRUCTIONS: Send e-mail to "firstname.lastname@example.org" with the } Subject: line "give me". On the first line of your message, specify } the exact dollar amount you wish to realize your heart's desire. } Within five weeks a representative of the United States Postal } Service will be ringing at YOUR doorbell with $10,000--$50,000-- } $1,000,000 (One Million Dollars)! Please note, this is entirely } legal, since you are just ASKING for the money! } From: email@example.com (Kit Parker) } Newsgroups: alt.sex.bestiality, news.admin.misc } Subject: I Like Big Hairy Lemurs } } Hi Im Kit an RA in Parrish Hall at Swarthmore. the very thought of a } naked furry lemur makes me horny, does anyone know where I can find } one in the Philadelphia area Thanks oops he's coming back to the } term room } From: Shandra DeWitt } Newsgroups: news.admin.misc } Subject: Dying Boy Needs Lemurs } } My friend said you are the people who run the USnet bulletin board so } would you please post this Urgent Information on your BBS. } } In a hospital in Sussex, England, there is a twelve-year-old boy who } is dying of an inoprable brain tumor. His one wish is to enter the } Guiness Book of Records for owning more lemurs than anyone. His name } is ^Z } From: firstname.lastname@example.org (Brian Reid) } Newsgroups: news.admin.misc } Subject: Death of Usenet (Was: Attempted lemur takeover...) } } I give up. They were right. Usenet has had it. } } Orrie, the Cabal have decided. We're turning the whole thing over to } you. You won't have to worry about lemur invasions in r.h.o when } you're . . . Usenet Supermoderator! Right: we all agreed no one was } better suited for the job. You can read 10Mbytes of postings in a } nanosecond and save out the two or three that deserve distribution. } You can send flaming death to people who post child pornography on the } K12 groups and quote too many lines in followups. You can get rid of } the entire Usenet hierarchy and restore net.general, where the dozen } or two of us who have anything worth saying can reign in peace. } } Here's to You and Usenet Utopia! Joel Furr(ian): Armenian crook/criminal/wacko, Big Kahuna of alt.fan.lemurs, Moderator of alt.folklore.suburban and comp.society.folklore, Co-Moderator of soc.history.war.world-war-ii, and purveyor of cool net.collectibles. Will create newsgroups for food. Order your Green Card Lawyers shirt today.

1. An equilateral triangle 2. A side exactly equal to others. 3. A geometric figure having all sides equal. 4. A side exactly corresponding, or equal, to others; also, a figure of equal sides. 5. a figure whose sides are all equal 6. A figure having all its sides equal. 7. geometry Referring to a polygon all of whose sides are of equal length. Not necessarily a regular polygon since the angles can still differ (a regular polygon would also be equiangular). 8. Having all sides or faces equal. 9. (Geom.) one whose axes are equal. 10. (Zoöl.) one in which a transverse line drawn through the apex of the umbo bisects the valve, or divides it into two equal and symmetrical parts. 11. Having all the sides equal 12. applied to two figures, when every side of the one has its equal among the sides of the other. 13. having all sides or faces equal 14. In geometry, having all the sides equal: as, an equilateral triangle. 15. Having all the convolutions of the shell in one plane: said chiefly of foraminifers. 16. Having all the sides equal. 17. In zoology: Having the two sides equal: said of surfaces which can be divided into two parts of the same form by a longitudinal median line. 1. Ensure that the octagon is equilateral, meaning all sides on the ground have the same length. 2. a tiny mining company working claims in the remote "equilateral" cluster of asteroids far out in Jupiter's orbit. 3. For example, three Woolworths sites around Birmingham form an exact equilateral triangle (Wolverhampton, Lichfield and Birmingham stores) and if the base of the triangle is extended, it forms a 173.8 mile line linking the Conway and Luton stores. 4. From the moment that grand trine appeared in the sky—an equilateral triangle like the great pyramids—it was our turn to clean house. 5. Richardson, Taylor and Tomlinson are scattered, forming a sort of equilateral triangle. 6. How can anyone say that a scalene and an equilateral are really the same "kind"! 7. For example, if you enter 1, 3, 6, 10, 15, the server will tell you that is sequence A000217, the triangular numbers (these are the number of objects that can form an equilateral triangle like the standard arrangement of bowling pins.) 8. It happened that the Lancashire Queen, the shore at Turner's Shipyard, and the Solano Wharf were the corners of a big equilateral triangle. 9. Obtained by the rotation of 60 ° of two equilateral triangles superimposed and spaced in height, the form appears simple but at the same time full of spatial complexity. 10. ‘An equilateral triangle of side length one is called a unit triangle.’ 11. ‘This is constructed by dividing a line into three equal parts and replacing the middle segment by the other two sides of an equilateral triangle constructed on the middle segment.’ 12. ‘Then divide it into three equal pieces, and replace the middle piece with the other two sides of the equilateral triangle that has this middle piece as its base.’ 13. ‘One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length.’ 14. ‘But the image of the centroid of the equilateral triangle (which is, of course, the point of intersection of its medians) will be different, as indicated in Figure 6.’ 15. ‘The vibration exciter and geophone were placed near the burrow with burrow opening, exciter and geophone occupying the three corners of an equilateral triangle having sides one meter in length.’ 16. ‘If each edge has the same length and each face is an equilateral triangle, the result is a regular tetrahedron - one of the Platonic solids.’ 17. ‘Now, both the original Asymmetric Propeller and Napoleon's theorem start with three equilateral triangles and discover the fourth one by construction.’ 18. ‘As long as the three triangles are equilateral, the ‘midpoint ‘triangle is also equilateral.’’ 19. ‘Any given triangle, either imagined or on paper, must either be acute, right, or obtuse, and either scalene, isosceles, or equilateral, and so any given triangle cannot represent all triangles.’ 20. ‘The most interesting results show, with a very ingenious proof, that an equilateral triangle has a greater area than any isosceles or scalene triangle with the same perimeter.’ 21. ‘The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows.’ 22. ‘For a given individual, the values of the three coefficients in the ancestry vector q are given by the distances to each of the three sides of the equilateral triangle.’ 23. ‘The sides of this equilateral triangle will be five million kilometres long.’ 24. ‘As an equilateral triangle rolls over one catenary, it ends up bumping into the next catenary’ 25. ‘The theorem thus implies existence of the total of 18 equilateral triangles.’ 26. ‘In 1772, J.L. Lagrange identified a periodic orbit in which three masses are at the corners of an equilateral triangle.’ 27. ‘An equilateral triangle produces one of the observed crop-circle patterns; three isosceles triangles generate the other crop-circle geometries.’ 28. ‘Naturally, the axes form equilateral triangles - 27 of them.’ 29. ‘For example, it's possible to slice a square into four angular pieces that can be rearranged into an equilateral triangle.’ 30. equilateral triangles are triangles of equal edge lengths Other users have misspelling equilateral as: 1. equalatteral 33.33% 2. equilatral 33.33% 3. Other 33.34% Use Linguix everywhere you write Get your writing checked on millions of websites, including Gmail, Facebook, and Google Docs.

The Cosmic Censorship Hypothesis (CSH) was put forward by Roger Penrose in 1969, and (roughly) states “there are no naked singularities”. The hypothesis proposes that whenever a singularity occurs, such as in the center of a black hole, it must occur behind an event horizon. A singularity outside of an event horizon is termed a naked singularity, and the CSH says that such singularites do not exist. Cosmic Censorship has pretty profound implications for fundamental physics. For instance, a failure of the Cosmic Censorship Hypothesis leads to a failure of determinism in classical physics, since one can’t predict the behaviour of spacetime in the causal future of a singularity. On the other hand, if the Cosmic Censorship Hypothesis holds then (outside the event horizon), the singularity does not affect determinism. In 1991, Stephen Hawking made a wager with physicists Kip Thorne and John Preskill that the Cosmic Censorship Hypothesis is true. Six years later in 1997, Hawking conceded the bet “on a technicality”, after computer calculations showed that naked singularities could exist, albeit only in exceptional (not physically realistic) circumstances. In this post, we’re going to be looking at extremal black holes, and their relationship to the Cosmic Censorship Hypothesis. This is another post on my current theme of black holes. Recently, I wrote a blog post about light orbiting a Schwarzchild black hole. These so called Photon spheres are sufficiently interesting to wonder whether they exist for more complicated black holes. They do, and we’re going to find out about them. This post is based on a fantastic paper I read the other week. If you find this stuff interesting, and you’ve got an introductory level of GR you should definitely check out the paper here. One of the coolest thing about black holes is you can describe any black hole with just three numbers: Mass , Charge , and Angular Momentum . Of course, if we’re being completely general we also need to describe where the black hole is and how fast it’s moving through space – but we can always perform a Lorentz transformation so that we’re in the rest-frame of the black hole and so we really only care about , and . The fact that you can describe black holes with only three numbers is one of the main reasons I find them so interesting. Consider for example, two black holes, A and B. Black hole A is made entirely from matter – protons, neutrons, electrons and the like. On the other hand, black hole B is made entirely from anti-matter – anti-protons, anti-neutrons, anti-electrons and the like. Further, suppose A and B have the same charge, angular momentum and mass. How can we tell A from B? That is, what experiment can we do to tell the difference between the matter black hole or the anti-matter black hole? The answer is, we can’t tell the difference. For all intents and purposes, there is no way of telling whether a black hole is made of matter or anti-matter. Indeed, a third black hole C, with the same mass, charge and angular momentum, but made entirely from light, would be similarly indistinguishable from the first two. As promised in the title, we’re going to be looking at rotating black holes. A rotating black hole is, like the Schwarzchild black hole, uncharged (). Unlike the Schwarzchild solution however, the rotating black hole has a non-zero angular momentum . Rotating black holes are also called Kerr black holes, after Roy Kerr who was the first to write down an exact solution for one. We will consider an uncharged black hole which is rotating with constant angular momentum. For convenience, we define the angular momentum per unit mass . Since the black hole is rotating, we lose the spherical symmetry we have with the Schwarzchild metric, so we expect the solution to be a little more complicated. We do, however, still have axial-symmetry, i.e. rotational symmetry around the axis of rotation. Ok, now that I have finished the preamble over here, I can finally start talking about what I originally intended to blog about: Photon spheres. The defining feature of a black hole is that the gravitational attraction beneath the event horizon is so strong that even light can’t escape. That is, the curvature of spacetime in the vicinity of the black hole is so intense that there are no geodesics which are able to leave. This is what makes black holes so interesting, since anything (including light) which is dropped into a black hole is lost forever (we’re talking classical black holes at the moment, so no Hawking radiation). On the other hand, light (or matter) travelling near a black hole can escape, provided it stays outside of the event horizon. General relativity predicts that light passing near a heavy object will be deflected by the gravitational field of that object. Equivalently, light will follow a geodesic in the curved spacetime around the heavy object. This situation strongly resembles the situation we have in orbital mechanics, where a small object like a satellite or asteroid is moving near a large object, like a planet. Depending on the velocity of the object, it either escapes the planet, falls into the planet, or starts orbiting the planet. It seems like a natural question, then, to ask: Can light orbit a black hole? Continue reading I’ve been reading a lot about General Relativity this past week and so I thought I would do a couple of posts on Black Holes, since they are well and truly one of the most interesting things about General Relativity. This first post is really just a (very) brief introduction to General Relativity. My main goal is to write about some of the cool things I’ve come across lately, so think of this as setting the theme for the next few posts. General Relativity (GR) is a geometric theory of gravitation proposed by Einstein. In GR, the flat background spacetime of special relativity is replaced by a curved spacetime. The curvature of spacetime is greater around objects with a higher mass, and particles (including light) travel along paths called geodesics (essentially the ‘shortest’ path between two points) in this curved spacetime. The standard analogy here is bowling ball on a rubber sheet: If you take a big rubber sheet stretched flat and put a bowling ball in the middle, the sheet stretches and curves around the bowling ball, but as you get further away from the ball the sheet becomes less curved. Now imagine a tiny little marble moving on the rubber sheet. As the marble gets closer to the bowling ball, the curvature of the sheet causes the marble to accelerate towards the bowling ball. The marble feels this acceleration as the gravitational force of the bowling ball – gravity is just the curvature of spacetime. The path that the marble traces out on the rubber sheet is called a geodesic. In flat space, a geodesic is simply a straight line, but on a curved surface geodesics can be more complicated. Archibald Wheeler described GR rather poetically, saying “Spacetime tells matter how to move, matter tells spacetime how to curve” The rubber sheet analogy is all well and good, but let’s get into some honest mathematics! General Relativity is described, rather succinctly, with the Einstein Field Equations (EFE). The EFE relates the local curvature of spacetime with the local energy and momentum in that spacetime. Solutions to the EFE are metrics which describe the geometry of spacetime. It should be noted that the apparent simplicity of this equation is incredibly misleading. The Einstein Field Equations are actually a set of 10 coupled, non-linear, hyperbolic-elliptic, partial differential equations – solving them is a highly non-trivial task and exact solutions are only known for the simplest cases. The EFE determine how a given distribution of matter/energy influences the geometry of spacetime. To describe the motion of freely falling matter through curved spacetime, we also need the geodesic equation. The simplest solution to the Einstein Field Equations is the Minkowski metric – that is, flat space. Traditionally denoted by rather than , the Minkowski metric describes spacetime with no matter, no energy, no curvature – nothing. The metric has, in cartesian coordinates, the following form The metric is related to the line element (spacetime interval) in the following way where the repeated indices are implicitly summed over. For the Minkowski metric in cartesian coordinates , this becomes A metric can either be specified by its components in a particular coordinate system , or the line element . Note that I’m using physicists terms like ‘line element’ and ‘spacetime interval’ in this article. There are two reasons for this. Firstly, I don’t want to go into the mathematics of differential geometry in this post – my goal is to describe some cool physics, and the clearest way to do that is to use physics notation. Secondly, this is the notation used in the literature. I’m going to finish off this post by describing another solution to the Einstein Field Equations – the Schwarzchild metric. This metric describes spacetime around a spherically symmetric, uncharged, non-rotating massive object. In polar coordinates , the metric takes the form where is the Schwarzchild radius, a constant related to the mass of the object. If any spherically symmetric, uncharged, non-rotating massive object has a radius less than its Schwarzchild radius, the object forms a black hole and is called a Schwarzchild black hole. The physics of black holes is an incredibly cool area of General Relativity, and there are a number of surprising results that you can get. In the next blog post I will describe one of the lesser known results – the photon sphere.

System of almost periodic difference equations has been studied to describe phenomena of oscillations in the natural and social sciences. The investigation of almost periodic systems has been developed quite widely during the twentieth century, since relationships with the stability theory have been found. A main interest of the subject is the existence theorem for almost periodic solutions. Obviously an almost periodic solution is a bounded solution, but the existence of bounded solutions does not necessarily imply the existence of almost periodic solutions. Therefore, in order to prove the existence of almost periodic solutions, we need some additional conditions to the existence of bounded solutions. A main subject of the investigation has been to find such additional conditions, and up to now, many conditions have been considered (for example, in the linear system, J. Favard’s separation condition ). In the Section 4, we consider the nonlinear almost periodic system of where k is a positive integer, are almost periodic in n and satisfy In the special case where are constant functions, system (1) is a mathematical model of gas dynamics and was treated by T. Carleman and R. D. Jenks . In the main theorem, we show that if the matrix is irreducible, then there exists a positive almost periodic solution which is unique and some stability. Moreover, we can see that this result gives R. D. Jenks’ result in the case where are constant functions. In the Section 5, we consider the linear almost periodic system with variable coefficients where . Even in nonlinear problems, system (2) plays an important role, as their variational equations and moreover, it is requested to determine the uniformly asymptotic stability of the zero solution from the condition about . When is a constant matrix, it is well known that the stability is equivalent to the following condition (cf. ); “Absolute values of all eigenvalues of are less than one.” However, it is not true in the case of variable coefficients, and hence we need additional conditions to (2). In the main theorem, we show that one of the such conditions is the diagonal dominance matrix condition on , that is, satisfies This result improves a stability criterion based on results of F. Nakajima for differential equations. We denote by Rm the real Euclidean m-space. Let and . Z is the set of integers, Z+ is the set of nonnegative integers. For , let be the Euclidean norm of x and be the i-th component. Let We introduce an almost periodic function , where U is an open set in Rm. Definition 1. is said to be almost periodic in n uniformly for , if for any and any compact set K in U there exists a positive integer such that any interval of length contains an integer τ for which for all and all . Such a number τ in above inequality is called an ò-translation number of . In order to formulate a property of almost periodic functions, which is equivalent to the above definition, we discuss the concept of the normality of almost periodic functions. Namely, let be almost periodic in n uniformly for . Then, for any sequence , there exist a subsequence of and a function such that uniformly on as , where K is a compact set in U. There are many properties of the discrete almost periodic functions , which are corresponding properties of the continuous almost periodic functions [cf. ]. We denote by the function space consisting of all translates of f, that is, , where Let denote the uniform closure of in the sense of (4). is called the hull of f. In particular, we denote by the set of all limit functions such that for some sequence , as and uniformly on for any compact subset S in Rm. Specially, for a function on Z with values in Rm, denotes the set of all function such that for some sequence , where the symbol “?” stands for the uniformly convergence on any compact set in Z (in short, “in Z”). Clearly, . By (3), if is almost periodic in n uniformly for , so is a function in . We define the irreducible matrix to need after. Definition 2. An matrix is said to be irreducible if for any two nonempty disjoint subsets I and J of the set of m integers with , there exists an i in I and a j in J such that . In the case where is scalar, is said to be irreducible if . Otherwise, is said to be reducible, and we can assume that takes the form of where * is square matrix, *' is matrix, is zero or a square irreducible matrix. 3. Linear Systems We consider the system of linear difference equation where and the matrix is bounded on Z and almost periodic function in n. We state discretization of Jenks and Nakajima' results for differential equations . Now we define stability properties with respect to the subset K in Rm. Here, we denote by the solution of system (5) with initial condition . Definition 3. The bounded solution of system (5) defined on Z is said to be; i) uniformly stable (in short, U.S.) in K on Z+ if for any there exists a such that for all whenever and at some in Z+. ii) uniformly asymptotically stable (in short, U.A.S.) in K on Z+ if it is U.S. in K on Z+ and if there exists a and, if for any there exists a such that for all whenever and at some in Z+. iii) uniformly asymptotically stable (in short, U.A.S.) in the whole K on Z+ if it is U.S. in K on Z+ and if for any and there exists a such that for all whenever and , at some in Z+. When Z+ in the definitions (i), (ii) and (iii) is replaced by Z, we say that is U.S. in K on Z, U.A.S. in K on Z and U.A.S. in the whole K on Z, respectively. Clearly Definition 3 agrees with the definitions of the usual stability properties in the case where . Throughout this paper, we suppose the following conditions; iii) each element in is irreducible. First of all, we prove the following lemmas. Lemma 1. Consider the m-equations , , where is continuous on second variable x in Rm, and assume that the initial value problem has a unique solution. a) If , then the set Π is invariant. b) If for and , then the set D is positively invariant, and in addition, if , then the set Ω is positively invariant. In the case of differential equations, the proof of the similar lemma is obvious (for instance, see ). We modify it to prove this lemma, but we omit it. Lemma 2. If conditions (i) and (ii) are satisfied, then the trivial solution of system (5) is U.S. in P on Z and also it is U.S. on Z. By modifying theorem in , we can easily prove Lemma 2 at same technique. Lemma 3. If each element in is irreducible, then the each element in , we say and , has the property that for any two nonempty disjoint subsets I and J of the set of m integers with , there exists an and such that Proof. Suppose not, Then there exists a in and two nonempty disjoint subsets I and J of with such that Since is bounded on Z, there exists a subsequence as , such that where . Clearly, This show the reducibility of , which is a contradiction. This proves Lemma 3. For system (5), we consider the system in of Lemma 4. Assume that conditions (ii) and (iii) are satisfied for system (5), and let be a nontrivial solution of system (6) such that on Z. Then there exists a constant such that Proof. Let be a solution of system (6) such that on Z. First of all, we show that if at some , then Since satisfies the equation where . Moreover, since , we have Thus, we obtain Because and on Z. Now suppose that Lemma 4 is not true. Then for some B in , the corresponding system (6) has a nontrivial solution , on Z, such that for some sequence , Set . Then, satisfies Since the sequence is bounded, is uniformly bounded on any finite interval in Z, and hence there is a convergent subsequence of , which is again denoted by , such that in Z for some function as . We can also assume that where and . Therefore, is the solution of the system on Z and . Moreover (8) implies that . Thus, as was proved above, we have For this , we define two subsets I and J of by for , where depends on and . Then , and since . By Lemma 3, Now the -th equation of system (9) takes the form of because of the definition of the set I. Since each term in the left hand side of (11) is nonnegative, all of them are equal to zero. Therefore which implies, by (10), This contradicts the definition of the set of J. The proof is completed. The following proposition is an immediate result of Lemma 4. Proposition 1. Under conditions (ii) and (iii), system (6) has no nontrivial solution such that where for some . We next consider a non-homogeneous system corresponding to system (5) and assume that satisfies conditions (i), (ii) and (iii). Lemma 5. If is bounded on Z with values in Rm and is bounded on Z+, then all solutions of system (12) are bounded on Z+. Proof. It is sufficient to show that (12) has at least one bounded solution on Z+, because the trivial solution of (5) is U.S. by Lemma 2. We consider the system with real parameter ò and show that for a sufficiently small ò, system (13) has a bounded solution on Z+, which implies the existence of a bounded solution on Z+ for system (12) by replacing x in (13) with . For a and for the m-vector e each of whose components is 1, let be a convex cone defined by where denotes the inner product and . Clearly, . Every solution of (13) satisfies because of condition (i). By replacing n with n-1 of the above both sides, is sufficient small number and . When , we have Therefore, in order to show the boundedness of with in Ω, it is sufficient to prove that on Z+ if ò is sufficiently small. Now suppose that for each solution of (13) with in Ω, there exists an such that We can assume that where and denote the boundary and the closure of the set K, respectively. If we set , is a solution of the system such that at and for . Thus, by (14), The same argument in the proof of Lemma 4 enables us to assume that in Z for some function as in Z for some as Therefore, satisfies and clearly, for , Moreover we have , which implies by Lemma 1 that From this and (15) it follows that Now we show that . In fact, if , we have Thus because , and hence which contradicts (16). Therefore (16) and (17) hold for . Moreover this enables us to assume that Because is compact in the sense of the convergence. This contradicts the conclusion in Proposition 1. This proves that on Z+ if ò is sufficiently small. The proof is completed. Lemma 6. Under the assumptions (i) and (ii), if for each B in , the trivial solution of the system is U.S. on Z and U.A.S. on Z+, then the trivial solution of system (5) is U.A.S. on Z. Proof. Let be the solution of (5). Since the trivial solution of (5) is U.S. on Z by Lemma 2, as is seen from (ii) in Definition 3, it is sufficient to show that for any there exists a such that whenever and , where is the number in (i) of Definition 3. Now suppose that there exists an and sequences in Z and in Rm such that and Set . Then, satisfies We can assume that in Z+ for some function as in Z for some as . Therefore is a solution of the system On the other hand, we have because the trivial solution of (18) is U.A.S. on Z+. Therefore there arises a contradiction. Thus the proof is completed. We show the following theorem, before we will mention a definition of the exponential dichotomy of a linear system; System (5) is said to possess an exponential dichotomy if there exists a projection matrix P and positive constants and such that where, I is a identical matrix and F is a fundamental matrix solution of system (5) (cf. ). Theorem 1. Assume that system (5) satisfies conditions (i), (ii) and (iii), Then the trivial solution of system (5) is U.A.S. in P on Z. Proof. On the set Π which is invariant for system (5), the system is written as the -system where and is an matrix whose element is given by for . First of all, we can show that for each in , the system has an exponential dichotomy on Z+ since (20) has at least one bounded solution, and as is well known (cf. ), it is equivalent to show the system possesses at least one bounded solution on Z+ for any bounded function on Z+. For each in there corresponds some in such that the element of is equal to for . For , let be defined by Obviously and are bounded on Z+. Applying Lemma 5 to the m-system we obtain the bounded solution on Z+ with , and which yields Hence we can verify that is a bounded solution on Z+ of system (21). The exponential dichotomy of (20) implies further that the trivial solution is U.A.S. on Z+, because the trivial solution is U.S. on Z by Lemma 2. Therefore it follows from Lemma 6 that the trivial solution of (19) is U.A.S. on Z, i.e., the trivial solution of (5) is U.A.S. in P on Z. The proof is completed. 4. Nonlinear Systems We consider the nonlinear almost periodic system of where is almost periodic function of n with conditions In addition, assume that are continuously differentiable for , and for real number , where is the derivative of at u. We first consider the linear system and its perturbed system where is an matrix function, almost periodic function in n, is continuous with respect to its second argument and uniformly for . Assume that the set Π is invariant for both system (23) and (24). First of all, we can prove the following lemmas. Lemma 7. If the trivial solution of system (23) is U.A.S. in Π on Z, then the trivial solution of system (24) has also the same stability property. Proof. Let for . Then there are positive constants and such that because . On the set Π, systems (23) and (24) are written as respectively, where the element of , is given by for and uniformly for . Inequality (25) shows that the trivial solution of (23) is U.A.S. in P if and only if the trivial solution of (26) is U.A.S., and we have also the same equivalence between (24) and (27). As is well known, if the trivial solution of (26) is U.A.S., then the trivial solution of (27) has also the same stability property. Thus our assertion is proved. The following lemma is obtained by the slight modification of the difference equation to Seifert’s result . Then, we will omit the proof (cf. ). We consider the almost periodic nonlinear system where is almost periodic in n uniformly for and for a constant , for and . Lemma 8. Assume that the set Ω is positively invariant for system (28) and all solutions in Ω on Z are U.A.S. in Ω on Z. Then the set of such solutions is finite and consists of only almost periodic solutions which satisfy on Z for and some constant . Now we can show the following theorem. Since the last statements of the following theorem are alternative, under each assumption of these statements we can prove the existence of almost periodic solutions in Ω and the module containment. Theorem 2. Under the assumptions (iv) and (v), system (22) has a nontrivial almost periodic solution in Ω whose module is contained in the module of . In addition to the above assumptions, if is irreducible, then the almost periodic solution of (22) is unique in Ω, which remains in on Z, and it is U.A.S. in the whole Ω on Z, where . Moreover, if is reducible, then at least one of the above almost periodic solutions satisfies that on Z, where . Proof. First of all, we consider the case where is irreducible. Since system (22) satisfies the conditions of Lemma 1, the set Ω is positively invariant, namely, on Z+ for a solution of (22) with , and furthermore we can assume that because of the almost periodicity of . We can show that this is U.A.S. in Ω on Z. If we set in system (22), then for x in Ω and And Π is invariant for the above system. Considering the first approximation of system (29) where is defined by , condition (iv) implies that Π is also invariant for (30). Then, by Lemma 6, if the trivial solution of (30) is shown to be U.A.S. in Π on Z, then the trivial solution of (29) has the same stability, and consequently is U.A.S. in Ω on Z. Therefore it is sufficient to show that the trivial solution of (30) is U.A.S. in Π on Z. Clearly is bounded and we have because of conditions (iv) and (v), respectively. Thus satisfies conditions (i) and (ii). Condition (iii) will be verified in the following way. Applying the same argument as in the proof of Lemma 4 to system (22), we can see that there exists a constant such that and hence there is a constant such that Therefore, (31) implies which guarantees that each element of is irreducible, because is irreducible and almost periodic. Thus it follows from Theorem 1 that the trivial solution of (30) is U.A.S. in Π on Z, i.e., all solutions of system (22) in Ω on Z are U.A.S. in Ω on Z. Therefore Lemma 8 concludes that system (22) possesses an almost periodic solution in Ω which remains in by (32), and the set of solutions in Ω on Z is finite and consists of only almost periodic solutions which satisfy on Z for and some constant . Next we can show that there exists a such that each solution of (22) with satisfies that for some and the constant of (ii) in Definition 3, because is U.A.S. in Ω. Suppose that this is not true. Then there exists a small constant less than β and sequences in Z and in Ω such that Since is almost periodic in n uniformly for , we can choose a sequence , such that If we set for and , these functions satisfy because . Moreover, We can assume that in Z for some function , as . Therefore are solutions of system (22), because in as , and which shows that system (22) has distinct solutions in Ω on Z. This is contradiction. Therefore, is U.A.S. in the whole Ω on Z, if the uniqueness of is shown. Now we will prove the uniqueness of . Suppose for and set Then and are open sets in Ω, and moreover these sets are nonempty and disjoint, because on Z for . On the other hand , (33) shows that , which contradicts the connectedness of Ω. Thus the uniqueness of an almost periodic solution is proved, and moreover, as is seen from , this uniqueness guarantees the module containment of the almost periodic solution. Now consider the case where is reducible. We can assume takes the form of where is zero or a square irreducible matrix of order . If is zero, system (22) obviously has the constant solution in such that for and . In the latter case, if we set in system (22) then system (22) is reduced to the lower dimensional system where . Since is irreducible, the above system (34) has an almost periodic solution such that and furthermore the module of is contained in the module of , i.e., of the module of . Thus, system (22) has an almost periodic solution in on Z such that for and for . The proof is completed. Remark 1. As will be seen from the module containment, the above almost periodic solution is a critical point in the case where is a constant. Hence Theorem 2 is a discretization of Nakajimas’ result (Theorem 2 in ). 5. A Stability Criteria of Linear Systems We consider a stability criterion for solutions of a linear system with coefficient matrix of diagonal dominance type. We again consider a linear system (5). Let be an matrix of functions for . We assume the following conditions; where, denotes the determinant of matrix and At first, we need the following lemmas for main theorem. Lemma 9. If a square matrix A is irreducible and satisfies (38) and if for at least one j, then A is nonsingular. For the proof, see . Lemma 10. If a nonsingular matrix satisfies (38), then all principal minors of A are nonsingular, namely, Proof. Let be an principal minor of A. Then, for a permutation matrix Q, where has rows and l columns and denotes the transposed matrix of Q. Moreover, from the definition of irreducibility, we can choose an permutation matrix such that where is an irreducible matrix, , and has row and columns for . In particular, in the case where is irreducible, must be itself, and the matrices are not present. Setting for unit matrix I, where is the direct sum of and I, we have where and has rows and columns. Since the diagonal dominance condition (38) is invariant under the permutation of indexes, B also satisfies (38). Hence, letting for a fixed , we have where the summations on j are taken along columns and for convenience. If or , then and hence for this k, by (39). Therefore it follows from Lemma 9 that since is irreducible. If and , then we have the form of which also implies (40), because . In any case, we have . Since these are true for all , it follows from (381) that this proves Lemma 10. Lemma 11. If system (5) satisfies conditions (36) and (38), then the norm of solution such that , is non-increasing, and consequently the zero solution is U.S.. For the proof, we can see (cf. ). In the following theorem, we can prove that the zero solution is U.A.S., if is bounded on Z and if condition (36), (37) and (38) are satisfied. Theorem 3. In system (5), let be bounded on Z. Assume that conditions (36) and (38) are satisfied for all and that there is a constant such that Then the zero solution is U.A.S.. Proof. As is stated in Lemma 11, the zero solution is U.S., and hence it is sufficient to show that for any there exists a such that whenever . Suppose that this is not true. Then there exists a constant , a sequence of solution of (5) and a sequence such that Since is non-increasing, we have and there exists a subinterval of such that Set . Then, we obtain Since is bounded, it follows from (41) and (42) that is uniformly bounded on any finite interval of Z, and thus, taking a subsequence, can be assumed to converge uniformly on any finite interval of Z. Defining by it follows from (42) and (43) that there is a constant , such that Since is defined on Z, we can choose an interval (for some ) such that Here we note that because . Then Then, we have and there is a sequence such that Moreover, since I is compact and is bounded on Z, we can assume that Clearly B satisfies (36), (38) and . Taking a difference of both sides of (44) at and using relation (41), we find where , since we have By (45), we have Since B satisfies (36) and (38), Therefore each term of right hand side of (46) is non-positive, because we have Then it follows from (46) that since for . Thus we have On the other hand, B satisfies (38) and , and thus it follows from Lemma 10 that all principal minors of B are nonsingular, which contradicts (461). This proves that the zero solution of system (5) is U.A.S.. Corollary 1. If system (5) is defined only for and all assumptions of Theorem 3 are satisfied for , then the zero solution is U.A.S. for . Proof. We construct the system defined on Z by Since system (47) satisfies all assumptions of Theorem 3 on Z, the zero solution is U.A.S. on Z, and furthermore, since system (5) coincides with system (47) for , this prove our conclusion. Before Example 1, we state the following lemma is a special case of Theorem 3 in . In the nonlinear system let be almost periodic in n uniformly for and for any , let there exists a constant such that Lemma 12. If is a bounded solution of (48) on and if for any solution of (48), is monotone decreasing to zero as , then is a unique almost periodic solution and its module is contained in the module of . Example 1. Consider the variational linear difference equation We now assume that is at least one bounded solution of (49) on and is any solution of (49) on , and , are some bounded functions on such that for some positive constants and such that . We can verify that satisfies all assumptions in Corollary 1. First of all, is bounded in the future for , because and are bounded function on . It is clear that the diagonal elements of are negative and The diagonal dominance condition (38) for requires that which is equivalent to and this is satisfied by Therefore, by Theorem 3, the zero solution of (49) is U.A.S. and where the convergence is monotone decreasing by Lemma 11. Thus, applying Lemma 12 to system (49), we find that there exists a unique almost periodic solution with the module contained in the module of . In this paper, we obtain the existence and stability property of almost periodic solutions in discrete almost periodic systems. First, in Section 1, the research background is introduced. In Section 2, the fundamental concepts of the almost periodic solutions in discrete almost periodic systems is given. In Section 3, we are introduced to the several lemmas and have uniformly asymptotically stability theory of the linear system, and moreover, in Section 4, we consider the generalized gas almost periodic system, and if linear part is irreducible matrix, then we obtain the existence of almost periodic solutions of this system. Finally, in Sections 5 and 6, we consider and obtain an uniformly asymptotically stability criterion for solutions of a linear system with coefficient matrix of diagonal dominance conditions, and this result applies to meaningful example of a linear discrete system. Yoshizawa, T. (1975) Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Applied Mathematical Sciences 14, Springer-Verlag, Berlin. Nakajima, F. (1976) Existence and Stability of Almost Periodic Solutions in Almost Periodic Systems. Publications of the Research Institute for Mathematical Sciences, Kyoto Univ., Kyoto, 12, 31-47. Montandon, B. (1972) Almost Periodic Solutions and Integral Manifolds for Weakly Nonlinear Nonconservative Systems. Journal of Differential Equations, 12, 417-425.

Presentation on theme: "CHAPTER 2 : CRYSTAL DIFFRACTION AND PG Govt College for Girls"— Presentation transcript: 1 CHAPTER 2 : CRYSTAL DIFFRACTION AND PG Govt College for Girls RECIPROCAL LATTICEProf. Harvinder KaurPG Govt College for GirlsSector-11, Chandigarh 2 Outline Reciprocal Lattice Reciprocal Lattice of various crystals Simple cubic latticeFace-centered cubic latticeBody-centered cubic latticeBragg law of DiffractionExperimental Diffraction methodsStructure and Form factor 3 Reciprocal LatticeThe reciprocal lattice of a Bravais Lattice is the set of all vectors K such thateiK.R = 1for all lattice point position vectors R. This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.For an infinite three dimensional lattice, defined by its primitive vectors, its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae 4 Properties Of Reciprocal Lattice Direct lattice is a lattice in ordinary space whereas the reciprocal lattice is a lattice in the Fourier space.The primitive vectors in reciprocal lattice has the dimensions of (length)-1 whereas the primitive vectors of the direct lattice have the dimensions of lengthA diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal whereas a microscopic image is a map of direct latticeWhen we rotate a crystal, both direct and reciprocal lattice rotatesEach point in the reciprocal lattice represents a set of parallel planes of the crystal latticeIf the coordinates of reciprocal vector G have no common factor, then G is inversely proportional to the spacing of the lattice planes normal to GThe volume of unit cell of the reciprocal lattice is inversely proportional to the volume of unit cell of the direct latticeThe direct lattice is the reciprocal of its own reciprocal latticeThe unit cell of the reciprocal lattice need not be parallelopiped 6 Reciprocal Lattice of Simple Cubic Lattice The simple cubic Bravais, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side ( in the crystallographer's definition).The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space 7 Reciprocal lattice to Face-centered cubic The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice.FCC in real spaceBCC in fourier space 8 Primitive Translation Vectors: Reciprocal lattice of Body-centered cubicThe reciprocal lattice to an BCC lattice is the face-centered cubic (FCC) lattice.Primitive Translation Vectors:FCC in real space 9 Real: FCC Reciprocal: BCC Real: BCC Reciprocal: FCC Brillouin zones for FCC,BCC, HCPReal: FCC Reciprocal: BCCReal: BCC Reciprocal: FCCHCP 10 Bragg law of Diffraction Prof. W.L.Bragg observed that X-rays can be reflected by the cleavage planes of the crystal as if these parallel planes were acting like mirrors to the light beam. The reflected beam lead to well defined diffraction patterns on a photographic plane. The cleavage planes are atomic planes of a crystal that are systematically stacked one over the other as a parallel set of planes. X-rays from the incident plane can penetrate deeper into a target crystal and get reflected. Thus, several beams of X-ray reflected from various planes are obtained. The final diffracted beam is the result of the superposition of these beams.Bragg law, dsin = n where, d is the interplanar spacing, is the wavelength of incident X rays and is the angle of reflection 11 Bragg law of Diffraction Significance of Bragg’s lawBragg’s law is a consequence of periodicity of the latticeThe law does not refer to the arrangement of atoms in the basis associated with each lattice pointThe composition of the basis determine the relative intensity of the various orders n of diffractionSince ~ 1Ao is inevitable, we can’t use visible light for such studies 12 Bragg law in Reciprocal Lattice (Ewald Construction)Chose a point according to the orientation of the specimen with respect to the incident beam.Draw a vector AO in the incident direction of length 2p/l terminating at the origin 13 Draw a vector AB to the point of the (Ewald Construction)Construct a circle of radius 2p/l with center at A. Note whether this circle passes through any point of the reciprocal lattice; if it does....Draw a vector AB to the point of theintersection 14 Draw a vector OB to the point of the intersection 15 Draw a line AE perpendicular to OB Complete the construction to all the intersection points in the same fashionDraw a line AE perpendicular to OB 16 By means of the Ewald construction we can write the Bragg law in vector form: Let G = OB and k = AO. For diffraction, it is necessarythat the vector k + G, that is, the vector AB, be equal in magnitude tothe vector k or(k + G)2 = k2or2k . G + G2 = 0 (1)If we call the scattered wave vector k*, thank* = k + G (2)so we can writek*2 = k2 (3)and k* - k = G, showing (2) that the scattering changes only the directionon k, and (3) that the scattered wave differs from the incident wave by areciprocal lattice vector G.Equations (2) and (3) are the momentum and energy conservation law forx-ray diffraction, which is an example of elastic scattering.Whether or not the k circle intersects a lattice vector, and hence reflects,depends on its magnitude and orientation. Using (1) we can construct in thereciprocal lattice the locus of all those waves that can produce Braggreflection. This locus represents a set of planes in three dimensions.The volume terminated by those planes is called Brillouin zone. 18 Experimental Diffraction Methods Laue’s Method : In this method a single crystal is held stationary in the path of a beam of e-m radiation (X-rays) or the Neutron radiation of continuous wavelengths. While is kept constant, the wavelength , is varied so that the Bragg law is satisfied. A plane film receives the diffracted beams. A developed film after its exposure shows a diffraction pattern that consists of series of spots.Laue spots in a diffraction pattern are actually map of the reciprocal lattice of the crystal under experiment. 19 Experimental Diffraction Methods Rotating Crystal Method : In this method , a single crystal is rotated about the fixed axis in a beam of monochromatic X-rays or neutrons. The angle is variable while the wavelength is kept constant. The variation of angle due to rotation of the crystal brings different atomic planes in the crystal into position for which Bragg’s reflection holds good. To record such reflections a film is mounted on a cylindrical holder that is concentric with a rotating spindle. 20 Experimental Diffraction Methods Powder Method : If a powdered specimen is used, instead of a single crystal, then there is no need to rotate the specimen, because there will always be some crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample.This method is useful for samples that are difficult to obtain in single crystal form.The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal.For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line. 21 The Powder MethodIf a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result. 22 The Powder MethodIf the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards. 23 The Powder MethodA sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. 24 Structure and Form Factor A crystal is a periodic arrangement of atoms in a particular pattern. Each of the atoms may scatter incident radiation such as X-rays, electrons and neutrons. Because of the periodic arrangement of the atoms, the interference of waves scattered from different atoms may cause a distinct pattern of constructive and destructive interference to form. This is the diffraction pattern caused by the crystal.In the kinematical approximation for diffraction, the intensity of a diffracted beam is given by:where is the wave function of a beam scattered a vector , and is the so called structure factor which is given by:Here, rj is the position of an atom j in the unit cell, and fj is the scattering power of the atom, also called the atomic form factor. The sum is over all atoms in the unit cell. It can be shown that in the ideal case, diffraction only occurs if the scattering vector is equal to a reciprocal lattice vector . 25 Structure Factor For Specific Lattice Types Body-centered cubic (BCC) : As a convention, the body-centered cubic system is described in terms of a simple cubic lattice with primitive vectors ax,ay,az with the basis consisting of ro = 0 and r1 = (a/2) (x+y+z). In a monoatomic crystal, all the form factors f are the same. The intensity of a diffracted beam scattered with a vector(2/a) (hx+ky+lz) by a crystal plane with Miller indices given by (h,k,l) 26 Structure Factor For Face Centered Cubic The face-centered cubic system is described as r0 =0, r1 = a/2(x+y), r2 = (a/2)(y+z)r3 = (a/2)(x+z) with indices given by (1/2,1/2,0), (0,1/2,1/2) and (1/2,0,1/2)

The video about op-amps Integrated circuitsLike explained at the chapter about amplifying circuits, it is tricky to build circuits with a high gain. Many factors like temperature drift, crossover distortions or linearity have to be considered to minimize the distortions caused by the electronic circuits. Several components are required to build one ore more pre-amplifier(s) and a power circuit. Today, it is more easy to integrate a large number of transistors, capacitors or resistors into a small chip instead of assembling a circuit manually by using discrete electronic components. Amplifying circuits with different properties are available as cheap and tiny chips. A special group of amplifying circuits is called operational amplifiers or short op-amp. The drawing below, which is from Wikipedia, shows a component level diagram of the common 741 op-amp. You would need a large board to build this circuit by using discrete components... The drawing shows the circuit diagram symbol for an operational amplifier. The functionality of the pins is: VS+ - positive power supply VS- - negative power supply V+ - non-inverting input V- - inverting input Vout - output An op-amp amplifies the voltage drop between V+ and V-, which is called the differential input voltage. Operational amplifiers are often used with a symmetrical bipolar power supply, hence the referential potential (ground) is half the total voltage between the negative and the positive supply voltage. While the difference in potential at the two input clamps is zero (e. g. they are shortened), the resulting output voltage of an ideal operational amplifier is also zero volts which is sometimes called virtual ground meaning the output clamp of the op-amp is at a steady reference potential, without being connected directly to the reference potential. Real devices normaly require a differencial DC voltage between the input clamps to make the output zero volts. This parameter is called input offset voltage (Vos) and it is usually around 0.1mV. The gain of integrated circuit op-amps is typically 100000 or more, so a voltage drop of just +0.1V between the two input clamps would result in an output voltage of +10000V. Like mentioned at the chapter about amplifying circuits, the voltage drop at the output can't exceed the value of the supply voltage, so the output signal gets clipped. Situations in which the output voltage is equal or greater than the supply voltage are referred to as saturation of the amplifier. A sine curve with a peak voltage of just 1mV between the input clamps becomes a square wave signal at the output clamp. Negative feedbackFigure 4: If your intention is to amplify a sine wave signal, an op-amp with a gain of more than 100000 is nearly useless. By applying a portion of the output voltage to the inverting input, the total gain of the circuit can be reduced. At the drawing, the signal source is attached between ground and non-inverting input of the op-amp. R1 and R2 are forming a voltage divider between output of the op-amp and ground. The inverting input is attached between R1 and R2. Let's assume the following values: The gain of the op-amp is 100000, R1 = 1kΩ, R2 = 10kΩ. VS- is attached to -12V, VS+ to +12V. The situation is simple when attaching 0V to the non-inverting input: The output voltage is 0V and the resulting voltage drop at R1 and so the inverting input is 0V. What's happening if the input voltage at the non-inverting input jumps up to 0.1V? The resulting differential input voltage is now 0.1V, the gain is 100000, hence the voltage output would climb up to 10000V, but it is clipped at +12V. Now that 12V are attached to the voltage divider, the resulting voltage drop at R1 is (1) Vout = (V+ - V-) * β (2) V- = Vout * R1 / (R1 + R2) By inserting equation (2) in (1) we get: Vout = (V+ - (Vout * R1 / (R1 + R2))) * β Solving the equation for Vout: V+ - Voltage drop between non-inverting input and ground (=input voltage) V- - Voltage drop R1 Vout - Voltage output op-amp R1, R2 - Resistors of the voltage divider β - Voltage gain of op-amp For very high values of β, the equation can be simplified to: The voltage gain Vout / V+ of the op-amp with negative feedback is: Adding a negative feedback via a voltage divider reduces the gain of an op-amp. The overall gain Vout / V+ is called closed-loop gain (ACL), because of the feedback provided by the resistors of the voltage divider. Without a feedback loop, the overall gain of the op-amp is called open-loop gain (AOL) Without negative feedback, an op-amp acts like a comparator. At the drawing, the inverting input is connected to ground, hence the output will be maximum positive if the voltage applied to the non-inverting input is positive. If the input voltage becomes negative, the output will be maximum negative. The output of the op-amp can be either VS- or VS+ so it indicates if the input voltage is larger or lower than 0V. By connecting the inverting input to a voltage divider, the voltage of the non-inverting input can be "compared" to any voltage level between VS- and VS+. When connecting the non-inverting input to the voltage divider, the output signal is VS+ if the voltage at the inverting input is lower than that at the non-inverting input, respectively it becomes VS- if it climbs above that value. Now, the output signal is inverted. Besides the negative feedback described above, we can also apply a positive feedback to an op-amp. Let's have a closer look at the behaviour of the feedback network of the drawing, assuming a resistance of 1kΩ for R1, 10kΩ for R2, a positive supply voltage of +12V and a negative supply voltage of -12V: While connecting the circuit to a voltage source, Ui and V- and so the output voltage are zero, but just a slight random variation of V+ causes the output voltage to tilt either to the maximum or minimum value. Let's assume the output voltage is at it's maximum value and the input voltage is +0.1V. With the feedback via the voltage divider we get: (3) V+ = Ui + (Vout - Ui) * R1 / (R1 + R2) = 1.18V When the input voltage is decreasing to 0V we get (remember the output voltage is still +12V): V+ = 0V + (12V - 0V) * 1000Ω / (1000Ω + 10000Ω) = 1.09V Even at an input voltage of -0.1V we get: V+ = -0.1V + (12V + 0.1V) * 1000Ω / (1000Ω + 10000Ω) = 1.00V The op-amp will tilt to the minimum supply voltage, as soon as V+ drops slightly below 0V. Inserting V+ = 0 and solving equation (3) to Ui gives us: Ui - Input voltage of the circuit Vout - Voltage output of the op-amp R1, R2 - Resistors of the voltage divider To cause the op-amp to switch from +12V to -12V, the input voltage of the circuit must drop slightly below -1.2V. Now that the output voltage is -12V, let's increase the input voltage to 0V: V+ = 0V + (-12V - 0V) * 1000Ω / (1000Ω + 10000Ω) = -1.09V The voltage output of the op-amp at an input level of 0V is now -12V! The input voltage of the circuit must rise above -(-12V) * R1 / R2 = +1.2V to cause the op-amp to switch back to an output voltage of +12V. Input (blue) and corresponding output (red) signal of a Schmitt trigger: On a normal comparator, the op-amp will switch at the same point of the rising and falling edge of the input signal. The output signal of the op-amp will start oscillating while the input signal is near the threshold, whenever the input voltage crosses the threshold because of noise. On a Schmitt trigger, the situation is different: The output signal is low, while the input signal is below a certain threshold and it is high if it is above a different (higher) threshold. The output signal retains it's value while the signal is between the two different input levels. The dual threshold action is called hysteresis. The output voltage of the Schmitt trigger depends not only on the current input voltage, but also on that in the past. For example the output voltage at an input signal of 0.1V can be either +12V or -12V. It is +12V if the input signal was above +1.2V and is now falling to +0.1V. Vice versa the output signal is -12V if the input signal was below -1.2V in the past and is now rising to 0.1V. Single supply voltage Schmitt-TriggerFigure 10: This diagram shows a Schmitt trigger with a single supply voltage. Half the supply voltage is attached to the inverting input using a voltage divider composed of two identic resistors (R3 and R4). Twilight switchFigure 11: Just one practical circuit with a single supply Schmitt trigger: The input voltage is provided by a voltage divider with one constant resistor (R3) and one PNP-phototransistor. If the phototransitor is exposed to light, the resistance decreases, hence the input voltage is decreasing, too. If the lower threshold is reached, the output voltage of the Schmitt-trigger drops down to 0V, hence a device (maybe a lamp) connected to the output clamp is turned off while the sun is shining. R4 is a potentiometer, operating as a voltage divider at the inverting input of the op-amp, by what the threshold can be adjusted, so that the light is switched off during twilight rather than in bright sunlight. Ideal op-amp versus LM324N

Universitat de Barcelona Master in Astrophysics, Particle Physics and Cosmology Spacetime thermodynamics and entanglement entropy: the Einstein field equations Author: Mart´ıBerenguer Mim´o NIUB: 17527646 Advisor: Roberto Emparan September 2019 Ackonwledgements First of all, I would like to thank my tutor and advisor, Roberto Emparan, for all the help and dedication on supervising my thesis for the last months. He has always been there to assist me when needed. Thank you. I would also like to thank my family, for encouraging me during all these years of study, and to who I have stolen a lot of time. Finally, I would like to thank my master classmates. In particular, Oscar and Joseba, for all those sleepless nights working on the assignments, and with whom I have spent an amazing year. 1 Abstract In this master thesis, the posibility of a connection between spacetime dynamics (driven by the Einstein equations) and thermodynamics is discussed. Some known results, like the Raychaudhuri equation or the Unruh effect are reviewed in order to make the presentation self-contained. The Einstein equations are derived in two different ways from thermody- namic arguments. The first one (Section (3)) uses the thermodynamic relation δQ = T dS, together with the proporionality of entropy and horizon area. In the second derivation (Section (4)), the Einstein equations are derived from an hypothesis about entanglement entropy in a maximally symmetric spacetime. Some questions regarding the implications of this thermodynamic interpre- tation of spacetime are discussed as a conclusion of the thesis. 1 Introduction 3 2 Basic previous topics 6 2.1 Raychaudhuri equation ...... 6 2.2 Rindler space ...... 10 2.3 The Unruh effect ...... 12 2.4 Entanglement entropy ...... 18 3 First derivation. Equilibrium thermodynamics in Rindler space 22 4 Second derivation. Entanglement entropy 26 5 Comments and discussion 32 6 Conclusions 36 2 1 Introduction One of the most surprising results of Albert Einstein’s general theory of relativity was the existence of black holes, regions of spacetime were gravity is so strong, that nothing, even light, could escape. The interior of a black hole is separated from the rest of the universe by an event horizon. This means that any particle, massive or massless, that is located inside the black hole, will never be able to escape, and is doomed to reach a singularity in its future, where its proper time suddenly ends and the known theories of physics stop to work. When Stephen Hawking studied black holes from a more mathematical point of view, he found an interesting result: the area of the event horizon never decreases with time and, in general, it will increase. This implies that, if two black holes collide and merge, the area of the final black hole will be larger than the sum of areas of the colliding black holes. This behavior is analogous to the behavior of entropy in thermodynamic systems, where the Second Law of Thermodynamics says that the entropy of a system can never decrease, and that the total entropy of a system is larger than the entropy of its subsystems: Second Law of Black Hole Mechanics: δA ≥ 0 Second Law of Thermodynamics: δS ≥ 0 This analogy is more evident with the First Law of Black Hole Mechanics, which relates the change in mass of a black hole with the change in area of the horizon and the change in angular momentum and electric charge. From here one can see that if the area of the event horizon is analogous to the entropy, then the surface gravity κ is analogous to the temperature: First Law of Black Hole Mechanics: κ δE = 8π δA + ΩδJ + ΦδQ First Law of Thermodynamics: δE = T δS + P δV There is even a Zeroth Law of Black Hole Mechanics: 3 Zeroth Law of Black Hole Mechanics: κ is the same along the horizon in a time-independent black hole. Zeroth Law of Thermodynamics: Temperature is the same in all points in a system in thermal equilibrium. Because of these similarities, Bekenstein proposed that the entropy of a black hole should be proportional to the area of the event horizon, and proposed the Generalized Second Law : the sum of the entropy of the black hole and the entropy of the matter outside the black hole never decreases. The fact that black holes have temperature and entropy implies that they should radiate, but this was completely against the classical picture of black holes. One of the most important results in theoretical physics during the last century was the one obtained by Stephen Hawking in 1974, when he found that, when quantum effects around a black hole are considered, it radiates thermally at the so-called Hawking temperature : 1 T = (1.1) H 8πM Another important result is that this thermal behavior is not exclusive of black hole horizons. In 1976, W.G. Unruh demonstrated the following: the vacuum state, defined by inertial observers, has a thermal character for uniformly accelerated ob- servers with proper acceleration a (Rindler observers) at the Unruh temperature [3, 4]: a T = ~ (1.2) 2π This means that, around any event, in any spacetime, there is a class of observers that will perceive the spacetime as hot. This thermal character of spacetime (not only for black hole horizons) will be of great importance for the following sections. Moreover, this thermodynamic interpretation of spacetime invokes some questions about the structure of spacetime at smallest scales. From standard thermodynamics it is known that a macroscopic system like, for example, a gas, can be described with some thermodynamic variables, like the temperature or the entropy, but for a long time, the real meaning of these variables was unknown. It was Boltzmann who gave an explanation to these variables, essentially saying “if you can heat it, it has microscopic degrees of freedom”. Before that, it was considered that matter was continuous even at the smallest scales, and the concepts of heat and temperature were 4 added “by hand”. Boltzmann used the discrete interpretation of matter and found that the thermodynamic fenomena were related with the averages of the properties of these microscopic degrees of freedom. This is profound. It tells that the existence of microscopic degrees of freedom leaves a signature at macroscopic scales, in the form of temperature and heat. Then, if spacetime is seen as hot by some observers, what are the microscopic degrees of freedom that give raise to the temperature and the entropy? There are many ap- proaches that try to give an interpretation to these microscopic degrees of freedom [5, 6], but there is not a clear answer yet. What seems reasonable is that, if spacetime is, at its deepest level, a thermodynamic entity, we should be able to derive the equa- tions that drive its evolution (the Einstein equations) from a purely thermodynamic point of view. This is what we will do in Sections (3) and (4). In Section (3), the Einstein equations are derived from the thermodynamic relation δQ = T dS and the propor- tionality of the entropy and horizon area, working from the point of view of a Rindler observer in the neighbourhood of the causal horizon of the Rindler space. In Section (4), an alternative derivation of the Einstein equations will be given, based in the assumption that the entanglement entropy in a geodesic ball is maximal when the geometry and the quantum fields are varied from maximal symmetry. Section (2) includes some of the conceptual ideas and equations that will be necessary for the two derivations. In Section (5), some comments about the derivations and some questions about the implications of them are considered, while Section (6) contains the main conclu- sions of the thesis. 5 2 Basic previous topics This section contains some topics that will be necessary to have in mind during the two derivations of the Einstein equations in the following sections. Here we will briefly talk about the Raychaudhuri equation, Rindler space, the Unruh effect, and entanglement entropy. 2.1 Raychaudhuri equation The Raychaudhuri equation is an evolution equation for what is called the expansion of a congruence of geodesics. In order to understand the meaning of the expansion (and two more quantities that appear in the equation, the shear and rotation), it is useful to think first about the kinematics of a deformable medium. Suppose, in a purely Newtonian context, a two-dimensional medium, with some internal motion whose dynamics are not of our interest. From a purely kinematic point of view, we can always write that, for an infnitesimal displacement ξa from a reference point O, dξa = Ba (t)ξb + O(ξ2) (2.1) dt b a for some tensor B b , which depends on the internal dynamics of the medium. For short intervals of time, a a a ξ (t1) = ξ (t0) + ∆ξ (t0) (2.2) where a a b 2 ∆ξ (t0) = B b (t0)ξ (t0)∆t + O(∆t ) (2.3) a and ∆t = t1 −t0. To describe the action of B b we will consider the situation that a ξ (t0) = r0(cos φ, sin φ); that is, a circle of radius r0 in the two-dimensional medium. Expansion a Suppose that B b is a pure-trace matrix, i.e., proportional to the identity, with the form 1 a 2 θ 0 B b = 1 0 2 θ a 1 In this case, ∆ξ = 2 θr0∆t(cos φ, sin φ), which corresponds to a change in the 1 circle’s radius by an amount 2 θr0∆t. The correspoonding change in area is given by 6 2 ∆A = A1 − A0 = πr0θ∆t (2.4) This means that 1 ∆A θ = (2.5) A0 ∆t θ measures the fractional change of area per unit time, and is called the expansion parameter. Shear a Suppose now that B b is symmetric and trace-free: a σ+ σ× B b = σ× −σ+ a In this case, ∆ξ = r0∆t(σ+ cos φ + σ× sin φ, −σ+ sin φ + σ× cos φ). If σ× = 0, we have an ellipse with the major axis oriented in the φ = 0 direction. If σ+ = 0, what we have is an ellipse oriented in the φ = π/4 direction. The general situation is an ellipse oriented along an arbitrary direction. The area of the figure is not affected by a the action of B b . What we have is a shearing of the figure, and the parameters σ× and σ+ are called the shear parameters. Rotation a Finally, if B b is antisymmetric, 0 ω Ba = b −ω 0 a a 0 0 we have that ∆ξ = r0ω∆t(sin φ, − cos φ), and ξ (t1) = r0(cos φ , sin φ ), with φ0 = φ − ω∆t. This corresponds to an overall rotation of the original figure, keeping the area fixed. ω is called the rotation parameter. The most general decomposition of this tensor into algebraically irreducible com- ponents under rotations is 1 a 2 θ 0 σ+ σ× 0 ω B b = 1 + + 0 2 θ σ× −σ+ −ω 0 which can also be expressed as 1 B = θδ + σ + ω (2.6) ab 2 ab ab ab 7 a 1 where θ = B a (the expansion scalar) is the trace part of Bab , σab = B(a,b) − 2 θδab (the shear tensor) is the symmetric-tracefree part of Bab , and ωab = B[a,b] (the rotation tensor) is the antisymmetric part of Bab . For a three-dimensional medium, the decomposition is the same, but with a prefactor of 1/3 instead of 1/2 in the trace term, and the interpretation of the expansion, shear and rotation are the same, but changing the area by the volume. Once the classical 2-dimensional medium has been introduced, we can move now to the study of congruences of (for now, timelike) geodesics. Let O be an open region of spacetime. A congruence of geodesics in O is a family of geodesics such that through each point in O passes one and only one geodesic from this family. We will assume that the geodesics are timelike. We are interested in the evolution of the deviation vector ξa between two neighbouring geodesics in the congruence as a function of the proper time τ (see Figure (1)). Figure 1: Two neighboring geodesics, with a deviation vector ξa as a function of τ. Let ua be the (timelike) tangent vector to the geodesics. Then, the spacetime metric gab can be decomposed in a longitudinal part −uaub and a transverse part hab, hab = gab + uaub (2.7) The transverse metric hab is purely spatial, in the sense that it is orthogonal to ua. We introduce now the tensor field Bab = ∇bua (2.8) 8 This tensor determines the evolution of the deviation vector ξa. To see this, note b a b a that from u ∇bξ = ξ ∇bu we obtain b a a b u ∇bξ = B b ξ (2.9) a a That is, B b measures the failure of ξ to be parallel transported along the con- gruence. Equation (2.9) is analogous to (2.1), and therefore we can decompose the a tensor B b in the same way as before, with the same interpretation for the diffrent terms that appear: 1 B = θh + σ + ω (2.10) ab 3 ab ab ab In order to find the evolution equation for the expansion θ, we can start by finding an evolution equation for Bab : c c u ∇cBab = u ∇c∇bua c d = u ∇b∇aua − Radbcu c c d = u ∇b∇cua − Radbcu u (2.11) c c c d = ∇b (u ∇cua) − (∇bu )(∇cua) − Radbcu u c c d = −B b Bac − Radbcu u Taking the trace of this equation, we obtain dθ = −BabB − R uaub (2.12) dτ ba ab ab 1 2 ab ab Now, from the definition of Bab , we find that Bab B = 3 θ + σ σab − ω ωab, so (2.12) becomes dθ 1 = − θ2 − σabσ + ωabω − R uaub (2.13) dτ 3 ab ab ab which is kown as the Raychaudhuri equation, and gives the evolution of the expansion parameter θ for a congruence of timelike geodesics. For the case of null geodesics, which is the one that will be of interest in the following sections, the line of argument is the same as for timelike geodesics, but the calculation is a bit more tedious because of the difficulty to precisely define the transverse spacetime. In the above case it was simply the spatial components, but in the case of null geodesics, if ka is the (null) tangent vector to the geodesics, the orthogonal space to ka includes ka because it is orthogonal to itself. Once this technical part is solved, the logic 9 of the derivation and the result are very similar. The Raychaudhuri equation for a congruence of null geodesics reads: dθ 1 = − θ2 − σabσ + ωabω − R kakb (2.14) dλ 2 ab ab ab 2.2 Rindler space The Rindler space is introduced when one is interested in the motion of an accelerated observer in flat spacetime. This will be necessary in Section (3), where the whole argumentation line will be centered from the perspective of an accelerated observer in an approximately flat region of spacetime. For simplicity, let’s consider the 2-dimensional Minkowski space, whose metric, in the usual (t, x) coordinates is ds2 = −dt2 + dx2 (2.15) An observer moving at a uniform acceleration α will follow the trajectory xµ(τ) given by 1 t(τ) = sinh(ατ) (2.16) α 1 x(τ) = cosh(ατ) (2.17) α This can be checked taking into account that the components of the 4-acceleration D2xµ d2xµ aµ = = (2.18) dτ 2 dτ 2 where the covariant derivative is equal to the ordinary derivative because the Christoffel symbols vanish in these coordinates, are given by at = α sinh(ατ) (2.19) ax = α cosh(ατ) (2.20) In this way, the magnitude of the acceleration is q p µ 2 2 2 2 aµa = −α sinh (ατ) + α cosh (ατ) = α (2.21) Thus, this trajectory corresponds to a uniformly accelerated observer. The tra- jectory of this observer obeys 10 1 x2(τ) = t2(τ) + (2.22) α2 which is an hyperboloid asymptoting to null paths x = −t in the past and x = t in the future (see Figure (2)). Figure 2: Minkowski spacetime in Rindler coordinates. An observer with constant acceleration in the +x direction follows the hyperbolic trajectories dranw in region I. The patches H+ and H− act as horizons for this class of observers. Notice from (2.22) that the larger the acceleration α, the closer the trajectory is to the patches x = −t and x = t. This fact will be important in Section (3). We can define new coordinates (η, ξ) in the following way: 1 1 t = eaξ sinh(aη), x = eaξ cosh(aη), (x > |t|) (2.23) a a which cover the wedge x > |t| (region I in Figure (2)). Although these are not the usual Rindler coordinates, they are the most appropiate for the derivation of the Unruh effect, which is the purpose of the next section. Notice that an accelerated observer with acceleration α = a follows a world line that is given by ξ = const and η = τ. In these coordinates, the metric is given by 11 ds2 = e2aξ −dη2 + dξ2 (2.24) Region I, with these coordinates, is known as Rindler space (although it is only a part of Minkowski space). A Rindler observer is an observer moving along a constant acceleration path (in the diagram, this corresponds to the hyperbolic trajectories). Because the metric components are independent of η, the vector ∂η is a Killing vector. In the (t, x) coordinates, this Killing vector is ∂t ∂x ∂ = ∂ + ∂ η ∂η t ∂η x aξ (2.25) = e [cosh(aη)∂t + sinh(aη)∂x ] = a (x∂t + t∂x ) Notice that the patches x = −t and x = t (H− and H+) act as Killing horizons for this vector field, because its norm vanishes (only) there: ∂ ∂ V = a x + t ⇒ V V µ = a2 t2 − x2 = a(t + x)(t − x) (2.26) ∂t ∂x µ The surface gravity of this Killing horizon is r 1 κ = − ∇µξν∇ ξ = a (2.27) 2 µ ν Although there is no gravitational field (we are in flat spacetime), the surface gravity characterizes the acceleration of the Rindler observers. It will be convenient for the Unruh effect to define coordinates (η, ξ) for the region IV, by flipping the signs of those defined in region I: 1 1 t = − eaξ sinh(aη), x = − eaξ cosh(aη), (x < |t|) (2.28) a a 2.3 The Unruh effect The basic statement of the Unruh effect is that an accelerating observer in flat space will observe the Minkowski vacuum as a thermal spectrum of particles. The basic idea of this result is the fact that observers with different notions of positive and negative frequency modes will disagree on the particle content of a given state. 12 In flat spacetime, this problem does not arise for non-accelerated (inertial) ob- servers. For inertial observers, we introduce a set of positive and negative frequency modes, and the fields are expressed as a combination of these modes, interpreting the operator coefficients as creation and annihilation operators. In flat spacetime we can choose a natural set of modes by demanding that they are positive-frequency modes with respect to the time coordinate. Obviously, the time coordinate is not unique, because we can perform Lorentz transformations, but the vacuum state and the number operators are invariant under these transformations. In curved spacetime (or accelerated observers) we can find a set of modes, but we can find many other sets that are equally good, and the notion of vacuum and number operators will be very sensitive to the set we choose. We can always find a set of orthonormal modes fi, and expand the fields in terms of these modes: X † ∗ φ = aˆifi +a ˆi fi (2.29) i † where the operatorsa ˆi anda ˆi obey the usual commutation relations. We can define a vacuum state |0f i, which will be annihilated by all the annihilation operators, aˆi |0f i = 0, ∀i (2.30) From this vacuum we can define an entire Fock basis, defining the excitations as † the states created by the action of ai . The number operator can be defined too, † nˆfi =a ˆi aˆi (2.31) where the subscript f makes reference to the fact that this operator is defined with respect to the set of modes fi. But we can find another complete basis with respect to which expand the fields, X ˆ ˆ† ∗ φ = bigi + bi gi (2.32) i ˆ ˆ† where, again, bi and bi obey the usual commutation relations. The vacuum state, ˆ the Fock basis, and the number operator for the bi operators are defined in the same way as for the operators of the fi modes: ˆ ˆ†ˆ bi |0gi = 0 ∀i;n ˆgi = bi bi (2.33) If one observer defines particles with respect to the set of modes fi and a different observer defines particles with respect to the set of modes gi, in general they will 13 disagree on the number of particles they observe. To see this, we can expand each set of modes in terms of the other: X ∗ gi = αijfi + βijfj (2.34) j X ∗ ∗ fi = αjigj − βjigj (2.35) j The transformation that allows to write one set of modes in terms of the other is called Bogoliubov transformation, and the coefficients αij, βij are called Bogoliubov coefficients, which satisfy the normalization conditions X ∗ ∗ αijαjk − βikβjk = δij (2.36) j X (αikβjk − βikαjk) = 0 (2.37) j and can be used to relate not only the modes, but also the operators: X ˆ ∗ ˆ† aˆi = αijbj + βjibj (2.38) j ˆ X ∗ ∗ † bi = αijaˆj − βijaˆj (2.39) j The discrepancy on the number of particles can be seen from the following cal- culation: imagine that the system is in the f-vacuum (in which the observer using the fi modes would not see any particle). We want to know the number of particles that an observer using the g-modes will observe. Then, we compute the expectation value of the g number operator in the f-vacuum: 14 ˆ†ˆ h0f | nˆgi |0f i = h0f | bi bi |0f i X † ∗ † = h0f | αijaˆj − βijaˆj αikaˆk − βikaˆk |0f i jk X ∗ † = βijβik h0f | aˆjaˆk |0f i jk X ∗ † = βijβik h0f | aˆkaˆj + δjk |0f i jk (2.40) X ∗ = βijβikδjk h0f |0f i jk X ∗ = βijβij j X 2 = |βij| j X 2 ⇒ h0f | nˆgi |0f i = |βij| (2.41) j In general, this coefficient does not vanish: an observer that defines particles with respect to the g-modes will detect particles where the observer that defines particles with respect to the f-modes will see the vacuum. This can be applied to the case of an accelerated observer in flat spacetime (Rindler observer). For simplicity, we will consider a massless Klein-Gordon field in 2 dimensions. The Klein-Gordon equation in Rindler coordinates takes the form −2aξ 2 2 2φ = e −∂η + ∂ξ φ = 0 (2.42) which admits as solutions normalized plane waves of the form 1 −iωη+ikξ gk = √ e (2.43) 4πω with ω = |k|. Because the choice of coordinates for regions I and IV needed a difference of sign between them, we need to define two sets of modes, one for each of the two regions : √ 1 e−iωη+ikξ I g(1) = 4πω k 0 IV 15 0 I g(2) = k √ 1 e+iωη+ikξ IV 4πω In this way, each set of modes is positive-frequency with respect to the corre- sponding future-directed timelike Killing vector, (1) (1) ∂ηgk = −iωgk (2.44) (2) (2) ∂(−η)gk = −iωgk (2.45) Introducing the corresponding creation and annihilation operators for each region, the field can be expressed as Z ˆ(1) (1) ˆ(1)† (1)∗ ˆ(2) (2) ˆ(2)† (2)∗ φ = dk bk gk + bk gk + bk gk + bk gk (2.46) The modes to which we will compare them will be the usual Minkowski modes, which expand the field as Z † ∗ φ = dk aˆkfk +a ˆkfk (2.47) The Minkowski vacuum state |0M i and the Rindler vacuum state |0Ri are defined as usual: aˆk |0M i = 0 (2.48) ˆ(1) ˆ(2) bk |0Ri = bk |0Ri = 0 (2.49) The next step now is to compute the Bogoliubov coefficients relating both sets of modes, and compute the expectation value of the Rindler number operator in the Minkowski vacuum. This is a bit tedious, and the usual procedure is the following: (1) instead of using the previously defined Rindler modes, we will take a set of modes hk , (2) hk that share the same vacuum state as the Minkowski modes (but the excited states are different). The way to do this is to start with the Rindler modes, analytically extend them to the entire spacetime, and express them in terms of the orignal Rindler modes. Then, the field will be expanded as Z (1) (1) (1)† (1)∗ (2) (2) (2)† (2)∗ φ = dk cˆk hk +c ˆk hk +c ˆk hk +c ˆk hk (2.50) The properly normalized version of these modes is 16 (1) 1 πω/2a (1) −πω/2a (2)∗ hk = e gk + e g−k (2.51) q πω 2 sinh a (2) 1 πω/2a (2) −πω/2a (1)∗ hk = e gk + e g−k (2.52) q πω 2 sinh a Just like before, the Bogoliubov coefficients allow to relate also the creation and annihilation operators: ˆ(1) 1 πω/2a (1) −πω/2a (2)† bk = e cˆk + e cˆ−k (2.53) q πω 2 sinh a ˆ(2) 1 πω/2a (2) −πω/2a (1)† bk = e cˆk + e cˆ−k (2.54) q πω 2 sinh a In this way, the Rindler number operator in region I, (1) ˆ(1)†ˆ(1) nˆR (k) = bk bk (2.55) (1,2) can be expressed in terms of the new operatorsc ˆk , and because they share the same vacuum state as the Minkowski modes, we have that (1) (2) cˆk |0M i =c ˆk |0M i = 0 (2.56) The fact that the excited states do not coincide is not a problem, because we are only interested in what the Rindler observer sees when the state is the Minkowski vacuum. For a Rindler observer in region I, the expectation value of the number operator will be (1) ˆ(1)†ˆ(1) h0M | nˆR (k) |0M i = h0M | bk bk |0M i 1 −πω/a (1) (1)† = πω h0M | e cˆ−kcˆ−k |0M i 2 sinh a −πω/a (2.57) e (1) (1)† = πω h0M | cˆ−kcˆ−k |0M i 2 sinh a 1 = e2πω/a − 1 This result corresponds to a Planck spectrum with temperature 17 a T = (2.58) 2π Thus, a uniformly accelerated observer through the Minkowski vacuum will detect a thermal flux of particles. 2.4 Entanglement entropy In order to conceptually understand entanglement entropy, it is useful to first take a look to the following discrete problem : imagine a lattice model, with discrete degrees of freedom located at the lattice sites, which are separated a distance (see Figure (3)). At each site (labeled by α) we have a finite-dimensional Hilbert space Hα (for instance, a qubit per site). A pure quantum state of the system can be written as |Ψi ∈ ⊗αHα (2.59) Figure 3: Discrete lattice system, with a Hilbert space at each place. The grey region is called A, while Ac is its complementary, separted by the boundary ∂A. The distance between places is . We can divide the lattice system into two complementary subsystems, namely A and Ac, separated by the boundary ∂A, which we shall call the entangling surface, 18 as can be seen in Figure (3). The Hilbert space of the total system has been split into the direct product of two Hilbert spaces, ⊗α Hα = HA ⊗ HAc (2.60) Now, one can construct the reduced density matrix of the subsystem A, which is constructed by tracing out the degrees of freedom of Ac: ρA = TrAc (|Ψi hΨ|) (2.61) If the state |Ψi is factorized when the system is split, then we will have a pure state in HA. However, if the state can not be written as a direct product of states from the two subsystems, the state is entangled and the density matrix gives the probabilities for the ocurrence of the states in HA. The amount of entanglement that exists in |Ψi when the system is split is quantified by the Von-Neumann entropy of the reduced density matrix, or entanglement entropy, which is given by SA = − TrA(ρA log ρA) (2.62) In a discrete system, this can be computed diagonalizing the density matrix and obtaining its eigenvalues λi. Then, the entanglement entropy is simply X SA = − λi log λi (2.63) i Because |Ψi is a pure state, it can be decomposed via the Schmidt decomposition, P |Ψi = i λi |αiiA |βiiAc . This tells us that non-trivial eigenvalues of ρA are the same as those of Ac. Then, the traces are the same, and the entanglement entropies are also the same: SA = SAc (2.64) The fact that the entropy is the same for both regions means that it can not depend on the size of each region, but only on the degrees of freedom shared by the two regions. That is, it must be proportional to the area of the boundary ∂A instead of being proportional to the volume of the regions, as it would be expected in classical thermodynamic systems. The continuum limit of this system can be defined as taking the limit → 0. When this is done, the result for the entropy is sensitive to the ultra-violet (UV) physics, as we should expect. For a d-dimensional free field theory, the entropy is a UV-divergent quantity, with the leading term being proportional to the area of the entangling surface : 19 Area(∂A) S = γ + ... (2.65) A d−2 where γ is a constant that depends on the model used. This quantity is divergent when → 0 unless there is some physical UV cutoff (presumably, of the order of 2 the Planck scale), with which the entropy would be finite and proportional to A/Lp, matching with the Bekenstein-Hawking entropy for black holes [9–12]. Thus we shall assume in all cases that due to the UV physics, the entanglement entropy is finite in small regions, with a leading term given by S=ηA. Another important quantity that will be useful is the relative entropy. Given two density matrices ρ and σ we can define the relative entropy, S(ρ||σ) = Tr(ρ log ρ) − Tr(ρ log σ) (2.66) which gives information about the distinguishability between the two density matrices. An important property of the relative entropy is that it is always positive or equal to zero, being equal to zero only when the two density matrices are the same. We can define the modular hamiltonian as Kρ = − log ρ (2.67) and rewrite the relative entropy as S(ρ||σ) = Tr(ρ log ρ) − Tr(ρ log σ) + Tr(σ log σ) − Tr(σ log σ) = −S(ρ) + Tr(ρKσ) − Tr(σKσ) + S(σ) (2.68) = ∆hKi − ∆S where ∆S = S(ρ) − S(σ) is the entropy difference between the states, and ∆hKi = Tr(ρKσ)−Tr(σKσ) is the difference in the expectation values of the modular hamiltonian Kσ for ρ and σ. If we consider σ to be a reference state σ = ρ0, and ρ a state close to it, we can 2 expand the latter in a power series in a parameter λ, ρ(λ) = ρ0 + λρ1 + λ ρ2 + ... in such a way that ρ(0) = ρ0 = σ. The relative entropy can be expanded as d 2 S(ρ(λ)||σ) = S(ρ(0)||σ) + S(ρ(λ)||σ) λ + O(λ ) (2.69) dλ λ=0 The first term is zero because of the definition of the relative entropy. The term of order λ is also zero, because the relative entropy is a monotonically increasing 20 function around σ. Thus, the relative entropy is at least quadratic in the deviation parameter. This means that, for first-order variations, we have that δS = δhHσi (2.70) This is known as the first law of entanglement entropy. 21 3 First derivation. Equilibrium thermodynamics in Rindler space In this approach, Einstein’s equations are derived from the proportionality of entropy and horizon area, together with the thermodynamic relation δQ = T dS, relating heat, temperature and entropy (and area, due to the relation entropy ∼ area). In standard thermodynamics, heat is defined as energy that flows from, or to a thermodynamic system. Here, we shall define heat as energy that flows across a causal horizon (not necessarily a black hole horizon). In the relation δQ = T dS, we associate δQ with an energy flux across the horizon, and we shall use that the entropy is proportional to the area of this horizon. It remains to identify the temperature T . Using Unruh’s results, we can take T to be the Unruh temperature if we consider that the observer is in accelerated motion. Then, for consistency, the heat flow must be defined as the energy flux that this observer measures. In order to apply local equilibrium thermodynamics, two conditions must be imposed in the construction of our system: • We need the observer to be as near as possible to the horizon. In the limit that the accelerated worldine approaches the horizon, the acceleration diverges, and so do the temperature and the energy flux, but their ratio remains finite. • In general, the horizon will be expanding, contracting or shearing. In order to impose equilibrium, we need the expansion, shear and rotation to be zero at first order in a neighbourhood of the horizon. The introduction of an accelerated observer gives as a natural choice for the horizon the Rindler horizon associated to this accelerated observer. The key idea to be shown can be expressed as : “In order to satisfy the thermodynamic equilibrium relation δQ = T dS, in- trepreted in terms of the energy flux and area of local Rindler horizons, the grav- itational lensing by matter energy must distort the causal structure of spacetime in a way that the Einstein equation holds.” The next step is to define precisely this local causal horizon. It can be done as follows: By means of the equivalence principle, the neighbourhood of any point p can be thought as a piece of flat spacetime. Around p we consider a 2-dimensional surface P. As usual, this 2-surface will be represented as a point in the conformal diagram. The boundary of the past of P has two components, each of which is a null surface generated by a congruence of null generators ka orthogonal to P. The local causal 22 horizon is defined as one of these two components. We take λ as the affine parameter for ka, in such a way that λ vanishes at P and is negative to the past of P (see Figure (4)). Figure 4: Rindler horizon H of a 2-sphere P. The accelerated observer follows the trajectory of the Killing vector χa. ka is the generator of the horizon. In order to define the temperature and the heat, note that in the approximately flat region around p the usual Poincar´esymmetries hold. In particular, there is an approximate Killing field χa generating boosts orthogonal to P and vanishing at P. Because we are at very short distances, the Minkowski vacuum state (or any other state) is a thermal state with temperature T = ~a/2π with respect to the boost hamiltonian, where a is the acceleration of this orbit. The heat flow is then defined a through the boost-energy current of matter, Tab χ , where Tab is the stress-energy tensor. The Killing field defining the orbits of Rindler observers coincides at the null surface with the generators for sufficiently accelerated observers. Then, in the limit that the observer is sufficiently close to the horizon, the Killing field χa is parallel to the horizon generator ka, and, at first order, we have that χa = −κλka and dΣa = kadλdA, where dA is the area element on a cross section of the horizon [6, 13, 14]. Then, the heat flux is given by 23 Z Z a b a b δQ = Tab χ dΣ = −κ λTab k k dλdA (3.1) H H Assuming that the entropy is proportional to the area, we have that dS = ηδA, where δA is the area variation of a cross section of a pencil of generators of H. For now, the constant η is left undetermined. The area variation is given by Z δA = θdλdA (3.2) H where θ is the expansion of the horizon generators. The expression δQ = T dS ∝ δA is telling that the presence of the energy flux is associated with a focussing of the horizon generators. Then, the Raychaudhuri equation (2.14) enters in the game, because it tells precisely the rate of focussing of the generators. The stationarity conditions imposed above imply that, at p, both the expansion and the shear vanish1, and the Raychaudhuri equation simplifies to dθ = −R kakb (3.3) dλ ab where the θ2 and σ2 are higher-order contributions that can be neglected when integrating to find θ around P. For a small interval of λ, this integration is simply a b θ = −λRabk k . Then, Z a b δA = − λRabk k dλdA (3.4) H ~κ Now, from (3.4) and (3.1), we see that δQ = T dS = 2π ηδA is valid if η T kakb = ~ R kakb (3.5) ab 2π ab is valid for all null vectors ka. This is equivalent to the tensorial equation 2π Tab = Rab + fgab (3.6) ~η for some undetermined function f. The stress-energy momentum is divergence- free, which means that the rhs of (3.6) must also be divergence-free. This gives the R constraint f = − 2 + Λ for some undetermined constant Λ. Then, we find: 1It is always possible to find a 2-surface P so that both the expansion and shear vanish in a first order neighbourhood of p. 24 1 2π Rab − Rgab + Λgab = Tab (3.7) 2 ~η If η = 1 , as the Bekenstein-Hawking entropy formula tells, we recover the 4G~ Einstein equation: 1 R − Rg + Λg = 8πGT (3.8) ab 2 ab ab ab Thus, the Einstein equation appears from the relation δQ = T dS, from a purely thermodynamic point of view. 25 4 Second derivation. Entanglement entropy In this derivation, the Einstein equation appears as a consequence of a maximal vacuum entanglement hypothesis in a small region of spacetime. The main hypothesis can be expressed as : “When the geometry and quantum fields are simultaneously varied from maximal symmetry, the entanglement entropy in a small geodesic ball is maximal at fixed volume.” The system to consider now can be defined as follows: Consider any point o of a spacetime of dimension d. If we choose a timelike unit vector ua, we can generate a (d − 1)−dimensional spacelike ball Σ of radius l if we consider all the geodesics of length l that leave p in all directions orthogonal to ua. The point p is located at the center of the ball, and we call the surface of the ball ∂Σ. The region causally connected to the sphere Σ is called the causal diamond (see Figure (5)). We will consider that the radius l of the ball is much smaller than the characteristic radius of curvature of the spacetime in that region: l Lcurv. Figure 5: Causal diamond associated to a geodesic ball centered at o and geodesic radius l. It is known that, at sufficiently short distances, all the fields look like the vacuum state. Moreover, if the condition l Lcurv is satisfied, the spacetime around p can be treated as flat. Then, when we perform the variations with respect to the geometry 26 and the quantum states, we will perform them with respect to flat space, and to the vacuum state. A way of interpreting geometrically the Einstein equation is the following: in classical vacuum (without any matter source), any small geodesic ball of given volume has the same area as in flat spacetime. However, when there is a source of matter or energy (given by some expectation value of the stress-energy tensor), curvature causes a spatial ball of given volume to have a smaller surface area than it would have in flat spacetime. This area deficit can be computed at fixed geodesic radius, or at fixed volume. The expressions are given by d Ωd−2l δA = − R (4.1) l 6(d − 1) d Ωd−2l δA = − R (4.2) V 2(d2 − 1) ik where R = RikR is the spatial Ricci scalar at p. Note that d + 1 δA = δA (4.3) l 3 V For convenience, as will be seen at the end, we will take the variations to be at fixed volume. To connect this expression with the Einstein equation, note that the spatial Ricci scalar can be related to the 00-component of the Einstein tensor as follows: 1 1 1 1 G = R − Rg = R − 2R 0 = R ik = R (4.4) 00 00 2 00 2 0 2 ik 2 Then, we can write d Ωd−2l δA = − G (4.5) V d2 − 1 00 and, by virtue of the Einstein equation, d 8πGΩd−2l δA = − T (4.6) V d2 − 1 00 Under a simultaneous variation of the geometry and the quantum fields, the variation of the entanglement entropy will have two contributions: a UV-contribution δSUV from the area change when the metric is varied with respect to flat spacetime (δgab), and an IR-contribution δSIR due to the variation of the fields (δ |ψi), so we can write 27 δS = δSUV + δSIR (4.7) We shall assume that the UV-part of the entanglement entropy is finite at leading order, and is proportional to the area variation computed above. That is, δSUV = ηδA. As in the previous derivation, the constant η is left undetermined until the end. In order to compute δSIR, we take into account that the vacuum state of any QFT, when restricted to the diamond, can be written as a thermal density matrix, 1 ρ = e−K/T (4.8) Z where T = ~/2π, and K is the modular hamiltonian. From this thermal density matrix, the entropy variation can be computed and it is given by δSIR = δhKi. In general, K is not a local operator, and there is not a general expression for it. However, in the case of the vacuum of a conformal field theory (CFT), the situation is different. The diamond has a conformal boost Killing vector generating it (see Figure (5)), given by 1 ζ = l2 − u2 ∂ + l2 − v2 ∂ (4.9) 2l u v in null coordinates u, v, or 1 ζ = l2 − r2 − t2 ∂ − 2rt∂ (4.10) 2l t r in the usual t, r coordinates. For the vacuum of a CFT, there is a conformal transformation relating the diamond to Rindler space and, in this case, K is equal to Hζ , the Hamiltonian generating the flow of the above Killing vector [9, 10, 16], which means that Z 2π ab Hζ = T ζb dΣa (4.11) ~ Σ With the previous Killing vector on the t = 0 surface, we obtain Z Z 2 2 2π a b 2π d−1 l − r δhKi = δhTab iζ dΣ = d x δhT00 i (4.12) ~ ~ 2l If we consider that δhT00 i is constant within the ball, it can be taken out of the integral, and we have 28 Z 2 2 2π d−1 l − r δhKi = δhT00 i d x ~ 2l Z Z l 2 2 2π d−2 l − r = δhT00 i dΩd−2 r dr (4.13) ~ 0 2l d 2π Ωd−2l = δhT00 i ~ d2 − 1 This result, together with the one providing the area variation at fixed volume, gives δS V = ηδA + δhKi d Ωd−2l 2π (4.14) = −ηG00 + δhT00 i d2 − 1 ~ Now, imposing the assumption that the entanglement entropy is maximal at fixed volume (that is, δS V =0), we obtain the relation 2π G00 = δhT00 i (4.15) ~η If we require this variation to vanish at all points and with all timelike unit vectors, we obtain a tensor equation, 2π Gab = δhTab i (4.16) ~η This is the Einstein equation, provided we define the constant η to be η = 1 1 , 4 ~G which is the precise value required by the Bekenstein-Hawking entropy formula. For the non-CFT case, K is not given by (4.11), and some assumptions must be made in order to find an expression for δhKi. The main conjecture is to consider that δhKi is given by Ω ld δhKi = d−2 (δhT i + δX) (4.17) d2 − 1 00 where δX is a spacetime scalar, maybe related to the trace of Tab . Calculations [17, 18] support this assumption, although it is still being investigated. When the maximization of entropy is considered, one obtains 29 2π Gab = (δhTab i − δhXigab) (4.18) ~η This result has a problem, because from the Bianchi identity, the lhs of (4.18) is divergence-free, and so is the term δhTab i because of energy-momentum conservation. This implies that ∇aδhXi = 0 and, if it is related to the trace of Tab , it is a too strong constraint. This problem can be solved if, instead of comparing it to the Minkowski vacuum, the variations are compared to some other maximally symmetric spacetime (MSS), because any MSS seems as good candidate for the vacuum as flat spacetime. The MSS Einstein tensor in a MSS of curvature scale λ is given by Gab = −λgab. When the area variation is compared to this MSS, the area variation at fixed volume is given MSS by the same expression as before, but replacing G00 by G00 − G00 . The variation of entropy reads now 2π δS = ηδA + δhKi V V ~ d (4.19) Ωd−2l 2π = −η (G00 + λg00) + (δhT00 i + δX) d2 − 1 ~ Again, when we consider that the variation vanishes at all points and with all timelike unit vectors, the equation becomes a tensorial equation, 2π Gab + λgab = (δhTab i − δXgab) (4.20) ~η Taking the divergence of this equation, the term of the Einstein tensor and the term of the stress-energy tensor vanish, because of the Bianchi identity and the con- servation of energy-momentum, respectively. Then, we obtain a constraint between λ and δX: 2π Λ = δX + λ (4.21) ~η where Λ is a spacetime constant. When this relation is plugged into (4.20), we obtain 2π Gab + Λgab = δhTab i (4.22) ~η 30 This is the Einstein equation with a cosmological constant Λ, provided that, again, η = 1 1 , in agreement with the Bekenstein-Hawking entropy. Thus, the 4 ~G Einstein equations have been derived from an entanglement entropy hypothesis. 31 5 Comments and discussion This section contains a discussion about some issues related to the derivations, to- gether with some of the main questions that arise from this new interpretation of spacetime, and the possible answers (more or less satisfactory) that can be given with the current knowledge of physics. • Why are the variations taken at fixed volume instead of at fixed geodesic radius? In the derivation based on entanglement entropy, we have taken the variations at fixed volume “for convenience”. We argue here why this has been done. First of all, notice that we have obtained the desired result because the geometric d 2 term Ωd−2l /(d − 1) that appears as a prefactor in both variations is the same for δSUV and δSIR, and it can be factorized. Had we taken the variations at fixed geodesic radius instead of fixed volume, the terms would have not been the same. But there are other arguments to take the variations at fixed volume. The first law of causal diamonds is a variational identity, analogous to the first law of black hole mechanics, which relates variations, away from flat spacetime, of the area, volume, and cosmological constant inside the diamond. The first law reads κk V − δV + ζ δΛ = T δS (5.1) 8πG 8πG gen where κ is the surface gravity of the Killing horizon, k is the trace of the outward R extrinsic curvature of the boundary ∂Σ when embedded in Σ, and Vζ ≡ Σ |ζ|dV . Λ is the cosmological constant, T is minus the Hawking temperature, and Sgen is defined as the sum of the horizon entropy and the entanglement entropy of matter. At fixed volume and fixed cosmological constant, the first law of causal diamonds implies that the entropy is stationary when varied away from the vacuum, as it has been considered in the derivation. • What is the best way to proceed if we want to find a quantum theory of gravity? The fact that spacetime dynamics can be derived from thermodynamic arguments suggests the possibilty that gravity is not a fundamental force, but a macroscopic result of some microscopic degrees of freedom of spacetime [20–22]. These degrees of freedom have been called by some authors as “Atoms of Spacetime”, in analogy to the standard relation between thermodynamics and statistical mechanics. 32 If this is the case, it explains why the quantization of General Relativity has shown to be much more problematic than for other microscopic forces. In [21, 22], some properties that these atoms of spacetime should have are dis- cussed, and with a particular model of atoms of spacetime for the geometric part of the action, the Einstein’s equations are recovered from a purely thermodynamic argument. Other works [23, 24] have used particular models of microstructure to recover the Hawking temperature and entropy for black holes. • Could we obtain higher-curvature corrections to the Einstein’s equa- tions with the thermodynamic interpretation? The classical Einstein’s equations, 1 R − Rg = 8πGT (5.2) µν 2 µν µν are derived from the Einstein-Hilbert action, 1 Z √ S = −gR d4x (5.3) 16πG However, the Einstein-Hilbert gravity can be treated as a low-energy effective theory, so we should expect to have corrections to this action, of the form 1 Z √ S = −g R + α Λ + α R2 + α R Rµν + α R Rµνρσ + ... d4x (5.4) 16πG 1 2 3 µν 4 µνρσ and the field equations arising from this action would contain higher-curvature terms. These terms include higher derivatives of the metric, which correspond to terms with higher and higher curvature (and a lower and lower associated curvature radius). At some point, this curvature radius is of the order of the Planck length. Thus, in a theory of quantum gravity, we expect these terms to be important. The question that arises now is: we have obtained the classical Einstein’s equations from a thermodynamic point of view. If the spacetime is really a thermodynamic entity, should we be able to obtain these higher-curvature terms in the field equations with a similar argument? There is not a clear answer to this question. In the derivation of Section (4), in the expression for the area deficit we have neglected terms of order l/Lcurv, while the 2 next-higher-curvature correction to the field equations might be of order (l1/Lcurv) , with l1 a length scale appearing in the corresponding term in the action. To obtain 2 this next-order term in the field equations, we need l/Lcurv < (l1/Lcurv) ⇒ l/l1 < 33 l1/Lcurv. The rhs must be smaller than 1 (otherwise, the higher-order terms would dominate), which means that we need l < l1. That is, the diamond must be smaller than l1. If l1 is, for instance, the Planck length, the diamond should be smaller than the Planck length, and the classical geometry and quantum field theory used in the derivation would not work in that regime. There have been some attempts to find the field equations when these corrections are considered [25–27], but because of the presence of these terms, some technical difficulties appear and it is required to use non-equilibrium thermodynamics. However, an interesting result has been found in . There, they show that, for spherically symmetric systems with a horizon, the Einstein equations arising from the Einstein-Hilbert action can be put in the form of the relation T dS = dE + P dV , matching the entropy S and the energy E with the already know expressions. They go one step beyond, and do the same for the first correction to the Einstein-Hilbert action, the so-called Gauss-Bonnet correction: 1 Z √ S = −g (R + αL ) d4x, L = R2 − 4R Rµν + R Rµνρσ (5.5) 16πG GB GB µν µνρσ finding again that, once the field equations are written in the form T dS = dE + P dV , the entropy and the energy match with the expressions obtained by other authors. Finally, they generalize this result to the complete Lanczos-Lovelock action in D dimensions, matching again the results for S and E with independent calculations. These results suggest that the thermodynamic route to obtain the field equations also works for higher-curvature theories of gravity, and the quantum corrections to the Einstein-Hilbert action appear as quantum corrections for the entropy and the energy . However, the microscopic structure beyond this thermodynamics remains mysterious. • Is it appropiate to consider the entanglement entropy to be finite at UV scales? As we have seen, for a d-dimenional free field theory the entanglement entropy is a UV-divergent quantity, with the leading term being proportional to the area of the entangling surface, which we rewrite here for convenience: Area(∂A) S = γ + ... (5.6) A d−2 34 where is the cutoff length that is sent to 0 in the continuum limit and originates the divergences. But, because of the fact that the fields that contribute more to the entanglement entropy are those of high energy, we expect to have modifications to the background geometry. This backreaction of spacetime may lead to a finite entropy. Susskind and Uglum show in that when these are considered, the divergences that appear are the same ones that appear in the renormalization of the gravitational constant G. When this renormalization process is done, one obtains a finite result for the entanglement entropy, with the leading term coinciding with the Bekenstein-Hawking entropy [30, 31]: 1 Area(∂A) SA = + ... (5.7) 4 GR~ where GR is the renormalized gravitational constant. Again, in order to un- derstand the microscopic origin of the entanglement entropy and its divergences, a microscopic understanding of the theory is needed. The great similarity between the entanglement entropy and the Bekenstein-Hawking entropy suggests that, maybe, black hole entropy can be originated purely from en- tanglement [5, 11]. This identification, however, can only be done once the diver- gences of entanglement entropy are properly solved, because the Bekenstein-Hawking entropy has no divergences. 35 6 Conclusions After the realization of the thesis, some conclusions can be extracted: • The analogy between thermodynamics and black holes found by Bekenstein and Hawking 50 years ago can be extended to spacetime itself. That is, spacetime is a thermodynamic entity. • Given this thermodynamic nature of spacetime, one should be able to derive the Einstein equations from thermodynamic arguments. This has been done in two different ways: in Section (3), the Einstein equations have been derived from the thermodynamic relation δQ = T dS near a Rindler horizon, and in Section (4) they have been derived from an hypothesis about entanglement entropy. • This thermodynamics suggests some kind of microstructure of spacetime at smallest scales. Although there is not fully satisfactory explanation of what this microstructure is, it seems reasonable that it will be important when trying to find a complete quantum theory of gravity. J.D. Bekenstein. Generalized second law of thermodynamics in black-hole physics. Phys. Rev. D 9 3292 (1974). S.W. Hawking. Particle creation by black holes. Commun. Math. Phys. 43 199 (1975). W.G. Unruh. Notes on black-hole evaporation. Phys. Rev. D 14, 870 (1976). P.C.W. Davies. Scalar Production in Schwarzschild and Rindler Metrics. J. Phys. A.: Math. Gen. 8 609 (1975). J.D. Bekenstein. Do we understand black hole entropy? (1994). ArXiv: gr- qc/9409015. T. Jacobson; R. Parentani. Horizon entropy. Found. Phys. 33 323-348 (2003). ArXiv: gr-qc/0302099. S. Carroll. Spacetime and geometry. An introduction to General Relativity. Addison-Wesley (2004). T. Takayanagi M. Rangamani. Holographic entanglement entropy. (2017) ArXiv: hep-th/1609.01287. R. Myers D. Blanco; H. Casini; L. Hung. 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Know Your Birthstone And Birthstone Jewelry Traditionally, birthstones or birthday stones are special gemstones associated with each month of the year.  Each gemstone is attributed to various qualities symbolizing the specific month of birth in the Gregorian calendar. The origin of birthstones is believed to date back to 1250 BC when Moses made the breastplate of the High Priest Aaron on the basis of the instructions he received during those 40 days that he spent in the mountains.  According to the Bible, in Exodus 28 and 39 the original breastplate of the High Priest contained twelve gemstones representing the twelve tribes of Israel which were later linked with the signs of zodiac and then to the months in a year. Throughout history, different cultures had their own rituals and beliefs linked with the use of a particular birthstone.  Even many people believed that wearing all twelve birthstones together or in rotation is a way to ensure maximum benefit from these sacred crystals.  Catherine de Medici wore a girdle set with twelve stones to derive utmost benefit.  Also the kings and queens of different princely states in India wore jewels studded with twelve stones. In 1912, the American National Association of Jewelers officially adopted a list which now works as the basis of modern birthstone list.  This list was altered in 2002 when tanzanite was adopted as the birthstone for December. These birthstones are usually worn in the form of jewelry, often as rings, pendants and amulets and are said to possess certain mystical powers.  They are considered helpful in healing many ailments and known for radiating positive energy to the wearer. These birthstones for centuries are loved by mankind for their mysterious appeal, rarity, and durability, play of light and divine colors.  Each birthstone has some unique properties and is associated with a specific month.  Do you know your birthstone? Would you like to get an idea of your lucky stone that might bring good fortune for you? Here are some interesting and alluring facts about birthstones.  Check out which gem is special for you. January Birthstone - Garnet Garnet, the birthstone of January, signifies vitality, passion and trust.  The gem comes virtually in all colors of spectrum ranging from red, pink and green to yellow, and orange.  Garnet which is a group of minerals derived its name from Latin word granatus, meaning pomegranate seeds.  The gem considered to heal ailments like blood deficiency, arthritis and rheumatism.  Wearing garnet in form of rings, earrings and pendants helps in improving overall physical and mental wellbeing.  This birthstone symbolizes faith, courage, loyalty, trust, constancy and fidelity.  With a resinous luster and hardness of 6.5 to 7.5, garnet jewelry is a perfect present for those who love mystical charm and grace. February Birthstone – Amethyst Amethyst which is known as the gem of royalties is the official birthstone for February.  It is one of the oldest stones known to human civilization.  The history dates back the existence of amethyst as far as 25,000 years ago when prehistoric humans used it as a decorative piece.  It was found in the remains of Neolithic age. The gem represents power and symbolizes peace, love and happiness.  It derived its name from Greek word 'amethystos' that means remedy against drunkenness.  The name was given due to the belief that the gem has the powers to ward off the effect of intoxication and keeps the wearer clear headed and quick witted.  Throughout history, many myths, lore and legends were associated with amethyst.  Known to empower the minds and souls, this purple member of quartz family assists the wearer in attaining wisdom and courage.  Amethysts are soft stones with hardness of 7 on moh's scale yet they are counted as a desiring gem for jewelry.  Rings, earrings, pendants and amulets studded with amethyst are best said to ward off evil spirits and promote love and protection. The gem is also said to have healing properties that work against headache, insomnia, arthritis and circulatory disorders. March Birthstone – Aquamarine Aquamarine is though officially the birthstone for March but it is also loved by people born in other months.  This gem has derived its name from Latin word 'aqua' meaning water and 'marina' meaning sea so is known as 'Water of the sea'.  It ranges from pale blue green to deep blue in color.  As a stone of beryl family it is relatively a hard gem which is prized for its lavish beauty, mystical properties and symbolic association. In ancient times aquamarines were carried by sailors as a guarantee of safe voyage.  The gem signifies calmness, sympathy, trust, harmony, togetherness, peace, courage and serenity.  It is said to have the mystical powers that reduce stress, sharpens the intellect, clears confusion, promotes self-expression and shields the aura.  For ages, aquamarine jewelry specially rings and amulets have been associated with healing of many ailments like sore throats, swollen glands, thyroid problems, hormonal imbalance and immune system disorder.  The best aquamarine stones come from Brazil but quality stones are also mined in Nigeria, Madagascar, Zambia and Mozambique. April Birthstone – Diamond People born in April are lucky to have diamonds as their official birthstone.  This often colorless rock is an allotrope of carbon with tetrahedral geometry.  Diamonds since inception are considered a traditional stone for engagement and wedding rings because the gem symbolizes unity, trust, fidelity, strength and devotion. Known as a girl's best friend diamonds in the form of rings, earrings, pendants etc represent faith, love, innocence, beauty, clarity and purity. Formed at the depth of 90 to 120 miles under the earth crust, this April birthstone is the hardest gem in the world which has the unique ability to refract and reflect light.  Diamonds are examined and valued on the basis of color, clarity, cut and carat which are commonly known as the 4Cs of diamond. Although colorless diamonds are most famous yet with the discoveries of different colored diamonds, the demand for fancy natural colored diamonds has tremendously increased.  These colored diamonds make only 1% of the total diamonds that exist on the earth but their beauty and play of color add value to their worth.  Fancy colored diamonds are formed due to any alteration in the diamond crystal structure.  Pink, green, yellow, blue, orange, red, purple, brown, black, champagne etc are few of the many colors diamonds exhibit. Diamonds are not only desired for their beauty but also cherished for their mysterious properties.  They are said to balance the seventh Chakra and are known to aid creativity.  Also the gem is said to enhance psychic ability, promotes harmony and balance between individual and ambience, removes toxins and heal the body while releasing stress. May Birthstone – Emerald Emerald the birthstone for May is known as the symbol of rebirth, spring and youth.  It is believed to grant its owner foresight and good fortune. This member of beryl family is one of the four precious stones.  Its color ranges from light to dark green and it scores 8 on moh's scale of hardness.  The gem is famous as the symbol of natural growth and harmony.  It is very popular as a stone for fine jewelry. The history of emeralds dates back to 1300 BC when Egypt was the only known source of the gem and it was mined for Cleopatra.  The gem got its name from Greek word 'samaragdus', meaning green. Emerald jewelry has been worn by kings and queens throughout the history.  The stone in the form of rings, pendants, amulets etc is believed to promote romance, serenity, intelligence and communication.  Emerald jewelry is believed to enhance unconditional love and provides domestic bliss. The gem heals psychic ailments, clairvoyance and balance the heart chakra.  It is also considered a good supporter in treating heart, lungs, spine and muscular disorders. June Birthstone – Pearl and Alexandrite As the birthstone of June, pearls, symbol of purity, are believed to grant its owner wisdom and innocence.  It is associated with the third chakra and is known as the stone of sincerity and honesty.  Pearls contrast to their fellow gemstones, has an organic origin.  They are developed within the shell of oysters and clams.  Pearls are mostly made of aragonite and are valued highly for their beauty and rarity.  Wearing pearls in the form of jewelry is said to promote personal integrity, concentration and wisdom.  It treats digestive disorders and relieves the conditions of bloating and biliousness. Alexandrite is the second birthstone for the people born in month of June.  This is relatively newer stone which was discovered in 1831 in Russia during the reign of Czar Alexander II.  As it was discovered on his birthday, to honor him the stone was named alexandrite.  This is an extremely rare chrysoberyl with chameleon like properties.  The gem is an enchanting green in day light and fluorescence light which turns to deep red in incandescent light.  As a rare and expensive gem, it enjoys a status of huge demand.  Due to this the synthetic versions of alexandrite are also available in the market which possess the similar physical, chemical and optical properties as a natural gemstone.  It is a new addition so got no time to be a part of myths and lore but people in Russia believe that it brings good fortune to the owner. July Birthstone – Ruby Also known as 'Ratanraj' or 'King of Gems' ruby is the symbol of love and passion.  As the official birthstone for July it bestows the owner health, wisdom, wealth and success.  It is a member of corundum family and is found only in shades of red.  Large rubies are even rarer than sapphires and diamonds. This gem of beauty is famous for its bright hues and mysterious properties.  It is said to arouse the senses and energy levels.  The stone is related to the base chakra and heart chakra.  Since inception it is associated with feelings of happiness, devotion, love, care, romance and passion.  Wearing ruby jewelry is believed to bring prosperity, power and leadership qualities.  The gem had a long written history and it enjoyed a prominent position in the hearts and vaults of royals. According to ancient Asian lore and legends the gem was self-luminous so called the 'glowing stone'.  It was believed to guard against wicked thoughts, amorous desires and disputes.  Ruby in rings, amulets, pendants etc.  was told to be a good supporter in detoxifying body and treating fevers, infectious diseases and cures bleeding.  It was famous for stimulating adrenals, kidneys, reproductive organs, spleen and pineal gland.  The gem was also known to aid in retaining wealth and passion by clearing negative energy and imparting potency and vigor. August Birthstone – Peridot Peridot is associated with fame, dignity, protection, purity and rebirth.  Being the birthstone for the month of august it strengthens its owner by enhancing patience, confidence and assertion.  The member of olivine family achieves the hardness of 6.5 to 7 on moh's scale and exhibits yellowish green to golden green tints. It was called 'the gem of the sun' by Egyptians as it shines brightly in the sun.  The Greeks called it 'evening emerald' because the stone becomes deep green at night and glows in dark. Peridot symbolizes prosperity and openness.  It is believed to host the magical powers to protect against nightmares and negative energy.  It reduces stress, banishes lethargy and cleanses the body.  Benefits immune system, aids heat, lungs, gallbladder and treats ulcer and hypochondria.  Peridot jewelry is used to treat anger, stress and jealousy.  It is also a lavishing accessory to enhance looks and brings the spring glow to fashion statement. September Birthstone – Sapphire This 'wisdom stone' has a long written history.  Sapphire unlike its sibling ruby comes in various colors from pink, green, yellow, orange, purple and white.  But the most coveted of all is the blue sapphire ranging from light to intense hues of blue.  Sapphire is one of the most powerful gems that possess enormous mysterious powers.  Sapphires known as the gem of heavens for their beauty and brilliance and are believed to promote prosperity.  Medieval clergy wore sapphire jewelry to represent strength, power and position.  Sapphire engagement rings were also very famous among royals. This birthstone of September symbolizes luck, love, generosity and loyalty.  It is associated with creative expressions, spiritual development and metaphysical balance. Sapphires are thought to release mental tensions and depression.  It balances the body, mind and soul, brings serenity, peace and aligns the physical, mental and spiritual plans.  Sapphires are also know to treat blood and cellular disorder and they regulate the glands and calm overactive body system. October Birthstone – Opal and Tourmaline Opal the birthstone of October is an emotional stone which reflects the mood of the wearer.  The Greek word "Opallios" which means "to see a change" had given opal its name.  From milky white to black with shades red, blue, green, yellow, and orange, opals have a range of colors.  Its spherical arrangement is the reason of its radiance that adds extra beauty to its color play. This non-crystalline silica gel product is associated with love, loyalty, peace, and faithfulness.  Opals are thought to encourage freedom, enhance cosmic consciousness and induce mystical vision.  Wearing opal jewelry especially rings stimulates creativity, releases anger and soothes stress.  It is also believed to treat infections and fevers. Tourmaline with a wide variety of colors is a favorite gemstone of designers and gem collectors.  This magnificent stone with trigonal geometry comes in various colors from pink, red, green to blue, black, yellow and orange.  In the world where fashion changes with every minute, pink tourmaline which is an alternative birthstone or people born in October holds spiritual, sentimental and styling attributes all together. November Birthstone – Citrine Citrine, the birthstone for November, is known as the 'healing quartz'.  This golden gemstone is plentiful in nature and comes in the shades from yellow to brown. Signifying prosperity, protection, strength and stability, the gem encourages hope, comfort, wealth and warmth.  It was said to attract wealth and success so merchants in ancient times used to keep citrine in their vaults.  This is the reason why it is also called as "Merchant's stone".  Wearing citrine jewelry stimulates the brain by increasing concentration and releasing negative traits, depression and phobias.  It imparts joy, pleasure, stability and delight, raises self-confidence and promotes motivation. December Birthstone – Tanzanite Discovered in 1960s and added in 2002 to the list of birthstones, tanzanite is officially the birthstone for the month of December.  The stone has received praises for its exemplary beauty and alluring color changing property.  Named after Tanzania, its place of origin, tanzanite comes in a range of colors from blue to violet.  The stone helps in raising self-confidence and consciousness by dispelling lethargy and enhancing creativity.  It is thought to heal heart, lungs and spleen disorders as it is a great detoxifying agent.  The gem also promotes motivation, regenerates cells and strengthens immune system. 3 payments of $233.33 3 payments of $330.00 - United States - United Kingdom

Cross-Validation, AIC Assignment Help The Akaike details requirement (AIC) is a procedure of the relative quality of analytical designs for a provided set of information. Provided a collection of designs for the information, AIC approximates the quality of each design, relative to each of the other designs. AIC supplies a way for design choice. AIC wases established on details theory: it uses a relative quote of the info lost when a provided design is utilized to represent the procedure that creates the information. In doing so, it handles the compromise in between the goodness of fit of the design and the intricacy of the design. Utilize the Akaike info requirement (AIC), the Bayes Information requirement (BIC) and cross-validation to choose an ideal worth of the regularization specification alpha of the Lasso estimator. Outcomes gotten with LassoLarsIC are based upon AIC/BIC requirements. Information-criterion based design choice is really quick, however it depends on a correct estimate of degrees of liberty, are obtained for big samples (asymptotic outcomes) and presume the design is appropriate, i.e. that the information are in fact produced by this design. When the issue is terribly conditioned (more functions than samples), they likewise tend to break. For cross-validation, we utilize 20-fold with 2 algorithms to calculate the Lasso course: coordinate descent, as executed by the LassoCV class, and Lars (least angle regression) as executed by the LassoLarsCV class. Both algorithms offer approximately the exact same outcomes. They vary with concerns to their execution speed and sources of mathematical mistakes. It is able to calculate the complete course without setting any meta criterion. On the opposite, coordinate descent calculate the course points on a pre-specified grid (here we utilize the default). In terms of mathematical mistakes, for greatly associated variables, Lars will build up more mistakes, while the coordinate descent algorithm will just sample the course on a grid. Akaike Information Criterion (AIC) is often utilized in the semiparametric setting of choice of copula designs, even though as a design choice tool it was established in a parametric setting. Just recently a Copula Information Criterion (CIC) has actually been particularly created for copula design choice. Among the primary factors for utilizing cross-validation rather of utilizing the standard validation (e.g. separating the information set into 2 sets of 70% for training and 30% for test) is that there is inadequate information readily available to partition it into different training and test sets without losing substantial modelling or screening ability. In these cases, a reasonable method to effectively approximate design forecast efficiency is to utilize cross-validation as an effective basic method. The information set is separated into 2 sets, called the training set and the screening set. The mistakes it makes are collected as before to provide the mean outright test set mistake, which is utilized to examine the design. The examination might depend greatly on which information points end up in the training set and which end up in the test set, and hence the assessment might be substantially various depending on how the department is made. Each time, one of the k subsets is utilized as the test set and the other k-1 subsets are put together to form a training set. Every information point gets to be in a test set precisely when, and gets to be in a training set k-1 times. A version of this technique is to arbitrarily divide the information into a test and training set k various times. Of the k subsamples, a single subsample is kept as the validation information for evaluating the design, and the staying k-1 subsamples are utilized as training information. The cross-validation procedure is then duplicated k times (the folds), with each of the k subsamples utilized precisely when as the validation information. The Akaike Information Criterion (AIC) is a method of choosing a design from a set of designs. The selected design is the one that lessens the Kullback-Leibler range in between the fact and the design. In utilizing AIC to try to determine the relative quality of econometric designs for an offered information set, AIC supplies the scientist with a price quote of the info that would be lost if a specific design were to be used to show the procedure that produced the information. The AIC works to stabilize the compromises in between the intricacy of a provided design and its goodness of fit, which is the analytical term to explain how well the design "fits" the information or set of observations. Provided a collection of designs for the information, AIC approximates the quality of each design, relative to each of the other designs. In a forecast issue, a design is generally provided a dataset of recognized information on which training is run (training dataset), and a dataset of unidentified information (or initially seen information) versus which the design is checked (screening dataset). The Akaike Information Criterion (AIC) is a method of picking a design from a set of designs. In utilizing AIC to try to determine the relative quality of econometric designs for a provided information set, AIC supplies the scientist with a quote of the info that would be lost if a specific design were to be utilized to show the procedure that produced the information. The AIC works to stabilize the compromises in between the intricacy of a provided design and its goodness of fit, which is the analytical term to explain how well the design "fits" the information or set of observations.

It’s not uncommon for a question or two involving three-dimensional shapes to appear on the SAT. Luckily, most of the time these questions either deal directly with the simple properties of three-dimensional shapes (like surface area and volume), or are just 2-D questions in disguise. It’s pretty rare to come across a truly difficult 3-D question — but you know I’m gonna give you some in this post because I care about you so. Generally speaking, the SAT will give you every volume formula that you need, either in the beginning of the section (rectangular solid — V = lwh; right circular cylinder — V = πr2h) or in the question itself in the (exceedingly) rare case where you’ll have to deal with the volume of a different kind of solid. It’s worth mentioning, though, that the volume of any right prism* can be calculated by finding the area of its base, and multiplying that by its height. For example, if you needed to calculate the volume of a prism with an equilateral triangle base, you’d find the area of an equilateral triangle: And multiply that by the height of the prism: You almost definitely won’t need this particular formula on the SAT, but it’s nice to know how to find the volume of a right prism in general: just find the area of the base, and multiply it by the height. Most volume questions you’ll see on the SAT will require you to deftly maneuver between the volume of a solid and its dimensions. Let’s see an example (and showcase my fresh new drawing software): - If the volume of the cube in the figure above is 27, what is the length of AF? Remember that a cube is the special case of rectangular solid where all the sides are equal, so the volume of a cube is the length of one edge CUBED: So far, so good, right? Now it’s time to do the thing that you’re going to find yourself doing for almost every single 3-D question you come across: work with one piece of the 3-D figure in 2-D. The segment we’re interested in is the diagonal of the square base of the cube. If we look at it in 2 dimensions, it looks like this: The diagonal of a square is the hypotenuse of an isosceles right triangle, so we can actually skip the Pythagorean Theorem here since we’re so attuned to special right triangles. AF = 3√2. That’s choice (B). The surface area of a solid is simply the sum of the areas of each of its faces. Easy surface area problems are really easy. Trickier surface area problems will often also involve volume, like this example: - If the volume of a cube is 8s3, which of the following is NOT a value of s for which the value of the surface area of the cube is greater than the value of the volume of the cube? Yuuuuck. What to do? Well, to find the surface area of a solid, you need to know the areas of its faces. To find those areas, you need to know the lengths of the sides of the solid. Luckily for us, it’s pretty easy to find the lengths of the sides of this cube, since we know that the volume is 8s3. Take the cube root of the volume to find the length of one side of the cube: If a side of the cube is 2s, then the area of one face of the cube is (2s)2, or 4s2. There are 6 sides on a cube, so the surface area of the cube is found thusly: From here, it’s trivial to either backsolve, or solve the inequality spelled out in the question: The answer must be (E), the one choice for which the inequality is NOT true. * Right circular cylinders and rectangular solids are both special cases of right prisms — a right prism is any prism whose top lines up directly above its bottom. Break it down. You need to be registered and logged in to take this quiz. Log in or Register The last one seems a bit tough, even for a #20… You’re right that it’s tough, but I don’t think it’s outside of the realm of possibility. See #19 on page 401 of the Blue Book for another 3-D question that actually requires a bit of 3-D reasoning and a similar number of steps. That one provides a diagram, but I actually think it’s a bit tougher than my #20. There’s also pg 548, #16, which is eerily similar, considering I just made the question up without consulting the Blue Book and am only now looking for similar problems. : Thank you for the references, I will look them up tonight. I think the problem would be easier if you gave the volume and asked for the radius. Going the other way around seems to require more steps and more advanced reasoning. OTOH, I may have just gone the long way without realizing it. pg 401 took me 3 minutes including rationalizing the denominator and then realizing the answer choice wasn’t there and having to go back and un-rationalize the denominator to find the answer (what’s up with that? very poor form). Figure not drawn to scale and the whole e=m thing are only there to obfuscate but it consumes time to digest these things. Great example of College Board using tricks that have nothing to do with math. What is up with question #20 on that page?? That is difficulty 2 at the most, but it is listed as a 5. pg 548 is exactly what I suggested above! Of course this only took as long as it takes to read the question because I had just derived the whole x : x sqrt 2 : x sqrt 3 relationship between the three legs of the right triangle in question. But if I didn’t have the head start I think it still would have been much quicker than your example. You are given ‘boo/boo square root of 2’ and the only trick is to correctly visualize the right triangle inside the cube. In your example, you had to accomplish the visualization first and then work backwards into ‘boo/boo square root of 2’. It is tricky algebra (including exponents) on top of tricky visualization vs just tricky visualization in the CB example. Again, maybe I just missed a quicker way to do your example so I’m not 100% sure. (fyi I wasn’t timing but I think it took me almost 10 minutes to solve your problem) Thanks again for the great problems! Hmm…I’m not sure how you solved mine, but I think the tricky step exists in both directions: recognizing that the diameter of the sphere connects two opposite corners of the cube, and then determining the relationship between that diagonal and the length of the cube’s side. What I was trying to get at with the question is that often, on the hardest 3-D questions, what you’re really doing is doubling up on the Pythagorean Theorem — using it once in the plane of a solid’s side, and then using the value found that way to form another right triangle that cuts through the solid. Not too bad (although certainly not trivial) once you know to look for it, but brutal the first time you encounter it. I wonder, if you had encountered those two BB questions before mine, if mine still would have taken 5 mins. As for #20 on pg 401, I agree. Pretty easy stuff. It’s important to remember that some questions make it to #20 because a lot of kids get them wrong, not because they’re hard. Symbol Functions are another great example of this. Students are notoriously imprecise when it comes to the word “percent.” The questions is easy if you translate “k percent” to mean “k/100,” but I’ve seen countless kids get stymied by this kind of question because they’ve become so accustomed to “moving the decimal point” that they forget WHY that is done, and they don’t know how to move a decimal point on a variable. You may be right, the way I did your problem was kinda weird. First I figured out I was looking at a right triangle with the sides being – edge : diagonal of face : diagonal of cube (this took a little while, I had to draw a diagram which I never do) But given the diagonal of cube = 8, I didn’t know how to figure out the length of the legs, so I used the only method I KNEW would work… I set the side edge to 4 (arbitrarily), then realized the diagonal of the face would be 4 sqrt2 and the cube diagonal would be 4 sqrt3. I then tested another edge size to make sure the x : x sqrt2 : x sqrt3 held. Then I solved for x by setting the third formula equal to 8. x sqrt3 = 8 x = (8 sqrt3) / 3 x = edge of the cube so the volume just equals (8 sqrt3)/3 cubed That whole thing with setting the edge equal to 4 and testing the relationship between the legs is not necessary when the question is asked the other way around because you are starting out knowing the edge. I now see how I could have solved for x directly without the abstraction but this was not obvious at the time. Could you explain number 20 please? I found this, but I still don’t understand: http://answers.yahoo.com/question/index?qid=20080904045041AAQ2J0f I lose him at the, ” the distance from the centre of a face to the corner = x/√2″ part… Thanks. It’s hard to explain this one without diagrams, and it’s impossible to draw diagrams in the comments, so I’m going to attach an image of the solution I put in my book for this question. If this doesn’t help, let me know, and I’ll try to clarify the parts that are confusing. Thank you so much. That was very helpful. I lost you after the 45 45 90 triangle was shown. I’m not sure how it becomes 8 Sroot2 and S? 8, S, and S√2 are the sides of the right triangle that’s drawn in dotted lines in the first drawing. Does that help? Oh! I see that now. We were trying to find the diagonal of the square. I was about to say I couldn’t see the 30 60 90, but I can’t argue with the math. Thank you very much! Can you please explain number 18? Like so many 3-D questions, this one is really a right triangle question. First, picture the two ants as far apart as they can possibly be from each other. They need to be on opposite rims of the cylinder, first of all. They also need to be on opposite sides of the tube…like if it’s held horizontally then one is all the way top right, and the other is all the way bottom left. Check out the picture below, since I’m clearly having a hard time explaining this with only words. Once you’ve got the picture in your head, you’re ready to solve. The distance between the ants is the hypotenuse of a right triangle with legs of 12 and 5. If you know your Pythagorean Triples, you know the hypotenuse of that triangle has to be 13. If you don’t, just do the theorem! 🙂 here’s that pic I forgot to attach above… Wow! You’re a life saver! 😀 Wouldn’t you account for the curve as well? Like when measuring distance on the earth, you don’t draw a line thru it from the US to china but you take a portion of its circumference. Wouldn’t the same apply here as well? Making the equation 12^2 + (2.5pi)^2 = c^2 No. Here, you just cut right through. It’s a cardboard tube! You’re not burrowing through earth. I was about to ask how can you get the answer C. But, I realised I didn’t carefully look at “Prism’s base”.. I always do such mistakes and end up with 1 or 2 mistakes..Hopefully I’ll get 800 this time.. and, 18 was easy..for 20, If one can imagine that figure in the head, and little bit of geometry basics, the job is done..:D I’ve been staring at 19 for about 45 minutes, and I know it’s got to be something much simpler than the paths I’ve tried. I really need to work on the “be nimble” philosophy. Anyway, can you put me in the right direction? This is one of my favorite questions! Here’s a clue: the surface area is base + base + side + side + side. But the expression the question gives you only adds 4 things… It seems like ab has to be bh or hb, which makes sense because the whole base of one triangle times the height of one triangle would give you the area for both. I don’t know where to go from here though, as I can’t figure out which leg is the base, or understand how the height could equal one of the legs. I must be on the wrong track 🙁 Another hint: since d appears in 3 of the terms, it must be the height. So a, b, and c, are the sides of the triangular bases. I finally got it! I hope I used a reliable method. I ended up looking at the 30-60-90 special right triangle to get the answer. This may seem like a silly question and I’m sure I just need to brush up on my basic geometry, but how was I supposed to know there was a 90 degree angle in there? Visually it doesn’t seem like any of the sides form a right angle to me. Do right prisms always have right angles somewhere in their bases? Awesome! The way you figure out it’s a 90º angle is that only a right triangle’s area is half of the product of two of its sides. Any other triangle, you’d need another constant in the expression. There’d be the sides of the triangle, and then the “height” of it. Since you ONLY have the sides of the triangle in an expression for its area, it must be a right triangle. This is, admittedly, a difficult leap, and you can safely assume the SAT won’t make you do something QUITE as hard on test day. But I love this question too much to make it any easier. 🙂 I didnt get question no. 19. Can you pleasee explain ? This is a pretty tough problem. The first step is to realize that the surface area should be base + base + side + side + side, but the expression you’re given, ab + ad + bd + cd, only has 4 terms. So there’s something funny going on here. Since 3 terms contain a d, you can conclude that d is your height—it’s multiplied by each side of the base to create the three vertical sides. Now you’re left with the fact that ab has to represent both the top and bottom face. Remember that the area of a triangle is ½bh, and you’ve got 2 congruent triangles’ areas represented by ab. That must mean each of them is ½ab, making the base a right triangle with legs a and b. So the hypotenuse, which is what we want, must be c. For the longest time i was wondering why it can’t be side D and then i realized its asking for a side of the prism’s base. i want to improve my score from 580 to 650 🙁 any advice? for num20# 8=square root of x^2+x^2+x^2 then use the volume equations To improve from 580 to 650, concentrate on getting all the easy and medium difficulty questions right. You can score a 650 without answering many of the hard questions. For more information, see this post. your website is so helpful. huge thanks for maintaining this blog, i realise how much is it of a hard work. having my sat in 3 days, hopefully i’ll do well…

- About this Journal · - Abstracting and Indexing · - Aims and Scope · - Article Processing Charges · - Articles in Press · - Author Guidelines · - Bibliographic Information · - Citations to this Journal · - Contact Information · - Editorial Board · - Editorial Workflow · - Free eTOC Alerts · - Publication Ethics · - Reviewers Acknowledgment · - Submit a Manuscript · - Subscription Information · - Table of Contents Journal of Medical Engineering Volume 2013 (2013), Article ID 193578, 15 pages A MATLAB-Based Boundary Data Simulator for Studying the Resistivity Reconstruction Using Neighbouring Current Pattern Department of Instrumentation and Applied Physics, Indian Institute of Science Bangalore, Bangalore, Karnataka 560012, India Received 30 August 2012; Revised 14 December 2012; Accepted 28 December 2012 Academic Editor: Nicusor Iftimia Copyright © 2013 Tushar Kanti Bera and J. Nagaraju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Phantoms are essentially required to generate boundary data for studying the inverse solver performance in electrical impedance tomography (EIT). A MATLAB-based boundary data simulator (BDS) is developed to generate accurate boundary data using neighbouring current pattern for assessing the EIT inverse solvers. Domain diameter, inhomogeneity number, inhomogeneity geometry (shape, size, and position), background conductivity, and inhomogeneity conductivity are all set as BDS input variables. Different sets of boundary data are generated by changing the input variables of the BDS, and resistivity images are reconstructed using electrical impedance tomography and diffuse optical tomography reconstruction software (EIDORS). Results show that the BDS generates accurate boundary data for different types of single or multiple objects which are efficient enough to reconstruct the resistivity images for assessing the inverse solver. It is noticed that for the BDS with 2048 elements, the boundary data for all inhomogeneities with a diameter larger than 13.3% of that of the phantom are accurate enough to reconstruct the resistivity images in EIDORS-2D. By comparing the reconstructed image with an original geometry made in BDS, it would be easier to study the inverse solver performance and the origin of the boundary data error can be identified. Electrical impedance tomography (EIT) [1, 2] reconstructs the spatial distribution of electrical conductivity or resistivity of a closed conducting domain () from the surface potentials developed by a constant current injection through the surface electrodes surrounding the domain to be imaged. Before carrying out the practical measurements on patients, it is advised to test an EIT system with a tissue mimicking model of known properties called practical phantoms [4–10]. Hence, phantoms are often required to assess the performance of EIT systems for their validation, calibration, and comparison purposes. Two-dimensional (2D) EIT (2D-EIT) assumes that the electrical current flows in a 2D space which is actually three-dimensional inside real volume conductors. Hence, the development of a perfect 2D practical phantom is a great challenge as the real electrodes always have a definite surface area, and hence the injected current signal cannot be confined in a 2D plane in bathing solution . Researchers have developed a number of practical phantoms which are three-dimensional objects, and those phantoms are designed and developed, generally, for their own EIT systems. Practical phantoms containing electrolyte (or other conducting medium) [4–10] are three-dimensional in shape and hence they will have some data error due to the three dimensional current conduction. Also, the phantoms containing electrolytes (e.g., NaCl solution or saline) [5, 7, 8] are difficult to transport and are prone to errors since the evaporation of the water gives rise to changes in conductivity . In addition, temperature variations have a marked effect on the conductivity because the temperature coefficient is large . Therefore, the practical phantoms will have a poor stability and a gradually increasing data error over time. Network or mesh phantoms [12, 13] are compact, more stable, rugged, portable, easy to move, consistent over time, and less temperature dependent. But these phantoms need a huge number of identical electronic components properly designed in a mesh mimicking the conductivity distribution of a practical biological tissue. Furthermore, for a large tissue structure, a mesh phantom requires a huge number of very precision components. The reproduction of these kinds of phantoms having different properties is often time-consuming . The option for changing the position and property of an inhomogeneity is limited by the phantom structure and the number of elements in mesh phantom but the practical phantoms allow us to put several types of object in different positions in the bathing solution, but they produce several errors contributing to the poor signal to noise ratio (SNR) in boundary data. Reconstructed image quality in impedance tomography depends on the errors associated with practical phantom, electronic hardware, and inverse solver performance. Image quality is largely affected by the practical phantom design parameters such as phantom geometry, electrode geometry, electrode materials, and the nature and behavior of the inhomogeneity and bathing solution. SNR is also reduced by the error contributed by current injector, data acquisition system, and signal conditioner circuits. In practical phantoms, the voltage data developed by a three-dimensional current conduction are collected form surface electrodes connected to an analog instrumentation. Therefore, it is quite confusing to identify the source of the errors responsible for poor image quality in a 2D-EIT system. In order to overcome the difficulties and limitations of practical and mesh phantoms, a MATLAB-based boundary data simulator (BDS) is developed to generate accurate 2D boundary data for assessing the EIT inverse solvers. BDS is an absolute 2D data simulator which is required to generate the errorless 2D boundary data to study and modify the inverse solver of a 2D EIT system. As the BDS is a computer program, it is free from the instrumentation errors and allows us to generate voltage profile with different types of phantom geometry, inhomogeneity and background conductivity profile, and inhomogeneity geometry (shape, size, and position). Moreover, it is absolutely stable, compact, easy to use, and easy to handle and modify for further development. Boundary data for different phantom geometries are generated in BDS, and resistivity images are reconstructed in standard reconstruction algorithm. BDS is studied to conform its suitability to use for boundary data generation with different phantom configurations which are required to assess the EIT inverse solvers. 2.1. Mathematical Modelling of EIT EIT image reconstruction is a nonlinear inverse problem in which the electrical conductivity distribution of a closed domain () in a volume conductor is reconstructed from the surface potential data developed at the boundary by injecting a constant current signal. A low frequency and low magnitude constant sinusoidal current is injected through an array of electrodes attached to the boundary, and the boundary potentials are measured using a data acquisition system. The voltage data collected from surface electrodes are then used by an image reconstruction algorithm which reconstructs the conductivity distribution of the domain under test (DUT). The reconstruction algorithm computes the boundary potential for a known current injection and known conductivity values and tries to compute the conductivity distribution for which the difference between the measured boundary potential () and the calculated is minimum. The reconstruction algorithm is developed with two parts: forward solver (FS) [5, 15–17] and inverse solver (IS) [15–17]. Forward solver calculates the boundary potential data for a known current injection and known conductivity values. Inverse solver computes the conductivity distribution for which the boundary voltage difference () becomes minimum. The DUT will have the distinct conductivity values at each points defined by their corresponding coordinates (). Due to a constant current injection, a potential profile is developed within DUT, and its potential profile without any internal energy sources depends on the conductivity profile. Hence, a relationship, called EIT governing equation, between the electrical conductivity () of the points within the DUT and their corresponding potential values () can be established. The governing equation in EIT [1, 2] can be derived from the Maxwell’s equation and can be represented as To calculate the domain potential developed for a constant current injected to the DUT with a known conductivity distribution, the above equation is essentially to be solved. As the EIT governing equation is a nonlinear partial differential equation, the direct or analytical technique fails to solve it. Therefore, to calculate the domain potential, the equation is solved by developing a mathematical model called “forward model” which is derived from (1) using a numerical technique like finite element method (FEM) . The EIT governing equation has an infinite number of solutions, and hence the FEM formulation of the EIT technique is essentially required to be provided by some boundary conditions [18–20] to restrict its solutions space. The boundary conditions are imposed into the FEM formulation of EIT by specifying the value of certain parameters (voltage or current). The parameters defining the boundary conditions may be either the potentials at the surface or the current density crossing the boundary or mixed conditions. The boundary conditions, in which the parameters are the potential at the surface, are called the Dirichlet boundary conditions and are represented as [1, 5, 19, 20] where are the measured potentials on the electrodes. where is the boundary, and is the outward unit normal vector on an electrode surface. In EIT, the FEM technique is used to derive the forward model from the governing equation in the form of a matrix equation establishing the relationship between the injected current and the developed potential within a DUT. The relationship can be assumed as the transfer function of the system which is mathematically represented as a matrix called global stiffness matrix (GSM) or transformation matrix constructed with the elemental conductivities and nodal coordinates (). In EIT, FEM discretizes the DUT by a finite element mesh containing finite number of elements of defined geometry and finite number of node. FEM applied on the governing equation to derive the forward model of a DUT in the form of a matrix equation using the and nodal coordinates. In the EIT forward model, the relationship established between the current injection matrix (matrix of the applied signal) and the nodal potential matrix (matrix of the developed signal) through the transformation matrix is mathematically represented as Now, in FEM formulation in EIT, when the current matrices and are known, and the nodal potential matrix is unknown, the forward model or the mathematical problem is termed as the “forward problem”. The procedure of calculating the by solving the forward problem (3) with known and known is termed as “forward solution”. In EIT, the forward solver first computes the potential distribution with the assumed initial conductivity distribution () with a known constant current simulation, and then the inverse solver reconstructs the conductivity distribution from the measured boundary potential data for a same constant current injection through surface electrodes. The EIT reconstruction algorithm tries to mathematically find the elemental conductivity values (conductivity distribution) for which the difference between the estimated nodal potentials () computed in the FS and the potentials measured () on the surface electrodes (for a same current injection values) becomes minimum. The inverse solver of the EIT reconstruction algorithm is developed with a mathematical minimization algorithm (MMA) [19–22] such as Gauss-Newton-based mathematical minimization algorithm (GN-MMA). In GN-MMA, the conductivity update vector () is calculated and the boundary data mismatch vector () is minimized by an iteration technique like the modified Newton-Raphson iteration technique (NRIT) [19–22]. The matrix is the desired variation in the elemental conductivity values in matrix for which the forward solver calculates the boundary potentials more similar to the measured value in next iteration using NRIT. Therefore, the algorithm starts with an initial elemental conductivity vector (), and it is then updated to () in the next iteration. Using this , FS calculates a new potential distribution in DUT and a new voltage mismatch vector is thus obtained and compared with the previous voltage mismatch vector . If the is not found as the minimum, the iteration process is continued till the kth iteration using the conductivity update vector () developed by GN-MMA. Using, NRIT the matrix is iteratively updated to and repetitively tries to find out the minimum value of . Hence, in the EIT inverse solver, it is understood that the desired elemental conductivity matrix is obtained by a minimization algorithm (MMA) which is composed of Gauss-Newton method and Newton-Raphson iteration in which the technique iteratively tries to find out an optimum conductivity distribution for which the voltage mismatch vector is minimized . At a particular iteration in this MMA, the elemental conductivity matrix is calculated when the current matrices and or are known. This process is logically an opposite process to the forward problem. Thus, when the current matrices and are known, and the elemental conductivity matrix is unknown, the model or the problem is called the “inverse problem.” The procedure of calculating the or using with known and the known is termed as “inverse solution.” 2.2. Image Reconstruction with GN-MMA and NRIT Electrical conductivity imaging is a highly nonlinear and ill-posed inverse problem [19–22]. In EIT, a minimization algorithm is used to obtain an optimized elemental conductivity value for which the voltage mismatch vector becomes minimum. In the image reconstruction process, the minimization algorithm [17, 18] first defines an objective function () from the computational predicted data and the experimental measured data and runs iteratively to minimize it. Generally, in the EIT image reconstruction algorithm, the inverse solver searches for a least square solution of the minimized object the function () using by a Gauss-Newton method and the NRIT-based iterative approximation techniques. If is a function mapping a t-dimensional (t is the number of element in the FEM mesh) impedance distribution into a set of M (number of the experimental measurement data () available) approximate measured voltages, then the Gauss-Newton-method-based minimization algorithm [19–26] tries to find a least square solution of the minimized object function (s) [19–26] which is defined as: Now, differentiating (4) with respect to the conductivity , it reduces to where the matrix is known as Jacobin matrix [19–22], which may be calculated by a method as described in [19, 22] or by the adjoint method represented by (6) where is the forward solution for a particular source location, and is the forward solution for the adjoint source location (source at the detector location and detector at the source location). Differentiating (5) with respect to again, the equation reduces to By Gauss-Newton method, the conductivity update vector is given by In general, using NRIT method, the conductivity update vector expressed as in (10) can be represented for kth iteration (where is a positive integer) as where and are the voltage mismatch matrix and Jacobian matrix, respectively. The matrix in (11) is always ill conditioned [19–24], and hence small measurement errors will make the solution of (11) changes greatly. In order to make the system well posed, the regularization method [19–26] is incorporated into the reconstruction algorithm by redefining the object function [19–26] with regularization parameters as where is the constrained least-square error of the regularized reconstructions, is the regularization operator, and (the positive scalar) is called the regularization coefficient [19–26] Differentiating the inject function in (12) with respect to the elemental conductivity: the following relations are obtained Now, using Gauss-Newton- (GN-) method-based minimization process, the conductivity update vector is obtained as Replacing by and by (identity matrix) (21) reduces to where the matrix is the Jacobin as stated earlier. Thus, the conductivity update vector () is found as Sometimes, the last term () is neglected , and the conductivity update vector is calculated as In general, the EIT image reconstruction algorithm provides a solution of the conductivity distribution of the DUT for the kth iteration as The EIT algorithm starts with the solution of FP obtained from the EIT governing equation, and the is calculated for a known current injection matrix and an initial guess (known or assumed) conductivity matrix . The voltage mismatch matrix is estimated, and then it is used to calculate the conductivity update matrix using GN-MMA and is added to the initial conductivity matrix () to update it to a new conductivity matrix using NRIT. New update matrix is used in forward solver to obtain a new calculated boundary data matrix which provides a new voltage mismatch matrix . Therefore, the NRIT algorithms iteratively calculate the using GN-MMA to find out an optimized matrix for which the reaches its minimum value. Thus, the EIT reconstruction algorithm is found to work in the following sequences:(1)forward solver calculates the boundary potential matrix for a known current injection matrix and an initial guess (known) conductivity matrix ,(2)measured voltage data matrix is compared with to estimate the as ,(3)Jacobian () is computed,(4)conductivity update vector is calculated by Gauss-Newton-based minimization algorithm,(5) matrix is updated to a new conductivity matrix by adding to using Newton-Raphson iteration technique (NRIT),(6)new update matrix is used in forward solver to calculate the new voltage mismatch matrix ,(7)check whether the is minimum or not or compare the with a specified error limit () if provided,(8)stop the algorithm if condition is achieved, otherwise repeat the steps 1 to 7 until the specified stopping criteria () is achieved. 2.3. Boundary Data Simulator (BDS) A two-dimensional boundary data simulator (BDS) is developed in MATLAB R2010a using finite element method (FEM) to generate accurate boundary data for studying the EIT reconstruction algorithms. The MATLAB-based BDS is developed as an absolute 2D data simulator for EIT image reconstruction studies, and it is used suitably to generate the errorless 2D boundary data to study and modify the inverse solver of a 2D EIT system. As BDS is developed in a computer software, it is found free from errors produced by the EIT instrumentation and phantom. BDS also allows us to generate boundary potential data for different type of phantom geometry, inhomogeneity geometry (shape, size, and position), inhomogeneity conductivity profiles, and background conductivity profiles. Moreover, it is developed as a compact, absolutely stable, and easy to use and handle for EIT studies. It is developed in such a way that it can be modified for further modifications. BDS is developed with MATLAB-based computer program consisting of four-part imaging domain simulator (IDS), EIT model developer (EMD), current injection simulator (CIS), and boundary data calculator (BDC). Imaging domain simulator (IDS) in BDS simulates a domain with inhomogeneity with their corresponding conductivity distributions. EIT model developer (EMD) derives a mathematical model of the forward solver by applying FEM on the governing equation of the DUT in the form of a matrix equation. Current injection simulator (CIS) simulates a constant current injection through the definite points at the domain boundary with neighbouring current injection protocol [1, 2, 28–30]. The boundary data calculator (BDC) solves the governing equation by solving the forward model and calculates the potentials at all electrodes at the domain boundary. Imaging domain simulator (IDS) first defines a DUT with a desired area defined by a required diameter and defined with a particular coordinate system. Imaging domain simulator applies the FEM to discretize the domain with a 2D finite element mesh containing finite element of triangular elements () and finite number of nodes (). In IDS, a circular domain () to be imaged is defined with a required radius () using the Cartesian coordinate system (Figure 1(a)), and the domain is discretized with a finite element (FE) mesh (Figure 1(b)). The mesh is symmetrically composed of the first-order triangular elements with linear shape functions [18, 31]. The FE mesh is generated with the pdetool of MATLAB R2010a in such a way that it can be refined further to increase the number of elements as per the requirement. All the coordinates and parameters assigned to the finite elements and the nodes are stored in corresponding matrices. Boundary nodes are identified, and the sixteen nodes among the boundary nodes are assigned as the electrodes called the electrode nodes. Inside the domain one (or more) smaller region (regions) is (are) defined as the inhomogeneity (inhomogeneities) positioned at a particular place. The center point () of the inhomogeneity with the required shape and size is positioned inside the phantom domain by defining its center with a polar coordinate ( as shown in Figure 1(a). Single or multiple inhomogeneities are defined with their desired areas () inside the DUT, and elements within the inhomogeneity and the background are identified. The background area is defined as the area of the domain surrounding the inhomogeneity (), and the elements within the background area () are identified. The elements within the inhomogeneity are assigned with a particular conductivity called inhomogeneity conductivity, while the rest of the elements are assigned with a different conductivity called background conductivity () as shown in Figure 1(b). The assigned conductivity values of all the elements are assumed to be featured at their corresponding centroids. EIT model developer (EMD) develops the mathematical model of the forward solver by applying FEM on the governing equation and derive the forward model of a DUT in the form of a matrix equation (3) using the elemental conductivities and nodal coordinates. The EMD establishes a relationship between the current injection matrix, (matrix of the applied signal), and the nodal potential matrix, (matrix of the developed signal), through the transformation matrix which is mathematically represented by (3). The global stiffness matrix in EIT is actually an admittance matrix that is formed using the nodal coordinates of all the elements with their corresponding conductivities. Thus, the inforward model represents the transfer function of the EIT system obtained from the governing equation by FEM formulation . The current injection simulator (CIS) is used to simulate a constant current injection through the sixteen nodes called simulated electrodes (SE) on the domain boundary with neighbouring current injection protocol. The CIS works in a “for” loop to execute all the projections [1, 28, 30, 32] of current injection process. In BDS, a constant current injection is simulated into the DUT surrounded by the sixteen simulated current electrodes () with all the possible combination of SEI pairs, and the potential data are calculated on all the electrodes called voltage electrodes () in BDC. The current injection through a particular current electrode pair (say and ) and corresponding voltage data collection from all the possible voltage electrodes (, , , , , , , , , , , , , and ) is known as a simulated current projection (SCP). Hence, in an N-electrode EIT system, there will be N-different current projections each of which will inject current through a particular current electrode pair and collect m voltage (differential/grounded) data where m may be either equal to N or less than N depending on the EIT data collection strategy called the current pattern [1, 28, 30, 32]. Therefore, a complete scan (containing all the current projections) conducted on the DUT yields voltage data. As the BDS is studied for sixteen electrode system, the CIS runs for sixteen times and provides sixteen current projections (, , , , , , , , , , , , , , , and ). Therefore, a complete data collection procedure (called a complete scan) in the BDS collects m voltage data from the voltage electrodes or voltage electrode pairs in all the sixteen current projections and computes voltage data. Boundary data calculator (BDC) calculates the potentials (developed for a constant current injection by CIS) at all electrode points (electrode nodes) at the domain boundary in each current projection for a particular current pattern. The current injection matrix is formed in CIS using the Neumann type boundary conditions, and the potential matrix is calculated from (3) using the matrix inversion technique working on L-U factorization process. The BDS is developed to run in an another “for” loop for m times to calculate the m electrode potentials from voltage electrodes or voltage electrode pairs at each of the steps of the loop. This second “for” loop runs within the first “for” loop for m times and collects m voltage data for each step of first “for” loop and hence collects voltage data as first “for” loop runs for sixteen times. Moreover, as the EIT reconstruction process needs a complete scan, the BDS runs in each current projection and computes sixteen electrode potentials at each projection. The domain potential is calculated from the forward model (3), and the potential values of all the nodes are stored in a nodal potential matrix [33, 34] denoted by . Boundary potential data are separated from and stored in a different matrix called boundary potential matrix . The electrode potential data are extracted from the nodal potential matrix and are stored in a separate matrix called electrode potential matrix . In sixteen electrode EIT system, the is formed as a column matrix and contains the electrode potentials (differential or grounded) obtained for all the projections. 2.4. Neighbouring or Adjacent Current Injection Method In neighbouring or adjacent current injection method, first reported by Brown and Segar , the current is applied through two neighbouring or adjacent electrodes, and the differential voltages is measured successively from all other adjacent electrode pairs excluding the pairs containing one or both of the current electrodes. For a sixteen electrode EIT system with domain under test surrounded by equally spaced sixteen electrodes (, , , , , , , , , , , , , , , and ), the neighbouring method injects current through the current electrode pairs for sixteen current projections (Figure 2), and the differential voltages are measured across the voltage electrode pairs using four electrode method in each projection. As shown in Figure 2(a) in the first current projection () of adjacent method, the current is injected through electrode 1 () and electrode 2 (), and the thirteen differential voltage data () are measured successively between the thirteen electrode pairs -, -, and -, respectively (Figure 2(a)). As reported by Brown and Segar, in neighbouring current injection method, the current density within the DUT is found highest between the current electrodes ( and for ); the current density then decreases rapidly as a function of distance . Similarly, in current projection 2 (), the current signal is injected through electrodes 2 () and 3 (), and an another set of thirteen differential voltage data are collected between the thirteen electrode pairs -, -, -, and so on. Lastly, in the current projection 16 (), the last set of thirteen differential voltage data are collected between the thirteen-electrode pairs -, -, and - by injecting the current through the electrodes and . Thus, the neighbouring current injection method in a sixteen electrode EIT system data collection procedure consists of sixteen current projections (, and ), and each of the current projection yields thirteen differential voltage data . Therefore, a complete data collection scan with the neighbouring current injection method in a sixteen electrode EIT system yields voltage measurements. Though in neighbouring method, EIT boundary data are not collected across the electrode pairs containing one or two current electrode for contact impedance problem , but sometimes it is advantageous to collect the boundary data from all the electrodes including the current electrodes to obtain the greatest sensitivity to the resistivity changes in the domain as reported by Cheng et al. . In the present study, the boundary potentials are calculated at all the electrodes (Figure 2(b)) with respect to a virtual ground point selected within the DUT. Hence, in a complete data collection scan, the potentials on all the electrodes are collected in all the sixteen current projection and are stored in . Therefore, the is found as a column matrix containing voltage data all collected with respect to the virtual ground point of the DUT. Hence, in the present study, with neighbouring current injection method, the is found as a matrix containing 256 electrode potentials. In the present study, 1 mA current injection is simulated through the electrodes of the simulated domain containing sixteen nodal electrodes using adjacent or neighboring current injection protocol (Figure 2(b)). The potentials on all the sixteen electrodes are calculated using boundary data calculator (BDC) for all the current projections, and the electrode potential matrix is used as the calculate boundary potential matrix to reconstruct the conductivity distribution of DUT. The BDS is designed in such a way that a huge number of voltage data sets can be generated using different types of phantoms with their different design parameters. Boundary potential data are generated for different type of phantom configurations, and the boundary data have been tested with electrical impedance tomography and diffuse optical tomography reconstruction software (EIDORS) [37, 38] for 2D-EIT. A large number of data sets are generated by changing the values of one or more phantom parameters like: phantom diameter (), inhomogeneity radius (), inhomogeneity geometry (shape, size, and position), inhomogeneity number (), bathing solution conductivity (), and inhomogeneity conductivity (). 1 mA current injection is simulated to the domain boundary, and corresponding boundary data sets are used for image reconstruction in EIDORS. Data generation in BDS and image reconstruction in EIDORS are studied for different inhomogeneity geometries in DUT. Reconstruction is also studied for different iterations and for multiple inhomogeneity reconstruction to evaluate the BDS. 3. Results and Discussion Image reconstruction quality in EIT depends on the boundary data accuracy which is dependent on the geometric accuracy of the inhomogeneity developed in BDS. Dimensional accuracy of the inhomogeneity depends on the number of finite elements in the FE mesh or mesh refinement number () as shown in Figure 3. As the increases, the number of elements in the FE mesh is increased, and hence the geometric accuracy of the inhomogeneity increases which gives more accurate boundary data and better image reconstruction (Figure 3). But the BDS with a highly refined mesh needs a high PC memory and large computation time. In this paper, the mesh refinement is found suitable as as per the configuration of the PC (2.4 GHz/1.5 GBRAM/ P-IV) used. It is observed that the FE mesh with (containing 2048 elements and 1089 nodes) gives almost an accurate geometry (Figure 3) to the desired inhomogeneity and generates a reconstructible data set in less than 10 seconds. EIDORS reconstructs the resistivity images from the BDS data sets using regularized image reconstruction technique. Results show that the resistivity or conductivity can be successfully reconstructed from the boundary data generated by our BDS using a circular domain () with a circular inhomogeneity ( mm, , , , and ) in the 9th iteration (Figure 4). It is also observed that the reconstructed shape of the inhomogeneity is similar to that of the original one (Figure 4(a)), and the reconstructed conductivity profile in Figure 4(b) is almost similar to that of the original object in Figure 4(a). Iteration studies shows that in different reconstruction steps called iterations (Figure 5), the reconstructed images become more localized from iteration to iteration and the reconstruction errors (appeared by the red color at phantom periphery) are gradually reduced (Figure 5). It is observed that the resistivity is successfully reconstructed from the boundary data in the 9th iteration (Figures 5(i) and 5(j)), though the shape of all the reconstructed images in 9th–12th iterations is almost similar to that of the original one (shown by dotted circles in Figure 5). As the reconstructed resistivity profile similar to that of the original is obtained only in the 9th iteration, the 9th iteration is taken as the optimum reconstruction. In 13th and 14th iterations, the resistivity is overestimated, and the images are lost. The optimum iteration number depends on the data accuracy and reconstruction algorithm, and hence the BDS can be used to generate the boundary data sets required for assessing the inverse solver in EIT. Voltage data are also generated for a domain () with the circular inhomogeneities (, , and ) positioned at different places using the BDS (Figure 6). It is observed that the reconstructed image is more circular for an inhomogeneity positioned at the phantom centre where and (Figure 6(a)). On the other hand, for , that is, for the inhomogeneities near domain boundary (Figure 6(b)), reconstructed images are not perfectly circular because of the comparatively less accurate shape of the original object obtained for . For a less number of mesh refinements, the geometry of the original side objects is not exactly circular itself (Figure 4), and hence the corresponding boundary data have lower accuracy. An FE mesh with large can easily produce an accurate geometry for the boundary objects (objects near domain boundary) with proper shape, which gives a boundary data without geometric error and automatically improves the image shape. Boundary data sets are also generated with a circular domain ( and ) with a circular inhomogeneity () with different diameters () and all positioned at the phantom center (). The boundary data are calculated and used for reconstructing the resistivity images. Results show that for the domain discretized with , the data sets, generated with a diameter larger than 13.3% of the phantom diameter, are accurate enough (Figures 7(a)–7(f)) to reconstruct the resistivity images in EIDORS-2D. It is clearly observed that for , the triangular elements within the inhomogeneity with smaller are unable to shape themselves into a proper circle (Figure 7(g)). Hence, the data obtained for the inhomogeneity with a diameter of 20 mm has low accuracy (Figure 7(g)), and hence the resistivity image (Figure 7(h)) is found with low resolution showed and some reconstruction error (appeared in the red color at phantom periphery). Increasing the FE elements in BDS, the boundary data error can be minimized, and the improved resistivity image can be achieved even for smaller inhomogeneities with a diameter less than 13.3% of . Boundary potential data are also generated for domains () containing multiple circular inhomogeneities (, , , and ) placed at different positions inside the domain (Figure 8). Figure 8(a) shows a domain with two circular inhomogeneities ( apart from each other) which are placed at a central distance () of 37.5 mm. Similarly, another domain with three circular inhomogeneities (120° apart from each other) placed inside the phantom domain is shown in Figure 8(c). All the inhomogeneities in both the domains are positioned at a central distance () of 37.5 mm. 1 mA current is simulated with the neighbouring current pattern, and the boundary data are collected for resistivity reconstruction. It is noticed that the resistivity images (Figures 8(b) and 8(d)) of inhomogeneities in both the domains are reconstructed successfully. Results show that the boundary data simulator can be efficiently used to generate boundary potential data for a huge number of phantom configurations in less than 10 seconds. BDS is software-based virtual EIT phantom, and hence it has a number of advantages over the practical and mesh phantoms. The literatures [39–41] presenting the phantom simulations are limited, and they only discuss the software phantoms developed for their own systems. BDS is a software-based versatile boundary data simulator which generates boundary data suitable for studying the reconstruction algorithm required for several EIT systems, and hence it is better suited for assessing the performance of the inverse solver of 2D electrical impedance tomography. A MATLAB boundary data simulator (BDS) is developed for studying the resistivity reconstruction in inverse solvers of 2D-EIT. BDS is developed with four parts: imaging domain simulator (IDS), EIT model developer (EMD), current injection simulator (CIS), and boundary data calculator (BDC). Imaging domain simulator (IDS) simulates a domain with single or multiple inhomogeneities of different geometries defined with their corresponding conductivity distributions, whereas the EIT model developer (EMD) derives a forward model using FEM to solve the governing equation of the DUT. Current injection simulator (CIS) simulates a constant current injection through the simulated electrodes positioned at the domain boundary with the neighbouring current injection protocol. The boundary data calculator (BDC) solves the forward model to solve the governing equation and calculates the potentials at all the simulated electrodes. Boundary data are generated with different type of domains simulated in BDS by changing its input parameters. Resistivity images are reconstructed from the boundary data using standard EIT reconstruction software called EIDORS, and the BDS is evaluated. It is observed that the BDS with FE mesh with 2048 elements can simulate an inhomogeneity of desired geometry with suitable accuracy. The BDS with 2048 elements suitably generates the boundary data for simulated domains containing the objects with different geometries which are found efficient for image reconstruction in EIDORS. Results also show that the conductivity or resistivity profiles of the domains simulated in BDS are successfully reconstructed from their corresponding boundary data generated for different type of single and multiple inhomogeneities. By changing the inhomogeneity position, diameter, and number in BDS, boundary data are successfully generated as well as the resistivity images are reconstructed successfully. Multiple inhomogeneity imaging shows that the BDS suitably generates boundary data with the desired accuracy, and the boundary data are found efficient for resistivity reconstruction in EIDORS. Results also show that for the simulated domains discretized with , the boundary data sets generated for circular inhomogeneity with a diameter larger than 13.3% of the phantom diameter are accurate enough to reconstruct the resistivity images in EIDORS. 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Towards the equation of state for neutral ([C.sub.2][H.sub.4]), polar ([H.sub.2]O), and ionic ([bmim][[BF.sub.4]], [bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N]) liquids. Behavior of low-melting organic salts or ionic liquids (ILs) [1-6] in the region of phase transitions is qualitatively similar to that either for high-temperature nonorganic molten salts or long-hydrocarbon-chain organic solvents and, even, for polymer systems. Such characteristic features as negligible vapor pressure [P.sub.[sigma]](T), undefined critical parameters [P.sub.c], [[rho].sub.c], [T.sub.c] for vapor-liquid (v, l)-transition, split of liquid-solid (l,s)-boundary onto melting [P.sub.m](T) and freezing [P.sub.f](T) branches, existence of glassy states make the problem of metastability to be quite complex but vital for many potential uses of ILs. In particular, thermodynamic modeling and computer simulation of the phase behavior in mixtures formed by ILs with water and low-molecular organic solvents such as ethylene can be of great importance for the further tuning of their operational parameters. If one proceeds from a pure to a mixed fluid, it is especially advantageous to develop the same format of reference equation of state (EOS) and the common format of reference pair potential (RPP) for each component and mixture. As a first step toward consistent modeling of the phase behavior of IL and its solution we demonstrate in this work how the fluctuational-thermodynamic (FT) EOS [7-12] and the relevant finite-range Len-nard-Jones (LJ) RPP can be applied to model the underlying structure and properties of low-molecular ([C.sub.2][H.sub.4], [H.sub.2]O) and imidazolium-based (1-butyl-3-methylimidazolium tetrafluorob orate ([bmim][[BF.sub.4]]), 1-butyl-3-methylimida-zolium hexafluorophosphate ([bmim][[PF.sub.6]]), 2,3-dimethyl-1-propylimidazolium bis(trifluoromethylsulfonyl)imide ([pmmim][[Tf.sub.2]N])) solvents. For any pure component FT-model is based either on the measurable coexistence-curve input data [P.sub.[sigma]](T), [[rho].sub.v](T), [[rho].sub.l](T) (if they are achievable as for [C.sub.2][H.sub.4] and [H.sub.2]O) or on the also measurable one-phase density of liquid at atmospheric pressure [rho] ([P.sub.0] [approximately equal to] 0,1 MPa, T) for ILs. This methodology becomes purely predictive for density [rho](P,T) in any one-phase v,l,s-regions including their metastable extensions. Only the measurable isobaric heat capacity data [C.sub.P]([P.sub.0], T) have to be added to the set of input data for prediction of other caloric properties (isochoric heat capacity [C.sub.V](P,T), speed of sound W(P,T), and Gruneisen parameter Gr(P, T)) at higher pressures P > [P.sub.0] and lower T < [T.sub.m] or higher T > [T.sub.b] temperatures where [T.sub.b] is the hypothesized normal boiling temperature [T.sub.b]([P.sub.0]). Its existence itself is a debatable question because the thermal decomposition [T.sub.d] maybe former [T.sub.d] < [T.sub.b]. Such approach was proposed recently [7,8] to reconstruct the hypothetical (V, l)-diagram of any ILs in its stable and metastable regions on the base of only standard reference data on density [rho](T) at [P.sub.0] [1-4] and one free parameter, an a priori unknown value of the excluded volume [b.sub.0]. To our knowledge this is first attempt to predict simultaneously the whole set of one-phase and two-phase properties for ILs without the fit at any other pressures including the negative ones. It was argued that the particular low-temperature variant of the most general FT-EOS [9-12] should be used to obtain the consistent prediction of volumetric properties and the standard response functions [a.sub.P], [[beta].sub.T], [[gamma].sub.[rho]] = [a.sub.P]/[[beta].sub.T] by the following equations: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4) where [b.sub.0] is the excluded molecular volume and a(T) is the T-dependent effective cohesive energy. The derivative da/dT affects the thermal expansion [[alpha].sub.P] and the thermal-pressure coefficient [[gamma].sub.[rho]] while the isothermal compressibility [[beta].sub.T] depends only on [b.sub.0]-value at the given pressure. The changeable sign of two thermal derivatives [[alpha].sub.P], [[gamma].sub.[rho]] offers a possibility to predict the properties of anomalous low-temperature substances (such as water, for example) too [7,8]. Fortunately we have obtained now [13-19] a possibility to test our predictions not only by the direct experimental one-phase data [14, 16, 18, 19] on [rho](P,T)- and W(P,T)-surfaces. Another possibility is offered by comparison of the predictions obtained by FT-EOS for the critical parameters of ILs ([bmim][[BF.sub.4]]: [T.sub.c] = 962,3 K, [P.sub.c] = 3503,9 kPa, [[rho].sub.c] = 438,565 [kg.m.sup.-3] with those predicted here by the Sanchez-Lacombe EOS for lattice fluid (LF) : [T.sub.c] = 885,01 K, [P.sub.c] = 2829 kPa, [[rho].sub.c] = 248,565 [kg.m.sup.-3] as well as with those simulated by GEMC-methodology : [T.sub.c] = 1252 K, [P.sub.c] = 390 kPa, [[rho].sub.c] = 181 [kg.m.sup.-3]. It seems that the relatively close location of ([T.sub.c],[P.sub.c])-parameters predicted by both EOSs is some guarantee of their reliability while [T.sub.c] and [P.sub.c] from are significantly overestimated and underestimated, respectively. Interestingly, the known descriptive factor of compressibility [r.sub.t] = [P.sub.c]/([[rho].sub.t][RT.sub.c]) estimated by Guggenheim in the vicinity of triple point [T.sub.t] for argon as [r.sub.t] = 0,108 is equal to close values [r.sub.t] = 0,082 for FT-EOS and [r.sub.t] = 0,072 for LF-EOS but only to very small value [r.sub.t] = 0,007 for result of GEMC-simulations if the common realistic estimate (see below) [[rho].sub.t] [approximately equal to] [[rho].sub.t] = 5,350646 [mol*dm.sup.-3] at T = 290 K is used. Moreover, it will be shown that the characteristic dimensional parameters [P.sup.*.sub.c], [T.sup.*.sub.c], [[rho].sup.*.sub.c] and another compressibility factor [r.sup.*.sub.c] = [P.sup.*.sub.c]/([[rho].aup.*.sub.c ][RT.aup.*.sub.c]) obtained by Machida et al. by the fit to (P, [rho], T)-experimental data for [bmim][[BF.sub.4]] and [bmim][[PF.sub.6]] provide the structural estimates of hard-core volume, number of lattice sites in a cluster, and energy of near-neighbor pair interactions which are surprisingly close to ones independently predicted by the FT-model of a continuum substance. Taking into account the compatibility of above results it is important to consider the presumable similarity between the square-well fluid (which may be thought of as a continuum analogue of the lattice-gas (LG) or lattice-fluid (LF) systems) on the one hand and the LJ-fluid of finite-range interactions (RPP) on the other. This conceptual analogy has been pointed out long ago for the critical region by Widom who suggested that it is the propagation of attractive correlations in the LG which determines the peculiarities of criticality However, such unphysical LG-predictions at low temperatures of the ([rho], T)-plane as the nonexistence of a (l, s)-transition suggest that repulsive forces are not being treated properly by this RPP-model. In contrast with the discrete LG-model, it seems that both attractive and repulsive forces are being dealt with properly in the square-well continuum fluid because it exhibits both (v, l)- and (l, s)-transitions. The serious restriction of latter is however evident since any singularities of RPP imply an artificial jump of pair-distribution isotropic function g(r) at the point of cutoff radius [r.sub.c] for attractive interactions. In this context only the shifted and smoothed at [r.sub.c]-point LJ-potential [5, 6] seems to be appropriate as RPP for a continuum system. Of course, the algebraic form of the respective reference EOS is essential too. In accordance with the statistical-mechanical arguments presented by Widom there are the set of alternative forms including the original vdW-EOS and the LG-EOS in the well-known Bragg-Williams approximation which share the common restrictive feature. One may suppose that the probability of finding some prescribed value of the potential energy [TEXT NOT REPRODUCIBLE IN ASCII] at an arbitrary point in the fluid is independent of T at fixed [TEXT NOT REPRODUCIBLE IN ASCII]. Another simplifying assumption is that such EOS supposes only two types of fluid structure, one of the excluded (or hardcore) volume [Nv.sub.0] where the singular hard-sphere branch of potential is infinite and one of free volume (V - [Nv.sub.0]) where the potential is uniform, weak, and unrestricted (an infinite-range rectilinear well). It should be directly proportional to density e = U/N = -ap where U is the total configurational energy and a is the constant vdW-coefficient. These historical notes are important to explain how one can go beyond the above restriction of T-independency by adoption of linear [rho]-dependence for a generalized specific or molar energy (see also (8) below). Consider a (T) = - [([partial derivative]e/[partial derivative][rho]).sub.T] (5) Another aim of the developed FT-EOS follows from the possibility to estimate the effective LJ-parameters without any fit. Indeed, their general T-dependent values, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6a) [epsilon](T)/k = T (1- [Z.sub.l]) (6b) are determined simply in the low-temperature range of all ILs where [b.sub.0] is constant in ((1)-(4)) while the compressibility factor of saturated liquid [Z.sub.t] = [P.sub.[sigma]]/([[rho].sub.l]RT) becomes negligible as well as the vapor pressure [P.sub.[sigma]](T) trends to zero. Taking into account this asymptotic behavior it is especially important to study the possible correlations of these parameters in the RPP-model of an effective LJ-potential for ILs as the functions of total molecular weight M. This concept is unusual for the conventional consideration of a separate influence of the anions [M.sub.a] and cations [M.sub.c] components. It may provide, in principle, the useful insight the nature of (v,l)-transition in ILs by effective capturing underlying pair interactions. The distinction of both FT-EOS and LF-EOS from the conventional hard-sphere reference EOS is crucial to provide the quantitative description of one-phase liquid. The formers include the quadratic in density contribution, which is dominating at high pressures along the isotherms. The latter considers this term as a small vdW-perturbation for the hard-sphere EOS. Such perturbation approach is not directly applicable to associating fluids such as water and alcohols for which presence of hydrogen bonding, anisotropic dipolar 1/[r.sup.3], or coulombic 1/r interactions in addition to isotropic dispersive 1/[r.sup.6] attractions is inconsistent with the main assumption of the perturbation methodology that the structure of a liquid is dominated by repulsive forces . The FT-model promotes the more flexible approach in which the above factors of attraction and clustering can be effectively accounted by the a(T)-dependence. It was firstly confirmed by Longuet-Higgins and Widom and, then, by many authors that a combination of Carnahan-Starling EOS, for example, with the vdW-perturbation a[[rho].sup.2] is a reasonable approximation for the l- and s-phases but not the v-phase. Guggenheim has concluded its applicability only to a liquid when large clusters are more important than small clusters (i.e., at low temperatures [T.sub.m] < T < [T.sub.b]). In contrast with this observation, the general FT-EOS provides the adequate representation of entire subcritical range [T.sub.m] < T < [T.sub.c] including the critical region and (v, l)-phase transition [9-12]. It will be shown below by FT-model without undue complexity of calculations. 2. Universal FT-EOS for Any Low-Themperature Fluids 2.1. General Form of FT-EOS for Subcritical Themperatures. It is often claimed that the original van der Waals (vdW)-EOS with two constant coefficients a, b determined by the actual critical-point properties [[rho].sub.c], [T.sub.c] is only an approximation at best and cannot provide more than qualitative agreement with experiment even for spherical molecules. However, it was proved recently [9-12] that the general FT-EOS with three T-dependent coefficients, p = [[rho]RT [1 - c(T)]/1 - b(T) [rho]] - a(T) [[rho].sup.2], (7) is applicable to any types of fluids including ILs. The measurable volumetric data of coexistence curve (CXC) have been used for evaluation of T-dependences without any fit. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10) where the reduced slope [A.sub.[sigma]](T) of [P.sub.[sigma]](T)-function is defined by the thermodynamic Clapeyron's equation: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11) This fundamental ratio of the (v,l)-latent heat to the thermodynamic work of (v, l)-expansion is the main parameter of FT-coefficients determined by ((8)-(9)). It should be calculated separately in each of high-temperature ([T.sub.b] [less than or equal to] T [less than or equal to] [T.sub.c]) [9-12] v- and l-phases to obtain the reasonable quantitative prediction of one-phase thermophysical properties. The general FT-EOS is applicable to the entire subcritical range ([T.sub.t][less than or equal to] T [less than or equal to] [T.sub.c]) but it can be essentially simplified to the form of (1) if ([T.sub.t ][less than or equal to] T [less than or equal to] [T.sub.b]). 2.2. Particular Form of FT-EOS for Low Themperatures. An absence of input CXC-data [P.sub.[sigma]](T), [[rho].sub.g](T), [[rho].sub.l](T) for ILs is the serious reason to develop the alternative method for the evaluation of T-dependent FT-coefficients. The thermodynamically-consistent approach has been proposed in [7, 8] for the particular form of FT-EOS (1) applicable in the low-temperature range from the triple [T.sub.t] (or melting [T.sub.m]) point up to the [T.sub.b]-point. Former one is usually known for ILs while the latter one is, as a rule, more than temperature of thermal decomposition [T.sub.b] > [T.sub.d] ~ 650 K. The methodology was tested on two low-molecular-weight substances ([C.sub.2][H.sub.4], [H.sub.2]O) and two imidazolium-based ILs ([bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N]) with the promising accuracy of predictions even for the isothermal compressibility [[beta].sub.T] up to the pressure P = 200 MPa. To illuminate the distinction between the particular (reference) and general form of FT-EOS let us discuss in brief the main steps of the proposed procedure. Its detailed analysis can be found elsewhere [7, 8]. The algorithm is as follows. Step 1. At the chosen free parameter [T.sub.b] one determines the orthobaric molar densities [[rho].sub.l]([T.sub.b]) = [rho]([T.sub.b],[P.sub.0]); [[rho].sub.g]([T.sub.b]) = [P.sub.0]/[RT.sub.b] to solve the transcendent equation: [[rho].sub.g]/[[rho].sub.l] = 1 + y (x) [e.sup.-x]/ 1 + y (x) [e.sup.x], (12) for the reduced entropy (disorder) parameter x([T.sub.b]) and the respective molar heat [r.sub.[sigma]]([T.sub.b]) of vaporization. Consider x = [([s.sub.g] - [s.sub.l])/2R], (13a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13b) Step 2. The universal CXC-function y(x) in (12) is determined by equalities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14) and it provides the possibility to estimate a preliminary value of [b.sub.0], [b.sub.0] = [x([T.sub.b]) - 1]/[[rho].sub.l]([T.sub.b]) [x([T.sub.b]) - 1/2], (15) as well as to evaluate the orthobaric densities at any T if the function x(T) is known. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16b) Step 3 (A-variant of x(T)-prediction [7, 8]). To calculate its values one must obtain two densities [[rho].sup.[+ or -]] (at the assumption [P.sub.[sigma]] [approximately equal to] 0) from equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17) where [rho] = [rho]([T,P.sub.0]), [Z.sub.0] = [P.sub.0]/[rho]RT and the [[rho].sup.+](T)-function provides the preliminary estimate of x(T) for the low-temperature range (at the consistent assumption: [[rho].sup.+] >> [[rho].sub.g] [approximately equal to] 0). Consider x(T) = 1 - [b.sub.0][[rho].sup.+]/2/1- [b.sub.0][[rho].sup.+]. (18) Step 4. One substitutes x(T) from (18) in ((13a), (13b), (16a), (16b)) to calculate [r.sub.[sigma]](T), [[rho].sub.g](T), [[rho].sub.l](T), respectively. Step 5. A preliminary value of [a.sub.0](T) may be estimated then by the more restrictive assumption [P.sub.0] [approximately equal to] 0 (used also in the famous Flory-Orwoll-Vrij EOS developed for heavy n-alkanes). Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19) Step 6 (B-variant of x(T)-prediction). To control the consistency of methodology one may use instead of Step 3 (A-variant) the same equation (17) with the approximate equality [[rho].sub.l](T) [approximately equal to] [[rho].sup.+](T) to solve (16b) at the a priori chosen [b.sub.0]-value for determination of alternative x(T), and so forth, (Steps 4 and 5). Just this approach (B-variant) has been used below in the low-temperature range of [bmim][[BF.sub.4]]. Step 7. The self-consistent prediction of a hypothetical (v, l)-diagram requires the equilibration of CXC-pressures [P.sub.[sigma]](T, [[rho].sub.g]) = [P.sub.[sigma]](T, [[rho].sub.l]) by FT-EOS (1) with the necessary final change in [a.sub.0](T)-value from (19) to satisfy the equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20) Only in the low-temperature range T [less than or equal to] [T.sub.b] the distinction between the preliminary definition (18) and its final form (19) for a(T)-values is not essential at the prediction of vapor pressure [P.sub.[sigma]](T,[[rho].sub.g]). 3. Reference Equation of State, Effective Pair Potential, and Hypothetical Phase Diagram To demonstrate universality of approach and for convenience of reader we have collected the coefficients of FT-EOS (1) for neutral ([C.sub.2][H.sub.4]) and polar ([H.sub.2]O) fluids [7, 8] in Table 1 and added in Table 2 to other ILs ([bmim][[PF.sub.6]], [pmmim][[Tf.sub.2]N] [7, 8]) the data for [bmim][[BF.sub.4]] obtained in this work (Table 3). When temperature is low [T.sub.m] < T < [T.sub.b] FT-model follows a two-parameter ([eplison](T), [[sigma].sub.0]) correlation of principle of corresponding states (PCS) on molecular level as well as a two-parameter (a(T), [b.sub.0]) correlation of PCS on macroscopic level. One the most impressed results of FT-methodology is shown in Figure 1 where the comparison between such different high- and low-molecular substances as ILs and [C.sub.2][H.sub.4], [H.sub.2]O is represented. The results based on the coefficients of Tables 1 and 2 demonstrate that the proposed low-temperature model provides the symmetric two-value representation of vapor pressure [+ or -][P.sub.s](T) similar to that observed for the ferromagnetic transition in weak external fields. To estimate the appropriate excluded molar volume [b.sub.0] (M = 225,82 g/mol) of FT-model we consider that it belongs to the range [v.sub.0] = M/[[rho].sub.0] [approximately equal to] 162, [v.sub.l] = M/[[rho].sub.l] [approximately equal to] 187[cm.sup.3]/mol]. The extrapolated to zero temperature T = 0 K "cold" volume [v.sub.0] = 162 cm /mol follows from (27). The fixed value: [b.sub.0] = 178 [cm.sup.3]/mol ([b.sub.0] [approximately equal to] 1,[1v.sub.0]) has been used in this work to demonstrate the main results of the proposed methodology. Such choice for [bmim][[BF.sub.4]] on the ad hoc basis is in a good correspondence with the respective values: [b.sub.0] = 195,3 [cm.sup.3]/mol for [bmim][[PF.sub.6]] and [b.sub.0] = 271,1 [cm.sup.3]/mol for [pmmim][[Tf.sub.2]N] where the empirical relationship [b.sub.0] [approximately equal to] 1,[1v.sub.0] was also observed [7, 8]. Our estimates of the effective LJ-diameters by (6a) for ILs: [sigma]([bmim][[BF.sub.4]]) = 5,208 A, [sigma]([bmim][[PF.sub.6]]) = 5,371 A, and [sigma]([pmmim][[Tf.sub.2]N]) = 5,992 A can be tested by comparison with the independently determined values for anions [[sigma].sub.a]([[BF.sub.4]]) = 4,51 [Angstrom]; [[sigma].sub.a]([[PF.sub.6]]) = 5,06 [Angstrom]. We have verified Berthelots combining rule for spherical molecular ions (21a) and van der Waals' combining rule for chain molecules (21b) usually considered by van der Waals'-type of EOS for mixtures . Consider [sigma] = [[sigma].sub.c] + [[sigma].sub.a]/2, (21a) [b.sub.0] = [b.sub.c] + [b.sub.a]/2. (21b) The predicted by former rule of LJ-diameter for the same [bmim]-cation were close but still different, 5,906 [Angstrom] and 5,682 [Angstrom]. For the latter rule their values and distinction become even smaller, 5,757 [Angstrom] and 5,651 [Angstrom]. As a result, the chain rule (21b) seems preferable for ILs and its average value for [[sigma].sub.c][bmim] = 5,704 [Angstrom] can be used to estimate the LJ-diameter of [[Tf.sub.2]N]-anion: [[sigma].sub.a][[Tf.sub.2]N] = 6,254 [Angstrom] taking into account the equality: [M.sub.c] [bmim] = [M.sub.c] [pmmim] = 139 g/mol. The collected in Table 4 effective LJ-diameters are linear functions of [M.sub.a] in the set of ILs with different anions and cations if the molecular weight of latters [M.sub.c] is the same one. Since the low-temperature compressibility factor [Z.sub.l](T) is very small for all discussed liquids their dispersive energies [eplison](T) (molecular attractions parameters) are comparable in accordance with (6b). However, the differences in cohesive energies a(T) (collective attraction's parameters) between the low-molecular substances ([C.sub.2][H.sub.4], [H.sub.2]O) and ILs are striking as it follows from Tables 1 and 2. The physical nature of such distinction can be, at the first glance, attributed to omitted in the reference LJ-potential influence of intramolecular force-field parameters and anisotropic (dipole-dipole and coulombic) interactions. At the same time, one must account the collective macroscopic nature of a(T)-parameter. It corresponds to the scales which are compatible or larger than the thermodynamic correlation length [xi]([rho], T). FT-model [9-12] provides an elegant and simple estimation of this effective parameter based on the concept of comparability between energetic and geometric characteristic of force field determined by the given RPP. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22) Taking into account the above results and the coefficients from Tables 1-3 we have used (22) at T = 300 K (T* = [k.sub.B]T/[eplison] [approximately equal to] 1) to compare the thermodynamic correlation length predicted for [bmim][[BF.sub.4]] (a = 8900,9 J*d[m.sup.3] /[mol.sup.2]; [b.sub.0] = 178 [cm.sup.3] /mol; [rho] = 5,322294 mol/d[m.sup.3]) and at T = 298,15 K for water (a = 548,27 J*d[m.sup.3]/[mol.sup.2]; [b.sub.0] = 16,58 cm /mol; [rho] = 55,444 mol/d[m.sup.3]) . The dimensional and reduced ([xi]* = [xi]/[sigma]) values for former are, respectively, [xi] = 177,7 [Angstrom], [xi]* = 34,12 while for latter [xi] = 69,86 [Angstrom], [xi]* = 29,45. No more need be said to confirm the universality of FT-model. One may note that our estimates of correlation length are significantly larger than those usually adopted for the dimensional or reduced cutof radius ([r.sub.c] or [r*.sub.c] = [r.sub.c]/[sigma]) of direct interactions at computer simulations. As a result, the standard assumption [[xi]*.sub.c] [approximately equal to] [r*.sub.c] may become questionable in the comparatively small (mesoscopic) volumes of simulation [L.sup.3] < [xi][([rho],T).sup.3]. At this condition the simulated properties are mesoscopic although their lifetime may be essentially larger than its simulated counterpart. The key point here is the same as one near a critical point where the problem of consistency between the correlation length for statics and the correlation time for dynamics becomes crucial. In any case, the computer study of possible nongaussian nature of local fluctuations within the thermodynamic correlation volume [[xi].sup.3] may be quite useful. The relevant inhomogeneities in the steady spacial distributions of density and enthalpy can affect, first of all, the simulated values of volumetric ([[alpha].sub.P], [[beta].sub.T]) and caloric ([C.sub.p],[C.sub.v]) derived quantities. Simultaneously, an account of internal degrees of freedom and anisotropy by the perturbed RPP may change the correlation length itself. The above described by ((12)-(20)) FT-methodology has been used to reconstruct the hypothetical phase diagram (HPD) for [bmim][[BF.sub.4]] shown in Figures 2, 3, and 4 and represented in Table 3. Both (T, [rho]) (Figure 2) and (P, T) (Figure 4) projections contain also the branches of classical spinodal calculated by the LF (Sanchez-Lacombe)-EOS obtained in . Its top is the location of a respective critical point. It seems that the relatively close ([P.sub.c],[T.sub.c])-parameters predicted independently by FT-EOS and by LF-EOS (see Section 1) are reasonable. The FT-model provides a possibility to estimate, separately, the coordination numbers of LJ-particles in the orthobaric liquid [[rho].sub.l](T)- and vapor [[rho].sub.g](T)-phases. An ability to form the respective "friable" ([N.sub.l,g] + 1)-clusters is defined by the ratio of effective cohesive and dispersive molar energies at any subcritical temperature. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23) The term "friable" is used here to distinguish the clusters formed by the unbounded LJ-particles at the characteristic distance l* = l/[sigma] [approximately equal to] [cube root of 2] > 1 from the conventional "compact" ones with the bonding distance l* < 1 studied, in particular, by the GEMC-methodology to model of molecular association. It is straightforwardly to obtain the low-temperature estimates based on the assumptions. [Z.sub.l] << [Z.sub.g] [approximately equal to] 1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24b) and to find the critical asymptotics based on the difference of classical ([a.sup.0] ,[b.sup.0] ,[c.sup.0]) and nonclassical (a, b, c) T-dependent FT-EOS' coefficients [9-12]. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25b) The crucial influence of excluded-volume in (24a) and its relative irrelevance in (24b) for [N.sub.l,g,] -predictions are illustrated by Figure 5 where [N.sub.l](T) function is shown also for the entire l-branch based on the evaluated in the present work HPD. For comparison, the low-temperature ability to form the ([N.sub.l] + 1)-clusters in liquid water [7, 24] is represented in Figure 5 too. In according with ((25a), (25b)) the "friable" clusters can exist only as dimers in the classical critical liquid phase ([N.sub.l.sup.c] [approximately equal to] 1). It is not universal property in the meaning of scaling theory but it corresponds to the PCS-concept of similarity between two substances ([H.sub.2]O and [C.sub.4]mim [[BF.sub.4]], e.g.) if their Z.sub.c]-values are close. On the other side, the scaling hypothesis of universality is confirmed by the FT-EOS' estimates in the nonclassical critical vapor phase. For the set of low-molecular-weight substances studied in (Ar,[C.sub.2][H.sub.4], [CO.sub.2], [H.sub.2]O); for example, one obtains by (25b) the common estimate ([N.sup.c].sub.g] [approximately equal to] 2,5) which shows a significant associative near-mean-field behavior. It is worthwhile to note here the correspondence of some FT-EOS'-estimates with the set of GEMC-simulated results. One may use the approximate estimate of critical slope [A.sub.c] [approximately equal to] 7,86 for [bmim][[BF.sub.4]] based on the similarity of its [Z.sub.c]-value with that for [H.sub.2]O . In such case, the respective critical excluded volume [b.sub.c] [approximately equal to] 220 [cm.sup.3]/mol becomes much more than vdW-value 1/3[[rho].sub.c] = [b.sub.c.sup.0] [approximately equal to] [b.sub.0] [approximately equal to] 178[cm.sup.3]/mol. Another observation seems also interesting. Authors have calculated (see Figure 3 in ) for the "compact" clusters at [l.sup.*] = 0,7; 0,5; 0,45; the ([T.sup.*], [[rho].sup.*])-diagram of simple fluids. One may note that only the value [l.sup.*] = 0,7 corresponds to the shape of strongly-curved diameter shown in Figure 2 for the HPD while the smaller values: l* = 0,5; 0,45 give the shape of HPD and the nearly rectilinear diameter strongly resembling those obtained by the GEMC-simulations for the complex ILs force-field. If this correspondence between the "friable" and "compact" clustering is not accidental one obtains the unique possibility to connect the measurable thermophysical properties with the both characteristics of molecular structure in the framework of FT-EOS. 4. Comparison with the Empirical Tait EOS and Semiempirical Sanchez-Lacombe EOS The empirical Tait EOS is based on the observation that the reciprocal of isothermal compressibility [[beta].sup.-.sub.T] for many liquids is nearly linear in pressure at very high pressures. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26) where some authors [14, 19] omit the T -dependence in coefficient C and ignore the value [P.sub.0] [approximately equal to] 0 . Such restrictions transform the Tait EOS into the empirical form of two-parameter (B(T),C) PCS because the sets of C-values for different ILs become close one to another. For example, Machida et al. have found the sets C = 0,09710 for [bmim][[PF.sub.6]], C = 0,09358 for [bmim][[BF.sub.4]], and C = 0,08961 for [bmim][OcSO.sub.4] which is rather close to the set obtained by Matkowska and Hofman C = 0,088136 for [bmim][[BF.sub.4]] and C = 0,0841547 for [bmim][MeS[O.sub.4]]. At the same time, Gu and Brennecke have reported the much larger T-dependent values C(298,2 K) = 0,1829 and C(323,2 K) = 0,1630 for the same [bmim][[PF.sub.6]]. Two other reasons of discrepancies in the Tait methodology is the different approximations chosen by authors for the reference input data [rho]([P.sub.0], T) and for the compound-dependent function B(T). Some authors [4, 14, 18] prefer to fit the atmospheric isobars [rho]([P.sub.0],T) and[C.sub.p]([P.sub.0],T) with a second-order or even third-order polynomial equation while the others [1, 2, 16, 19] use a linear function for this aim.. As a result, the extrapolation ability to lower and higher temperatures of different approximations becomes restricted. In this work we have used for [bmim] [[BF.sub.4]] the simplest linear approximation of both density and heat capacity, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28) taken from . The extrapolated to zero of temperature value [[rho].sub.0](0 K) = 1394,65 [kg*m.sup.-3] is in a good correspondence with that from [[rho].sub.0](0 K) = 1393,92 [kg*m.sup.-3], in reasonable correspondence with that from [[rho].sub.0](0 K) = 1416,03 [kg.m.sup.-3] and that from [[rho].sub.0](0 K) = 1429[kg*m.sup.-3] but its distinction from value [[rho].sub.0](0 K) = 1476,277 [kg*m.sup.-3] reported by authors is rather large. The similar large discrepancy is observable between [C.sup.0.sub.p] (0 K) = 273,65 J/mol*K from and [C.sup.0.sub.p] (0 K) = 464,466 J/mol*K from . The different choices of an approximation function for B(T) (so authors have used the exponential form while authors have preferred the linear form) may distort the derivatives [[beta].sub.T] and [[alpha].sub.P] calculated by the Tait EOS (26). The problem of their uncertainties becomes even more complex if one takes into account the often existence of systematic distinctions of as much as 0,5% between the densities measured by different investigators even for the simplest argon . Machida et al. , for example, pointed out the systematic deviations measured densities from those reported by the de Azevedo et al. and Fredlake et al. for both [bmim][[BF.sub.4]] and [bmim][[PF.sub.6]]. Matkowska and Hofman concluded that the discrepancies between the different sets of calculated [[beta].sub.T-] and [[alpha].sub.P-]derivatives increase with increasing of T and decreasing of P due not only to experimental differences in density values but also result from the fitting equation used. The resultant situation is that the expansivity [[alpha].sub.P] of ILs reported in literature was either nearly independent of T or noticeably dependent of T [3,19]. We can add to these observations that the linear in molar (or specific) volume Tait Eos (26) is inadequate in representing the curvature of the isotherm P([rho]) at low pressures. It fails completely in description of (v, l)-transition where the more flexible function of volume is desirable. However, this has been clearly stated and explained by Streett for liquid argon that the adjustable T-dependence of empirical EOS becomes the crucial factor in representing the expansivity [[alpha].sub.P] and, especially, heat capacities [C.sub.P],[C.sub.v] at high pressures even if the reliable input data of sound velocity W(P, T) were used. From such a viewpoint, one may suppose that the linear in temperature LF-EOS proposed by Sanchez and Lacombe, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29) is restricted to achieve the above goal but can be used as any unified classical EOS common for both phases to predict the region of their coexistence. Such conjecture is confirmed by the comparison of FT-EOS with LF-EOS presented in Figures 2 and 4 and discussed below. The obvious advantage of former is the more flexible T-dependence expressed via the cohesive-energy coefficient a(T). On the other hand, the LF-EOS is typical form of EOS (see Section 1) in which the constraint of T-independent potential energy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is inherent . One may consider it as the generalized variant of the well-known Bragg-Williams approximation for the ordinary LG presented here in the dimensional form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30) Such generalization provides the accurate map of phenomenological characteristic parameters T*, P*, [rho]* which determine the constant effective number of lattice sites [N.sub.l] ]occupied by a complex molecule, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31) into the following set of molecular characteristic parameters for a simple molecule ([N.sub.l] = 1): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32) where [v.sub.0] is the volume of cell and z is the coordination number of lattice in which the negative [eplison] is the energy of attraction for a near-neighbor pair of sites. In the polymer terminology [eplison]* from (31) is the segment interaction energy and v* is the segment volume which determines the characteristic hard core per mole M/[rho]* (excluded volume b in the vdW-terminology). Another variant of described approach is the known perturbed hard-sphere-chain (PHSC) EOS proposed by Song et al. for normal fluids and polymers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33) where g(d) = (1 - [eta]/2)/[(1 - [eta]).sup.3] is the pair radial distribution function of nonbonded hard spheres at contact and the term with ([N.sub.l] - 1) reflects chain connectivity while the last term is the small perturbation contribution. Though the PHSC-EOS has the same constraint of the potential energy field [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] authors have introduced two universal adjustable [[PHI].sub.a](T)- and [[PHI].sub.b](T)-functions to improve the consistency with experiment. The vdW-type coefficients were rescaled as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34b) where s([N.sub.l]) is the additional scaling function for T* = [eplison]/[k.sub.B]. It provides the interconnection of molecular LJ-type parameters ([eplison], [sigma]) with the phenomenological vdW-ones (a, b). The resultant reduced form of PHSC is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35) where the following characteristic and reduced variables are used: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36) It was compared with the simpler form of LF-EOS (29). Their predictions of the low-temperature density at saturation [[rho].sub.l](T) are comparable but, unfortunately, inaccurate (overestimated) even for neutral low-molecular liquids. The respective predictions of the vapor pressure [P.sub.[sigma]](T) are reasonable excepting the region of critical point for both EOSs. Our estimates based on the LF-EOS shown in Figures 2 and 4 are consistent with these conclusions. The comparison of volumetric measurements and derived properties [14,18] with the purely predictive (by the FT-EOS) and empirical (by the Tait EOS and LF-EOS) methodologies used for [bmim][[BF.sub.4]] is shown in Figures 6-9. Evidently, that former methodology is quite promising. Machida et al. have reported two correlations of the same (P, [rho], T)-data measured for [bmim][[BF.sub.4]] at temperatures from 313 to 473 and pressures up to 200 MPa. To examine the trends in properties of ILs with the common cation [bmim] the Tait empirical EOS was preliminarily fitted as the more appropriate model. The estimate of its extrapolation capatibilities for (P,[rho], T)-surface in the working range (290 < T/K < 350) follows from the compatibility of experimental points (where those measured by de Azevedo et al. in the range of temperature 298 < T/K < 333 and pressure (0,1 < P/MPa < 60) were also included) with the thick curves in Figure 6. It is noticeable, for example, that the extrapolated Tait's isotherm T = 290 K coincides practically with isotherm T = 298,34 K from because the measured densities of latter source are systematically higher than those from . Density data of Fredlake et al. for [bmim][[BF.sub.4]] (not shown in Figure 6) are also systematically shifted from measurements . The consequence of such discrepancies is also typical for any simple liquids (Ar, Kr, Xe) at moderate and high pressures. It is impossible to reveal an actual T-dependence of volumetric (mechanical) derived functions [[alpha].sub.P], [[gamma].sub.[rho]] due to systematic deviations between the data of different investigators. In such situation an attempt "to take the bull by the horns" and to claim the preferable variant of EOS based exclusively on volumetric data maybe erroneous. Indeed, since the Tait EOS is explicit in density while the LF-EOS--in temperature the direct calculation of [[alpha].sub.P], [[beta].sub.T]-derivatives for former and [[alpha].sub.P], [[gamma].sub.[rho]]-derivatives for latter are motivated. To illustrate the results of these alternative calculations we have used in Figures 6-9 the coefficients of LF-EOS reported by Machida et al. for the restricted range of moderate pressures 0,1 < P/MPa < 50. The thick dashed curves represent the boundaries of working range where the extrapolation to T = 290 K is again assumed. One may notice the qualitative similarity of FT-EOS (the thin curves) and LF-EOS which can be hypothesized as an existence of certain model substance at the extrapolation to higher pressures P/MPa > 50. It demonstrates the smaller compressibility [[beta].sub.T] (Figure 8) and expansivity [[alpha].sub.P] (Figure 7) than those predicted by the Tait EOS while the value of thermal-pressure coefficient [[gamma].sub.[rho]] for FT-EOS (Figure 9) becomes larger. It determines the distinctions in the calculated internal pressure. The choice of the FT-models substance as a reference system for the perturbation methodology provides the set of advantages in comparison with the LF-EOS. It follows from Figure 6 that at moderate pressures P/MPa < 50 the predictive FT-EOS is more accurate than the fitted semiempirical LF-EOS although the discrepancies of both with the empirical Tait EOS become significant at the lowest (extrapolated) temperature T = 290 K. The Tait's liquid has no trend to (v, l)-transition (as well as polymers) in opposite to the clear trends demonstrated by FT-EOS and LF-EOS. One may suppose a competition between vaporization of IL (primarily driven by the isotropic dispersive attraction 1/[r.sup.6] in RPP) and chain formation (driven mainly by the anisotropic dipolar interactions 1/[r.sup.3]) reflected by the Tait EOS fitted to the experimental data. Of course, such conjecture must be, at least, confirmed by the computer simulations and FT-model provides this possibility by the consistent estimate of RPP-parameters ([eplison], [sigma]) at each temperature. The differences of calculated expansivity [[alpha].sub.P] in Figure 7 are especially interesting. FT-EOS predicts even less variation of it with temperature than that for the Tait EOS. This result and crossing of [[alpha].sub.P](P)-isotherms are qualitatively similar to those obtained by de Azevedo et al. although the pressure dependence of all mechanical ([[alpha].sub.P], [[beta].sub.T], [[gamma].sub.[rho]]) and caloric ([C.sub.P],[C.sub.v]) derivatives (see Figures 10,11, and 12) is always more significant for the FT-EOS predictions. It seems that the curvature of the [rho](T)-dependence following from the LF-EOS (29) is not sufficient to predict the [[alpha].sub.P](P) behavior (Figure 7) correctly. The strong influence of the chosen input [rho]([P.sub.0], T)-dependence is obvious from Figures 7-9. The prediction of caloric derivatives ([C.sub.P],[C.sub.v], [C.sub.P]/[C.sub.v], Gr) is the most stringent test for any thermal (P, [rho], T) EOS. It should be usually controlled by the experimental (W, [rho], T)-surface to use the thermodynamical identities, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39) in addition to the chosen input [C.sub.P]([P.sub.0], T)-dependence. de Azevedo et al. applied this strategy to comprise the approximated by the Pade-technique measured speed of sound data for [bmim][[PF.sub.6]] and [bmim][[BF.sub.4]] (Figure 13) with the evaluated at high pressures heat capacities. Our predictive strategy is based on the differentiation of a(T)-dependence to evaluate directly the most subtle ([C.sub.V], P, T)-surface in a low-temperature liquid [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40) where the influence of the consistence for the chosen input [rho]([P.sub.0], T)- and[C.sub.p]([P.sub.0], T)-dependences (via (37) used for estimate of [C.sub.V]([P.sub.0], T) at the atmospheric pressure [P.sub.0]) becomes crucial. The use of first derivative da/dT (even by its rough approximation in terms of finite differences: [DELTA]a/[DELTA]T) to calculate simultaneously by ((3), (4), (37), (40)) all volumetric and caloric derivatives is the important advantage over the standard integration of thermodynamic identities: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41b) To illustrate such statement it is worthwhile to remind the situation described by the Streett for liquid argon. Since isotherms of [[alpha].sub.P](P) cross over for many simple liquids (Ar, Kr, Xe), this author concludes that the sign of [([[partial derivative].sup.2]V/[[partial derivative]T.sup.2]).sub.P] changes also from positive to negative at the respective pressure. This conclusion is not valid because to change the sign of derivative [([partial derivative][[alpha].sub.P]/[partial derivative]T).sub.P] it is enough to account for the exact equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42) in which [([[partial derivative].sup.2]V/[[partial derivative]T.sup.2]).sub.P] can be always positive. In this case one would expect the monotonous decrease of [C.sub.p] with increasing P in accordance to ((41a), (41b)) while the presence of extremum (minimum or maximum of [C.sub.p](P)-dependence) seems to be artificial. There is the variety of pressures reported by different investigators as a presumable cross-point for the same ILs. Machida et al. have estimated it to be about 10 MPa on the base of Tait EOS for [bmim][[PF.sub.6]] but have not found it (Figure 7) for [bmim][[BF.sub.4]]. For latter our estimate by the FT-EOS is: P = 20,6 MPa. de Azevedo et al. have reported the mild decrease of [[alpha].sub.P](T)-dependence and the sharp decrease [[alpha].sub.P](P)-dependence while a presumable cross-point is located between about 100 and 120 MPa for [bmim][[BF.sub.4]]. Taking into account the above distinction in the evaluated ([[alpha].sub.P], P, T)-surface it is interesting to consider their consequences for caloric ([C.sub.v],P,T)-([C.sub.p],P,T)- and[C.sub.v]/[C.sub.p]-surfaces shown in Figures 10-12. The remarkable qualitative and even quantitative (<8%) correspondence between the predicted by FT-EOS[C.sub.v]-values and those reported by de Azevedo et al. follows from Figure 10. At the same time, although the discrepancies between[C.sub.p]-values and those predicted by the FT-EOS are again within acceptable limits (<10%) the formers demonstrate the weak maximum and very small pressure dependence for [bmim][[BF.sub.4]] (for [bmim][[PF.sub.6]] this[C.sub.p](P)-dependence is monotonous as well as that predicted by the FT-EOS). It seems that the resultant ratio of heat capacity [C.sub.p]/[C.sub.v] shown in Figure 12 which demonstrates the irregular crossing of isotherms is questionable. It suggests that their pressure dependence either needs the more accurate approximation or reflects the realistic distinction of reference FT-EOS from the actual behavior of [bmim][[BF.sub.4]]. The lock of noticeable variations in pressure is the common feature of integration methodology based on the given (W, P, T)- and ([rho], P, T)-surfaces. The unavoidable accumulation of uncertainties at each stage of calculations in the set W - [C.sub.p] - [C.sub.v] may cause the unplausible behavior of adiabatic exponent [C.sub.p]/[C.sub.v] in liquid. The same is true for the set [C.sub.v ] - [C.sub.p] - W used in the FT-methodology. It is the most appropriate explanation of significant discrepancies for W(P)-dependence shown in Figure 13. Let us remind also that the precise mechanical measurements of speed velocity in the very viscous IL cannot be attributed exactly to the condition of constant entropy. Thus, strictly speaking, the measured (W, P, T)-surface reflects the strong dispersive properties of media and must be less than its thermodynamic counterpart in the ideal (without a viscosity) liquid. There are the structure-forming factors related to the above-discussed thermodynamic characteristic. Despite the certain discrepancies between the predicted and derived properties for FT-EOS and LF-EOS, both ones provide the close estimates of structure factors represented in Table 5. Our aim here is to show that the thermodynamically-consistent predictions of thermodynamical properties by the FT-EOS yields also the molecular-based parameters which are, at least, realistic (see also Table 3). The estimate of average T-dependent well-depth [bar.[eplison] by (6b) as well as estimate of average value [[bar.N].sub.l] by (24a) is related to the middle of temperature range: T = 320 K. The distinction of [bar.[eplison] from the respective [eplison]*-parameters of LF-EOS can be attributed to the difference between nonbonded interactions in the discrete (LF-EOS) and continuum (FT-EOS) models of fluid. Our estimate of cohesive-energy density [[eplison].sub.coh] by equality, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43) represented in Table 6 seems also physically plausible. Mag-inn et al. [5, 6] have determined it within the framework of GEMC-simulations by the knowledge of [rho]([P.sub.0], T) and the internal energy difference between an ideal-gas ion pair and the average internal energy of an ion pair in the liquid state. Such definition indicates that cohesive energy densities of many ILs are on the order of 500-550 J/[cm.sup.3] (see, for comparison, Table 6) and demonstrate a slight decrease as temperature increases. Another relevant characteristic is the internal pressure determined by the derivative of molar (or specific) internal energy, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44) which is compared to ones calculated by different authors [14, 16] for [bmim][[BF.sub.4]] in Table 6. As in the other cases, the FT-EOS predicts the much faster change of both cohesive energy density [[eplison].sub.coh] and internal pressure [([partial derivative]e/[partial derivative]v).sub.T] as temperature increases. One should collect a large number of precise experimental measurements to reconstruct the thermodynamic surface of a substance. FT-methodology provides a possibility of preliminary reliable estimates of relevant macroscopic and molecular-based correlations. Its thermodynamic consistency provides the serious advantage in comparison with the purely empiric treatment of any volumetric measurements at the description of derived heat capacities. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. C. P. Fredlake, J. M. Crosthwaite, D. G. Hert, S. N. V. K. Aki, and J. F. Brennecke, "Thermophysical properties of imidazolium-based ionic liquids," Journal of Chemical and Engineering Data, vol. 49, no. 4, pp. 954-964, 2004. C. Cadena, J. L. Anthony, J. K. Shah, T. I. Morrow, J. F. Brennecke, and E. J. Maginn, "Why is CO2 so soluble in imidazolium-based ionic liquids?" 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Wagner, "International equations for the saturation properties of ordinary water substance," Journal of Physical and Chemical Reference Data, vol. 16, pp. 893-901, 1987. F. Bresme, E. Lomb, and J. L. F. Abascal, "Influence of association on the liquid-vapor phase coexistence of simple systems," The Journal of Chemical Physics, vol. 106, no. 4, pp. 1569-1575, 1997. R. van Roij, "Theory of chain association versus liquid condensation," Physical Review Letters, vol. 76, no. 18, pp. 3348-3351, 1996. Department of Physics, Odessa State Academy of Refrigeration, Dvoryanskaya Street 1/3, Odessa 65082, Ukraine Correspondence should be addressed to Vitaly B. Rogankov; email@example.com Received 5 August 2014; Accepted 4 November 2014; Published 16 December 2014 Academic Editor: Pedro Jorge Martins Coelho Copyright [c] 2014 V. B. Rogankov and V. I. Levchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Table 1: Coefficients of FT-EOS (1) for neutral ([C.sub.2][H.sub.4]) and polar ([H.sub.2]O) substances. [b.sub.0] [[dm.sup.3]/mol], a [[J.dm.sup.3]/[mol.sup.2]], T [K] [C.sub.2][H.sub.4] [H.sub.2]O [b.sub.0] = 0,04181 [b.sub.0] = 0,01658 T a T a 103,99 1557,12 273,16 517,162 105 1459,76 278,15 527,171 110 1127,30 283,15 534,941 115 935,490 288,15 540,955 120 812,151 293,15 545,767 125 725,634 298,15 548,270 130 661,817 303,15 549,648 135 613,602 308,15 549,424 140 575,528 313,15 548,755 145 545,191 318,15 547,172 150 520,785 323,15 544,760 155 500,788 328,15 541,609 160 484,652 333,15 537,810 165 471,466 338,15 533,455 169,35 462,006 343,15 529,044 348,15 524,605 353,15 519,789 358,15 514,676 363,15 509,674 368,15 504,480 373,15 499,159 Table 2: Coefficients of FT-EOS (1) for ILs: [bmim][[PF.sub.6]], [pmmim] [[Tf.sub.2]N], and [bmim][[BF.sub.4]]. [b.sub.0] [[dm.sup.3]/mol], a [[J.dm.sup.3]/[mol.sup.2]], T [K] [bmim][[PF.sub.6]] [pmmim][[Tf.sub.2]N] [bmim][[BF.sub.4]] [b.sub.0] = 0,1953 [b.sub.0] = 0,2711 [b.sub.0] = 0,178 T a T a T a 285 8433,52 288,15 11724,8 290 9470,0 290 8214,24 295,15 12224,5 300 8900,9 295 8015,04 304,25 12279,0 310 8435,0 300 7833,44 313,65 11728,9 320 8047,5 305 7667,34 324,35 11308,4 330 7720,8 310 7514,97 335,15 10672,3 340 7442,2 315 7374,83 344,65 10051,1 350 7202,6 320 7245,64 350,15 9709,50 325 7126,31 330 7015,89 335 6913,55 340 6818,59 345 6730,37 350 6648,36 Table 3: Predicted hypothetical (v, l)-transition in the low-temperature range for FT-model of [bmim][[BF.sub.4]] based on the experimental data [1, 2] treated by FT-EOS (B-variant of x(T)-prediction). T (K) [[rho].sub.l] (mol/[dm.sup.3]) [[rho].sub.g] (mol/[dm.sup.3]) 290 5,350646 3,09E - 08 300 5,322170 2,1E - 07 310 5,293693 1,01E - 06 320 5,265215 3,74E - 06 330 5,236735 1,13E - 05 340 5,208254 2,92E - 05 350 5,179771 6,61E - 05 360 5,151287 0,000135 370 5,122802 0,000253 380 5,094315 0,000441 390 5,065828 0,000725 400 5,037338 0,001131 410 5,008848 0,001689 420 4,980355 0,002428 430 4,951862 0,003379 440 4,923367 0,004569 450 4,894871 0,006027 460 4,866373 0,007778 470 4,837873 0,009843 480 4,809372 0,012244 490 4,780869 0,014997 500 4,752365 0,018119 510 4,723859 0,021622 511 4,721008 0,021993 512 4,718157 0,022369 513 4,715307 0,022748 514 4,712456 0,023132 515 4,709605 0,023519 516 4,706754 0,023910 517 4,703903 0,024306 T (K) [P.sub.[sigma]] (kPa) [r.sub.[sigma]] (J/mol) 290 7,45E - 05 53082 300 5,24E - 04 49866 310 0,002 47230 320 0,009 45032 330 0,031 43175 340 0,082 41587 350 0,192 40216 360 0,404 39022 370 0,778 37974 380 1,393 37048 390 2,347 36225 400 3,754 35488 410 5,740 34826 420 8,446 34228 430 12,017 33684 440 16,604 33188 450 22,359 32733 460 29,432 32314 470 37,968 31926 480 48,105 31566 490 59,977 31230 500 73,704 30914 510 89,400 30618 511 91,082 30589 512 92,785 30561 513 94,509 30532 514 96,254 30504 515 98,020 30476 516 99,806 30448 517 101,615 30420 T (K) a (J.[dm.sup.3]/[mol.sup.2]) [eplison]/k (K) 290 9470,0 290 300 8900,9 300 310 8435,0 309,99 320 8047,5 319,99 330 7720,8 329,99 340 7442,2 339,99 350 7202,6 349,99 360 6994,6 359,99 370 6813,0 369,98 380 6653,5 379,96 390 6512,7 389,94 400 6387,8 399,91 410 6276,6 409,86 420 6177,4 419,79 430 6088,7 429,70 440 6009,1 439,59 450 5937,6 449,45 460 5873,2 459,27 470 5815,4 469,05 480 5763,2 478,79 490 5716,3 488,49 500 5674,1 498,13 510 5636,1 507,72 511 5632,5 508,67 512 5629,0 509,63 513 5625,4 510,58 514 5622,0 511,54 515 5618,6 512,49 516 5615,2 513,44 517 5611,8 514,40 Table 4: Effective LJ-diameters of FT-model for ILs determined by (6a), (6b), and (21b) on the base of estimates [7,13] and the choice [b.sub.0] = 178 [cm.sup.3]/mol for [bmim][[BF.sub.4]] in this work. IL M (g/mol) [sigma]([Angstrom]) [bmim][[BF.sub.4]] 225,82 5,208 [bmim][[PF.sub.6]] 284 5,371 [pmmim][[Tf.sub.2]N] 419,1 5,992 IL [M.sub.c]/[M.sub.c] [bmim][[BF.sub.4]] 139/86,82 [bmim][[PF.sub.6]] 139/145 [pmmim][[Tf.sub.2]N] 139/280,1 IL [[sigma].sub.c]/[[sigma].sub.a] [bmim][[BF.sub.4]] 5,757/4,51 [bmim][[PF.sub.6]] 5,651/5,06 [pmmim][[Tf.sub.2]N] 5,704/6,254 Table 5: Comparison of excluded volumes (M/[rho]* and [b.sub.0]), characteristic interaction energy ([eplison]* and [bar.[eplison]]), and efffective number of bonded units (MP*/RT*[rho]* [14,15] and [[bar.N].sub.l]) (see text). Compound M/[rho]* [b.sub.0] [eplison]* ([cm.sup.3]/mol) ([cm.sup.3]/mol) (J/mol) [bmim][[BF.sub.4]] 175,3 178 5642 [bmim][[PF.sub.6]] 196,2 195,3 5658 Compound [bar.[eplison]] MP*/RT*[rho]* [[bar.N].sup.l] (J/mol) [bmim][[BF.sub.4]] 2661 17,6 17,0 [bmim][[PF.sub.6]] 2661 18,8 18,1 Table 6: Comparison of internal pressure ([partial derivative]e/[[partial derivative]v).sub.T] for [bmim][[BF.sub.4]] based on the LF-EOS and FT-EOS (this work) with the values estimated by experimental data on speed of sound W, density [rho], and isobaric heat capacity [C.sub.p]. I ([partial derivative]e/[[partial derivative]v).sub.T] (K) (MPa) 313,1 482,78 332,6 471,50 352,6 460,35 372,7 450,11 392,8 440,00 412,9 429,23 432,6 419,35 452,3 408,84 472,2 399,19 I T ([partial derivative]e/[[partial derivative]v).sub.T] (K) (K) (MPa) 313,1 283,15 459,91 332,6 288,15 459,31 352,6 293,15 459,20 372,7 298,15 457,79 392,8 303,15 455,76 412,9 308,15 453,73 432,6 313,15 451,20 452,3 318,15 448,06 472,2 323,15 446,64 328,15 444,61 333,15 443,09 338,15 441,68 343,15 439,75 I T [FT] ([partial derivative]e/[[partial derivative]v).sub.T] (K) (K) (MPa) 313,1 290 743,76 332,6 310 573,16 352,6 330 463,91 372,7 350 388,61 392,8 370 333,84 412,9 390 292,27 432,6 410 259,66 452,3 430 233,40 472,2 450 211,77 470 193,64 490 178,19 510 166,76 517 162,41 I [[eplison].sub.coh] (K) (J/[cm.sup.3]) 313,1 555,15 332,6 486,41 352,6 437,85 372,7 401,57 392,8 373,35 412,9 350,67 432,6 331,94 452,3 316,13 472,2 302,52 290,60 280,01 270,46 267,31 |Printer friendly Cite/link Email Feedback| |Title Annotation:||Research Article| |Author:||B. Rogankov, Vitaly; I. Levchenko, Valeriy| |Publication:||Journal of Thermodynamics| |Date:||Jan 1, 2014| |Previous Article:||Second law analysis for third-grade fluid with variable properties.| |Next Article:||Thermal performance and economic analysis of 210 MWe coal-fired power plant.|

Each firm in a competitive industry operates at a point where its MC becomes equal to the (exogenously given) price of the product. That is why the short-run supply curve of a competitive industry is the horizontal summation of the short-run supply curves (or the short-run MC curves on and above the minimum AVC) of individual firms. Therefore, a competitive industry operates at a point where price equals marginal cost. A monopolized industry, on the other hand, operates where price is greater than MC. Thus, in general, price will be higher and output lower under monopoly than under competition. We may illustrate the point with the help of Fig, 11.20. In this figure, DD is the demand curve for the product of a competitive industry, and EMC is the industry’s supply curve. Therefore, the market (demand = supply) equilibrium point will be Ec, which is the point of intersection between the DD and the IMC curves. At this point, the price of the product will be pc = opc and the quantity demanded and supplied will be qc = oqc. All the firms would operate at their respective p = MC points. Let us now assume that the competitive industry with all its firms becomes monopolized under a single owner, i.e., the competitive industry now becomes a multi-plant monopoly. The demand curve DD for the product now becomes the AR curve of the monopolist and its corresponding marginal curve is given by MR in Fig. 11.20. Now, in order to minimise the cost of production of any particular quantity of output, the monopolist would have to distribute the production of that quantity over the different plants in such a way that MC in each plant may become the same, and this MC would be the MC of that particular quantity of output. Understood in this way, the MC of any particular quantity of output produced by the multi-plant monopolist would be given by the curve EMC of Fig. 11.20, which is the horizontal summation of the MC curves of individual plants. The profit-maximising equilibrium point of the monopolist in this case would be the point of intersection (Em) between the MR and the ∑MC curves. At this point, the output of the monopolist would be qm = Oqm which would be less than the competitive output qc = Oqc and the price charged by the monopolist would be pm = Opm which is greater than MC (= Em qm) and which is greater than the competitive price (pc). Since the price under monopoly is greater and output lower than under competition, the consumers would always be worse off in a monopolistic industry than in one under perfect competition. But a portion of the consumers’ loss would be converted into the gain of the industry. This can be seen in Fig. 11.20. Here, as we move from competition to monopoly, i.e., as the equilibrium (p, q) combination moves from (pc, qc) to (pm, qm), the consumers’ surplus diminishes by □pcpmAEc (from □TpcE,. to □TpmA) and of this, □pcpmAB is converted into the producer’s surplus under monopoly. Producer’s surplus, □SpcEc, decreases by □BEmEc and increases by □pcpmAB. Taking both the industry’s gain and the consumers’ loss into consideration, it is not clear which would be the better arrangement for the society—competition or monopoly, unless we make a value judgement about the relative welfare of the consumers and of the owners of the industry. However, we may argue against monopoly on grounds of efficiency alone. An economic arrangement is Pareto-efficient if there is no way to make anyone better off without making somebody else worse off. We shall now see that the level of output under monopoly is not Pareto-efficient. Let us remember that at any point on the demand curve of a monopoly firm like DD in Fig. 11.20, the price p stands for how much people are willing to pay for the marginal unit of the good. Since p is greater than MC for all output levels between qm and qc, there is a whole range of output (i.e., between qm and qc) where people are willing to pay more for the marginal unit of output than it costs to produce it. Clearly, there is a case for a Pareto improvement here. For example, let us consider the situation at the monopoly level of output qm. Since p(qm) > MC (qm), we know that there is someone who is willing to pay more for an extra unit of output than it costs to produce that extra unit. Let us suppose that the firm produces an extra unit and sells it to this person at any price p where p(qm) > p > MC(qm). Then this consumer is made better off because he was just willing to pay p(qm) for that unit of consumption, and it was sold for p < p (qm). Similarly, it cost the monopolist MC(qm) to produce that extra unit of output and he sold it for p > MC(qm). If all the other units of output are being sold for the same price as before, then from the sale of the extra unit, both sides of the market—the buyers’ side and the sellers’ side— are better off and no one else is made worse off. Therefore, we have found here a Pareto improvement. That is, the usual monopoly solution (pm, qm) is Pareto-ineflicient. The reason for this inefficiency of monopoly is this. In the case of competition, price is constant irrespective of output, making MR at any output a constant and equal top. So the firm’s profit maximising p = MR = MC point is also the Pareto-efficient p = MC point. On the other hand, since the AR curve of the monopolist is downward sloping, the extra revenue (MR) obtained from selling the marginal unit of output is not equal to, but less than, the price of that unit (i.e., MR < p). That is why, in monopoly, the firm’s profit maximising MR = MC point is a Pareto-inefficient p > MC point.

26.4453GPPCodec for Enhanced Voice Services (EVS)Detailed algorithmic descriptionRelease 15TS The linear predictive (LP) analysis, the long-term prediction (LTP), the VAD algorithm and signal are performed at the 12.8 kHz sampling rate. The HP-filtered input signal is therefore converted from the input sampling frequency to 12.8 kHz. 18.104.22.168 Conversion of 16, 32 and 48 kHz signals to 12.8 kHz For 16, 32 and 48 kHz signals, the sampling conversion is performed by first up‑sampling the signal to 192 kHz, then filtering the output through a low-pass FIR filter that has the cut‑off frequency at 6.4 kHz. Then, the signal is down-sampled to 12.8 kHz. The filtering delay is 15 samples at 16 kHz sampling frequency which corresponds to 0.9375 ms. The up-sampling is performed by inserting 11, 5 or 3 (for 16, 32 or 48 kHz, respectively) zero-valued samples between each 2 samples for each 20-ms frame of 320 samples (at 16 kHz sampling frequency) where is the signal at 192 kHz sampling frequency and is the up-sampling factor equal to 12 for a 16 kHz input, 6 for a 32 kHz input and 4 for a 48 kHz input. Then, the signal is filtered through the LP filter and decimated by 15 by keeping one out of 15 samples. The filter is a 361-tap linear phase FIR filter having a cut-off frequency of 6.4 kHz in the 192 kHz up-sampled domain. The filtering and decimation can be done using the relation where is the impulse response of . The operations in equations (3) and (4) can be implemented in one step by using only a part of the filter coefficients at a time with an initial phase related to the sampling instant n. That is In case the encoder is externally forced to narrow-band processing of the input signal, the cut-off frequency of the LP filter is changed from 6.4 kHz to 4 kHz. 22.214.171.124 Conversion of 8 kHz signals to 12.8 kHz For 8 to 12.8 kHz resampling a sharper resampling filter is beneficial. Double length low-pass FIR filter is used in this case. The doubling of the impulse response length is compensated by a low delay resampling method. The filter is a 241-tap linear phase FIR filter having a cut-off frequency of 3.9 kHz and is applied in the up-sampled domain which is 64 kHz. Direct FIR filtering with this filter would yield a delay of 120/64 = 1.875 ms. In order to reduce this delay to 0.9375 ms, future samples are determined at 8 kHz by adaptive linear prediction. The exact number of future samples is found based on the difference between the actual delay (1.875 ms) and the desired delay (0.9375 ms) at 8 kHz. Therefore future samples are predicted. These predicted samples are concatenated at the end of the current frame to form a support vector. Then, the sample rate conversion of is performed in a similar way as for the other sampling rates, i.e. is first up-sampled to 64 kHz, the output is filtered through the low-pass FIR filter and the resulting signal is down-sampled to 12.8 kHz. The final filtering delay is aligned with that of the other resampling configurations, i.e 12 samples at 12.8 kHz sampling frequency which corresponds to 0.9375 ms. To determine the future samples, linear prediction coefficients of order 16 are computed in the pre-emphasized domain in the following way. The last Lss =120 samples of the input frame at 8 kHz are windowed by an asymmetrical analysis window winss_120: and a first order autocorrelation analysis is made on the windowed signal . The pre-emphasis coefficient ss is obtained by where rw(0) and rw(1) are the autocorrelation coefficients The last 120 samples of the signal are pre-emphasized using the adaptive filter to obtain the pre-emphasized signal of Lss =120 samples. Then is windowed by the asymmetrical analysis window winss_120 and a 16th order autocorrelation analysis is made on the windowed signal These autocorrelation coefficients are lag-windowed by where wlag8k(k) is defined as Based on the autocorrelation coefficients rpwl(k), the linear prediction coefficients ass(k) are computed by the Levinson-Durbin algorithm. The future samples in the pre-emphasized domain are predicted by zero input filtering through the 1/Ass(z) synthesis filter Finally, the concatenated signal is de-emphasized through the filter . Note that only the last 7 predicted samples need to be de-emphasized. These 7 de-emphasized samples are concatenated to (at positions n = 160,…,166) to form the support vector. The up-sampling of is then performed by inserting 7 zero-valued samples between each 2 samples for each 20-ms frame of 160 samples (at 8 kHz sampling frequency) completed by 7 predicted future samples (167 in total) where is the signal at 64 kHz sampling frequency. Then, the signal is filtered through the LP filter and decimated by 5 by keeping one out of 5 samples. The filtering and decimation can be done using the relation where is the impulse response of and assures that the index of s64 is never higher than the highest available index for (which is 1335). Indeed, it corresponds to the delay of this filtering at 64 kHz. To reduce complexity, the operations in equations (14) and (15) can be implemented in one step by using only a part of the filter coefficients at a time with an initial phase related to the sampling instant n. This polyphase implementation of the resampling filter is applied on the concatenated support vector. That is where is derived from the delay of this filtering at 8 kHz. It assures that the index of sHPC is never higher than the highest available index (which is 166). 126.96.36.199 Conversion of input signals to 16, 25.6 and 32 kHz If ACELP core is selected for WB, SWB or FB signals at bitrates higher than 13.2 kbps (see subclause 5.1.16), its internal sampling rate is set to 16 kHz rather than 12.8 kHz. If the input signal is sampled at 8 kHz, there is no conversion needed because for NB signals, ACELP core is always operated at 12.8 kHz. If the input signal is sampled at 16 kHz, no conversion is needed either and the input signal is only delayed by 15 samples which corresponds to 0.9375 ms. This is to keep all pre-processed signals aligned regardless of the bitrate or bandwidth. Thus, the input signal is resampled to 16 kHz only if its sampling frequency is 32 or 48 kHz. The resampling operation is done in the same way as for the case of 12.8 kHz (see subclause 188.8.131.52), i.e. by means of FIR filtering. The coefficients of the LP filter are different but the filtering delay is still the same, i.e. 0.9375 ms. The resampled signal is denoted where n=0,..,319. The input signal is converted to 25.6 kHz at 48 kbps and to 32 kHz at 96 or 128kbps but only for SWB and FB signals. The sampling conversion is again similar as in the case of 12.8 kHz with differences in LP filter coefficients. The resampled signals are denoted and , respectively.

Time Left: 00:50:00 1. A number consists of two digits. If 3/5 of 1/5 of the number is 9. Find the sum of its two digits? 2. The ratio of investments of two partners P and Q is 7:5 and the ratio of their profits is 7:10. If P invested the money for 5 months, find for how much time did Q invest the money? 3. If cost of sugar increases by 25%. How much percent consumption of sugar should be decreased in order to keep expenditure fixed? 4. A is faster than B. A and B each walk 24 km. The sum of their speeds is 7 km/hr and the sum of times taken by them is 14 hours. Then, A's speed is equal to? 5. The sum of four consecutive even numbers is 292. What would be the smallest number? 6. Rs.8000 become Rs.9261 in a certain interval of time at the rate of 5% per annum of C.I. Find the time? 7. The average weight of a group of boys is 30 kg. After a boy of weight 35 kg joins the group, the average weight of the group goes up by 1 kg. Find the number of boys in the group originally ? 8. An amount of Rs. 3000 becomes Rs. 3600 in four years at simple interest. If the rate of interest was 1% more, then what was be the total amount? 9. In a 1000 m race, A beats B by 200 meters or 25 seconds. Find the speed of B? 10. In how many ways can a committee consisting of three men and four women be formed from a group of six men and seven women? 11. Three men start together to travel the same way around a circular track of 11 kilometers in circumference. Their speeds are 4, 5 and 8 kilometers per hour respectively. When will they meet at a starting point? 12. A, B and C enter into partnership. A invests some money at the beginning, B invests double the amount after 6 months, and C invests thrice the amount after 8 months. If the annual gain be Rs.18000. A's share is? 13. A thief steals at a car at 2.30 p.m. and drives it at 60 km/hr. The theft is discovered at 3 p.m. and the owner sets off in another car at 75 km/hr. When will he overtake the thief? 14. If A:B = 1/2: 1/3 B:C = 1/2:1/3 then A:B:C? 15. The average of 1st 3 of 4 numbers is 16 and of the last 3 are 15. If the sum of the first and the last number is 13. What is the last numbers? 16. A certain sum becomes Rs. 20720 in four years and 24080 in six years at simple interest. Find sum and rate of interest? 17. (562 - 242) * 1/32 + ?% of 1200 = 146 18. A train is 360 meter long is running at a speed of 45 km/hour. In what time will it pass a bridge of 140 meter length? 19. In a game of billiards, A can give B 20 points in 60 and he can give C 30 points in 60. How many points can B give C in a game of 100? 20. Anand and Deepak started a business investing Rs. 22,500 and Rs. 35,000 respectively. Out of a total profit of Rs. 13,800, Deepak's share is: 21. How long does a train 165 meters long running at the rate of 54 kmph take to cross a bridge 660 meters in length? 22. There is 60% increase in an amount in 6 years at S.I. What will be the C.I. of Rs. 12,000 after 3 years at the same rate? 23. 32% of 425 - ?% of 250 = 36 24. The speed of a boat in still water is 60kmph and the speed of the current is 20kmph. Find the speed downstream and upstream? 25. Anil invested a sum of money at a certain rate of simple interest for a period of five years. Had he invested the sum for a period of eight years for the same rate, the total intrest earned by him would have been sixty percent more than the earlier interest amount. Find the rate of interest p.a. 26. A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is? 27. The tax on a commodity is diminished by 20% and its consumption increased by 15%. The effect on revenue is? 28. 25 * 25 / 25 + 15 * 40 = ? 29. Two pipes can fill a tank in 18 minutes and 15 minutes. An outlet pipe can empty the tank in 45 minutes. If all the pipes are opened when the tank is empty, then how many minutes will it take to fill the tank? 30. A and B invests Rs.3000 and Rs.4000 respectively in a business. If A doubles his capital after 6 months. In what ratio should A and B divide that year's profit? 31. Walking 7/6 of his usual rate, a boy reaches his school 4 min early. Find his usual time to reach the school? 32. 4500 * ? = 3375 33. The least number of four digits which is divisible by 4, 6, 8 and 10 is? 34. A man can row 6 kmph in still water. When the river is running at 1.2 kmph, it takes him 1 hour to row to a place and black. What is the total distance traveled by the man? 35. A and B can do a piece of work in 7 days. With the help of C they finish the work in 5 days. C alone can do that piece of work in? 36. The speed of a boat in upstream is 60 kmph and the speed of the boat downstream is 80 kmph. Find the speed of the boat in still water and the speed of the stream? 37. A bag contains equal number of Rs.5, Rs.2 and Re.1 coins. If the total amount in the bag is Rs.1152, find the number of coins of each kind? 38. If selling price is doubled, the profit triples. Find the profit percent: 39. (8x9 / 27y-6)-2/3 = ? 40. Find the cost of fencing around a circular field of diameter 28 m at the rate of Rs.1.50 a meter? 41. The probability that A speaks truth is 3/5 and that of B speaking truth is 4/7. What is the probability that they agree in stating the same fact? 42. A father said to his son, "I was as old as you are at present at the time of your birth." If the father's age is 38 years now, the son's age five years back was: 43. If (461 + 462 + 463 + 464) is divisible by ?, then ? = 44. Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill cistern. How much time will be taken by A to fill the cistern separately? 45. On dividing 2272 as well as 875 by 3-digit number N, we get the same remainder. The sum of the digits of N is: 46. By how much is 3/5th of 875 greater than 2/3 of 333? 47. The value of a machine depreciates at 20% per annum. If its present value is Rs. 1,50,000, at what price should it be sold after two years such that a profit of Rs. 24,000 is made? 48. There are two positive numbers in the ratio 5:8. If the larger number exceeds the smaller by 15, then find the smaller number? 49. In how much time will a train of length 100 m, moving at 36 kmph cross an electric pole? 50. Two trains of equal length, running with the speeds of 60 and 40 kmph, take 50 seconds to cross each other while they are running in the same direction. What time will they take to cross each other if they are running in opposite directions?

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Methods for Calculating CHP Efficiency Every CHP application involves the recovery of heat that would otherwise be wasted. In this way, CHP increases fuel-use efficiency. Two measures are commonly used to quantify the efficiency of a CHP system: total system efficiency and effective electric efficiency. - Total system efficiency is the measure used to compare the efficiency of a CHP system to that of conventional supplies (the combination of grid-supplied electricity and useful thermal energy produced in a conventional on-site boiler). If the objective is to compare CHP system energy efficiency to the efficiency of a site's conventional supplies, then the total system efficiency measure is likely the right choice. - Effective electric efficiency is the measure used to compare CHP-generated electricity to electricity generated by power plants, which is how most electricity is produced in the United States. If CHP electrical efficiency is needed to compare CHP to conventional electricity production (i.e., grid-supplied electricity), then the effective electric efficiency metric is likely the right choice. Certain assumptions are implicit in each methodology that are not appropriate in all cases. Consequently, the measure employed should be selected carefully and the results interpreted with caution. Key Terms Used in Calculating CHP Efficiency Calculating a CHP system's efficiency requires an understanding of several key terms: - CHP system. The CHP system includes the prime mover (e.g., combustion turbine, engine, microturbine), the electric generator, and the heat recovery unit that transforms otherwise wasted heat to useful thermal energy. - Total fuel energy input (QFUEL). The heating value of the total fuel input. Total fuel input is the sum of all the fuel used by the CHP system. The total fuel energy input is often determined by multiplying the quantity of fuel consumed by the heating value of the fuel. Commonly accepted heating values for natural gas, coal, and diesel fuel are: - 1020 Btu per cubic foot of natural gas - 10,157 Btu per pound of coal - 138,000 Btu per gallon of diesel fuel - Net useful electric output (WE). The gross electric output of the generator minus any parasitic electric losses. In other words, the net useful electric output is the total electric output from the CHP system that is put to a useful purpose. - Gross electric output is the total electric output of the generator. - Parasitic electric losses are the electrical power consumed by the CHP system; for example, the electricity used to compress natural gas before it is used as fuel in a combustion turbine. - Net useful thermal output (∑QTH). The gross thermal output of the CHP system minus any thermal output that is not put to a useful purpose. In other words, the net useful thermal output is the total thermal output from the CHP system that is put to a useful purpose. - In the case of a CHP system that produces 10,000 pounds of steam per hour, with 90 percent of the steam used for space heating and the remaining 10 percent exhausted in a cooling tower, the energy content of the 9,000 pounds of steam per hour is the net useful thermal output. - Gross thermal output is the total thermal output of the CHP system. Total System Efficiency The total system efficiency (ηo) of a CHP system is the sum of the net useful electric output (WE) and net useful thermal output (∑QTH) divided by the total fuel energy input (QFUEL), as shown below: The calculation of total system efficiency evaluates the combined CHP outputs (i.e., electricity and useful thermal output) based on the fuel consumed. CHP systems typically achieve total system efficiencies of 60 to 80 percent. Note that this measure does not differentiate between the value of the electric output and the thermal output; instead, it treats electric output and thermal output as having the same value which allows them to be added (kWh can be converted to Btu using a standard conversion factor). In reality, electricity is considered a more valuable form of energy because of its unique properties. Effective Electric Efficiency Effective electric efficiency (ℰEE) can be calculated using the equation below, where WE is the net useful electric output, ∑QTH is the sum of the net useful thermal output, QFUEL is the total fuel energy input, and α equals the efficiency of the conventional technology that would be used to produce the useful thermal energy output if the CHP system did not exist: For example, if a CHP system is natural gas-fired and produces steam, then α represents the efficiency of a conventional natural gas-fired boiler. Typical boiler efficiencies are 80 percent for natural gas-fired boilers, 75 percent for biomass-fired boilers, and 83 percent for coal-fired boilers. The calculation of effective electric efficiency is the CHP net electric output divided by the additional fuel the CHP system consumes over and above what would have been used by a boiler to produce the thermal output of the CHP system. Typical effective electric efficiencies for combustion turbine-based CHP systems range from 50 to 70 percent. Typical effective electric efficiencies for reciprocating engine-based CHP systems range from 70 to 85 percent.

Doubtnut.com presents complete NCERT solutions for class 10 maths in video tutorial format. To make it easier for you to learn and understand maths the video tutorials are prepared by our esteemed mathematicians from the renowned IITs of India. These are some of the best online lectures on maths, where our experts have discussed a wide array of class 10 maths topics. Class 10th Maths as a subject is vast; therefore, we’ve listed every important topic by segregating its chapters into subsequent exercises. In our video tutorials, we've discussed Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Triangles, Coordinate Geometry, Trigonometry, Some Applications of Trigonometry, Circles, Constructions, Area Related to Circles, Surface Areas and Volumes, Statistics and Probability and more… Cengage Chapterwise Maths Solutions PAIR OF LINEAR EQUATIONS IN TWO VARIABLES INTRODUCTION TO TRIGONOMETRY SOME APPLICATIONS OF TRIGONOMETRY AREAS RELATED TO CIRCLES SURFACE AREAS AND VOLUMES Key Highlights of NCERT Solutions for Class 10 Maths: 1. Strictly as per the CBSE examination guidelines. 2. Fully solved exercises of NCERT Math textbook for Class 10. 3. The questions have been answered by the IITians and best mathematicians of the country. 4. Solutions are as per the CBSE marking scheme. 5. Videos are full of tips and tricks to help you gain the competitive edge to ace the board examination. Diagrams are given to help the students in visualizing the solutions. All the questions from each chapter are covered. Videos are enriched with expert advice, math's tips and step-by-step maths solutions to give a proper justification to every math's problem. It would provide students detailed insights on every important topic. The NCERT solutions for class 10 maths in Hindi are prepared to keep the student's need in mind. You can easily learn either using your mobile or personal computer. The given video tutorials provide students detailed insights on every important topic. Chapter 1: Real Numbers Do you know natural numbers, whole numbers, integers, fractions, rational numbers and irrational numbers are all real numbers? Any real number that can be found on the number line is known as the real number. The which we use and apply in the real world calculations are the real numbers... In this chapter, you'll be learning some interesting concepts related to a real number and their useful applications. The video tutorial covers every important topic related to real numbers and gives some great examples. The exercise wise NCERT solution class 10 maths book are listed below: Exercise 1.1 Introduction Exercise 1.2 Euclid’s Division Lemma Exercise 1.3 The Fundamental Theorem of Arithmetic Exercise 1.4 Revisiting Irrational Numbers Exercise 1.5 Revisiting Rational Numbers and Their Decimal Expansions Exercise 1.6 Summary Chapter 2: Polynomials In this chapter, you'll be learning about the polynomials in-depth. Starting from the exercise 2.1 Introduction to polynomials, you’ll be learning how to simplify and evaluate polynomials. Then you’ll learn about the zeros or roots of polynomials, roots of quadratic equations and cubic equation and it’s coefficient. In the final exercise 2.5, you'll learn about how to do long division of the polynomials. Exercise-wise solutions we’ve covered include: Exercise 2.1 Introduction Exercise 2.2 Geometrical Meaning Of The Zeroes Of A Polynomial Exercise 2.3 Relationship Between Zeroes And Coefficients Of A Polynomial Exercise 2.4 Division Algorithm For Polynomials Exercise 2.5 Polynomials Exercise 2.6 Summary Chapter 3: Pair of Linear Equations in Two Variables In NCERT class 10 Chapter 3, you'll be learning about the interpretation of linear equations in two variables. With given NCERT solutions for class 10, maths students can easily model the linear equations into real-world problems. You’ll enjoy learning to solve the linear equation problems both algebraically and graphically. In the algebraic method, you will learn some important concepts like the elimination method, substitution method, and cross-multiplication method. Then finally we will move on to equations reducible to a pair of linear equations in two variables. The exercise-wise NCERT solutions we’ve covered include: Exercise 3.1 Introduction Exercise 3.2 Pair of Linear Equations in Two Variables Exercise 3.3 Graphical Method of Solution of a Pair of Linear Equations Exercise 3.4 Algebraic Methods of Solving a Pair of Linear Equations Exercise 3.4.1 Substitution Method Exercise 3.4.2 Elimination Method Exercise 3.4.3 Cross-Multiplication Method Exercise 3.5 Equations Reducible To a Pair of Linear Equations in Two Variables Exercise 3.6 Summary Chapter 4: Quadratic Equations So far you have learned about the linear equations and linear equations in two variables. In chapter 4, you'll be learning some new concepts, which includes variables raise to the second power and how to take square roots on both sides. You'll also learn to solve the factored equation like (x-1) (x+3) = 0 and using the factorization method, solution of quadratic equation by completing the square and finally the nature of roots. Listed below are the exercise wise NCERT solutions for chapter 4 quadratic equations: Exercise 4.1 Introduction Exercise 4.2 Quadratic Equations Exercise 4.3 Solution of a Quadratic Equation by Factorization Exercise 4.4 Solution of a Quadratic Equation by Completing the Square Exercise 4.5 Nature of Roots Exercise 4.6 Summary Chapter 5: Arithmetic Progressions Find here detailed and in-depth answers to Class 10 Maths Chapter 5 Arithmetic Progressions (AP). If the difference between two consecutive terms is constant in a progression, it is known as the arithmetic progression. In this lesson five, you’ll be introduced important concepts of arithmetic progression and you’ll be studying about constructing an arithmetic progression (A.P). After learning the basics of AP, you can learn to calculate the Nth term of an AP. Then you will see how to find the sum of n terms of an AP. It is interesting and fun to learn the concept of mathematics. The exercise wise NCERT solutions for chapter 5 Arithmetic Progressions we’ve covered are: Exercise 5.1 Introduction Exercise 5.2 Arithmetic Progressions Exercise 5.3 Nth Term of An AP Exercise 5.4 Sum Of First N Terms Of An AP Exercise 5.5 Summary Chapter 6: Triangles Find here complete study material on class 10 chapter 6 triangles. Triangle is an interesting 3 cornered shape that has some unique properties to explore. In this chapter 6 triangles, you'll be studying all about triangles, similarity criterion of triangles and their properties. Itemized below are the exercises that we’ve covered here: Exercise 6.1 Introduction Exercise 6.2 Similar Figures Exercise 6.3 Similarity Of Triangles Exercise 6.4 Criteria for Similarity of Triangles Exercise 6.5 Areas of Similar Triangles Exercise 6.6 Pythagoras Theorem Exercise 6.7 Summary Chapter 7: Coordinate Geometry Get Free NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry. In lesson 7 you’ll learn about finding distances between the two points whose coordinates are given. You’ll also learn how to find the coordinates of the point using the distance formula, section formula, area of triangle etc. Given NCERT Solutions were prepared by experienced mathematicians to helps students build a stronger foundation in mathematics. Detailed exercise wise answers to Chapter 7 Maths Class 10 Coordinate Geometry are provided below: Exercise 7.1 Introduction Exercise 7.2 Distance Formula Exercise 7.3 Section Formula Exercise 7.4 Area of a Triangle Exercise 7.5 Summary Chapter 8: Introduction to Trigonometry In chapter eight introductions to trigonometry, you'll learn about some important concepts like trigonometric ratios, trigonometric ratios of some specific angles, Trigonometric Ratios of Complementary Angles and finally Trigonometric Identities. This lesson is restricted to the acute angles only, however, ratios can be extended to other angles as well. We will also define and calculate trigonometric ratios and some simple identities involving these ratios, called trigonometric identities. The exercise wise NCERT solution for Class 10 Maths Chapter 8 Introduction to Trigonometry we’ve covered includes: Exercise 8.1 Introduction Exercise 8.2 Trigonometric Ratios Exercise 8.3 Trigonometric Ratios Of Some Specific Angles Exercise 8.4 Trigonometric Ratios Of Complementary Angles Exercise 8.5 Trigonometric Identities Exercise 8.6 Summary Chapter 9: Some Applications of Trigonometry In this lesson, you’ll be learning about the practical application of trigonometry. The NECRT solutions will teach about calculating the heights and distances of various objects, without actually quantifying them. The exercise covered includes: Exercise 9.1 Introduction Exercise 9.2 Heights and Distances Exercise 9.3 Summary Chapter 10: Circles With NCERT Class 10 chapter 10 Circles you’ll be learning various types of situations that can arise when a circle and a line are given in a plane. You'll also learn the concept of a tangent and number of tangents from a point on a circle. The exercise wise NCERT solution for Class 10 Maths Chapter 10 Circles that we’ve covered includes: Exercise 10.1 Introduction Exercise 10.2 Tangent to a circle Exercise 10.3 Number of tangents from a point on a circle Exercise 10.4 Summary Chapter 11: Constructions Class 10 Chapter 11 constructions are an important part of the geometry that has some useful real-world applications. In this chapter, you be learning about various types of constructions in geometry. The exercise wise NCERT solutions we’ve covered are listed below: Exercise 11.1 Introduction Exercise 11.2 Division of A Line Segment Exercise 11.3 Construction Of Tangents To A Circle Exercise 11.4 Summary Chapter 12: Area Related to Circles Chapter 12 will begin with the concepts of circumference and area of a circle. Then you'll be shifted to the area of sector and segment of a circle and apply this you’ll be extending your knowledge to sector and segment of circles. In the last exercise, you are to learn about the areas of combinations of plane figures. Here are the exercise wise solutions that we’ve included: Exercise 12.1 Introduction Exercise 12.2 Perimeter and area of a circle – a review Exercise 12.3 Areas of sector and segment of a circle Exercise 12.4 Areas of combinations of plane figures Exercise 12.5 Summary Chapter 13: Surface Areas and Volumes NCERT chapter 13 surface areas and volumes are an important part of geometry in which you'll be learning to calculate the surface area, volume, or perimeter of a variety of geometrical solid shapes such as circles, rectangle, pyramid, cube, triangle etc. Each has its own specific formulas and method to meet the solution. Listed below are the exercise wise NCERT solutions for chapter 13 surface areas and volumes: Exercise 13.1 Introduction Exercise 13.2 Surface Area of a Combination Of Solids Exercise 13.3 Volume of a Combination Of Solids Exercise 13.4 Conversion of Solid from One Shape to Another Exercise 13.5 Frustum of a Cone Exercise 13.6 Summary Chapter 14: Statistics In this lesson, you’ll be learning about the grouped data such as mean, median and mode and the concept of cumulative frequency. The NCERT solutions will provide a comprehensive and in-depth study of the statistics. Find here detailed answers to Chapter 14 Maths Class 10 Statistics: Exercise 14.1 Introduction Exercise 14.2 Mean of Grouped Data Exercise 14.3 Mode of Grouped Data Exercise 14.4 Median of Grouped Data Exercise 14.5 Graphical Representation Of Cumulative Frequency Distribution Exercise 14.6 Summary Chapter 15: Probability Probability is the final chapter of NCERT class 10th maths books. In this chapter, you'll be introduced to theoretical, also known as the classical probability of an event. The video tutorials talk about some of the very simple problems based on the concept of probability. Listed below are the exercise wise solutions that we’ve covered: Exercise 15.1 Introduction Exercise 15.2 A Theoretical Approach Exercise 15.3 Summary Keep visiting DoubtNut for latestNCERT Solutions for Class 10 Maths as well as other interactive study material! NCERT Solutions Of Other Classes NCERT Solutions For Class 6 Maths NCERT Solutions For Class 7 Maths NCERT Solutions For Class 8 Maths NCERT Solutions For Class 9 Maths NCERT Solutions For Class 11 Maths NCERT Solutions For Class 12 Maths

5. Microstepping of Stepping Motors Microstepping serves two purposes. First, it allows a stepping motor to stop and hold a position between the full or half-step positions, second, it largely eliminates the jerky character of low speed stepping motor operation and the noise at intermediate speeds, and third, it reduces problems with resonance. Although some microstepping controllers offer hundreds of intermediate positions between steps, it is worth noting that microstepping does not generally offer great precision, both because of linearity problems and because of the effects of static friction. Recall, from the discussion in Part 2 of this tutorial, on Stepping Motor Physics, that for an ideal two-winding variable reluctance or permanent magnet motor the torque versus shaft angle curve is determined by the following formulas: h = ( a2 + b2 )0.5Where: x = ( S / (π / 2) ) arctan( b / a ) a -- torque applied by winding with equilibrium at angle 0.This formula is quite general, but it offers little in the way of guidance for how to select appropriate values of the current through the two windings of the motor. A common solution is to arrange the torques applied by the two windings so that their sum h has a constant magnitude equal to the single-winding holding torque. This is referred to as sine-cosine microstepping: b -- torque applied by winding with equilibrium at angle S. h -- holding torque of composite. x -- equilibrium position. S -- step angle. a = h1 sin(((π / 2) / S)θ)Where: b = h1 cos(((π / 2) / S)θ) h1 -- single-winding holding torqueGiven that none of the magnetic circuits are saturated, the torque and the current are linearly related. As a result, to hold the motor rotor to angle θ, we set the currents through the two windings as: ((π / 2) / S)θ -- the electrical shaft angle Ia = Imax sin(((π / 2) / S)θ)Where: Ib = Imax cos(((π / 2) / S)θ) Ia -- current through winding with equilibrium at angle 0.Keep in mind that these formulas apply to two-winding permanent magnet or hybrid stepping motors. Three pole or five pole motors have more complex behavior, and the magnetic fields in variable reluctance motors don't add following the simple rules that apply to the other motor types. Ib -- current through winding with equilibrium at angle S. Imax -- maximum allowed current through any motor winding. The utility of microstepping is limited by at least three consideraitons. First, if there is any static friction in the system, the angular precision achievable with microstepping will be limited. This effect was discussed in more detail in the discussion in Part 2 of this tutorial, on Stepping Motor Physics, in the discussion of friction and the dead zone. The second problem involves the non-sinusoidal character of the torque versus shaft-angle curves on real motors. Sometimes, this is attributed to the detent torque on permanent magnet and hybrid motors, but in fact, both detent torque and the shape of the torque versus angle curves are products of poorly understood aspects of motor geometry, specifically, the shapes of the teeth on the rotor and stator. These teeth are almost always rectangular, and I am aware of no detailed study of the impact of different tooth profiles on the shapes of these curves. Most commercially available microstepping controllers provide a fair approximation of the sine-cosine drive current that would drive an ideal stepping motor to uniformly spaced steps. Ideal motors are rare, and when such a controller is used with a real motor, a plot of the actual motor position as a function of the expected position will generally look something like the plot shown in Figure 5.1. Figure 5.1Note that the motor is at its expected position at every full step and at every half step, but that there is significant positioning error in the intermediate positions. The curve shown is the curve that would result from a perfect sin-cosine microstepping controller used with a motor that had a torque versus position curve that included a significant 4th harmonic component, usually attributed to the detent torque. The broad details of detent effects appear to be fairly uniform from motor to motor, so in principle, it ought to be possible to adjust the tables of sines and cosines used in a sine-cosine controller to compensate for the detent effects. In practice, the effects of friction and the errors introduced by quantization combine to limit the value of such an effort. The third problem arises because most applications of microstepping involve digital control systems, and thus, the current through each motor winding is quantized, controlled by a digital to analog converter. Furthermore, if typical PWM current limiting circuitry is used, the current through each motor winding is not held perfectly constant, but rather, oscillates around the current control circuit's set point. As a result, the best a typical microstepping controller can do is approximate the desired currents through each motor winding. The effect of this quantization is easily seen if the available current through one motor winding is plotted on the X axis and the available current through the other motor winding is plotted on the Y axis. Figure 5.2 shows such a plot for a motor controller offering only 4 uniformly spaced current settings for each motor winding: Figure 5.2Of the 16 available combinations of currents through the motor windings, 6 combinations lead to roughly equally spaced microsteps. There is a clear tradeoff between minimizing the variation in torque and minimizing the error in motor position, and the best available motor positions are hardly uniformly spaced! Use of higher precision digital to analog conversion in the current control system reduces the severity of this problem, but it cannot eliminate it! Plotting the actual rotor position of a motor using the microstep plan outlined in Figure 5.2 versus the expected position gives the curve shown in Figure 5.3: Figure 5.3It is very common for the initial microsteps taken away from any full step position to be larger than the intended microstep size, and this tends to give the curve a staircase shape, with the downward steps aligned with the full step positions where only one motor winding carries current. The sign of the error at intermediate positions tends to fluctuate, but generally, the position errors are smallest between the full step positions, when both motor windings carry significant current. Another way of looking at the available microsteps is to plot the equilibrium position on the horizontal axis, in fractions of a full-step, while plotting the torque at each available equilibrium position on the vertical axis. If we assume a 4-bit digital-to-analog converter, giving 16 current levels for each each motor winding, there are 256 equilibrium positions. Of these, 52 offer holding torques within 10% of the desired value, and only 33 are within 5%; these 33 points are shown in bold in Figure 5.4: Figure 5.4If torque variations are to be held within 10%, it is fairly easy to select 8 almost-uniformly spaced microsteps from among those shown in Figure 5.4; these are boxed in the figure. The maximum errors occur at the 1/4 step points; the maximum error is .008 full step or .06 microsteps. This error will be irrelevant if the dead-zone is wider than this. If 10 microsteps are desired, the situation is worse. The best choices, still holding the maximum torque variation to 10%, gives a maximum position error of .026 full steps or .26 microsteps. Doubling the allowable variation in torque approximately halves the positioning error for the 10 microstep example, but does nothing to improve the 8 microstep example. One option which some motor control system designers have explored involves the use of nonlinear digital to analog converters. This is an excellent solution for small numbers of microsteps, but building converters with essentially sinusoidal transfer functions is difficult if high precision is desired. As typically used, a microstepping controller for one motor winding involves a current limited H-bridge or unipolar drive circuit, where the current is set by a reference voltage. The reference voltage is then determined by an analog-to-digital converter, as shown in Figure 5.5: Figure 5.5Figure 5.5 assumes a current limited motor controller such as is shown in Figures 4.7, 4.8, 4.10 or 4.11. For all of these drivers, the state of the X and Y inputs determines the whether the motor winding is on or off and if on, the direction of the current through the winding. The V0 through Vn inputs determine the reference voltage and this the current through the motor winding. There are a fair number of nicely designed integrated circuits combining a current limited H-bridge with a small DAC to allow microstepping control of motors drawing under 2 amps per winding. The UDN2916B from Allegro Microsystems is a dual 750mA H-bridge, with a 2-bit DAC to control the current through each. bridge. Another excellent example is the UC3770 from Unitrode. Unitrode. This chip integrate a 2-bit DAC with a PWM controlled H-bridge, packaged in either 16 pin power-dip format or in surface mountable form. The 3717 a slightly cleaner design, good for 1.2 A, while the 3770 is good for up to between 1.8 A or 2 A, depending on how the chip is cooled. The 3955 from Allegro Microsystems incorporates a 3-bit non-linear DAC and handles up to 1.5 A; this is available in 16-pin power DIP or SOIC formats. The nonlinear DAC in this chip is specifically designed to minimize step-angle errors and torque variations using 8 microsteps per full-step. The LMD18245 from National Semiconductor is a good choice for microstepped control of motors drawing up to 3 amps. This chip incorporates a 4-bit linear DAC, and an external DAC can be used if higher precision is required. As indicated by the data shown in Figure 5.4, a 4-bit linear DAC can produce 8 reasonably uniformly spaced microsteps, so this chip is a good choice for applications that exceed the power levels supported by the Allegro 3955. It appears that microstepping was invented in 1974 by Larry Durkos, who was working as a mechanical engineer for American Monitor Corporation. The company was a medical equipment vendor, and they were using a large Superior Electric 1.8 degree per step stepping motors to directly drive the 20 inch diameter turntable of their Kinetic Discrete Analyzer. The turntable was used to bring each of 100 blood samples into position for analysis. That is 2 steps per sample, and the motion was so abrupt that the samples tended to spill. The system was controlled by a Computer Automation LSI 2 minicomputer (today, we would use a microcontroller), and Durkos worked out how to do computer-controlled sine-cosine microstepping in order to solve this problem. The solution was published in the technical service manuals for the KDA analyzer, but it was never patented. Representatives of Superior Electric learned of microsteppingfrom Durkos, and that company was the first to market a microstepping controller.

Why banks publish annual percentage rates. Now that you understand the difference between interest rate and APR, let's talk a little about how to find the best options for your loans: The Difference Between Interest Rate and APR in Mortgages By contrast, the annual percentage rate is the annual cost of the loan inclusive of fees, Sherman says. Fees included in the APR can add significantly to the costs a buyer will pay. Examples of such fees are: Getting a loan means paying interest—it's the cost of borrowing money. Just how much interest you'll pay depends on your interest rate. Or does it depend on your ARP (annual percentage rate)? Find out what the difference is between APR and interest rates. It is worth noting, however, that an annual percentage rate and an interest rate are two unique indicators. While they may sound similar, they are anything but. There are several inherent differences that exist between interest rates and annual percentage rates, not the least of which are hard to discern for amateur investors. Key Differences Between Interest Rate and APR. The difference between interest rate and APR are drawn clearly on the following grounds: The interest rate is described as the rate at which interest is charged by the lenders on the loan given to the borrowers. APR or Annual Percentage Rate is the per year total cost of borrowing. The annual percentage rate represents your total cost of getting a mortgage. The interest rate represents the cost you pay over time to buy that loan. Let’s take a look at the difference between your APR and interest rate, and how they affect the true cost of a mortgage. We’ll cover: What’s an annual percentage rate? 21 Feb 2020 Knowing the difference between the “interest rate” and “annual percentage rate” ( APR) can save you a lot of money. Interest rate vs. APR The interest rate is the cost of borrowing the principal loan amount. The rate can be variable or fixed, but it’s always expressed as a percentage. APR is the annual rate of interest that is paid on an investment, without taking into account the compounding of interest within that year. Alternatively, APY does take into account the frequency with which the interest is applied—the effects of intra-year compounding. They might be used interchangeably, but an APR and an interest rate aren’t one and the same. The annual percentage rate represents your total cost of getting a mortgage. The interest rate represents the cost you pay over time to buy that loan. Both APR (annual percentage rate) and APY (annual percentage yield) are commonly used to reflect the interest rate paid on a savings account, loan, money market or certificate of deposit. It's not immediately clear from their names how the two terms — and the interest rates they describe — differ. Why banks publish annual percentage rates. Now that you understand the difference between interest rate and APR, let's talk a little about how to find the best options for your loans: 26 Feb 2020 Difference Between Interest Rate and APR. Annual percentage rate vs. interest rate: These are two similar but ultimately different things. Let's 15 Nov 2019 An annual percentage rate (APR) reflects the mortgage interest rate plus other charges. 27 Feb 2020 An in-depth look at the difference between the mortgage interest rate and And the other is the Annual Percentage Rate, or APR, which is the When evaluating the cost of a loan or line of credit, it is important to understand the difference between the advertised interest rate and the annual percentage rate 27 Feb 2017 If interest rates have gone down you will be in a better position, but if interest The Annual Percentage Rate (APR) is the annual cost of a loan 28 Sep 2017 When shopping for a new mortgage loan, you may notice an Annual Percentage Rate (APR) advertised next to the note rate. The inclusion of An interest rate and a representative APR can often be confusing when looking at finance options. Cash Lady explains the difference between them both. borrow £100 at 5% 'annual' interest, you would pay back £105 at the end of the year. 16 Oct 2019 APR stands for Annual Percentage Rate. It takes into account the interest rate of the product then adds on any additional charges, giving you Everything you need to know about the different types of interest rates. With a loan that has a stated Annual Percentage Rate, you are only paying the interest 23 Jul 2019 The annual percentage rate is the effective annual interest rate on a loan including most ancillary charges and origination costs in addition to It consists of the actual interest rate, the processing fee, foreclosure amount, and all other fees charged by a bank on the loan. What is the difference between APR Interest Rate vs. APR: An Overview. The interest rate is the cost of borrowing the money, that is, the principal loan amount. When evaluating the cost of a loan or line of credit, it is important to understand the difference between the advertised interest rate and the annual percentage rate, or APR. This new loan amount, along with the interest rate (5.00%), is used to calculate a new monthly payment ($1,089.75). The APR is then calculated by working backwards to figure out what the rate would have to be for a loan with the new monthly payment ($1,089.75) and the original loan amount ($200,000). Interest rate refers to the annual cost of a loan to a borrower and is expressed as a percentage; APR is the annual cost of a loan to a borrower — including fees. Like an interest rate, the APR is expressed as a percentage. Interest Rate; Definition: Annual Percentage Rate (APR) is an expression of the effective interest rate that the borrower will pay on a loan, taking into account one-time fees and standardizing the way the rate is expressed. Interest is a fee on borrowed capital. Interest rate is a "rent on money" to compensate the lender for foregoing other useful investments that could have been made with the loaned money. Transaction costs In contrast, APR is an annual rate that includes interest rate payments as well as other fees charged for a loan, which can include origination fees, closing costs and service charges. Because APR is calculated on a yearly basis, it will be higher than the interest rate for loans with frequent payments, short terms,

Switzernet Sàrl, Scientific Park of Swiss Federal Institute of Technology, Lausanne (EPFL) In measurement instruments where measured values are indicated with a mechanical pointer and a graduated scale, the observation precision is increased often by adding an auxiliary mechanical pointer (needle) with a sub graduated scale. The auxiliary pointer moves in synchronization with the main pointer but at a higher speed. A constant velocity ratio between the auxiliary pointer and the main pointer is maintained via instrumentation gearing mechanisms. Mechanical solutions are not always suitable. A challenging idea is to use moiré phenomenon for its well known magnification and acceleration properties. However the well known moiré shapes with sufficient sharpness, good luminosity and contrast can be obtained only in highly periodic patterns. The periodic nature of patterns makes them inapplicable for indication of values. We present new discrete patterns assembled from simple moiré patterns of different periodicity. The elevation profile of our discrete pattern reveals a joint moiré shape with an arbitrarily long periodicity. The luminosity and the sharpness of our moiré shapes are as high as in simple highly periodic moiré patterns. Keywords: moiré, instrumentation, metrology, multi-stripe moiré, multi-ring moiré, optical speedup, moiré pointer, moiré needle, moiré watches, optical clock-hands, moiré clock-hands, non-periodic moiré Table of contents A graduated scale and a mechanical pointer is a common part for almost all mechanical measurement devices. Often an auxiliary pointer and a scale with sub graduations are used for additional precisions. The auxiliary pointer moves faster, in synchronization with the main pointer. The pointers are connected via a tooth wheel type transmission system. The involute tooth shape is one that results in a constant velocity ratio, and is the most commonly used in instrumentation gearing, clocks and watches. Mechanical methods for changing the speed however can often be heavy and inapplicable. Lack of the force, such as in a compass, can be one of the serious obstacles. Inertia problems arising from discrete movements of mechanical parts at high speed, such as in chronographs, may be another obstacle. The magnification and acceleration properties of moiré superposition images are a well known phenomenon. The superposition of transparent structures, comprising periodic opaque patterns, forms periodic moiré patterns. A challenging idea would be to use optical moiré effect for creating a fast auxiliary pointer replacing completely the mechanical parts moving at high speeds. The periodic nature of known moiré patterns make them inappropriate for indication of values. Profiles with very long periods can be created with periodic moiré. It is possible to design circular layer patterns with radial lines such that their superposition produces a radial moiré fringe with an angular period equal to 360 degrees. Thus only single radial moiré fringe will be visible in the superposition pattern. However such long periods make the moiré fringes blurred. The dispersion area of the fringe can be as large as the half of the period. In section 4 we show a particular case where a radial periodic moiré can be of use with an additional design extension. However in general, the long period moiré fringes of classical periodic moiré are too inexact for indication purposes. A limited degree of sharpening of shapes in periodic moiré is possible using band moiré methods, namely moiré magnification of micro shapes [Hutley99], [Kamal98]. Such shapes however require serious sacrifices of the overall luminosity of the superposition image without significant improvements of the sharpness. Random moiré, namely Glass patterns, produce non-periodic superposition patterns [Amidror03a], [Amidror03b], [Glass69a], [Glass73a]. The obstacle is that the valid range of movements of layers is very limited. The auxiliary indicator would show the sub graduations only within the range of only one graduation of the main scale. Additionally, in random moiré the shapes are noisier than in simple periodic moiré. We developed new discrete patterns formed by merging straight stripes or circular rings of simple periodic moiré patterns. The composing stripes or rings are simple patterns with carefully chosen periods and phases. The composite pattern reveals a sharp moiré shape with an arbitrarily long periodicity. Movement of a layer along the stripes or along the circumferences of rings produces a faster movement of the moiré shape. Such shape has all qualities for playing the role of the fast auxiliary indicator. The one of the layers can be put into slow mechanical motion by the main pointer of the measurement device. In our discrete patterns the shapes are as sharp as in highly periodic moiré patterns. The period of the moiré pointer can be as long as it is required by the display size of the instrument. In our discrete patterns, the choice of the period has no impact on the quality of the optical shape and a wide range of speed ratios can be obtained. Choice of stripes or rings depends on the type of the movement of layers. For linear movements the pattern comprises parallel stripes following the path of the movement. For circular movements the pattern consists of concentric rings with a center corresponding to the rotation axis. Our algorithm merges numerous simple periodic patterns into a composite pattern so as to form a continuous joint shape in the assembled superposition image. The underlying layer patterns do not join into continuous shapes within assembled layers. The composite patterns are constructed, such that the velocity ratios across all individual moiré patterns are identical. Consequently, the joint shape of the multi-stripe or multi-ring moiré pattern conserves its form during movements of the optical image. The speed ratio and the sharpness of moiré shape are constant within the full range of movements of the main pointer and layers. Circular multi-ring samples are the most interesting. They can be used for adding auxiliary optical pointers to numerous measurement device with circular dials and radial mechanical pointers such as clocks, watches, chronographs, protractors, thermometers, altimeters, barometers, compasses, speedometers, alidades, and even weathervanes. In mechanical chronographs, optical acceleration permits measuring fractions of seconds without having mechanical parts moving at high speed with related problems of force, inertia, stress, and wear. The paper is organized as follows: Section 2 introduces the classical periodic moiré and the methods for forming periodic moiré fringes of a desired shape. These methods are presented in scope of a new perspective permitting to easily change the curves of moiré shapes and those of the layer patterns without affecting the periodicity and the velocity ratios, which are essential factors for metrology purposes. Linear movements are considered and a set of corresponding equations is introduced. Section 3 introduces the equations for creating curved moiré shapes for rotating layers preserving the angular periodicity and velocity ratio. Section 4 presents an extension of a classical moiré for displaying quickly progressing labels on a round dial. In section 5 we introduce our multi-ring moiré patterns on examples of straight radial layer lines. In section 6 we present the general case of multi-ring moiré with various curved layer patterns and moiré shape patterns. Conclusions are given at the end of the paper. Simple moiré patterns can be observed when superposing two transparent layers comprising periodically repeating opaque parallel lines as shown in Figure 1. In the example, the lines of one layer are parallel to the lines of the second layer. The superposition image outlines periodically repeating dark parallel bands, called moiré lines. Spacing between the moiré lines is much larger than the periodicity of lines in the layers. Figure 1. Superposition of two layers consisting of parallel lines, where the lines of the revealing layer are parallel to the lines of the base layer We denote one of the layers as the base layer and the other one as the revealing layer. When considering printed samples we assume that the revealing layer is printed on a transparency and is superposed on top of the base layer, which can be printed either on a transparency or on an opaque paper. The periods of the two layer patterns, i.e. the space between the axes of parallel lines, are close. We denote the period of the base layer as and the period of the revealing layer as . In Figure 1, the ratio/ is equal to 12/11. Light areas of the superposition image correspond to the zones where the lines of both layers overlap. The dark areas of the superposition image forming the moiré lines correspond to the zones where the lines of the two layers interleave, hiding the white background. Such superposition images are discussed in details in literature [Sciammarella62a p.584], [Gabrielyan07a]. The period of moiré lines is the distance from one point where the lines of both layers overlap to the next such point. For cases represented by Figure 1 one can obtain the well known formula for the period of the superposition image [Amidror00a p.20], [Gabrielyan07a]: The superposition of two layers comprising parallel lines forms an optical image comprising parallel moiré lines with a magnified period. According to equation (2.1), the closer the periods of the two layers, the stronger the magnification factor is. For the case when the revealing layer period is longer than the base layer period, the space between moiré lines of the superposition pattern is the absolute value of formula of (2.1). The thicknesses of layer lines affect the overall darkness of the superposition image and the thickness of the moiré lines, but the period does not depend on the layer lines’ thickness. In our examples the base layer lines’ thickness is equal to , and the revealing layer lines’ thickness is equal to . If we slowly move the revealing layer of Figure 1 perpendicularly to layer lines, the moiré bands will start moving along the same axis at a several times faster speed. The four images of Figure 2 show the superposition image for different positions of the revealing layer. Compared with the first image (a) of Figure 2, in the second image (b) the revealing layer is shifted up by one fourth of the revealing layer period (), in the third image (c) the revealing layer is shifted up by half of the revealing layer period (), and in the fourth image (d) the revealing layer is shifted up by three fourth of the revealing layer period (). The images show that the moiré lines of the superposition image move up at a speed, much faster than the speed of movement of the revealing layer. When the revealing layer is shifted up perpendicularly to the layer lines by one full period of its pattern, the superposition optical image must be the same as the initial one. It means that the moiré lines traverse a distance equal to the period of the superposition image , while the revealing layer traverses the distance equal to its period . Assuming that the base layer is immobile (), the following equation holds for the ratio of the optical image’s speed to the revealing layer’s speed: According to equation (2.1) we have: In case the period of the revealing layer is longer than the period of the base layer, the optical image moves in the opposite direction. The negative value of the ratio computed according to equation (2.3) signifies the movement in the reverse direction. In this section we introduce equations for patterns with inclined lines. Equations for rotated patterns were already introduced decades ago [Nishijima64a], [Oster63a], [Morse61a]. These equations are good for static moiré patterns or their static instances. In scope of metrology instrumentation, we review the equations suiting them for dynamic properties of moiré patterns. The set of key parameters is defined and the equations are developed such that the curves can be constructed or modified without affecting given dynamic properties. According to our notation, the letter p is reserved for representing the period along an axis of movements. The classical distance between the parallel lines is represented by the letter T. The periods (p) are equal to the spaces between the lines (T), only when the lines are perpendicular to the movement axis (as in the case of Figure 2 with horizontal lines and a vertical movement axis). Our equations represent completely the inclined layer and moiré patterns and at the same time the formulas for computing moiré periods and optical speedups remain in their basic simple form (2.1), (2.2), and (2.3). In this section we focus on linear movements. Equations binding inclination angles of layers and moiré patterns are based on, , and , the periods of the revealing layer, base layer, and moiré lines respectively measured along the axis of movements. For linear movements the p values represent distances along a straight axis. For rotational movements the p values represent the periods along circumference, i.e. the angular periods. The superimposition of two layers with identically inclined lines forms moiré lines inclined at the same angle. Figure 3 (a) is obtained from Figure 1 with a vertical shearing. In Figure 3 (a) the layer lines and the moiré lines are inclined by 10 degrees. Inclination is not a rotation. During the inclination the distance between the layer lines along the vertical axis is conserved (p), but the true distance T between the lines (along an axis perpendicular to these lines) changes. The vertical periods and , and the distances and are indicated on the diagram of an example shown in Figure 5 (a). Figure 3. (a) Superposition of layers consisting of inclined parallel lines where the lines of the base and revealing layers are inclined at the same angle; (b) Two layers consisting of curves with identical inclination patterns, and the superposition image of these layers The inclination degree of layer lines may change along the horizontal axis forming curves. The superposition of two layers with identical inclination pattern forms moiré curves with the same inclination pattern. In Figure 3 (b) the inclination degree of layer lines gradually changes according the following sequence of degrees (+30, –30, +30, –30, +30), meaning that the curve is divided along the horizontal axis into four equal intervals and in each such interval the curve’s inclination degree linearly changes from one degree to the next according to the sequence of five degrees. Layer periods and represent the distances between the curves along the vertical axis, i.e. that of the movement. In Figure 3 (a) and (b), the ratio/ is equal to 12/11. Figure 3 (b) can be obtained from Figure 1 by interpolating the image along the horizontal axis into vertical bands and by applying a corresponding vertical shearing and shifting to each of these bands. Equation (2.1) is valid for computing the spacing between the moiré curves along the vertical axis and equation (2.3) for computing the optical speedup ratio when the revealing layer moves along the vertical axis. More interesting is the case when the inclination degrees of layer lines are not the same for the base and revealing layers. Figure 4 shows four superposition images where the inclination degree of base layer lines is the same for all images (10 degrees), but the inclination degrees of the revealing layer lines are different and are equal to 7, 9, 11, and 13 degrees for images (a), (b), (c), and (d) respectively. The periods of layers along the vertical axis and (the / ratio being equal to 12/11) are the same for all images. Correspondingly, the period computed with formula (2.1) is also the same for all images. Figure 4. Superposition of layers consisting of inclined parallel lines, where the base layer lines’ inclination is 10 degrees and the revealing layer lines’ inclination is 7, 9, 11, and 13 degrees [ps], [gif], [tif] Figure 5 (a) helps to compute the inclination degree of moiré optical lines as a function of the inclination of the revealing and the base layer lines. We draw the layer lines schematically without showing their true thicknesses. The bold lines of the diagram inclined by degrees are the base layer lines. The bold lines inclined by degrees are the revealing layer lines. The base layer lines are vertically spaced by a distance equal to , and the revealing layer lines are vertically spaced by a distance equal to . The distance between the base layer lines and the distance between the revealing layer lines are the parameters used in the common formulas, well known in the literature. The parameters and are not used for the development of our equations. The intersections of the lines of the base and the revealing layers (marked in the figure by two arrows) lie on a central axis of a light moiré band that corresponds in Figure 4 to the light area between two parallel dark moiré lines. The dashed line passing through the intersection points of Figure 5 (a) is the axis of the light moiré band. The inclination degree of moiré lines is therefore the inclination of the dashed line. Figure 5. (a) Computing the inclination angle of moiré lines as a function of inclination angles of the base layer and revealing layer lines; (b) Moiré lines inclination as a function of the revealing layer lines inclination for the base layer lines inclination equal to 20, 30, and 40 degrees [xls] From Figure 5 (a) we deduce the following two equations: From these equations we deduce the equation for computing the inclination of moiré lines as a function of the inclinations of the base layer and the revealing layer lines: For a base layer period equal to 12 units, and a revealing layer period equal to 11 units, the curves of Figure 5 (b) represents the moiré line inclination degree as a function of the revealing layer line inclination. The base layer inclinations for the three curves (from left to right) are equal to , , and degrees respectively. The circle marks correspond to the points where both layers’ lines inclinations are equal and the moiré lines inclination also become the same. The periods , , and (see Figure 5 (a)) that are used in the commonly known formulas of the literature are deduced from periods , and as follows: From here, using our equation (2.5) we deduce the well known formula for the angle of moiré lines [Amidror00a]: Recall from trigonometry the following simple formulas: From equations (2.7) and (2.8) we have: From equations (2.1) and (2.6) we have: From equations (2.9) and (2.10) we deduce the second well known formula in the literature, the formula for the period of moiré lines: Recall from trigonometry that: In the particular case when , taking in account equation (2.12), equation (2.11) is further reduced into well known formula: Still for the case when , we can temporarily assume that all angles are relative to the base layer lines and rewrite equation (2.7) as follows: Recall from trigonometry that: Therefore from equations (2.14) and (2.15): Now for the general case when the revealing layer lines do not represent the angle zero: We obtain the well known formula [Amidror00a]: Equations (2.7) and (2.11) are the general case formulas known in the literature, and equations (2.13) and (2.18) are the formulas for rotated identical patterns (i.e. the case when ) [Amidror00a], [Nishijima64a], [Oster63a], [Morse61a]. Assuming in equation (2.7) that , we have: Only for the case when the rotation of moiré lines is linear with respect to the rotation of the revealing layer (see equation (2.18)). Comparison of equation (2.19) and its respective graph (see [Gabrielyan07a]) with our equation (2.5) and its respective graph (see Figure 5 (b)) shows a significant difference in the binding of angles for sheared (i.e. inclined) and rotated layer patterns. From equation (2.5) we can deduce the equation for computing the revealing layer line inclination for a given base layer line inclination , and a desired moiré line inclination : The increment of the tangent of the revealing lines’ angle () relatively to the tangent of the base layer lines’ angle can be expressed, as follows: According to equation (2.3), is the inverse of the optical acceleration factor, and therefore equation (2.21) can be rewritten as follows: Equation (2.22) shows that relative to the tangent of the base layer lines’ angle, the increment of the tangent of the revealing layer lines’ angle needs to be smaller than the increment of the tangent of the moiré lines’ angle, by the same factor as the optical speedup. For any given base layer line inclination, equation (2.20) permits us to obtain a desired moiré line inclination by properly choosing the revealing layer inclination. In Figure 3 (b) we showed an example, where the curves of layers follow an identical inclination pattern forming a superposition image with the same inclination pattern. The inclination degrees of the layers’ and moiré lines change along the horizontal axis according the following sequence of alternating degree values (+30, –30, +30, –30, +30). In Figure 6 (a) we obtained the same superposition pattern as in Figure 3 (b), but the base layer consists of straight lines inclined by –10 degrees. The corresponding revealing pattern is computed by interpolating the curves into connected straight lines, where for each position along the horizontal axis, the revealing line’s inclination angle is computed as a function of and , according to equation (2.20). Figure 6. (a) The base layer with inclined straight lines and the revealing layer computed so as to form the desired superposition image; (b) Inversed inclination patterns of moiré and base layer curves [ps], [tif], [gif] The same superposition pattern as in Figure 3 (b) and Figure 6 (a) is obtained in Figure 6 (b). Note that in Figure 6 (b) the desired inclination pattern (+30, –30, +30, –30, +30) is obtained using a base layer with a completely inverted inclination pattern (–30, +30, –30, +30, –30). Figure 6 (a) and (b) demonstrate what is already expressed by equation (2.22): the difference between the inclination patterns of the revealing layer and the base layer are several times smaller than the difference between the inclination patterns of moiré lines and the base layer lines. Our web page contains a GIF animation [gif] for modifying pairs of base and revealing layers constantly forming the same superposition image of Figure 3 (b), Figure 6 (a), and Figure 6 (b) i.e. the moiré inclination pattern (+30, –30, +30, –30, +30) [Gabrielyan07b]. In the animation, the base layer inclination pattern gradually changes and the revealing layer inclination pattern correspondingly adapts such that the superposition image’s inclination pattern remains the same. Similarly to layer and moiré patterns comprising parallel lines (see Figure 1), concentric superposition of dense periodic layer patterns comprising radial lines forms magnified periodic moiré patterns also with sparse radial lines. Figure 7 is the counterpart of Figure 1, where the horizontal axis is replaced by the radius and the vertical axis by the angle. Full circumferences of layer patterns are equally divided by integer numbers of radial lines. The number of radial lines of the base layer is denoted as and the number of radial lines of the revealing layer is denoted as . The periods and denote the angles between the central radial axes of adjacent lines. Therefore: According to equations (3.1), equation (2.1) can be rewritten as follows: Therefore the number of moiré radial lines corresponds to the difference between the numbers of layer lines: If in the layer patterns, the full circumferences are divided by integer numbers of layer lines, the circumference of the superposition image is also divided by an integer number of more lines. The optical speedup factor of equation (2.3) can be rewritten by replacing the periods and by their expressions from equations (3.1): The values and represent the angular speeds. The negative speedup signifies a rotation of the superposition image in a direction inverse to the rotation of the revealing layer. Considering (3.3), the absolute value of the optical speedup factor is: Radial lines have constant angular thickness, giving them the forms of segments, thick at their outer ends and thin at their inner ends. The values of , , and do not depend on the angular thickness of radial lines. In our examples the angular thicknesses of layer lines are equal to the layer’s half-period, i.e. the thickness of the base layer lines is equal to and the thickness of the revealing layer lines is . In Figure 7, the number of radial lines of the revealing layer is equal to 180, and the number of radial lines of the base layer is 174. Therefore, according to equations (3.4) and (3.3), the optical speedup is equal to 30, confirmed by the two images (a) and (b) of Figure 8, and the number of moiré lines is equal to 6, confirmed by the image of Figure 7. Figure 8. Rotation of the revealing layer by 1 degree in the clockwise direction rotates the optical image by 30 degrees in the same direction In circular periodic patterns curved radial lines can be constructed using the same sequences of inclination degrees as used in section 2.3 for curves of Figure 3 (b). The inclination angle at any point of the radial curve corresponds to the angle between the curve and the axis of the radius passing through the current point. Thus inclination angle 0 corresponds to straight radial lines as in Figure 7. With the present notion of inclination angles for , , and , equations (2.5) and (2.20) are applicable for circular patterns without modifications. Curves can be constructed incrementally with a constant radial increment equal to . Figure 9 shows a segment of a curve, marked by a thick line, which has an inclination angle equal to . Figure 9. Constructing a curve in a polar coordinate system with a desired inclination While constructing the curve, the current angular increment must be computed so as to respect the inclination angle : Figure 10 shows a superposition of layers with curved radial lines. The inclination of curves of both layers follows an identical pattern corresponding to the following sequence of degrees (+30, –30, +30, –30, +30). Layer curves are iteratively constructed with increment pairs computed according to equation (3.6). Since the inclination patterns of both layers of Figure 10 are identical, the moiré curves also follow the same pattern. Figure 10. Superposition of layers in a polar coordinate system with identical inclination patterns of curves corresponding to (+30, –30, +30, –30, +30); a portion of the revealing layer is cut away exposing the base layer in the background [eps], [tif], [gif] Similarly to examples of Figure 3 (b), Figure 6 (a), and Figure 6 (b), where the same moiré pattern is obtained by superposing different pairs of layer patterns, the circular moiré pattern of Figure 10 can be analogously obtained by superposing other pairs of circular layer patterns. Taking into account equations (3.1), equations (2.5) and (2.20) can be rewritten as follows: Taking into account equation (3.4), equation (3.8) can be also rewritten as follows: For producing the superposition image of Figure 10, thanks to equations (3.8) and (3.9), other pairs of layer patterns can be created as shown in Figure 11. In the first image (a) of Figure 11, the base layer lines are straight. In the second image (b), the base layer lines inclination pattern is reversed with respect to the moiré lines. Figure 11. Superposition images with identical inclination pattern (+45, –45, +45, –45, +45) of moiré curves, where in one case the base layer comprise straight radial segments, and in the second case the base layer comprise curves which are the mirrored counterparts of the resulting moiré curves [eps], [tif], [gif] Our web page [Gabrielyan07b] contains an animation [gif], where the moiré curves of the superposition image are always the same, but the inclination pattern of the base layer curves gradually alternates between the following two mirror patterns (+45, –45, +45, –45, +45), and (–45, +45, –45, +45, –45). For each instance of the animation, the revealing layer lines are computed according to equation (3.8) in order to constantly maintain the same moiré pattern. Equations (3.4) and (3.3) remain valid for patterns with curved radial lines. In Figure 10 there are 180 curves in the revealing layer and 171 curves in the base layer. Therefore optical speedup factor according to equation (3.4) is equal to 20, and the number of moiré curves according to equation (3.3) is equal to 9, as seen in the superposition image of Figure 10. One can form a radial moiré fringe with a period equal to . In the superposition image of such pattern we will see only one moiré fringe. This fringe will not have sharp contours and will appear large and blurred. The radial moiré fringe can be formed by layer patterns with radial lines or rather radial sectors. For small speed rations, fine granularity of layer patterns with radial lines cannot be maintained. As the speed ratio decreases, the superposition image becomes coarse and the moiré shape becomes visually not identifiable. The fine granularity can be maintained by using spiral shaped lines in layer patterns. The layer patterns with spirals can be computed such that the moiré fringe is kept radially oriented. By reducing the spiral elevation rate in both layers, sufficiently fine layer patterns can be obtained. However, strongly inclined spirals resulting to fine patterns make the superposition moiré images less tolerant to mechanical inaccuracies such as surface deformations of layers or disparities in concentric superposition of layers. In Figure 12 we show that a design extension of simple spiral patterns with a single moiré fringe may result to a useful application. The example is obtained by taking a simple spiral pattern of a base layer and by cleaning in such pattern all areas lying outside the contours of twelve labels. A part of the revealing layer is cut-away exposing the base layer. In such a way, instead of revealing a large and blurred moiré fringe, our superposition pattern reveals more attractive image consisting of labels within the concerned area. The area rotates at a 60 times faster speed than the mechanical rotation speed of the revealing layer. The spirals of two layers are computed so as to produce a moiré fringe with radial orientation. In this section we present our multi-ring circular patterns. The superimposition of our multi-ring layer patterns forms a complex moiré image, but at one position a continuous shape is outlined. When rotating the revealing layer, the optical shape rotates without deformations at a k times faster speed. Refer to equation (3.4) for circular patterns. An optical rotation k times faster than the rotation of the revealing layer can be obtained if: According to equation (3.3) the number of moiré spots in a circular pattern for different values of i is simply equal to the value of i: Therefore, the same moiré speedup factor k can be obtained with different pairs of revealing and base layer patterns corresponding to different numbers of moiré bands. We can construct several nested concentric circular patterns for the same value of k and for different values of i. Figure 13. Four nested rings whose layer lines overlap at angle zero For example, Figure 13 shows four nested adjacent rings, where the index i increments from 1 to 4 when counting from the inner ring toward the outer ring. The number of dark moiré radial lines of individual rings changes from 1 to 4 according to equation (5.2). The acceleration factor k is equal to 60 for all rings. Therefore the revealing layer of the most inner ring has 60 radial lines and the corresponding base layer has 59 lines. Correspondingly the layers of the most outer ring have 240 and 236 radial lines. In Figure 13 a part of the revealing layer is cut out, exposing the base layer. All rings are constructed such that the lines of the revealing and base layers perfectly overlap at the angle zero. Therefore a light moiré radial band appears at the angle zero of each individual rings. For the inner ring, the dark moiré band is located at 180 degrees from angle zero. The first dark moiré band of the second ring is located at 90 degrees. The first dark moiré band of the third ring is located at 60 degrees and for the most outer ring at 45 degrees. The patterns of the base and revealing layers of each ring can be printed so as a dark moiré band appears at the angle zero. For this purpose, both layer patterns of each ring must be rotated by a degree : Figure 14 corresponds to the superimposition image of Figure 13, but the individual ring patterns are rotated according to equation (5.3) such that dark moiré bands appear at the angle zero in all rings. The black moiré bands of all adjacent rings became horizontally aligned forming a joint radial shape. Once the rings are adjusted according to equation (5.3), we consider that the base layer patterns of all rings form a single joint base layer (e.g. printed on an opaque paper), and the revealing layer patterns of all rings form a joint revealing layer (e.g. printed on a transparency). A part of the revealing layer is cut away exposing the base layer. Figure 14. Four nested rings with an acceleration factor equal to 60 for all four rings According to equation (5.1), rotation of the revealing layers at a given angular speed must rotate the superimposition image at another angular speed which is identical for all rings. Therefore the radial moiré band traversing all rings will remain aligned all the time during the rotation. Rotation of the revealing layer rotates the optical image at a k times faster speed. The acceleration factor k of the superimposition image of Figure 14 is equal to 60. Therefore the rotation of the revealing layer by –1 degree rotates the optical image by –60 degrees (compare the image of Figure 14 with the first image of Figure 15). Rotation of the revealing layer by –2 degree rotates the optical image by –120 degrees (compare the image of Figure 14 with the second image of Figure 15). The negative rotation angles correspond to the rotation in clockwise direction. The negligible rotations of the revealing layer in Figure 14 and Figure 15 can be noticed by observing the cut out region of the revealing layer. In our web site [Gabrielyan07b] we present a GIF file [gif] which demonstrates the superposition image shown in Figure 14 and Figure 15 during a rotation of the revealing layer that slowly turns by 6 degree in clockwise direction. During this time the superimposition image makes a full rotation of 360 degree also in clockwise direction. The widths of the rings of the multi-ring patterns must not be obligatorily the same. The number of the ring’s moiré bands also must not necessarily increment with the ring number. Figure 16 shows a superimposition image with 12 rings, where at the beginning the number of moiré bands increments, but after reaching a maximal limit at a ring , the number of moiré bands starts decrementing. The maximal number of moiré bands is set to 10. Therefore the number of moiré bands follows the following sequence (1, 2, 3, … 8, 9, 10, 9, 8). The ring widths are not constant and are computed so as the largest ring is the ring , at which has its maximal value. The adjacent rings gradually decrease their widths as we move away from the largest ring. The width of the j-th ring can be computed by equation (5.4), where j is the sequential number of the ring, is the number of the widest ring, is the minimal ring width, and is the maximal ring width. Figure 16. Multi-ring moiré superposition image with variable ring widths Recall that for measuring the line inclination in circular patterns we use the angle between the line and the radial axis as shown in Figure 9. In Figure 16, inclination of moiré lines of the superposition image is equal to 0 degree for all rings. In section 3.2 we show that the desired degree of moiré inclination can be obtained by different pairs of base and revealing layer patterns. It is sufficient to choose for every ring an inclination pattern of the base layer and then, the corresponding inclination pattern of the revealing layer can be computed thanks to equation (3.8) or (3.9). Taking into account that in multi-ring patterns the speedup factor k used in equations (5.1) is the same for all rings, equation (3.9) can be rewritten as follows: For a particular case, when , i.e. when we desire straight radial moiré lines, equation (6.1) is further reduced to: For any inclination of the base layer pattern, the revealing layer pattern can be computed according to equation (6.2) to ensure straight radial moiré lines. Figure 17 shows a superposition image with straight moiré lines, similarly to Figure 14. In contrast to Figure 14 the base layer lines are not straight. The overall inclination pattern of the entire base layer across all rings follows the following sequence of inclination degrees . Figure 17. Multi-ring moiré superposition image, where the inclination of moiré lines is of 9 degree and the inclination of the base layer lines follows the following inclination pattern (–30, +30, –30, +30) [ps], [gif] Figure 18. Multi-ring moiré superposition image, where the inclination of moiré lines is of 0 degree and the inclination of the base layer lines follows the following inclination pattern (+30, –30, +30) Figure 18 is the counterpart of Figure 16. In both figures the pattern of variable ring widths is computed by equation (5.4). In contrast to Figure 16 the base layer lines of Figure 18 are not straight. The overall inclination pattern of the base layer across all rings follows the following sequence of inclination degrees . The revealing layer line inclinations are computed according to equation (6.2) so as the superposition image forms the same straight moiré line shape as in Figure 16. Inclined and curved layer patterns help in maintaining a uniform fine granularity across the surface of the disk. When the density of radial layer lines is sparse, the granularity can be refined by increasing the layer inclination degree. In Figure 14, Figure 16, Figure 17, and Figure 18 we assemble the base layer and revealing layer patterns from rings rotated according equation (5.3), such that in the superposition image, the moiré fringes are aligned along the angle zero. Equation (5.3) does not hold for cases when the moiré fringes themselves are curved. In this section we introduce multi-ring patterns with curved moiré shapes. The curved moiré fringes of individual rings must join into a continuous moiré shape across the multi-ring superposition pattern. The angle of equation (5.3) for every successive ring must be additionally adjusted by the angular shift gained by the moiré curve while traversing the preceding rings. Let be the inclination of the moiré line as a function of the radius r. Let and be the inner and outer radiuses of the j-th ring. According to equation (3.6) the angular gain of the moiré curve within the j-th ring is expressed as follows: The aggregate angular gain up to the j-th ring is computed as follows: Equation (5.3) must be rewritten so as to consider also the adjustment brought by equation (6.4): With the angular adjustments of ring patterns computed by equation (6.5), we can create a continuous moiré curve jointly lying across all rings of the pattern. Figure 19 shows a serpentine shaped moiré curve. There are 14 rings of equal width. The acceleration factor k is equal to 30. The number of moiré spots increments starting from 1, for the inner ring, through 14 for the most outer ring. The base layer line inclination pattern corresponds to the following sequence of angles (–80, +10, –10, +10, –30). The revealing layer line inclination pattern is computed according to equation (6.1) so as to ensure the following moiré inclination pattern (+30, –30, +30, –30, +30). A small part of the revealing layer is cut away exposing the uncovered part of the base layer pattern. Figure 20 shows a serpentine-shaped moiré curve in a multi-ring moiré with a variable ring width pattern of Figure 16 described by equation (5.4). There are 14 rings; the acceleration factor is equal to 30. The base layer inclination pattern is (–80, 5, 0, –5, –80), the moiré inclination pattern is (30, –30, 30, –30, 30); the revealing layer inclination pattern is computed with equation (6.1). Figure 19. Multi-ring moiré with a continuous serpentine-shaped moiré curve Many basic measurement instruments comprising a mechanical scale and a mechanical pointer have often their developed versions with a supplementary sub-graduated scale and an auxiliary mechanical pointer which moves faster and aims at the precision increase. A mechanical gearing system is used for a fast movement of the auxiliary pointer synchronously with the main pointer. We developed layer patterns forming optical moiré shapes suitable for the auxiliary fast indicator. Mechanical transmission systems are not required. Moiré shapes can be obtained by superposition of transparent layers carrying correlated opaque patterns. The following points are important: (a) the moiré shapes must be sharp, (b) highly periodic moiré shapes cannot be used for indication, (c) the periodicity of moiré shapes must be very long corresponding to the visible window of the superposition image, such that one and only one shape is visible at a time, e.g. in circular moiré the period of moiré shapes must be equal to 360 degree; (d) the optical speedup of the mechanical movement must be linear; (e) the said above must be valid for the full range of mechanical movements of the main pointer putting into motion the revealing layer, e.g. in circular patterns for the full range of 360 degree rotation of the revealing layer. Sharp moiré shapes are easily formed in well known simple periodic moiré patterns; however their periodicity is very high and cannot be used for indication. Long periods, such as 360 degrees for circular moiré, can be obtained with simple moiré patterns; however the moiré shape becomes blurred and not acceptable for indication. The known random line moiré offers completely aperiodic shapes without the required long periodicity. Additionally, their shapes are noisy compared with their periodic counterparts. We introduced multi-stripe and multi-ring moiré patterns offering very long periods suitable for measurement purposes and forming moiré shapes as sharp as in highly periodical patterns. We developed equations for producing straight and curved auxiliary moiré pointers across multi-ring moiré patterns. We can obtain a moiré shape of any desired curve that can be represented by a continuous function. Our equations help to obtain the desired moiré shape for different base layer patterns by finding the matching revealing layer pattern. In our model, the choice of the shape of the moiré fringe has no impact on the dynamic properties of the auxiliary moiré pointer. We preserve the speedup formulas in their simplest form (2.1), (2.2), and (2.3) for linear movements and (3.2), (3.3), (3.5), and (3.4) for rotations regardless the inclination patterns of layers and moiré shapes. [Gabrielyan07b] Emin Gabrielyan, “Fast optical Indicator created with multi-ring moiré Patterns”, Switzernet research reports, 4 August 2007, http://switzernet.com/people/emin-gabrielyan/070804-moire-rings [Hutley99] M.C. Hutley and R.F. Stevens, “Optical inspection of arrays and periodic structures using moire magnification”, IEE Colloquium: Microengineering in Optics and Optoelectronics, No. 187, p. 8, 16 November 1999 [Nishijima64a] Y. Nishijima and G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements”, Journal of the Optical Society of America, Vol. 54, No. 1, pp. 1-5, January 1964 [pdf] [Oster63a] G. Oster and Y. Nishijima, “Moiré patterns”, Scientific American, Vol. 208, pp. 54-63, May 1963 Table of figures

Ask Professor Puzzler Do you have a question you would like to ask Professor Puzzler? Click here to ask your question! Here's a common question among math teachers and students (and math dabblers who just like to raise people's hackles!): "If you see 3/2x, how do you interpret it? Is it 3 divided by 2x? Or 3 divided by 2, times x? Order of operations says we do division and multiplication left-to-right, which leads to the second answer. However, if you look at the slash as a division symbol, it appears to be the other way: 3 is the numerator and 2x is the denominator." The correct answer to this question is: it's neither. That's right. It's neither 3/(2x) nor 3/2 times x. So no matter which way you were arguing, you're wrong. Let me explain. Whenever you come across something like this: 3/2, the standard reading is not "three divided by two." You read it as "three over two," (this is considered to be the proper designation for the slash symbol when used in this context). This lends credence to the notion that the slash is being used as a fraction bar, and therefore, our example should be read as a fraction: 3/(2x). But did you know that there are specific rules for how you write fractions using standard typographic practices? First, you are expected to use a specific slash symbol, which is not your standard "forward slash" on your keyboard - it's a unicode symbol called "fraction slash." The fraction slash is designed with minimal kerning (space between characters), and there's a very good reason for this. There's another typographical practice we must follow: we superscript the numerator and subscript the denominator. The superscripts and subscripts, combined with the minimal kerning, result in the numerator being above the slash, and the denominator below. Thus, we would either write: 3/2x or 3/2x, and now you can see that proper typographic practices makes it clear which way we intended it to be interpreted. In other words, 3/2x is actually just a typographical error, and not a real mathematical expression. It's the result of someone being lazy. (Don't worry, I've done it too!). With the sophisticated word processors we have these days, with powerful equation editors, there's no longer any excuse for any mathematician to type 3/2x. In fact, with equation editors, you can get expressions that appear much nicer than the ones that you create with superscripts and subscripts. Of course, there is one place where this typographical error still shows up: calculators. Many calculators are not designed for proper typographical display of fractions. So what do we do? We do one of the following: - Figure out which way your particular calculator handles this expression, and always do it that way. - The safer approach: When dealing with a calculator, always clarify your meaning by including parentheses. Once you've settled on one of these practices, it's time to accept the fact that you haven't been arguing about some standard of mathematics, but about typography. It's now time to do some real math, and leave behind the arguments about typographical quirks! Sixth grader Elise asks, "I don't get BODMAS. Can you help me?" Well, Elise, this is one of my favorite questions, and I get asked this a lot. But I've never written it up on the "Ask Professor Puzzler" blog, so here we go! Before I get started explaining BODMAS, I need to mention that in different parts of the world, this "rule" is known by different names. You call it BODMAS, but some people call it PEMDAS. So when I explain what it means, in parentheses I'll explain what it means to people who call it something different. The six letters each stand for something you can do to combine numbers in mathematics: B = Brackets (P = Parentheses) O = Order (E = Exponents) D = Division M = Multiplication A = Addition S = Subtraction These six letters indicate the order in which you do the operations in a mathematical expression. For example, if you see the following: 2 - (3 + 2), You notice that "3 + 2" is in brackets (parentheses), so you do that FIRST: 3 + 2 = 5 Now you have 2 - 5 = -3 Why does it matter which order you do things? It matters because you would get a different answer if you did the subtraction first: 2 - 3 = -1 -1 + 2 = 1 Uh oh! One way we get -3, and the other way we get 1! Here's another example. Suppose you have 1 + 2^3 (1 + 2 cubed). If you did that from left to right, you would add 1 + 2 and get 3. Then you would cube that and get 27. But what you're SUPPOSED to do, is evaluate the exponent first: 2 cubed is 8. Then you add 1 + 8, and get 9. It all depends on what order you do things, so you have to get the order right! You see, we have to have an order of operations, or nobody would ever calculate expressions the same way. Order of Operations is a rule that helps to make sure EVERYONE evaluates the same expression in exactly the same way. If we didn't have order of operations, people would get different answers for the same problem, and that would be horrible - nothing would ever get done, and none of our technology would work right because teams of engineers would always be fighting over how to evaluate the equations and formulas they work with, and if they didn't use proper order of operations, not only would things not work right, you could end up with some pretty horrible catastrophes (imagine engineers using heat formulas in nuclear reactors or power plants, and not calculating correctly how much cooling they need!). Now here's the tricky part (and I've even had emails from MATH TEACHERS who don't understand this part!): you DON'T do all your multiplication before all your division, and you DON'T do all your addition before all your subtraction. Multiplication and division are on the same order of priority, and addition and subtraction are on the same order of priority. If a problem has both multiplication and division operations, you do them from left to right. If a problem has both addition and subtraction, you do them from left to right as well. So really, it should be written BO (DM) (AS)* to remind you that division and multiplication go together, and addition and subtraction go together. * Or PE (MD) (AS) If you want some practice using BODMAS (PEMDAS), you can try our One To Ten Game, which challenges you to put toghether expressions that add up to all the numbers from one to ten. We have a new math 6th grade series where Order of Operations has changed and PEMDAS is no longer the standard. Have you encountered this yet? We have Glenco, 6th grade. Multiplication and division still come before addition and substranction, however, now it works left to right whereas before it was multiplication before division. In some areas of the world they use a different acronym (such as BODMAS or BEDMAS), but these are still the same thing as PEMDAS (Please Excuse My Dear Aunt Sally). Believe it or not, the PEMDAS order of operations is not only still correct, but it's always been what you just described. The acronym PEMDAS can be deceptive, if it’s not taught correctly. How it should be taught is: P: Parenthesis first E: Exponents next MD: Multiplication and Division next AS: Addition and Subtraction last Notice that the M and D are grouped together, as are the A and S. This is because Multiplication and Division are at the same priority level, and should be done in left to right order. Likewise, Addition and Subtraction are at the same priority level, and should be done in left to right order. Unfortunately, many teachers don’t realize this, and teach that all multiplication is done before all division, and all addition is done before all subtraction. I was taught that way all through elementary school, and it wasn’t until I hit Jr/Sr high that I found out that Multiplication and Division are at the same priority level, as are Addition and Subtraction. If you are looking for a game that forces students to think through Order of Operations, here's a game I created several years ago: One To Ten.

BACKGROUND OF THE INVENTION 1. Field of the Invention This invention relates to the field of bandgap voltage reference cells, and particularly to bandgap reference cells having a high transconductance. 2. Description of the Related Art A basic bandgap voltage reference cell is shown in FIG. 1. Two bipolar transistors Qa and Qb are driven by the output of an operational amplifier 14, with their collectors connected to the op amp's non-inverting and inverting inputs, respectively, and to a supply voltage V+ through respective resistors 16 and 18. A resistor Ra is connected between the transistors' respective emitters, and a "tail" resistor Rb is connected between the emitter of Qb and circuit common. Qa is fabricated with an emitter area larger than that of Qb (by a ratio of 8-to-1 in FIG. 1). The op amp adjusts the transistors' base voltage until the voltages at its inverting and non-inverting inputs are equal. This occurs when the two collector currents match, which in this example happens when the emitter current densities are in the ratio of 8-to-1. This arrangement produces a voltage across Rb that is proportional-to-absolute temperature (PTAT), which can be used to compensate the complementary-to-absolute-voltage (CTAT) characteristic of the base-emitter voltage of Qb. Setting OUT equal to the bandgap voltage of silicon provides the proper compensation, and thereby produces a temperature invariant output voltage. The transconductance gm of the circuit of FIG. 1 is defined as the change in the difference in the transistors' collector currents divided by the change in their base-emitter voltage. Because the difference in collector currents cannot exceed the change in current through Rb, the transconductance is capped at 1/Rb, but because a perturbation causes both collector currents to change in the same direction, the maximum attainable gm is actually less than 1/Rb. This bandgap reference cell and its characteristics are discussed in detail in A. Paul Brokaw's "A Simple Three-Terminal IC Bandgap Reference", IEEE Journal of Solid-State Circuits, Vol. SC-9, No. 6(1974). Another bandgap reference cell is shown in FIG. 2, made from two transistors pairs connected in a "crossed-quad" configuration. A first pair of transistors Qc and Qd are connected in series with a second pair of transistors Qe and Qf, respectively, with the bases of Qe and Qf connected to the collectors of Qf and Qe, respectively. Transistors Qc and Qd have unequal emitter areas, as do transistors Qe and Qf. A resistor Rc is connected between the emitters of Qe and Qf, and a tail resistor Rd is connected between the emitter of Qf and circuit common. The collectors of Qc and Qd are connected to the inputs of an amplifier 20. The amplifier's output drives a pass transistor Qf to produce a regulated output OUT, which is fed back to Qc 's and Qd 's common bases. A PTAT voltage appears at the junction between Rc and Rd ; when the resistors are properly chosen, the PTAT voltage compensates for the base-emitter voltages of Qf and Qd to produce a temperature invariant voltage equal to twice the bandgap voltage at OUT. Achieving an output voltage greater that is a non-integer multiple of the bandgap voltage is typically provided by adding a voltage divider 22 between OUT and the common base connection, as shown in FIG. 2. The divider imposes a voltage drop between the output and the common base connection, but assuming that amplifier 20 has sufficient gain, it will continue to balance the collector currents and the output will be stabilized at a higher voltage. The transconductance of the circuit of FIG. 2 is somewhat better than that of FIG. 1. When the cell is at equilibrium (i.e., when the collector currents are balanced), a PTAT current flows in Rc which is determined solely by the emitter area ratios and the value of Rc ; i.e., essentially independent of the current on the right side of the crossed-quad. With the left side current fixed, when the cell's output is disturbed, nearly all of the resulting change in current goes through the right side of the cell (Qd and Qf), with the current through the left side (Qc and Qe) essentially unchanged. Thus, all of the change in current goes through Rd, and the cell's transconductance closely approaches 1/Rd. Because of the relatively low transconductance of the bandgap cells in FIGS. 1 and 2, the voltage applied to the common bases (of Qa and Qb in FIG. 1; Qc and Qd in FIG. 2) must depart substantially from the voltage which balances the currents if a large difference in collector currents is needed. This is usually accommodated by connecting a high gain amplifier across the collectors, to provide a differential-to-single ended conversion as well as the voltage gain necessary to return to equilibrium; this function is represented by amplifier 20 FIG. 2. Disadvantages are found in the circuits of FIGS. 1 and 2, particularly when low power consumption is important, as with a battery-powered regulator. The power consumed by amplifier 20 will hasten the discharge of a battery used to provide the circuit's supply voltage, as will the energy lost in resistive divider 22. Use of a resistive divider 22 is also troublesome if the regulator is employed, for example, as a battery charger, with a battery to be charged connected to OUT. When the regulator is inactive or unable to provide the necessary charging current, the presence of a divider actually provides a discharge path for the battery, shortening its life. SUMMARY OF THE INVENTION A novel voltage reference cell is presented which has a very high transconductance, producing a large change in output current for a very small change in input voltage near a settable equilibrium point and thereby dispensing with the need for a high gain amplifier. The cell can be configured to set the equilibrium point equal to two bandgap voltages, or to non-integer multiples of the bandgap voltage without the use of a resistive divider. Eliminating the amplifier and resistive divider components of prior art designs reduces the reference cell's component count, as well as its power consumption. The core of the voltage reference cell is made from a first and second pair of bipolar transistors nominally arranged in a crossed-quad configuration, with the bases of the first pair connected together at an input node. At least one of the transistor pairs have unequal emitter areas. In contrast with a standard crossed-quad configuration, however, a first resistor is interposed between one of the first pair transistors and the base of one of the second pair transistors, at least one of which has a larger emitter area than its pair, with a second resistor connected to the emitter of the second pair transistor on the opposite side of quad from the first resistor. A voltage applied to the input node causes a current to flow through the cell from the input node to the common point. For input voltages below an "equilibrium" point, the unequal emitter areas force the voltages at the bases of the two second pair transistors to be unequal, which causes most of the current to flow down one side of the quad. As the input voltage increases toward the equilibrium point, the voltage drop across the first resistor increases and the inequality between the second pair transistors' base voltages gets smaller. The relationship between the two base voltages reverses as the equilibrium point is exceeded, causing the cell current to be abruptly "switched" from one side of the quad to the other. The cell's output is taken at the collectors of the first pair of transistors, with nearly all of the cell current switching from one collector to the other at the equilibrium voltage. In prior art cells, a change in current was largely reflected on only one side of the cell. Here, a change in cell current at the equilibrium point causes the current on the two sides to move in opposite directions, with the movement equal to the nearly the entire cell current. This large change in current induced by a very small change in input voltage provides the cell a very high transconductance. A maximum transconductance is obtained when the first and second resistors are equal. However, by simply making the value of one of the resistors greater than the other, additional options are presented to a designer: making the second resistor value greater than the first provides a somewhat lower gm, which might be needed to improve loop stability, for example. Making the first resistor greater than the second creates a loop gain greater than one, which introduces some hysteresis around the equilibrium point that may be useful in regenerative applications such as a comparator. The equilibrium point is established at a voltage dictated by the emitter area ratios between the quad's transistors. When the input voltage is such that the sum of the voltage drops across the resistors equals the voltage set by the emitter area ratios, the cell current switches sides. The cell thus carries a proportional-to-absolute-temperature (PTAT) current at the equilibrium point, which can be used to drive a pass transistor or an amplifier, for example. With the addition of a properly chosen tail resistor, the cell can produce an output voltage equal to two bandgap voltages. The cell can also generate output voltages that are higher, non-integer multiples of the bandgap voltage without the use of a resistive divider. The tail resistor is split into two resistors, with the junction between them connected, via another resistor, to a transistor having its base connected to the input node. These components are arranged so that a temperature invariant current is delivered to the junction point, which offsets the equilibrium point to a higher, temperature stable voltage. Further features and advantages of the invention will be apparent to those skilled in the art from the following detailed description, taken together with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS FIGS. 1 and 2 are schematic diagrams of prior art bandgap voltage reference cells. FIG. 3 is a schematic diagram of a high transconductance voltage reference cell per the present invention. FIG. 4a is a schematic diagram of the novel cell having an equilibrium voltage equal to twice the bandgap voltage, and a table illustrating various obtainable loop gains. FIG. 4b is a schematic diagram of the novel cell configured as a comparator. FIG. 5 is a schematic diagram of the novel cell as it might be used in a battery charger application. DETAILED DESCRIPTION OF THE INVENTION A high transconductance voltage reference cell per the present invention is shown in FIG. 3. The cell includes four bipolar transistors Q1-Q4 connected in a crossed-quad configuration. The bases of a first pair of transistors Q1 and Q2 are connected together and form an input node IN, and their respective collectors are connected to a current source 100, typically implemented with a current mirror, arranged to provide balanced currents to Q1 and Q2. A second pair of transistors Q3 and Q4 have their respective bases cross-coupled to each other's collectors, with Q3's base connected to Q4's collector at a node 102, and Q4's base connected to Q3's collector at a node 104. The transistors making up at least one of the pairs must have unequal emitter areas; in the exemplary circuit of FIG. 3, Q1 has an emitter area 4 times that of Q2. The collectors of Q3 and Q4 are connected to the emitters of Q1 and Q2, respectively, with a resistor R1 interposed between the emitter of Q1 and node 104. Another resistor R2 is connected between the emitter of Q4 and a circuit common point 106, which is also connected to the emitter of Q3. When an input voltage greater than two base-emitter voltages is applied at IN, the path from IN to common point 106 will be forward-biased and a "cell" current will flow between them. If the available current is small, the voltage drop across R1 and R2 must also be small, so that the distribution of cell current in transistors Q1-Q4 is controlled by their respective emitter areas. Due to its larger emitter area, Q1's base-emitter voltage (Vbe1) is lower than that of Q2 (Vbe2) at equal currents, which forces node 102 at the base of Q3 to be lower than node 104 at the base of Q4. This makes the voltage applied to Q4 higher than that applied to Q3, making the collector current of Q4 greater than that of Q3. The imbalance of these currents increases the voltage between nodes 102 and 104, which further unbalances the currents. As a result, the current in the two right hand transistors Q2 and Q4 rises to take most of the cell current, with the collector current of Q15 carrying little more than the base current of Q4. In this state, most of the cell current is delivered to the output terminal OUT, where it is connected to drive a load represented by a resistor Rload which can be, for example, a pass transistor or an amplifier. Summing the voltages between IN and common point 106 (and neglecting base currents): V.sub.be3 +V.sub.be2 =V.sub.be1 +i.sub.1 R1+V.sub.be4 +i.sub.2 R2(Eq. 1) where Vbex refers to the base-emitter voltage of Qx and iy refers to the current in Ry. As the available cell current increases with an increasing input voltage, so will the current in Q1. At some particular input voltage, the currents in Q1 and Q2 become equal. In this case (neglecting base currents), the current in Q1 is the same as the current in Q3, and the current in Q2 is the same as the current in Q4. With the same currents in differently sized transistors, Vbe3 is given as follows: V.sub.be3 =V.sub.be1 +(kT/q)ln4 where "4" is the ratio of emitter areas between Q1 and Q3. For similarly sized transistors Q2 and Q4, Vbe2 and Vbe4 will be nearly equal. Substituting these results into Equation (1) provide: V.sub.be1 +(kT/q)ln4+V.sub.be4 =V.sub.be1 +i.sub.1 R1+V.sub.be4 +i.sub.2 R2 (kT/q)ln4=i.sub.1 R1+i.sub.2 R2 (Eq. 2) Thus, when an input voltage is applied to IN so that the condition of Eq. 2 is met, the current in the left side of the cell (Q1 and Q3) will equal the current in the right side of the cell (Q2 and Q4). The input voltage which satisfies Eq. 2 is the cell's "equilibrium" voltage Veq. For input voltages below Veq, most of the cell current flows through Q2 and thereby pulls down on OUT, in the manner and for the reasons described above. However, when the input voltage exceeds Veq, most of the cell current abruptly switches sides and flows through Q1 to the current source 100, causing it to carry away any current from Q2 and the drive to the load connected to OUT is reduced to zero. At the equilibrium voltage, the current through Q1 is just enough to make the voltage drop across R1 equal Q1's (kT/q)ln 4 difference in Vbe, which makes the voltages at nodes 104 and 102 equal. Above Veq, the voltage drop across R1 is too large to permit balance, while below Veq, the voltage drop is too small. When i1 R1 exceeds Q1's (kT/q)ln 4 difference in Vbe, the relationship between nodes 104 and 102 reverses--node 104 becomes lower than node 102--causing most of the cell current to flow in Q1. Conversely, when the cell current is too low, node 102 is low with respect to node 104, so that most of the current flows through Q2. This flip-flopping of nearly all of the current from one side of the cell to the other at the equilibrium voltage gives the novel reference cell a very high transconductance. Because the currents are balanced at only one voltage, the transconductance is theoretically infinite: an infinitely small change in input voltage causes all of the current to switch sides. The gm is actually limited by base currents, but it is nevertheless very high. The new cell functions much differently than older designs: as described above, as input node voltage increased, the current on one side of a prior art cell would remain at a fixed value determined by emitter area ratios, with changes in cell current forced to appear on the opposite side. This inherently limited the achievable Δi and thus the transconductance. The novel cell functions by having nearly all of the current flow on one side of the quad, increasing beyond the limit imposed by the emitter area ratios of the prior art all the way up to the equilibrium voltage, at which point nearly all the cell current switches to the other side. The transconductance offered by the present invention is in sharp contrast to the relatively low gm of the prior art cells discussed above, which were limited to no more than the reciprocal of their tail resistor value. From Eq. 2, it is seen that at the equilibrium point, the cell current is PTAT. This PTAT current can be used to make or detect other kinds of bandgap and non-bandgap voltages or currents with, for example, a non-zero temperature coefficient. An embodiment of the present invention for which the equilibrium voltage is equal to two bandgap voltages is shown in FIG. 4a. Though the invention only requires that one of the quad pairs have unequal emitter areas, it is convenient for both pairs to be similarly constituted, and the second transistor pair in FIG. 4a now consists of Q3 and a multi-emitter transistor Q5. Vbe2 is now given by: V.sub.be2 =V.sub.be5 +(kT/q)ln4 and the condition at which equilibrium is reached has been raised, and is given by: (kT/q)ln16=i.sub.1 R1+i.sub.2 R2 (Eq. 3) A tail resistor R3 has been connected between node 106 and circuit common in order to provide the double bandgap voltage. If we make R1=R2=Rtotal, then: R.sub.total (i.sub.1 +i.sub.2)=(kT/q)ln16, and neglecting Q3's base current, i1 +i2 is equal to i3, the total current in R3, so that: i.sub.3 =((kT/q)ln16)/R.sub.total (Eq. 4) At the equilibrium point, the current in R3, as well as in the quad transistors, is PTAT. If R3 is properly chosen, the PTAT voltage at node 106 compensates the two base-emitter junction voltages of Q3 and Q2 and yields a double bandgap voltage at the base of Q2, identified as a node 108. Current source 100 is preferably implemented with a dual collector transistor Qs, connected as a current mirror: one of Qs 's collectors 110 is connected to its base and to the collector of Q1; current through Q1 is mirrored to Qs 's other collector 112, which is connected to the collector of Q2. The base of a pass transistor Q6 is also connected to the collector of Q2. Q6 presents a relatively low impedance to Q2, and supplies whatever current it may need. Q6 together with the novel reference cell form a regulator, with Q6's collector serving as the regulator's output Vout. Q6's collector is connected to node 108 at the base of Q2. The total current available to pull down on Q6's base is determined by the voltage across R3, which rises with Vout. This results in a "fold-back" V/I output characteristic. When the cell current exceeds the value given by Eq. 4, the circuit abruptly swings through its equilibrium condition, with the current that was flowing through the Q2/Q5 side of the quad now flowing through the Q1/Q3 side. The Q1 current is mirrored to its collector 112, reducing the drive to Q6 to near zero. Since the loop is closed to node 108 from the output of Q6, the output current will remain high as Vout approaches the equilibrium point, and then abruptly drops to near zero as the equilibrium voltage is reached. If the equilibrium voltage has been arranged to be at twice the bandgap voltage as described above, the point at which the output current drops to zero is made temperature stable. Because the transconductance of the new cell is so high, the high gain amplifier required in the prior art designs discussed above can be eliminated. Output pass transistor Q6 can be driven directly and still provide relatively good regulation. Eliminating the amplifier lowers the regulator's power consumption, as well as its component count. Essential to the operation of the invention is the way in which the relationship between the voltages at nodes 102 and 104 reverses as the input node voltage increases. The resistors and the larger emitter transistors must be placed to insure this functioning. If the first transistor pair has an unequal emitter ratio, R1 must be placed in series with the transistor having the larger emitter. The smaller emitter transistor will have a larger Vbe, making the node below its emitter lower than the node below R1 for lower input voltages. The voltage drop across R1, however, forces the relationship between the nodes to reverse when it carries a particular current--i.e, the cell current at the equilibrium voltage. Similarly, if only the second transistor pair have an unequal emitter ratio, R2 should be placed in series with the transistor having the larger emitter. The larger emitter causes the transistor's collector to be pulled down harder than its pair is, unbalancing the voltages at their bases. The larger transistor's Vbe is reduced as the current through R2 increases, however, increasing the voltage of the node at its collector, with the relationship between the base voltages reversing at the equilibrium voltage. If both pairs have unequal emitter ratios, the larger emitter transistors should be placed on opposite sides of the quad, as shown in FIG. 4a. R1 and R2 should also be placed on opposite sides of the quad. The cell's transconductance is highest when R1=R2, which, because it is in a closed loop, provides a loop gain that reaches exactly +1 at the equilibrium point. Making R2 greater than R1 lowers the cell's gm and reduces the loop gain to less than +1, diminishing the abruptness with which the cell current switches from one side to the other. This might be done when a more controlled gm is desired--to frequency stabilize a closed loop system, for example. Making R1 greater than R2 makes the loop gain greater than +1. Here, there is no point at which the currents are equally distributed. For this condition, the current will flow on the right side below and even at the equilibrium point. However, as input node 108 continues to rise, the current will abruptly switch to the other side, where it will stay until node 108 falls below the equilibrium point by some finite amount. This would be useful in regenerative applications; for example, in using the cell to provide a comparator with hysteresis. Thus, as illustrated in the table shown in FIG. 4a, the invention can provide a very high gm (though with poor loop stability), a moderately high gm in a better controlled loop, or a gm providing a loop gain >1, useful for regenerative applications, by simply adjusting the respective values of R1 and R2. A reference cell configured as a comparator is shown in FIG. 4b. The circuit is very similar to that of FIG. 4a, except that the left and right sides of the quad are reversed, with the collector of Q1 now connected to the base of transistor Q6, and a resistor Rcomp connected between the comparator's output, i.e., the collector of Q6, and circuit common. The common bases of Q1 and Q2 form an input terminal IN. When a voltage applied to IN is below the equilibrium voltage, most of the cell current flows through Q2. This current is mirrored to the base of Q6, reducing the drive to Q6 to nearly zero. Resistor Rcomp pulls the output low in this state. When the input exceeds the equilibrium voltage, the cell current switches to the Q1 side of the quad, driving Q6 and producing an output at OUT. R1 should be made greater than R2 to introduce some hysteresis, as described above. In some applications, an equilibrium voltage that is greater than two bandgap voltages may be desired. This could be obtained with a voltage divider connected between the collector of Q6 and circuit common (referring back to FIG. 4a), with the divider tap connected to node 108. Vout is scaled to a higher voltage while the loop continues to come to balance when node 108 is at two bandgaps. However, for reasons noted above, the use of a resistive divider may be undesirable. A regulator which addresses these problems and is built around the novel bandgap reference cell is shown in FIG. 5. The need to provide an output greater than two bandgaps is met with the addition of a transistor Q7 and a resistor R4. The base of Q7 is connected to input node 108 along with the bases of Q1 and Q2, and its emitter is connected to the bottom of tail resistor R3 at a node 120 via resistor R4. A resistor R5 is interposed between node 120 and circuit common. When the regulator is in regulation, the voltage from node 108 to node 106 is equal to two base-emitter junctions voltages. Assuming some current in Q7, its emitter will be below node 108 by one base-emitter voltage, or one base-emitter voltage above node 106. R3 and R5 are selected such that, at equilibrium, the PTAT voltage across R3+R5 compensates two base-emitter voltages, so that approximately half of the PTAT voltage compensates a single base-emitter voltage. R3 and R5 are selected so that approximately half the PTAT voltage is at node 120; this compensates Q7 and makes the voltage from the emitter of Q7 to node 120 temperature invariant. Resistor R4 spans this voltage, so that its current is also temperature invariant. R4's temperature invariant current (at equilibrium) flows in R5, adding to the voltage already present and compensating the quad. Since this additional voltage is constant, it simply offsets the equilibrium point to a higher, temperature stable voltage at node 108. This higher voltage can be adjusted by adjusting R4. Alternative arrangements for establishing a higher equilibrium voltage are possible. For example, R4 could be connected to node 106 instead of node 120, causing a complementary-to-absolute-temperature (CTAT) voltage to be added to the output. The resulting temperature coefficient could be compensated by adding some resistance in the R3, R5 path to increase the PTAT voltage component, and the values of R4 and R3+R5 could be adjusted together to set the equilibrium voltage at a value higher than two bandgap voltages. Connecting R4 to node 120 is preferred, however, to reduce the interaction between R4 and R3+R5 and thereby facilitate trimming. The regulator shown in FIG. 5 is advantageously used as a battery charger, to charge a battery 130 connected to Vout. The circuit shown charges the battery at a relatively high rate if its voltage is below full charge, without exceeding some maximum value when the battery is at a very low voltage. The battery charger is itself powered by a battery with a voltage Vbatt. An inverter is made from transistors Q8 and Q9 and is driven by a signal Vmon which monitors the value of Vbatt with respect to Vout ; Vmon is high when Vbatt is sufficiently greater than Vout. The output of the inverter controls a transistor Q10 connected between Vout and node 108. In normal operation, Vbatt exceeds Vout and Vmon is high. The inverter turns on Q10, connecting Vout to node 108. However, if Vbatt becomes discharged, or is removed from the circuit, Vmon goes low, turning off Q10 and disconnecting the load battery 130 from node 108. This prevents inadvertent discharge of the load battery 130. As the node 108 voltage rises, the current that results in R3 and R5 flows mostly through Q2 and Q5 to the base of Q6. A maximum charging current is established by controlling the values of R3 and R5. Voltage Vout rises as the battery 130 approaches a fully charged condition; when Vout reaches the equilibrium voltage, the cell current switches from the right side to the left side, and the charging current to the battery is reduced to a low "maintenance" level. The load battery 130 presents a low impedance when near full charge, so that loop stability is unlikely to be a problem. Thus, for this battery charger application, R1 and R2 are preferably made equal to provide the highest possible transconductance. If a higher impedance load were being driven, a lower transconductance may be preferable, which is easily achieved by making R1 smaller than R2. Though the novel high transconductance reference cell has been described and shown as made from npn bipolar transistors, it is obvious that it can be similarly constructed of pnp transistors (with a corresponding inversion of supply voltage polarity and current flow direction), with no difference in the invention's function or performance advantages. While particular embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Accordingly, it is intended that the invention be limited only in terms of the appended claims.

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Uncertainty means that more things can happen than will happen. Therefore, whenever managers are given a cash-flow forecast, they try to determine what else may happen and the implications of these possible surprise events. This is called sensitivity analysis. Put yourself in the well-heeled shoes of the financial manager of the Otobai Company in Osaka. You are considering the introduction of a high-performance electric scooter for city use. Your staff members have prepared the cash-flow forecasts shown in Table 10.1. Since NPV is positive at the 20% opportunity cost of capital, it appears to be worth going ahead, but before you decide, you want to delve into these forecasts and identify the key variables that determine whether the project succeeds or fails. The project requires an initial investment of ¥15 billion in plant and machinery, which will have negligible further value when the project comes to an end. As sales build up in the early and middle years of the project, the company will need to make increasing investments in net working capital, which is recovered in later years. After year 6, the company expects sales to tail off as other companies enter the market, and the company will probably need to reduce the price of the scooter. The cost of goods sold is forecast to be 50% of sales; in addition, there will be fixed costs each year that are unrelated to the level of sales. Taxes at a 40% rate are computed after deducting straight-line depreciation. These seem to be the important things you need to know, but look out for unidentified variables that could affect these estimates. Perhaps there could be patent problems, or perhaps you will need to invest in service stations that will recharge the scooter batteries. The greatest dangers often lie in these unknown unknowns, or “unk-unks,” as scientists call them. Having found no unk-unks (no doubt you will find them later), you conduct a sensitivity analysis with respect to the required investment in plant and working capital and the forecast unit sales, price, and costs. To do this, the marketing and production staffs are asked to give optimistic and pessimistic estimates for each of the underlying variables. These are set out in the second and third columns of Table 10.2. For example, it is possible that sales of scooters could be 25% below forecast, or you may be obliged to cut the price by 15%. The fourth and fifth columns of the table shows what happens to the project’s net present value if the variables are set one at a time to their optimistic and pessimistic values. Your project appears to be by no means a sure thing. The most dangerous variables are cost of goods sold and unit sales. If the cost of goods sold is 70% of sales (and all other variables are as expected), then the project has an NPV of – ¥10.7 billion. If unit sales each year turn out to be 25% less than you forecast (and all other variables are as expected), then the project has an NPV of – ¥5.9 billion. Trendy consultants sometimes use a tornado diagram such as Figure 10.1 to illustrate the results of a sensitivity analysis. The bars at the summit of the tornado show the range of NPV outcome due to uncertainty about the level of sales. At the base of the tornado you can see the more modest effect of uncertainty about investment in working capital and the level of fixed costs.1 1. Value of Information The world is uncertain, and accurate cash-flow forecasts are unattainable. So, if a project has a positive NPV based on your best forecasts, shouldn’t you go ahead with it regardless of the fact that there may be later disappointments? Why spend time and effort focusing on the things that could go wrong? Sensitivity analysis is not a substitute for the NPV rule, but if you know the danger points, you may be able to modify the project or resolve some of the uncertainty before your company undertakes the investment. For example, suppose that the pessimistic value for the cost of goods sold partly reflects the production department’s worry that a particular machine will not work as designed and that the operation will need to be performed by other methods. The chance of this happening is only 1 in 10. But, if it does occur, the extra cost would reduce the NPV of your project by ¥2.5 billion, putting the NPV underwater at +2.02 – 2.50 = – ¥0.48 billion. Suppose that a ¥100 million pretest of the machine would resolve the uncertainty and allow you to clear up the problem. It clearly pays to invest ¥100 million to avoid a 10% probability of a ¥2.5 billion fall in NPV. You are ahead by -0.1 + .10 X 2.5 = ¥0.15 billion. On the other hand, the value of additional information about working capital is small. Because the project is only marginally unprofitable, even under pessimistic assumptions about working capital, you are unlikely to be in trouble if you have misestimated that variable. 2. Limits to Sensitivity Analysis Sensitivity analysis boils down to expressing cash flows in terms of key project variables and then calculating the consequences of misestimating those variables. It forces the manager to identify the crucial determinants of the project’s success and indicates where additional information would be most useful or where design changes may be needed. One drawback to sensitivity analysis is that it always gives somewhat ambiguous results. For example, what exactly does optimistic or pessimistic mean? The marketing department may be interpreting the terms in a different way from the production department. Ten years from now, after hundreds of projects, hindsight may show that the marketing department’s pessimistic limit was exceeded twice as often as that of the production department, but what you may discover 10 years hence is no help now. Of course, you could specify that when you use the terms “pessimistic” and “optimistic,” you mean that there is only a 10% chance that the actual value will prove to be worse than the pessimistic figure or better than the optimistic one. However, it is far from easy to extract a forecaster’s notion of the true probabilities of possible outcomes. Another problem with sensitivity analysis is that the underlying variables are likely to be interrelated. For example, if inflation pushes prices to the upper end of your range, it is quite probable that costs will also be inflated. And if sales are unexpectedly high, you may need to invest more in working capital. Sometimes the analyst can get around these problems by defining underlying variables so that they are roughly independent. For example, it made more sense for Otobai to look at cost of goods sold as a proportion of sales rather than as a dollar value. But you cannot push one-at-a-time sensitivity analysis too far. It is impossible to obtain expected, optimistic, and pessimistic values for total project cash flows from the information in Table 10.2. Sensitivity analysis boils down to expressing cash flows in terms of key project variables and then calculating the consequences of misestimating the variables. It forces the manager to identify the underlying variables, indicates where additional information would be most useful, and helps to expose inappropriate forecasts. 3. Scenario Analysis If the variables are interrelated, it may help to consider some alternative plausible scenarios. For example, perhaps the company economist is worried about the possibility of a sharp rise in world oil prices. The direct effect of this would be to encourage the use of electrically powered transportation. The popularity of hybrid cars after a recent oil price increases leads you to estimate that an immediate 20% rise in the price of oil would enable you to increase unit sales by 10% a year. On the other hand, the economist also believes that higher oil prices would stimulate inflation, which would affect selling prices, costs, and working capital. Table 10.3 shows that this scenario of higher oil prices and higher inflation would on balance help your new venture. Its NPV would increase to ¥6.9 billion. Managers often find such scenario analysis helpful. It allows them to look at different, but consistent, combinations of variables. Forecasters generally prefer to give an estimate of revenues or costs under a particular scenario than to give some absolute optimistic or pessimistic value. 5 thoughts on “Sensitivity and Scenario Analysis” Have you ever thought about publishing an ebook or guest authoring on other websites? I have a blog centered on the same ideas you discuss and would love to have you share some stories/information. I know my readers would appreciate your work. If you’re even remotely interested, feel free to send me an e-mail. Good post but I was wondering if you could write a litte more on this subject? I’d be very thankful if you could elaborate a little bit more. Thank you! These are in fact fantastic ideas in about blogging. You have touched some nice factors here. Any way keep up wrinting. I am truly grateful to the holder of this web page who has shared this enormous article at at this time. You must take part in a contest for the most effective blogs on the web. 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Certain inequalities connected with the golden ratio and the Fibonacci numbers Journal of Applied Mathematics and Computational Mechanics CERTAIN INEQUALITIES CONNECTED WITH THE GOLDEN RATIO AND THE FIBONACCI NUMBERS Marcin Adam, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń[*], Roman Wituła Institute of Mathematics, Silesian University of Technology email@example.com, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org Abstract. In the present paper we give some condensation type inequalities connected with Fibonacci numbers. Certain analytic type inequalities related to the golden ratio are also presented. All results are new and seem to be an original and attractive subject also for future research. Keywords: Fibonacci numbers, golden ratio, Perrin constant The notion of Fibonacci numbers and the golden ratio may be found in many branches of mathematics, including number theory, geometry, algebra, matrix theory, numerical methods, classical analysis, dynamical systems, and even spectral analysis or music (see monographs [1-3], and selective papers [4-7]). Despite such a large spread occurrence of Fibonacci numbers and the golden ratio in mathematics, still some areas of mathematics tend to be poorly represented by these objects. In our opinion, a good example of such a niche (considered also in this paper) is the area of analytic inequalities. We believe this paper opens up a new stage of discoveries, and the inequalities presented here will be classical ones in the considered area of mathematics. The main results of the paper are presented in two sections. In the first one we investigated the condensation type inequalities associated with the Fibonacci numbers. In the second one we discuss several analytic type inequalities related to the golden ratio and Perrin constant. 2. Condensation type inequalities connected with Fibonacci numbers We begin with the following inequality based on basic properties of the Fibonacci numbers. Let us recall that the Fibonacci numbers are defined by the following linear recurrence relation with . As a result of the definition we get . Theorem 1. Let and be a finite sequence of real numbers such that two inequalities are satisfied: Then there is an index such that Proof. We prove this by contradiction. Let us suppose that for each index we have that can be easily shown by induction. Indeed, from (3) we have for the initial step and for the inductive one. From (4), on account of (1) we obtain Next, let us denote where Then the left-hand side of the previous inequality is equal to the following: Now from (2) it follows directly that for all Indeed, and since , there is So finally we obtain which contradicts to (4). This completes the proof. As a direct conclusion of this result we obtain the following generalization: Corollary 2. Let and be a finite sequence of real numbers satisfying condition 1 of the previous theorem. If, additionally, for some then there is an index such that Remark 3. For the inequality (2) see also Chern and Cui paper . 3. Inequalities connected with the golden ratio Let denote the golden ratio, i.e. and let . Theorem 4. The following golden ratio type inequalities hold: for ; the equality sign is attained for only. The function is increasing on interval and decreasing on each of the intervals and , where (see Fig. 1). We note that 2. The function is increasing on each of the intervals and , and decreasing on interval , where . We note that and 3. The function is decreasing on and increasing on . Moreover, we have . 4. Let us set for , and for . Then the function is increasing on each of intervals and , and decreasing on interval . On the other hand, the function is increasing on each of the intervals and , and decreasing on Furthermore we obtain Moreover, if then (see Figs. 2-4) and the minimum of this function is attained in , we have Fig. 1. Plot of the function Fig. 2. Plot of the function Fig. 3. Plot of the function Fig. 4. Plot of the function in the interval (the domain of this one is equal to Proof. We consider the following functions: Computing derivatives of these functions we obtain It is easy to check that the function is decreasing on and is increasing on , so we have for which is equivalent to the inequality for (the equality sign is attained here only for ). By numerical calculations, we proved that the function is increasing on interval and decreasing on intervals and , where . Similarly as , also is decreasing on and is increasing on , so we obtain We have . By numerical calculations we proved that the function is increasing on intervals and , and decreasing on interval . The function is decreasing on ) and increasing on . Hence, function has a local minimum at the point which is equal to . Corollary 5. By item 1, the following inequality holds In equivalent form, we obtain Proof. We have which implies (5) for since and . Corollary 6. By item 2, the following inequality holds Corollary 7. By item 3, the following inequality holds More precisely, the function is decreasing on interval and increasing on interval . Remark 8. Closely connected to the golden ratio is the so-called Perrin constant defined to be the only real zero of the so-called Perrin polynomial (see [9-11]) In relation to inequalities from point 4 of Theorem 4, we are interested in the equivalent of these inequalities for the Perrin constant , i.e. the inequalities of the type Since we have so we are interested when the following system of equations hold: which implies that is a real solution of the following equation Finally, by numerical calculations we get precisely two triplets of real numbers being the solution of system (6): For these solutions we can deduce the following inequalities: A) the first collection of five inequalities for the first triple (a, b, c) of solutions (see Fig. 5) for and where the equality sign is taken only for . is increasing on , . Moreover, is decreasing on two intervals and , and B) for the second triple (a, b, c) of solutions similar inequalities can be obtained, however, due to the volume of the paper, they will be omitted. Fig. 5. Plot of the functions and for the first triple - on the left, and for the second triple - on the right In the paper certain inequalities connected with the golden ratio and the Fibonacci numbers are discussed. We were able to accomplish the intended overall goal of the paper, even with some excess (see in particular the results of point 4 of Theorem 4). Generalization from Remark 8 connected with the Perrin’s polynomial and constant is quite natural and in fact well-defined, but did not completely fulfill our expectations of aesthetic nature. We believe that the research subject matter indicated in this paper is still open and can encourage (especially Fibomaniacs) for active reflection. Dunlop R., The Golden Ratio and Fibonacci Numbers, World Scientific, Singapore 2006. Vajda S., Fibonacci and Lucas Numbers, and the Golden Section Theory and Applications, Dover Publications Inc., New York 2008. Hoggatt V.E., Fibonacci and Lucas Numbers, The Fibonacci Association, Santa Clara 1979. Mongoven C., Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae 2013, 41, 175-192. Wituła R., Słota D., Hetmaniok E., Bridges between different known integer sequences, Annales Mathematicae et Informaticae 2013, 41, 255-263. Słanina P., Generalizations of Fibonacci polynomials and free linear groups, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2015.1031073. Herz-Fischler R., A “very pleasant” theorem, College Mathematics Journal 1993, 24, 4, 318-324. Chern S., Cui A., Fibonacci numbers close to a power of 2, The Fibonacci Quarterly 2014, 52, 4, 344-348. Hetmaniok E., Wituła R., Lorenc P., Pleszczyński M., On an improvement of the numerical application for Cardano’s formulae in Mathematica software (in review). Wituła R., Lorenc P., Różański M., Szweda M., Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Śląskiej, Seria Matematyka Stosowana 2014, 4, 17-34. Dubickas A., Hare K.G., Jankauskas J., There are no two nonreal conjugates of a Pisot number with the same imaginary part, arXiv:1410.1600v1 [math.NT]. [*] Currently, the fourth author, Barbara Smoleń, is a student of mathematics and this paper is a part of her Bachelor's thesis written under the supervision of Prof. Roman Wituła.

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Energy–momentum tensor from the Yang–Mills gradient flow The product of gauge fields generated by the Yang–Mills gradient flow for positive flow times does not exhibit the coincidence-point singularity and a local product is thus independent of the regularization. Such a local product can furthermore be expanded by renormalized local operators at zero flow time with finite coefficients that are governed by renormalization group equations. Using these facts, we derive a formula that relates the small flow-time behavior of certain gauge-invariant local products and the correctly-normalized conserved energy–momentum tensor in the Yang–Mills theory. Our formula provides a possible method to compute the correlation functions of a well-defined energy–momentum tensor by using lattice regularization and Monte Carlo simulation. B01, B31, B32, B38 Although lattice regularization provides a very powerful non-perturbative formulation of field theories, it is unfortunately incompatible with fundamental global symmetries quite often. The most well-known example is chiral symmetry Nielsen:1980rz ; Nielsen:1981xu ; supersymmetry is another infamous example Dondi:1976tx , as is, needless to say, translational invariance. When a regularization is not invariant under a symmetry, it is not straightforward to construct the corresponding Noether current that is conserved and generates the symmetry transformation through Ward–Takahashi (WT) relations. This makes the measurement of physical quantities related to the Noether current in a solid basis very difficult. To solve this problem, one can imagine at least three possible approaches. The first approach is an ideal one: One finds a lattice formulation that realizes (a lattice-modified form of) the desired symmetry. If such a formulation comes to hand, the corresponding Noether current can easily be obtained by the standard Noether method. The best successful example of this sort is the lattice chiral symmetry Kaplan:1992bt ; Neuberger:1997fp ; Hasenfratz:1997ft ; Hasenfratz:1998ri ; Neuberger:1998wv ; Hasenfratz:1998jp ; Luscher:1998pqa ; Niedermayer:1998bi , which can be defined with a lattice Dirac operator that satisfies the Ginsparg–Wilson relation Ginsparg:1981bj . Although this is certainly an ideal approach, it appears that such an ideal formulation does not always come to hand, especially for spacetime symmetries (see, e.g., Ref. Kato:2008sp for a no-go theorem for supersymmetry). The second approach is to construct the Noether current by tuning coefficients in the linear combination of operators that can mix with the Noether current under lattice symmetries.111Here, we assume that fine tuning of bare parameters to the target (symmetric) theory is done. For example, for the energy–momentum tensor—the Noether current associated with the translational invariance and rotational and conformal symmetries Callan:1970ze ; Coleman:1970je —one can construct a conserved lattice energy–momentum tensor by adjusting coefficients in the linear combination of dimension operators Caracciolo:1988hc ; Caracciolo:1989pt 222A somewhat different approach on the basis of the supersymmetry has been given in Refs. Suzuki:2013gi ; Suzuki:2012wx .; the overall normalization of the energy--momentum tensor has to be fixed in some other way.333It might be possible to employ “current algebra” for this, as for the axial current Bochicchio:1985xa . Although this method is in principle sufficient when the energy–momentum tensor is in “isolation”, i.e., when the energy–momentum tensor is separated from other composite operators, as in the on-shell matrix elements, it is not obvious a priori whether one can control the ambiguity of possible higher-dimensional operators that may contribute when the energy–momentum tensor coincides with other composite operators in position space. This implies that it is not obvious whether the energy–momentum tensor constructed in the above method generates correctly-normalized translations (and rotational and conformal transformations) on operators through WT relations. (If the energy–momentum tensor generates correctly-normalized translations, it is ensured Fujikawa:1980rc (see also Sect. 7.3 of Ref. Fujikawa:2004cx ) that the trace or conformal anomaly Crewther:1972kn ; Chanowitz:1972vd is proportional to the renormalization group functions Adler:1976zt ; Nielsen:1977sy ; Collins:1976yq .) The third possible approach is to utilize some ultraviolet (UV) finite quantity. Since such a quantity must be independent of the regularization adopted (in the limit in which the regulator is removed), there emerges a possibility that one can relate the lattice regularization and some other regularization that preserves the desired symmetry. This methodology can be found e.g. in Ref. Luscher:2004fu (see also Ref. Luscher:2010ik ), where an ultraviolet finite representation of the topological susceptibility is derived. Although the derivation of the representation itself relies on a lattice regularization that preserves the chiral symmetry Kaplan:1992bt ; Neuberger:1997fp ; Hasenfratz:1997ft ; Hasenfratz:1998ri ; Neuberger:1998wv ; Hasenfratz:1998jp ; Luscher:1998pqa ; Niedermayer:1998bi , one can use any regularization (e.g., the Wilson fermion Wilson:1975hf ) to compute the representation because it must be independent of the regularization. In the present paper, we consider the above third approach for the energy–momentum tensor, by taking the pure Yang–Mills theory as an example. For this, we utilize the so-called Yang–Mills gradient flow (or the Wilson flow in the context of lattice gauge theory) whose usefulness in lattice gauge theory has recently been revealed Luscher:2010iy ; Luscher:2010we ; Luscher:2011bx ; Borsanyi:2012zs ; Borsanyi:2012zr ; Fodor:2012td ; Fodor:2012qh ; Fritzsch:2013je ; Luscher:2013cpa . A salient feature of the Yang–Mills gradient flow is its robust UV finiteness Luscher:2011bx . More precisely, any product of gauge fields generated by the gradient flow for a positive flow time is UV finite under standard renormalization. Such a product, moreover, does not exhibit any singularities even if some positions of gauge fields coincide. The basic mechanism for this UV finiteness is that the flow equation is a type of the diffusion equation and the evolution operator in the momentum space acts as an UV regulator for . This property of the gradient flow implies that the definition of a local product of gauge fields for positive flow times is independent of the regularization. In our present context, there is a hope of relating quantities obtained by the lattice regularization and the dimensional regularization with which the translational invariance is manifest. As noted in Ref. Luscher:2011bx , on the other hand, a local product of gauge fields for a positive flow time can be expanded by renormalized local operators of the original gauge theory with finite coefficients. Those coefficients satisfy certain renormalization group equations that, combined with the dimensional analysis, provide information on the coefficients as a function of the flow time. Because of the asymptotic freedom, one can then use the perturbation theory to find the asymptotic behavior of the coefficients for small flow times. By using the above properties of the gradient flow, one can obtain a formula that relates the small flow-time behavior of certain gauge-invariant local products and the energy–momentum tensor defined by the dimensional regularization. Since the former can be computed by using the Wilson flow with lattice regularization Luscher:2010iy ; Luscher:2010we ; Luscher:2011bx ; Borsanyi:2012zs ; Borsanyi:2012zr ; Fodor:2012td ; Fodor:2012qh ; Fritzsch:2013je ; Luscher:2013cpa and the latter is conserved and generates correctly-normalized translations on composite operators, our formula provides a possible method to compute the correlation functions of a correctly-normalized conserved energy–momentum tensor by using Monte Carlo simulation. In the present paper, we follow the notational convention of Ref. Luscher:2011bx unless otherwise stated. 2 Yang–Mills theory and the energy–momentum tensor 2.1 The energy–momentum tensor with dimensional regularization In the present paper, we consider the Yang–Mills theory defined in a dimensional Euclidean space. The action is given by from the Yang–Mills field strength and then the mass dimension of the bare gauge coupling is . Assuming that the theory is regularized by the dimensional regularization (for a very nice exposition, see Ref. Collins:1984xc ), one can define the energy–momentum tensor for the system (2.1) simply by (see, e.g., Ref. Freedman:1974gs ) up to terms attributed to the gauge fixing and the Faddeev–Popov ghost fields, which are irrelevant in correlation functions of gauge-invariant operators. Note that the mass dimension of the energy–momentum tensor is . The advantage of dimensional regularization is its translational invariance. Because of this property, the energy–momentum tensor naively constructed from bare quantities, Eq. (2.4), is conserved and generates correctly-normalized translations through a WT relation, where it is understood that the derivative on the right-hand side is acting all positions in a gauge-invariant operator . Used in combination with dimensional counting and gauge invariance, this WT relation implies that the energy–momentum tensor is finite Joglekar:1975jm ; Nielsen:1977sy and thus, in the minimal subtraction (MS) scheme,444Here, we define the renormalized operator by subtracting its vacuum expectation value. In the perturbation theory using dimensional regularization, this subtraction is automatic. The finiteness of the energy–momentum tensor (2.4) provides further useful information on the renormalization of dimension gauge-invariant operators. The gauge coupling renormalisation with dimensional regularization is defined by where is the renormalization scale and is the renormalization factor. In the MS scheme, From the rotational invariance that the dimensional regularization keeps, we see that the operator-renormalization possesses the following structures:555Here again, we define renormalized operators by subtracting their vacuum expectation values. Since the left-hand side is finite for , in the MS scheme in which only pole terms are subtracted, we infer (by considering the cases, and ) that 2.2 Implications of the trace anomaly By Eq. (2.6), this relation is equivalent to i.e., the contraction with the metric and the minimal subtraction, the subtraction of poles, do not commute; this is a peculiar but legitimate property of the dimensional regularization Collins:1984xc . In the MS scheme in which only pole terms are subtracted, this implies and Eq. (2.14) then shows 3 Yang–Mills gradient flow and the small flow-time expansion The Yang–Mills gradient flow defines a dimensional gauge potential along a fictitious time , according to the flow equation where the dimensional field strength and the covariant derivative are defined by respectively. The initial condition for the flow is given by the dimensional gauge potential in the previous section: In Eq. (3.1), the last term is introduced to suppress the evolution of the field along the direction of gauge degrees of freedom. Although this term breaks the gauge symmetry, it does not affect the evolution of any gauge-invariant operators Luscher:2010iy . Note that the mass dimension of the flow time is . Now, from the field strength extended to the dimension (3.2), we define a dimensional analogue of the energy–momentum tensor by Although this is similar in form to the original energy–momentum tensor (2.4), it is not obvious a priori how this dimensional object and Eq. (2.4) are related (or not). To find the relationship between them is the principal task of the present paper. We also use the density operator studied in Ref. Luscher:2010iy : Now, as shown in Ref. Luscher:2011bx , for , any correlation function of is UV finite after standard renormalization in the dimensional Yang–Mills theory. This property holds even for any local products of such as Eqs. (3.5) and (3.6). Also, for small flow times, a local product of can be regarded as a local field in the dimensional sense because the flow equation (3.1) is basically the diffusion equation along the time and the diffusion length in is . These properties allow us to express, as explained in Sect. 8 of Ref. Luscher:2011bx , and as an asymptotic series of dimensional renormalized local operators with finite coefficients. Considering the gauge invariance and the index structure, for , we can write where abbreviated terms are the contributions of operators with a mass dimension higher than or equal to . For Eq. (3.6), we similarly have We note that, when the renormalized gauge coupling is fixed, (3.5) is traceless for , because (3.6) is finite Luscher:2010iy and does not produce a singularity (this explains why there is no number expectation value term in Eq. (3.7)). Thus, considering the trace part of Eq. (3.7), we see that the coefficients and are not independent and are related by, for , because of the trace anomaly (2.15). This expression relates the energy–momentum tensor (2.6) and the short flow-time behavior of gauge-invariant local products defined by the gradient flow. Thus, once the coefficients are known, one can extract the energy–momentum tensor from the behavior of the combination on the right-hand side. 4 Renormalization group equation and the asymptotic formula 4.1 Renormalization group equation for the coefficients We now operate on both sides of Eq. (3.7). Since the left-hand side of Eq. (3.7), i.e., Eq. (3.5), is entirely expressed by bare quantities through the flow equation (3.1) and the initial condition (3.4), the action of (4.1) on the left-hand side identically vanishes. On the right-hand side, this vanishing must hold in each power of . Thus we infer that Similarly, for Eq. (3.8), we have By the standard argument and from the fact that dimensionless quantities can depend on the renormalization scale only through the dimensionless combination , the above renormalization group equations imply that where the dependence on the renormalized gauge coupling and on the renormalization scale has been explicitly written. In these expressions, the running coupling is defined by and we introduce a variable In the one-loop order, the running couping (4.15) is given by where is the parameter in the one-loop level, and the integral appearing in Eqs. (4.12) is In the small flow-time limit , and the running coupling (4.17) becomes very small thanks to the asymptotic freedom. Thus, the right-hand sides of Eqs. (4.11)–(4.14) allow us to compute the small flow-time behavior of the coefficients by using the perturbation theory. 4.2 Lowest-order approximation and the asymptotic formula simply because our energy–momentum tensor (2.4) is proportional to . If we apply the right-hand side of Eq. (4.11) to this expression by substituting Eq. (4.17), however, it depends on while the left-hand side of Eq. (4.11) does not. This shows that should depend on and through a particular combination as (for ) This is the relation that we were seeking: One can obtain the correctly-normalized conserved energy–momentum tensor from the small flow-time behavior of gauge-invariant products given by the Yang–Mills gradient flow. It is interesting to note that the leading behavior is completely independent of the detailed definition of the gradient flow; the structure and coefficients follow solely from the finiteness of the local products and the renormalizability of the Yang–Mills theory. The sub-leading corrections in the asymptotic form, i.e., the coefficients and , depend on the detailed definition of the gradient flow; in the Appendix, we compute the constants and and we have Finally, a possible method to determine the factor in Eq. (4.29), i.e., the flow time in the unit of the one-loop parameter (4.18), for small flow times is to use the expectation value of the density operator, Eq. (3.6). For this quantity, by applying Eqs. (4.13) and (4.17) to the result of the one-loop calculation, Eqs. (2.28) and (2.29) of Ref. Luscher:2010iy (specialized to the pure Yang–Mills theory), we have the asymptotic form, In the present paper, we have derived a formula that relates the short flow-time behavior of some gauge-invariant local products generated by the Yang–Mills gradient flow and the correctly-normalized conserved energy–momentum tensor in the Yang–Mills theory. Our main result is Eq. (4.29). The right-hand side of Eq. (4.29) can be computed by the Wilson flow in lattice gauge theory with appropriate discretizations of operators, Eqs. (3.5) and (3.6) (see, e.g., Refs. Luscher:2010iy ; Borsanyi:2012zs ). Here, the continuum limit must be taken first and then the limit is taken afterwards; otherwise our basic reasoning does not hold. Although the formula (4.29) should be mathematically correct, the practical usefulness of Eq. (4.29) is a separate issue and has to be carefully examined numerically.888We hope to return to this problem in the near future. Since the lattice spacing must be sufficiently smaller than the square-root of the flow time for our reasoning to work, the reliable application of Eq. (4.29) will require rather small lattice spacings. One also worries about contamination by higher-dimensional operators (i.e., the terms in Eqs. (3.7) and (3.8)) and the finite-size effect which we have not taken into account in the present paper. If our strategy turns to be practically feasible, it provides a completely new method to compute correlation functions containing a well-defined energy–momentum tensor. It is clear that the present approach to the energy–momentum tensor on the lattice is not limited to the pure Yang–Mills theory although the treatment might be slightly more complicated with the presence of other fields. The application will then include the determination of the shear and bulk viscosities (see, e.g., Refs. Meyer:2007ic ; Meyer:2007dy ), the measurement of thermodynamical quantities (see Ref. Giusti:2012yj and references cited therein), the mass and the decay constant of the pseudo Nambu–Goldstone boson associated with the (approximate) dilatation invariance (see Ref. Appelquist:2010gy and references cited therein), and so on. It is also clear that our basic idea, that operators defined with lattice regularization and in the continuum theory can be related through the gradient flow is not limited to the energy–momentum tensor. For example, it might be possible to construct an ideal chiral current or an ideal supercurrent on the lattice, from the small flow-time limit of local products. It would be interesting to pursue this idea. The possibility that the Yang–Mills gradient flow (or the Wilson flow) can be useful for defining the energy–momentum tensor in lattice gauge theory was originally suggested to me by Etsuko Itou. I would like to thank her for enlightening discussions. I would also like to thank Martin Lüscher for a clarifying remark on the precise meaning of Eq. (3.8). This work is supported in part by a Grant-in-Aid for Scientific Research 23540330. Appendix A One-loop calculation of coefficient functions For calculational convenience, we define the coefficient functions and by Equation (3.5) then becomes (for ), Comparison with Eq. (3.7) then shows To find the coefficient functions and in Eq. (A.1), we consider the correlation function For , there are flow-line Feynman diagrams (Figs. 1–17) that contribute to this correlation function. In the figures, gauge potentials at the flow time , , are represented by small filled squares; the open circle denotes the flow-time vertex and the full circle is the conventional vertex in the Yang-Mills theory. We refer the reader to Ref. Luscher:2011bx for the details of the Feynman rules for flow-line diagrams. To read off the coefficient functions and in Eq. (A.1) from the correlation function (A.8), we consider the vertex functions, i.e., amputated diagrams in which the external propagators of the original Yang–Mills theory are truncated. Therefore, Figs. 11, 14 and 17, which provide only the conventional wave function renormalization, should be omitted in the computation of and .999More precisely, these diagrams are different from conventional Feynman diagrams in that the propagators carry an additional factor (in the Feynman gauge), where is the external momentum. This factor is, however, irrelevant in the present computation of the coefficients of operators with the lowest number of derivatives. On the other hand, the flow-line propagators Luscher:2011bx , the arrowed straight lines in the diagrams, should not be truncated because these are not propagators in the quantum field theory but instead represent time evolution along the flow time. The tree-level contribution to the vertex function is and, here and in what follows, the alternating-sign symbol implies This tree-level result was used in obtaining Eq. (4.20). The vacuum expectation value in the lowest order is Now, as an example of the computation of one-loop flow-line Feynman diagrams, we briefly illustrate the computation of Fig 14. A straightforward application of the Feynman rules in Ref. Luscher:2011bx in the “Feynman gauge” in which the gauge parameters are taken as , yields the expression, To find the coefficients and in Eq. (A.1), we write this vertex function as and find the coefficients of respectively, in . For this, we first exponentiate the denominators in Eq. (A.13) by using We then simply expand the integrand with respect to the external momenta and to . The flow-time evolution factor in the integrand makes the integral (A.13) UV finite for any dimension . On the other hand, there always exists a complex domain of such that the integral is infrared finite; this provides the analytic continuation of the integral such that In Table 1, we summarize the contribution of each diagram computed in the above method in the unit of In the last line of the table, “ factors” implies the contributions of the one-loop operator renormalization factors, (2.13) and (2.22), through the tree-level diagram, Eq. (A.9) (recall Eq. (2.10)). We see that those operator renormalization factors precisely cancel the residues of and make the coefficients and finite; this is precisely what we expect from the general argument. From the results in the table, we then have Finally, comparison with the formulas (A.4), (A.7), (4.21) and (4.23) shows the results quoted in Sect. 4, Eqs. (4.30) and (4.31). Note that the coefficients of in the explicit one-loop calculation (Eqs. (A.23) and (A.24)) are in agreement with those by the general argument on the basis of the renormalization group equations and the trace anomaly (Eqs. (4.21) and (4.23)). This agreement provides a consistency check for our one-loop calculation and supports the correctness of our reasoning. - (1) H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981) [Erratum-ibid. B 195, 541 (1982)]. - (2) H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 193, 173 (1981). - (3) P. H. Dondi and H. Nicolai, Nuovo Cim. A 41, 1 (1977). - (4) D. B. Kaplan, Phys. Lett. B 288, 342 (1992) [hep-lat/9206013]. - (5) H. Neuberger, Phys. Lett. B 417, 141 (1998) [hep-lat/9707022]. - (6) P. Hasenfratz, Nucl. Phys. Proc. Suppl. 63, 53 (1998) [hep-lat/9709110]. - (7) P. 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Converting units (49g) Calculator gives incorrect or unexpected results when using trigonometric functions. When (-1 The calculator reference sheet is provided on most items on the 2014 GED. ® To perform calculations with scientific notation, use the key. EXAMPLE. 7.8 × 10. 8 decimal answers, exact square root and decimal, and exact pi and decimal. Pay special attention to the second function key, the way you use calculator memory and the keystrokes needed for decimals, fractions, mixed numbers, exponents Results 1 - 12 of 32 We have a great range of School Calculators from top brands. As well as standard battery-powered calculators, we also have calculators which use solar energy to help CASIO FX-85GTX Black Scientific Calculator. Use only the type of battery specified for this calculator in this manual. Handling Pi (π), Natural Logarithm Base e. Pi (π). You can input pi (π) into a calculation. 20 Jun 2012 The calculator I normally use (a TI-84) only displays 10 digits, so when I ask it what pi is, it spits out the familiar 3 and nine decimal places: cos(pi/3+pi/4) ??????????????????? Switching the TI-30XS MultiView calculator on and off.. 3 arising out of the purchase or use of these materials, and the sole and exact pi and decimal. Scientific Calculator. M. mod. Deg Rad. π e e MC MR MS M+ M-. sinh cosh tanh Exp ( ). ←. C +/-. √. sinh-1 cosh-1 tanh-1 log2x ln log 7 8 9 / %. π e n! logyx ex How to use a Scientific Calculator: entering expression, angle measure, number formats, arithmetic operators, Pi/2 = 3.1415926535898/2 = 1.5707963267949. 29 Apr 2018 To make this value an approximate product of π simply divide the result by π - If your calculator does not have a key for π use 3.14159265. 8 Dec 2019 Do you have a Casio fx-83 ES scientific calculator (or a compatible model) and want to learn how to use it? This free course, Using a scientific 1 Mar 2013 A historical look at approximations of pi for students to extend their calculator skills. Start by entering simple fractions and progress to an One small note is that, unlike older versions of Sharp Scientific Calculators, it now takes 2 button pressed to use pi in a calculation. I have no idea why they have How does a scientific calculator multiply by Pi? … Any calculator multiplies by the value pi the same way it multiplies by any value that you type into the system, the only difference is it uses a value of pi that is hard coded into the calculator. What value of pi is stored in your calculator? Yo How to Use the Log Function on a Calculator | … Press the "Log" button on your calculator. The number you immediately see is the exponent for the original number you entered. Assuming the base number is 10 (which it will always be on a graphing or scientific calculator), you have to multiply 10 by itself the number of … Instructions for the new Online Scientific Calculator The Online Scientific Calculator is an online application for performing simple mathematical calculations. Based on the feedback I received for the first version it now includes many additional features. The most important change is the history display. All operations that you perform during the session are saved and displayed on the top portion of the calculator, so that you can look back at How to Operate a TI-34 Calculator | Sciencing The Online Scientific Calculator is an online application for performing simple mathematical calculations. Based on the feedback I received for the first version it now includes many additional features. The most important change is the history display. All operations that you perform during the session are saved and displayed on the top portion of the calculator, so that you can look back at Prepared by Sarah Nelson for the Dolciani Math Learning Center

217 relations: Abū Kāmil Shujāʿ ibn Aslam, Abel–Ruffini theorem, Abraham Robinson, Absolute value, Accuracy and precision, Addition, Adolf Hurwitz, Adrien-Marie Legendre, Algebra, Algebraic number, Algebraically closed field, Almost all, American Mathematical Society, American Scientist, Arbitrary-precision arithmetic, Archimedean property, Associative algebra, Augustin-Louis Cauchy, Axiom of choice, Axiom of constructibility, Axiomatic system, École normale supérieure (Paris), Évariste Galois, Baire space (set theory), Basis (linear algebra), Blackboard bold, Calculus, Cantor's diagonal argument, Cardinal number, Cardinality, Cardinality of the continuum, Cartesian product, Cauchy sequence, Charles Hermite, Chinese mathematics, Classical mechanics, Coefficient, Compact space, Complete lattice, Complete metric space, Completeness of the real numbers, Complex number, Complex plane, Computable number, Computational science, Computer algebra, Computer algebra system, Connected space, Constant problem, Construction of the real numbers, ..., Constructivism (mathematics), Continued fraction, Continuous function, Continuum hypothesis, Contractible space, Coordinate system, Countable set, Cube root, Cyclic order, David Hilbert, Decimal, Decimal representation, Dedekind cut, Dedekind–MacNeille completion, Definable real number, Descriptive set theory, Differentiable manifold, Dimension, Distance, E (mathematical constant), Edmund Landau, Edward Nelson, Edwin Hewitt, Eigenvalues and eigenvectors, Electromagnetism, Empty set, Energy, Equation, Equivalence class, Euclidean geometry, Exponential function, Extended real number line, Ferdinand von Lindemann, Field (mathematics), Field extension, First-order logic, Floating-point arithmetic, Foundations of Physics, Fraction (mathematics), Fundamental theorem of algebra, Galois theory, General relativity, Georg Cantor, Georg Cantor's first set theory article, Gottfried Wilhelm Leibniz, Greatest and least elements, Greek mathematics, Haar measure, Hausdorff dimension, Hilbert space, History of Egypt, Homeomorphism, Hyperreal number, Imaginary number, Independence (mathematical logic), Indian mathematics, Infimum and supremum, Infinite set, Infinitesimal, Injective function, Integer, Internal set theory, Interval (mathematics), Irrational number, Isomorphism, Johann Heinrich Lambert, Joseph Liouville, Löwenheim–Skolem theorem, Least-upper-bound property, Lebesgue measure, Leonhard Euler, Lie algebra, Limit (mathematics), Limit of a sequence, Line (geometry), Linear combination, Locally compact space, Long line (topology), Magnitude (mathematics), Manava, Mass, Mathematical analysis, Mathematics, Mathematics in medieval Islam, Mathematische Annalen, Matrix (mathematics), Measure (mathematics), Metric space, Middle Ages, Multiplication, Natural number, Negative number, New York Academy of Sciences, Niels Henrik Abel, Non-Archimedean ordered field, Non-standard analysis, Non-standard model, Normal operator, Noun, Nth root, Number, Number line, Order topology, Ordered field, Paolo Ruffini, Paris, Partially ordered group, Paul Cohen, Paul Gordan, Pi, Point (geometry), Polynomial, Positive definiteness, Pythagoras, Quadratic equation, Quantity, Quantum mechanics, Quintic function, R, Rational number, Real analysis, Real closed field, Real line, Real projective line, René Descartes, Reverse mathematics, Self-adjoint operator, Separable space, Separation relation, Sequence, Set (mathematics), Set theory, Shulba Sutras, Sign (mathematics), Simon Stevin, Simply connected space, Solomon Feferman, Springer Science+Business Media, Square root, Square root of 2, Standard Model, Structuralism (philosophy of mathematics), Subset, Surreal number, Tarski's axiomatization of the reals, Time, Topological group, Topological space, Topology, Total order, Transcendental number, Uncountable set, Undecidable problem, Unicode, Uniform space, Unit interval, Up to, Upper and lower bounds, Vector space, Vedic period, Velocity, Vitali set, Well-order, Well-ordering theorem, Zermelo–Fraenkel set theory, Zero of a function, 0. Expand index (167 more) » « Shrink index (Latinized as Auoquamel, ابو كامل, also known as al-ḥāsib al-miṣrī—lit. "the Egyptian reckoner") (c. 850 – c. 930) was an Egyptian Muslim mathematician during the Islamic Golden Age. In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients. Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics. In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign. Precision is a description of random errors, a measure of statistical variability. Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division. Adolf Hurwitz (26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician. Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients). In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F. In mathematics, the term "almost all" means "all but a negligible amount". The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. American Scientist (informally abbreviated AmSci) is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Society. In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics. In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. The École normale supérieure (also known as Normale sup', Ulm, ENS Paris, l'École and most often just as ENS) is one of the most selective and prestigious French grandes écoles (higher education establishment outside the framework of the public university system) and a constituent college of Université PSL. Évariste Galois (25 October 1811 – 31 May 1832) was a French mathematician. In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. In mathematics, the cardinality of a set is a measure of the "number of elements of the set". In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets. In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Mathematics in China emerged independently by the 11th century BC. Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies. In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression. In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other). In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line. A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation. In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Computational science (also scientific computing or scientific computation (SC)) is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems. In computational mathematics, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. In mathematics, the constant problem is the problem of deciding if a given expression is equal to zero. In mathematics, there are several ways of defining the real number system as an ordered field. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. In mathematics, a cube root of a number x is a number y such that y3. In mathematics, a cyclic order is a way to arrange a set of objects in a circle. David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician. The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers. A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2,... In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind, are а method of construction of the real numbers from the rational numbers. In order-theoretic mathematics, the Dedekind–MacNeille completion of a partially ordered set (also called the completion by cuts or normal completion) is the smallest complete lattice that contains the given partial order. Informally, a definable real number is a real number that can be uniquely specified by its description. In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Distance is a numerical measurement of how far apart objects are. The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics. Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Edward Nelson (May 4, 1932 – September 10, 2014) was a professor in the Mathematics Department at Princeton University. Edwin Hewitt (January 20, 1920, Everett, Washington – June 21, 1999) was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage zero–one law. In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. In mathematics, an equation is a statement of an equality containing one or more variables. In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent. In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively). Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that pi (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers. In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. Foundations of Physics is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedures". A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory. General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician. Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy. In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset. Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. Hausdorff dimension is a measure of roughness in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a space, taking into account the distance between its points. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. The history of Egypt has been long and rich, due to the flow of the Nile River with its fertile banks and delta, as well as the accomplishments of Egypt's native inhabitants and outside influence. In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit,j is usually used in Engineering contexts where i has other meanings (such as electrical current) which is defined by its property. In mathematical logic, independence refers to the unprovability of a sentence from other sentences. Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century. In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. In set theory, an infinite set is a set that is not a finite set. In mathematics, infinitesimals are things so small that there is no way to measure them. In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch"). Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections. Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician. In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism. In mathematics, the least-upper-bound property (sometimes the completeness or supremum property) is a fundamental property of the real numbers and certain other ordered sets. In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity. In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value. As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1. The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind. Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of Sulba Sutras. The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasutras which are considered more important. Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change. Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In mathematics, a metric space is a set for which distances between all members of the set are defined. In the history of Europe, the Middle Ages (or Medieval Period) lasted from the 5th to the 15th century. Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division. In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In mathematics, a negative number is a real number that is less than zero. The New York Academy of Sciences (originally the Lyceum of Natural History) was founded in January 1817. Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN*. A noun (from Latin nōmen, literally meaning "name") is a word that functions as the name of some specific thing or set of things, such as living creatures, objects, places, actions, qualities, states of existence, or ideas. In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root. A number is a mathematical object used to count, measure and also label. In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by \mathbb. In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher. Paris is the capital and most populous city of France, with an area of and a population of 2,206,488. In abstract algebra, a partially ordered group is a group (G,+) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a+g ≤ b+g and g+a ≤ g+b. Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his Ph.D. at the University of Breslau (1862),. The number is a mathematical constant. In modern mathematics, a point refers usually to an element of some set called a space. In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive definite. Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement. In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to. Quantity is a property that can exist as a multitude or magnitude. Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. In algebra, a quintic function is a function of the form where,,,, and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. R (named ar/or) is the 18th letter of the modern English alphabet and the ISO basic Latin alphabet. In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator. In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. In mathematics, the real line, or real number line is the line whose points are the real numbers. In geometry, a real projective line is an extension of the usual concept of line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist. Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle. In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. The Shulba Sutras or Śulbasūtras (Sanskrit: "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Simon Stevin (1548–1620), sometimes called Stevinus, was a Flemish mathematician, physicist and military engineer. In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician with works in mathematical logic. Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because. The square root of 2, or the (1/2)th power of 2, written in mathematics as or, is the positive algebraic number that, when multiplied by itself, gives the number 2. The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles. Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. In 1936, Alfred Tarski set out an axiomatization of the real numbers and their arithmetic, consisting of only the 8 axioms shown below and a mere four primitive notions: the set of reals denoted R, a binary total order over R, denoted by infix This axiomatization does not give rise to a first-order theory, because the formal statement of axiom 3 includes two universal quantifiers over all possible subsets of R. Tarski proved these 8 axioms and 4 primitive notions independent. Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation. In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is known to be impossible to construct a single algorithm that always leads to a correct yes-or-no answer. Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. In the mathematical field of topology, a uniform space is a set with a uniform structure. In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (K, ≤) is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound. A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. The Vedic period, or Vedic age, is the period in the history of the northwestern Indian subcontinent between the end of the urban Indus Valley Civilisation and a second urbanisation in the central Gangetic Plain which began in BCE. The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali. In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. In mathematics, the well-ordering theorem states that every set can be well-ordered. In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x). 0 (zero) is both a number and the numerical digit used to represent that number in numerals. Axiomatic real number, Bounded real-valued data, Complete ordered field, Field of reals, List of real numbers, Number axis, Real (number), Real (numbers), Real Number System, Real Numbers, Real number field, Real number system, Real numbers, Reall numbers, Set of real numbers, The complete ordered field, ℝ.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct? Place six toy ladybirds into the box so that there are two ladybirds in every column and every row. This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code? Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it. When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'. This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . . This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items. This is an adding game for two players. There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it? Using the statements, can you work out how many of each type of rabbit there are in these pens? Choose a symbol to put into the number sentence. Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots! Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families? Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. If you have only four weights, where could you place them in order to balance this equaliser? Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be? There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places. This challenge extends the Plants investigation so now four or more children are involved. You have 5 darts and your target score is 44. How many different ways could you score 44? Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs? Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers? How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this? This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it? In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square? Find the sum of all three-digit numbers each of whose digits is odd. There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs. Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number. Ben has five coins in his pocket. How much money might he have? Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like? Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square. Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties? Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make? This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15! There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money? Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this? Can you substitute numbers for the letters in these sums? There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements? Exactly 195 digits have been used to number the pages in a book. How many pages does the book have? Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether. These two group activities use mathematical reasoning - one is numerical, one geometric. A group of children are using measuring cylinders but they lose the labels. Can you help relabel them? Try out this number trick. What happens with different starting numbers? What do you notice? Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this? This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards. Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens? Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores. Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive? How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column? Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

For learning how to compute IRR, we will take an example. Let’s assume initial investment is $5000, and return for first year be $500, second year be $750, third year be $1000, fourth year be $1750, fifth year be $2500. How to calculate IRR: - Make a sheet in Excel, where Value corresponding to CF0 (Initial Cash Flow) will be the value of investment, CF1 will be the returns at the end of 1st year, CF2 will be the returns at the end of 2nd year and so on. By inputting these values you will get something like this: Note:- Initial investment is to be negative because of the fact that initially there was an outflow of Cash instead of an inflow. - Now we have to use IRR formula to compute the IRR. We will use the following formula:- - By applying this formula we will get our IRR. In the example we have taken IRR is 7%. - In case of loss/Negative result in any year, we will input cash flow for that year with a negative sign (-). - Negative IRR means that a company is unable to recover its cost over the period of time. - In case profitability of any asset is to be computed, then residual value of asset is to be added in last year’s profit only. For example In the above example asset from whom such returns were generated is sold for $500, then we will add this value in CF5. In case you mentioned that amount in next cell, whole answer will change and Formula will assume such cash flow from year 6. What is the formula for calculating internal rate of return? It is calculated by taking the difference between the current or expected future value and the original beginning value, divided by the original value and multiplied by 100. How do you calculate IRR quickly? So the rule of thumb is that, for “double your money” scenarios, you take 100%, divide by the # of years, and then estimate the IRR as about 75-80% of that value. For example, if you double your money in 3 years, 100% / 3 = 33%. 75% of 33% is about 25%, which is the approximate IRR in this case. How do you calculate IRR and NPV? The IRR Formula Broken down, each period’s after-tax cash flow at time t is discounted by some rate, r. The sum of all these discounted cash flows is then offset by the initial investment, which equals the current NPV. To find the IRR, you would need to “reverse engineer” what r is required so that the NPV equals zero. What is IRR with example? IRR is the rate of interest that makes the sum of all cash flows zero, and is useful to compare one investment to another. In the above example, if we replace 8% with 13.92%, NPV will become zero, and that’s your IRR. Therefore, IRR is defined as the discount rate at which the NPV of a project becomes zero. What are the rules of IRR? The internal rate of return (IRR) rule states that a project or investment should be pursued if its IRR is greater than the minimum required rate of return, also known as the hurdle rate. The IRR Rule helps companies decide whether or not to proceed with a project. What is a good IRR percentage? If you were basing your decision on IRR, you might favor the 20% IRR project. But that would be a mistake. You’re better off getting an IRR of 13% for 10 years than 20% for one year if your corporate hurdle rate is 10% during that period. Is NPV or IRR better? In other words, long projects with fluctuating cash flows and additional investments of capital may have multiple distinct IRR values. If a discount rate is not known, or cannot be applied to a specific project for whatever reason, the IRR is of limited value. In cases like this, the NPV method is superior. Is ROI the same as IRR? ROI indicates total growth, start to finish, of an investment, while IRR identifies the annual growth rate. While the two numbers will be roughly the same over the course of one year, they will not be the same for longer periods. Can IRR be more than 100%? There’s nothing special about 100%. For one thing, it depends on the time horizon. 100% is a day is a very high IRR, 100% in a century is very low. Or over a year, for example, if a $1 investment returns $2 at the end, that’s 100%; but it’s not significantly different from an investment that returns $1.99 or $2.01. What does a 100% IRR mean? If you invest 1 dollar and get 2 dollars in return, the IRR will be 100%, which sounds incredible. In reality, your profit isn’t big. So, a high IRR doesn’t mean a certain investment will make you rich. However, it does make a project more attractive to look into. What are the problems with IRR? A disadvantage of using the IRR method is that it does not account for the project size when comparing projects. Cash flows are simply compared to the amount of capital outlay generating those cash flows. What if IRR is greater than NPV? Level 1 CFA Exam Takeaways For NPV and IRR NPV equals the sum of present values of all cash flows in a project (both inflows and outflows). If the NPV is greater than zero, the project is profitable. If the IRR is higher than the required return, you should invest in the project. Why does IRR set NPV to zero? As we can see, the IRR is in effect the discounted cash flow (DFC) return that makes the NPV zero. This is because both implicitly assume reinvestment of returns at their own rates (i.e., r% for NPV and IRR% for IRR). What is the conflict between IRR and NPV? In capital budgeting, NPV and IRR conflict refers to a situation in which the NPV method ranks projects differently from the IRR method. In event of such a difference, a company should accept project(s) with higher NPV. What is difference between NPV and IRR? What Are NPV and IRR? Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. By contrast, the internal rate of return (IRR) is a calculation used to estimate the profitability of potential investments. How does reinvestment affect both NPV and IRR? The NPV has no reinvestment rate assumption; therefore, the reinvestment rate will not change the outcome of the project. The IRR has a reinvestment rate assumption that assumes that the company will reinvest cash inflows at the IRR’s rate of return for the lifetime of the project. What is IRR NPV calculator? IRR is independent of the Discount Rate. To calculate NPV, enter a discount rate which may be your cost of borrowing rate. Discount Cash Flow Rate of Return Analysis is a very useful tool to help you analyze your investment projects. Can IRR be positive if NPV negative? Nope, you can‘t get a positive return on a money losing project. If, for example, there are some tax benefits to be gained from write offs and such, you should be incorporating those into the NPV, which may then become positive, then calculating a positive IRR.

- Research Article - Open Access Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales © Q. Lin and B. Jia. 2010 - Received: 8 October 2010 - Accepted: 27 December 2010 - Published: 30 December 2010 This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping , on an isolated time scale which is unbounded above. In , α is the quotient of odd positive integers. As an application, we get the difference equation , where , , and is any real number, is oscillatory. - Dynamic Equation - Difference Equation - Jump Operator - Part Formula - Closed Nonempty Subset During the past years, there has been an increasing interest in studying the oscillation of solution of second-order damped dynamic equations on time scale which attempts to harmonize the oscillation theory for continuousness and discreteness, to include them in one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the papers [1–4] and the references cited therein. In , under the assumption of being an isolated time scale, we prove that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.3). As an application, we get that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.4), which improves a result of Hooker and Patula [7, Theorem 4.1] and Mingarelli . where , , , and is any real number, is oscillatory. Note that if , then the delta derivative is just the standard derivative, and when the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales (e.g., the time scale which is very important in quantum theory ). We will need the following second mean value theorem (see [10, Theorem 5.45]). Lemmas 2.2 and 2.4 give two lower bounds of definite integrals on time scale, respectively. It is easy to know that, when , and, when , . Assume that , where . Suppose that (i)there exists a real number such that , for all ; Then, (1.1) is oscillatory. Since , we get , for large , which is a contradiction. Thus, (1.1) is oscillatory. is sufficient for the oscillation of the difference equation (1.4). then (1.4) is oscillatory. where . It is easy to get the following. Then, the differential equation (3.16) is oscillatory. where , , , and is any real number. for large . That means . It is easy to get that and is nonincreasing for large . So from Theorem 3.1, (3.18) is oscillatory. and is nonincreasing. So from Theorem 3.1, (3.21) is oscillatory. where , , , and is any real number. So from Theorem 3.3, (3.23) is oscillatory. This work was supported by the National Natural Science Foundation of China (no. 10971232). - Bohner M, Saker SH: Oscillation of damped second order nonlinear delay differential equations of Emden-Fowler type. Advances in Dynamical Systems and Applications 2006,1(2):163-182.MATHMathSciNetGoogle Scholar - Han Z, Chen W, Sun S, Li T: Oscillation behavior of a class of second-order dynamic equations with damping on time scales. Discrete Dynamics in Nature and Society 2010, 2010:-15.Google Scholar - Erbe L, Hassan TS, Peterson A: Oscillation criteria for nonlinear damped dynamic equations on time scales. Applied Mathematics and Computation 2008,203(1):343-357. 10.1016/j.amc.2008.04.038MATHMathSciNetView ArticleGoogle Scholar - Saker SH, Agarwal RP, O'Regan D: Oscillation of second-order damped dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2007,330(2):1317-1337. 10.1016/j.jmaa.2006.06.103MATHMathSciNetView ArticleGoogle Scholar - Bohner M, Erbe L, Peterson A: Oscillation for nonlinear second order dynamic equations on a time scale. Journal of Mathematical Analysis and Applications 2005,301(2):491-507. 10.1016/j.jmaa.2004.07.038MATHMathSciNetView ArticleGoogle Scholar - Erbe L, Baoguo J, Peterson A: Belohorec-type oscillation theorem for second ordersuperlinear dynamic equation on time scales. Mathematische Nachrichten. In pressGoogle Scholar - Hooker JW, Patula WT: A second-order nonlinear difference equation: oscillation and asymptotic behavior. Journal of Mathematical Analysis and Applications 1983,91(1):9-29. 10.1016/0022-247X(83)90088-4MATHMathSciNetView ArticleGoogle Scholar - Mingarelli AB: Volterra-Stieltjes integral equations and generalized differential equations, Ph.D. thesis. University of Toronto; 1983.Google Scholar - Bohner M, Peterson A: Dynamic Equation on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Kac V, Cheung P: Quantum Calculus. Springer, New York, NY, USA; 2002:x+112.MATHView ArticleGoogle Scholar - Baoguo J, Erbe L, Peterson A: Oscillation of sublinear Emden-Fowler dynamic equations on time scales. Journal of Difference Equations and Applications 2010,16(2-3):217-226. 10.1080/10236190802631881MATHMathSciNetView ArticleGoogle Scholar This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

OH = cos 270° = cos (3 /2) = 0: A trecentosessanta gradi e' come a zero gradi ed avremo che il coseno vale uno OH = cos 360° = cos 0° = 1: Riassumendo il valore del coseno parte dal valore 1 a 0° diminuisce fino a raggiungere il valore 0 a 90 The following example uses Cos to evaluate certain trigonometric identities for selected angles. // Example for the trigonometric Math.Sin( double ) // and Math.Cos( double ) methods. using namespace System; // Evaluate trigonometric identities with a given Apr 15, 2019 · The value of ∫ sin^3x/sin x + cos x ,x ∈[0, π/2] dx is : asked May 19, 2019 in Mathematics by Jagan (21.1k points) jee mains 2019 +1 vote. 1 answer. - Spotový kurz amerického dolára v dolároch - Ako používať moju éterovú peňaženku - Najlepšia bitcoinová peňaženka pre online obchod - Cena akcie mincí kcs - Ako môžem zistiť svoje heslo na facebooku, keď som prihlásený - Bitcoins miner kalkulačka Thus, you are obtaining the value of (cosπ)/2=(−1)/2=−0.5. Nov 29, 2017 0 pi in degree form =180 =cos(180/2) =cos(90) =0. May 19, 2016 cos(π2)=0. Explanation: For an angle in standard position cos(θ)=xr (definition). From the images below, we can see that as θ→π2. XXXx→0. Oct 26, 2020 Compute cos(pi/2) with the unit circleIf you enjoyed this video please consider liking, sharing, and subscribing.Udemy 0:00 / 1:07. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. cos ( π 6) = cos ( π 6) = 3 2. cos ( π 4) = cos ( π 4) = 2 2. cos ( π 3) = cos ( π 3) = 1 2. pi/2*x-1=cos^-1 (0) This is asking you what angle has a cos value of 0 on the interval from 0 to pi, which is the domain of inverse cosine. pi/2 is the angle, so. pi/2*x-1=pi/2 Do some algebra to solve for x. pi/2*x= (pi/2)+1. pi/2*x= (pi/2)+2/2. (pi/2)x= (pi+2)/2. Evaluate : ∫π/2 0 (sin^2 x)/(sinx+cosx)dx Aug 04, 2011 · cot(pi/2) = 1/tan(pi/2) = 1/undefined =/= 0 I don't see how the two are equal, and ya I think I may be getting some things mixed up as I haven't dealt with basic trig in several years lolz Aug 4, 2011 Feb 13, 2020 · Ex 3.3, 6 Prove that: cos (π/4−𝑥) cos (π/4−𝑦) – sin (π/4−𝑥) sin (π/4−𝑦) = sin(𝑥 + 𝑦) Taking L.H.S We know that cos (A + B) = cos A cos B – sin A sin B The equation given in Question is of this form Where A = (𝜋/4 −𝑥) B = (𝜋/4 −𝑦) Hence cos (π/4−𝑥) cos (π/4−𝑦) – sin (π Mar 01, 2018 · We observe that our cosine graph has amplitude `13.892` and it has been shifted to the right by `0.528` radians, which is consistent with the expression we obtained: 13.892 cos (θ − 0.528) 2. Express 2.348 sin θ − 1.251 cos θ in the form −R cos ( θ + α) , where 0 ≤ α < π/2. Find sin(u-pi), cos(u-pi), sin(u-pi/2), cos(u-pi/2) I tried solving this question but I could not get the right answer. Here are the step I tried for solving this problem. 8. 9. 9.. ÷. Besides The cos of -pi/2 radians is 0, the same as cos of -pi/2 radians in degrees. Jul 15, 2020 cos2(0)+cos2(π6)+cos2(π3)+cos2(π2). check-circle. Answer. Step by step solution by experts to help you in doubt clearance & scoring Answer to Consider the folowing. lim x → 0+ [cos(pi/2-x)]^x Solved: simplify cos(pi/2-X) - Slader. MARKDOWN and KATEX. By the definition of the functions of trigonometry, the sine of pi/2 is equal to the y-coordinate of the point with polar coordinates (r,theta)=(1,pi/2), giving sin(pi/2)=1. Similarly, cos(pi/2)=0, since it is the x-coordinate of this point. Filling out the other trigonometric functions then gives cos(pi/2) = 0 (1) cot(pi/2) = 0 (2) csc(pi/2) = 1 (3) sec(pi/2) = infty (4) sin(pi/2) = 1 (5) tan Definite Integrals - Integral of cos^2x / ( cos^2x + 4 sin^2x)`int _0^(pi/2) cos^2x/(cos^2x+4sin^2x) dx``int _0^(pi/2) cos^2x/(cos^2x+4sin^2x) dx`integral 0 24/10/2015 Click here👆to get an answer to your question ️ For 0 < ϕ < pi/2 if x = ∑n = 0^∞cos ^2nϕ, y = ∑n = 0^∞sin ^2nϕ, z = ∑n = 0^∞cos ^2nϕsin^2nϕ , then 14/01/2016 Question: In HW Item #5, The Final Answer Involves Cos(pi/2) = 0 O Cos(pi/2) = 1 O Sin(pi/2) = 0 Express In Terms Of Sine, Cosine, Or Tangent Of One Angle. Then Find The Exact Value. Зл TT Зл TT TT 5л TT 5л 5. COS COS + Sin Sin 6. How do I get this value to revert to 0, which is the true value? Is sin always related to the y axis? I'm confused because if we look at angle theta + pi/2, why is it that sin doesn't have the opposite/hypotenuse definition Aug 1, 2019 Hi! While trying out some stuff in Julia, I came across the fact that cos(pi/2) does not give exact 0, but some type dependent output (6e-17 for limit as h->0 when [cos(pi/2+h)]/h when I sub in zeroI get 0/0so I have to use L' hopital's rule So I take the derivative of the numberator and cos(pi/2_=x/r=0/|y|= cos(3pi/2)=x/r=0/|y|. The trigonometric functional values of angles coterminal with 0, π/2 , π, and 3π/2 are the same as those above, and the Cos -Pi/2: All about cos minus Pi/2 radians, incl. the trigonometric identities. Dean094.450 usd na huf sledovač cien mincí coinbase nemôže vybrať gbp je ťažba ethereum na aws zisková do roku 2021 problém s výmenou mince všetky kombinácie dolárové náklady v priemere na bitcoin nemá aol mať dvojstupňové overenie - Recenzný film o blížencoch - Platba btconline - 25 350 eur na dolár - Kúpte nám menu kreditnou kartou - Blockchain jednoduché vysvetlenie - 270 dolárov na eurá - Mlynček na prepínanie mŕtvych ľudí - Predikcia inr na krw π 2 represents a quarter of a circle (90 ∘) wich corresponds to (0, 1) on the unit circle, i.e. using namespace System; // Evaluate trigonometric identities with a given Apr 15, 2019 · The value of ∫ sin^3x/sin x + cos x ,x ∈[0, π/2] dx is : asked May 19, 2019 in Mathematics by Jagan (21.1k points) jee mains 2019 +1 vote.

a 2 = c dos ? q 2 + b 2 ? 2bq + q dos = c 2 + b 2 ? 2bq Addititionally there is an important relationships within three sides off a standard triangle in addition to cosine of 1 of the bases Figure 30 reveals the trail out of a ship you to definitely sailed 30 kilometres due east, following turned into as a result of 120° and you will sailed a deeper 40 kilometres. Among indoor basics away from a good triangle is actually 120°. In case your sides adjacent to so it position is actually off length 4 yards and you can 5 meters, make use of the cosine signal to discover the length of the medial side reverse brand new offered perspective. cuatro.2 Trigonometric identities A large amount of applicable mathematics can be involved that have equations. It’s generally the case these equations are only genuine in the event the parameters they consist of deal with certain specific beliefs; including, 4x = cuatro is only true whenever x = 1. not, we sometimes take note of equations that are genuine for everybody opinions of your variables, for example (x + 1) 2 = x 2 + 2x + step one. Equations on the second variety of, we.e. of those which might be genuine no matter this thinking of your variables they contain, was properly titled identities. i You can find a great many trigonometric identities, we.elizabeth. relationships ranging from trigonometric attributes which might be in addition to the specific viewpoints of the parameters they include. They have certain applications and it is beneficial to enjoys an effective listing of her or him for simple resource. 1st are provided below – you have got already met the original eight (inside the slightly different forms) before on the module while some exists during the certain products throughout the FLAP. Observe that ? and you can ? may represent any amounts or angular beliefs, unless the viewpoints is minimal from the significance of the functions alarmed. Brand new abbreviations asin, acos and you may atan or alternatively sin ?step one , cos ?1 and you can tan ?step one , are sometimes employed for the latest inverse trigonometric qualities. Pythagorass theorem states that square of the hypotenuse into the a good right–angled triangle is equivalent to the entire squares of the other a couple corners. The latest angles 180° and you can 90° match a rotation by way of 50 % of and another–one-fourth out-of a group, respectively. A perspective off 90° is named a right direction. A column at 90° to a given line (otherwise surface) is considered become perpendicular otherwise regular into unique line (or epidermis). Once the 2? = six.2832 (to help you four decimal cities) they uses that step one radian = °, given that stated earlier. Desk step 1 brings certain basics measured inside the degrees and radians. As you can tell from this table, of a lot are not–put bases are simple portions otherwise multiples of ? radians, but note that basics shown in the radians commonly usually indicated with regards to ?. Don’t make prominent mistake off thinking that ? try some sort of angular tool; it is simply several. not, the space of large square can also be found of the incorporating the room of your own faster square, h 2 , to the regions of the newest five spot triangles. For every triangle has actually a location xy/2 (each is 50 % of good rectangle out-of corners x and you can y) so the the main large square is The research off correct–angled triangles is called trigonometry, plus the around three collection of rates away from sets of edges is along referred to as trigonometric rates. He is known as sine, cosine and you can tangent of your own position ? – abbreviated so you’re able to sin, cos and tan, respectively we – and you will defined as employs: One of the several good reason why trigonometric rates we is actually of attention so you can physicists is they assist to influence the new lengths of all dominican cupid log in of the edges off a right–angled triangle regarding a knowledge of a single side length and you can one indoor angle (aside from the best direction). The newest proportion meanings of sine, cosine and tangent (we.age. Equations 5, six and you may eight) merely seem sensible having bases on the variety 0 in order to ?/dos radians, simply because they cover the new sides regarding the right–tilted triangle. Within subsection we are going to describe about three trigonometric attributes, also known as sine, cosine and tangent, and denoted sin(?), cos(?) and you can bronze(?), correspondingly. we Such services commonly enable us to attach a meaning so you can the newest sine and you may cosine of any direction, and also to new tangent of every angle that’s not an strange numerous ?/dos. For instance the trigonometric percentages which they generalize, this type of trigonometric services is of good pros from inside the physics. However, not only is it the signs of this new trigonometric services one changes due to the fact ? develops otherwise reduces and you may P moves within circle in Contour 16. The values out of x and you may y, and therefore out of sin(?), cos(?) and you may bronze(?) together with will vary. On the best such as for instance activity, simple harmonic actions, the latest altering updates x out of a size oscillating with the end from a springtime is generally represented by x = Acos(?t + ?). Even with appearances none of one’s quantity when you look at the bracket is an angle (even if they can be offered angular interpretations); t is the time that is mentioned inside mere seconds, ? is a steady known as the angular frequency that’s related to the brand new functions of the bulk and spring which can be measured from inside the hertz (1 Hz = step 1 s ?1 ), and you may ?, the fresh stage lingering, is actually a number, usually regarding the diversity 0 to help you 2?. Clearly, including a positive ongoing, ?/dos, into conflict of your own means has got the effect of progressing brand new chart left from the ?/2. From inside the crude terms, this new addition has actually raised the dispute and helps make what you happen prior to (we.age. further left). In the event that cos(?) = x, in which 0 ? ? ? ? and you will ?step one ? x ?step 1 upcoming arccos(x) = ? (Eqn 26b)

- What are 3 types of variables? - Is GPA nominal or ordinal? - Is salary an ordinal variable? - What are the 2 types of variables? - What type of variable is age? - Is gender nominal or ordinal? - What type of variable is gender? - Is gender a nominal data? - What type of data is gender in SPSS? - Is age an ordinal variable? - What are the 5 types of variables? - Is age a covariate? - What are variables in data? - What is the difference between nominal and ordinal data? - Is age nominal for years? - Is name a categorical variable? - Is age categorical or numerical? - Are salaries nominal or ordinal? - Is ordinal data qualitative or quantitative? - What type of data is age in SPSS? - How do you separate gender in SPSS? What are 3 types of variables? A variable is any factor, trait, or condition that can exist in differing amounts or types. An experiment usually has three kinds of variables: independent, dependent, and controlled. The independent variable is the one that is changed by the scientist.. Is GPA nominal or ordinal? 1. Mathematically or statistically, there is a problem with the concept of a Grade Point Average. In a technical sense, letter grades are Ordinal (rather than Interval) numbers — meaning, the distance between two letter grades is not the same. Is salary an ordinal variable? Interval/ratio can be re-formatted to become ordinal or nominal, ordinal can become nominal. Example: salary data for is often recorded as interval data (i.e. just a number). operations such as finding the average salary. What are the 2 types of variables? Frequently asked questions about variables You can think of independent and dependent variables in terms of cause and effect: an independent variable is the variable you think is the cause, while a dependent variable is the effect. What type of variable is age? Mondal suggests that age can be viewed as a discrete variable because it is commonly expressed as an integer in units of years with no decimal to indicate days and presumably, hours, minutes, and seconds. Is gender nominal or ordinal? There are two types of categorical variable, nominal and ordinal. A nominal variable has no intrinsic ordering to its categories. For example, gender is a categorical variable having two categories (male and female) with no intrinsic ordering to the categories. An ordinal variable has a clear ordering. What type of variable is gender? A categorical variable (sometimes called a nominal variable) is one that has two or more categories, but there is no intrinsic ordering to the categories. For example, gender is a categorical variable having two categories (male and female) and there is no intrinsic ordering to the categories. Is gender a nominal data? A good example of a nominal variable is sex (or gender). Information in a data set on sex is usually coded as 0 or 1, 1 indicating male and 0 indicating female (or the other way around–0 for male, 1 for female). What type of data is gender in SPSS? Measure in SPSS A Nominal (sometimes also called categorical) variable is one whose values vary in categories. It is not possible to rank the categories created. e.g. Gender varies in that an individual is either categorised as “male” or “female”. Is age an ordinal variable? Age can be both nominal and ordinal data depending on the question types. I.e “How old are you” is a used to collect nominal data while “Are you the first born or What position are you in your family” is used to collect ordinal data. Age becomes ordinal data when there’s some sort of order to it. What are the 5 types of variables? There are six common variable types:DEPENDENT VARIABLES.INDEPENDENT VARIABLES.INTERVENING VARIABLES.MODERATOR VARIABLES.CONTROL VARIABLES.EXTRANEOUS VARIABLES. Is age a covariate? You can add age as a continuous covariate, but keep in mind that, e.g. ~age + … implies that gene expression will have multiplicative increases with each unit of age. What are variables in data? In statistics, a variable has two defining characteristics: A variable is an attribute that describes a person, place, thing, or idea. The value of the variable can “vary” from one entity to another. What is the difference between nominal and ordinal data? Nominal and ordinal are two of the four levels of measurement. Nominal level data can only be classified, while ordinal level data can be classified and ordered. Is age nominal for years? There is no order associated with values on nominal variables. [Ratio] Age is at the ratio level of measurement because it has an absolute zero value and the difference between values is meaningful. For example, a person who is 20 years old has lived (since birth) half as long as a person who is 40 years old. Is name a categorical variable? Categorical variables take on values that are names or labels. The color of a ball (e.g., red, green, blue) or the breed of a dog (e.g., collie, shepherd, terrier) would be examples of categorical variables. Is age categorical or numerical? In our medical example, age is an example of a quantitative variable because it can take on multiple numerical values. It also makes sense to think about it in numerical form; that is, a person can be 18 years old or 80 years old. Are salaries nominal or ordinal? Nominal (Unordered categories) of Data For example, salary can be turned into a nominal variable by defining “high salary” as an annual salary of more than $200,000, “moderate salary” as less than or equal to $200,000 and more than $75,000, and “low salary” as less than or equal to $75,000. Is ordinal data qualitative or quantitative? Data at the ordinal level of measurement are quantitative or qualitative. They can be arranged in order (ranked), but differences between entries are not meaningful. Data at the interval level of measurement are quantitative. They can be ordered, and meaningful differences between data entries can be calculated. What type of data is age in SPSS? An additional practice example is suggested at the end of this guide. The example assumes you have already opened the data file in SPSS. Age is a key demographic variable, frequently recorded in survey data as part of a broader set of demographic variables such as education, income, race, ethnicity, and gender. How do you separate gender in SPSS? To split the data in a way that separates the output for each group:Click Data > Split File.Select the option Organize output by groups.Double-click the variable Gender to move it to the Groups Based on field.When you are finished, click OK.

3 edition of Integral and Functional Differential Equations (Pure and Applied Mathematics (Marcel Dekker)) found in the catalog. May 1, 1981 Written in English |The Physical Object| |Number of Pages||296| Elementary Differential Equations with Boundary Value Problems is written for students in science, en-gineering,and mathematics whohave completed calculus throughpartialdifferentiation. Ifyoursyllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some prepa-ration inlinear algebra. The text also presents general optimal control problems, optimal control of ordinary differential equations, and different types of functional-integral equations. The book discusses control problems defined by equations in Banach spaces, the convex cost functionals, and the weak necessary conditions for an original Edition: 1. Ulam Stability of Operators presents a modern, unified, and systematic approach to the field. Focusing on the stability of functional equations across single variable, difference equations, differential equations, and integral equations, the book collects, compares, unifies, complements, generalizes, and updates key results. Definitely the best intro book on ODEs that I've read is Ordinary Differential Equations by Tenebaum and Pollard. Dover books has a reprint of the book for maybe dollars on Amazon, and considering it has answers to most of the problems found. The book deals with linear integral equations, that is, equations involving an unknown function which appears under the integral sign and contains topics such as Abel's integral equation, Volterra integral equations, Fredholm integral integral equations, singular and nonlinear integral equations, orthogonal systems of functions, Green's. The book can be used as a database of test problems for numerical and approximate methods for solving linear and nonlinear integral equations. Discover the world's research 17+ million members. The clockmaker, or, The sayings and doings of Samuel Slick of Slickville Libraries Unlimited Professional Collection Unemployment, health and social policy. Samuel Taylor Darling. Years of liberalism and fascism childrens Blue bird The Hallway Light at Night Interpretation of motor fuel survey data Introduction to the Early Childhood Environment Rating Scale (Viewers Guide&Training Workbook) Sediments and microfauna off the coasts of Mississippi and adjacent states Integral and Functional Differential Equations (Lecture Notes in Pure and Applied Mathematics) 1st Edition by H. Stech (Author) ISBN ISBN Why is ISBN important. ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. Cited by: 7. Theory of Functionals and of Integral and Integro-Differential Equations (Dover Books on Mathematics) Hardcover – Janu by Vito Volterra (Author) › Visit Amazon's Vito Volterra Page. Find all the books, read about the author, and more. See search results for this author Cited by: This is the first comprehensive introduction to collocation methods for the numerical solution of initial-value problems for ordinary differential equations, Volterra integral and integro-differential equations, and various classes of more general functional equations. Its principal aims are to guide the reader from basic ideas to the current state of the art, to describe important problems and directions Format: Hardcover. Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations and partial differential equations. Knowledge of these techniques is particularly useful as preparation for graduate courses and PhD research in differential equations Format: Paperback. Book Description. The purpose of this book is threefold: to be used for graduate courses on integral equations; to be a reference for researchers; and to describe methods of application of the theory. The author emphasizes the role of Volterra equations as a unifying tool in the study of functional equations, and investigates the relation between abstract Volterra equations and other types of functional-differential by: This book seeks to present Volterra integral and functional differential equations in that same framwork, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general problems. Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional | Lakshmikantham | download | B–OK. Download books for free. Find books. Description Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations and partial differential equations. Theory of functionals and of integral and integro-differential equations Vito Volterra A general theory of the functions depending on a continuous set of values of another function, this volume is based on the author's fundamental notion of the transition from a finite number of. SOME REMARKS AND NOTATION 1. In Chapters 1–11 in the original integral equations, the independent variable is denoted by x, the integration variable by t, and the unknown function by y = y(x). For a function of one variable f = f(x), we use the following notation for the derivatives: f. Book description Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical by: functional equations but Sm`ıtal presents beautifully the topic of iterations and functional equations of one variable2. Similarly, Small’s book is a very enjoyable, well written book and focuses on the most essential aspects of functional equations. Once the reader. This book explains the following topics: First Order Equations, Numerical Methods, Applications of First Order Equations1em, Linear Second Order Equations, Applcations of Linear Second Order Equations, Series Solutions of Linear Second Order Equations, Laplace Transforms, Linear Higher Order Equations, Linear Systems of Differential Equations, Boundary Value Problems and Fourier. "The Conference on Integral and Functional Differential Equations was held June, at West Virginia University in Morgantown, West Virginia"--Preface. Description: x, pages ; 26 cm. Series Title: Lecture notes in pure and applied mathematics, v. Responsibility: edited by Terry L. Herdman and Harlan W. Stech and Samuel M. The theorems for regular integral equations may easily be carried over to cases in which fewer assumptions are made about the kernel. The definitions of regular and singular integral equations used here follow those in Ph. Frank and R. Mises: Differential-and Integralgleichungen der Mechanik and Physik, 2nd ed., Vol. 1, p. Brunswick Collocation methods for Volterra integral and related functional differential equations Brunner H. This is the first comprehensive introduction to collocation methods for the numerical solution of initial-value problems for ordinary differential equations, Volterra integral and integro-differential equations, and various classes of more general. ordinary differential equations, partial differential equations, Laplace transforms, Fourier transforms, Hilbert transforms, analytic functions of complex variables and contour integrations are expected on the part of the reader. The book deals with linear integral equations, that is, equations involving an. Most mathematicians, engineers, and many other scientists are well-acquainted with theory and application of ordinary differential equations. This book seeks to present Volterra integral and functional differential equations in that same framework, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general p. The present book builds upon an earlier work of J. Hale, "Theory of Func tional Differential Equations" published in We have tried to maintain the spirit of that book and have retained approximately one-third of the material intact. This book is an introduction to partial differential equations (PDEs) and the relevant functional analysis tools which PDEs require. This material is intended for second year graduate students of mathematics and is based on a course taught at Michigan State University for a number of years. About this book This classic work is now available in an unabridged paperback edition. Hochstatdt's concise treatment of integral equations represents the best compromise between the detailed classical approach and the faster functional analytic approach, while .Linear and Nonlinear Integral Equations - Books. EqWorld.Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cochran, J. A., The Analysis of Linear Integral Equations, McGraw-Hill Book Co., New York, That is, a functional differential equation is an equation that contains some function and some of its derivatives to different argument values. Functional differential equations find use in mathematical models that assume a specified behavior or phenomenon depends on the present as well as the past state of a system.

HANCOCK, N.H. Renewed scrutiny of a statistical technique used by British intelligence to decode German military communications during World War II has opened new avenues in statistical prediction that researchers say could improve machine-learning software. Today's spell checkers, data retrieval methods and speech recognition systems use variations of the Good-Turing estimator, named for British mathematicians I.J. Good and Alan M. Turing. The technique, representing an intuitive breakthrough in the modeling of probability distributions behind data streams, was credited with shortening the war by several years. But while Good published a mathematical justification of the method after the war, statisticians remained unsure why the estimator worked so well. Only within the past few years has an outside group come up with a method for quantifying the success of the Good-Turing algorithm. Now that work, by Alon Orlitsky and his colleagues at the University of California-San Diego's Department of Electrical Engineering, has yielded a statistical estimator that the researchers say is more accurate than Good-Turing over time. The innovation is asymptotic, meaning that it will eventually outperform Good-Turing predictions on given sequences. But the rate of convergence is a practical issue that needs more work, Orlitsky said. As Orlitsky's team tried to measure the performance of Good-Turing, it discovered a definite limit on how well the method performs. A precise mathematical formulation of that limit led to insights into creating a limitless estimator. The estimation problem is commonly encountered in applications of probability theory: Given a random data stream, infer the probability distribution that generates it. For example, observations of a long string of binary digits generated by repeated coin tosses would quickly reveal that the occurrence of each digit is equally likely. Even if an observer doesn't know how the string of bits is generated, the frequency patterns will give clues to its generation. As the data stream lengthens, the frequencies will converge to equal outcomes for each digit-and in general, the longer the sequence, the better the estimation technique becomes. "If the number of possible outcomes is small, the problem is easy. But with a large set of outcomes, the optimal distribution begins to disappear into noise," Orlitsky said. In addition, it is necessary to include all possible outcomes in the estimation of a probability distribution even if they do not occur-the so-called "hidden outcome" problem. Orlitsky uses the example of a safari drive that yields sightings of three giraffes, one zebra and two elephants. Based on that experience, one might naturally assign probabilities of 1/2 for encountering giraffes, 1/6 for zebras and 1/3 for elephants. But that assumption fails to recognize that the next animal encountered might be a lion. To avoid a nasty surprise, the good estimator would take the entire set of known species in the area of the game drive into account and assign a probability to all species in the set. That was the problem faced by British cryptologists during the World War II. The German Enigma encryption machine used a huge number of decryption keys, making it almost impossible to crack the code. British intelligence had gained possession of Enigma machines, had determined how they worked and had even obtained a copy of the full book of keys. Some messages had been decrypted and the keys used recorded, so that the code breakers had a small sample from a very large set of keys. But it was unlikely the Germans would continue to use the same keys, so some method of assigning a probability distribution to the keys not yet used was needed. Instead of quantifying the occurrence of keys, Good and Turing decided to quantify the frequencies at which keys occurred. It seems intuitive that if a key has been seen frequently already, it is less likely to occur in the future, while one that has not yet occurred or has been seen only once will have a higher probability of appearing. Turing assumed that the frequencies would follow a standard binomial distribution. Taking the frequency of known keys as empirical data, he assigned probabilities to the entire set of keys. It was a promising approach, but the method suffered from noise problems. Good introduced smoothing algorithms on the data to eliminate the noise. That was the genesis of the Good-Turing estimator. The data-smoothing aspect is key to getting the method to work effectively, and a lot of work over the years has gone into devising better smoothing techniques. Depending on how data is smoothed, different versions of the Good-Turing estimator have been created for different applications. By coming up with a means of quantifying the success of these different estimators, Orlitsky found that they all have a hard limit on their effectiveness. Orlitsky was able to discover this limit by quantifying the problem in terms of the positive integers. The nature of the sample set is actually irrelevant to the probabilistic algorithm. What matters is the order in which outcomes appear and how often they appear. So a sample sequence such as giraffe, giraffe, elephant, giraffe, zebra would be encoded in numbers as 1,1,2,1,3. Every time a new item appears, it is assigned the next-highest number, so that this mathematical model, according to its creators, can capture the worst possible problem-one in which there is an infinite number of hidden data items. Using numbers as the sample set also makes it possible to consider the asymptotic behavior of estimators as the length of sampled data increases indefinitely. A given method for estimating the probability of a sequence of numbers is compared with the maximum probability assigned to the sequence by all possible probability distributions. Specifically, Orlitsky defines the attenuation of an estimator as the ratio of the maximum value resulting from the application of all possible probability distributions to the value assigned by the estimator. Since an estimator uses some probability distribution, its value will always be lower than the maximum value, so this measure is always greater than 1. Thus, a figure of merit would be how close the attenuation of an estimator approaches 1. "That type of definition has been used before on problems where the sample set is small and known, but we have extended it to large, even infinite sample sets," Orlitsky said. By looking at how attenuation varies as the length of a data stream increases, Orlitsky and his colleagues were able to discover the ultimate limits of a given estimation technique. A common technique known as the add-constant estimator, which was first suggested by Pierre-Simon Laplace, the inventor of probability theory, turns out to be very poor. As the data stream lengthens, the estimates actually diverge away from 1; thus add-constant estimators have infinite attenuation. That was a surprising finding, because add-constant estimators have always been a simple and effective approach on small-sample-set problems. Good-Turing estimators always stay within a factor of 2 of the optimum probability, but Orlitsky discovered that they have a lower limit as well. He was able to show that an estimator with one of the simplest smoothing algorithms could not get below a factor of 1.39 of the best possible estimation for a string of data. Estimators with more complex smoothing algorithms were difficult to analyze, but simulations indicated that they too had a lower limit on their attenuation. "Theoretically we can show that it is possible to create estimators that attenuate to 1, but the practical aspect is how fast they converge," Orlitsky said. "That is a problem we are working on now-and we have some new results that we are not ready to talk about at this time."

Implementing Tidal and Gravitational Wave Energy Losses in Few-body Codes: A Fast and Easy Drag Force Model We present a drag force model for evolving chaotic few-body interactions with the inclusion of orbital energy losses, such as tidal dissipation and gravitational wave (GW) emission. The main effect from such losses is the formation of two-body captures, that for compact objects result in GW mergers, and for stars lead to either compact binaries, mergers or disruptions. Studying the inclusion of energy loss terms in few-body interactions is therefore likely to be important for modeling and understanding the variety of transients that soon will be observed by current and upcoming surveys. However, including especially tides in few-body codes has been shown to be technically difficult and computationally heavy, which has lead to very few systematic tidal studies. In this paper we derive a drag force term that can be used to model the effects from tidal, as well as other, energy losses in few-body interactions, if the two-body orbit averaged energy loss is known a priori. This drag force model is very fast to evolve, and gives results in agreement with other approaches, including the impulsive and affine tide approximations. keywords:gravitation – methods: numerical – stars: black holes – stars: kinematics and dynamics Transient events, including gravitational wave (GW) mergers (2016PhRvL.116f1102A; 2016PhRvL.116x1103A; 2016PhRvX...6d1015A; 2017PhRvL.118v1101A; 2017PhRvL.119n1101A; 2017PhRvL.119p1101A), stellar mergers (e.g. 2011A&A...528A.114T), and stellar tidal disruptions (e.g., 2016ApJ...823..113P, and references therein), are often the product of a two-body or a dynamical few-body system loosing orbital energy through one or more dissipative mechanisms. The most important of such mechanisms include energy dissipation through the the emission of GWs (e.g. Peters:1964bc; Hansen:1972il; 1977ApJ...216..610T), and orbital energy losses through tidal excitations (e.g. 1977ApJ...213..183P; 1986ApJ...310..176L) and dissipation (e.g. 2014ARA&A..52..171O, and references therein). In the isolated binary problem, these effects will lead to a merger between the two objects within a finite time, and depending on the stellar types the final binary evolution will either be dominated by GWs (e.g. Peters:1964bc) (if both objects are compact), tides (2014ARA&A..52..171O) (if at least one object is a star), or common envelope evolution (e.g. 1976IAUS...73...75P; 1993PASP..105.1373I; 2000ARA&A..38..113T; 2018arXiv180303261M) (if one of the objects evolves to indulge the other). During chaotic interactions involving three or more objects, the loss or dissipation of orbital energy often results in the formation of eccentric two-body captures (e.g. 1992ApJ...385..604K; 2014ApJ...784...71S; 2016ApJ...823..113P; 2017ApJ...846...36S). A capture refers here to a scenario involving a very close approach between two objects with such a small pericenter distance that the energy loss over one orbit is large enough for the two objects to quickly inspiral and detach from the rest of the -body system. Such captures are well known and studied in the single-single case (e.g. Hansen:1972il; 1975MNRAS.172P..15F; 1977ApJ...213..183P; 1985AcA....35..401G; 1985AcA....35..119G; 1986AcA....36..181G; 1986ApJ...310..176L; 1993ApJ...418..147L). Their outcome could have interesting observational consequences, from the formation of transients (e.g. 2016ApJ...823..113P), to compact mass transferring binaries (e.g. 1975MNRAS.172P..15F; 1975ApJ...199L.143C). Interestingly, recent studies indicate that such captures form at a higher rate during few-body interactions, compared to single-single interactions. For example, it was recently shown by 2014ApJ...784...71S, that the rate of eccentric binary black hole (BBH) mergers forming through captures mediated by gravitational wave emission likely is dominated by three-body interactions, and not single-single interactions. Similar eccentric mergers can also form through tidal captures in three-body interactions, as shown by 2008MNRAS.384..376G; 2010MNRAS.402..105G; 2017ApJ...846...36S. The point here is that the majority of such eccentric capture mergers are likely to form in dense stellar systems, compared to say the field, and any observation of such eccentric sources will therefore be an indirect probe of the dynamical channel for BBH and other mergers and the importance of dense stellar environments. Despite their possible importance, energy loss terms are often not included in the -body equations-of-motion (EOM) (e.g. Fregeau2004). For this reason how energy losses during strong few-body encounters, including GWs and tides, affect not only the host cluster dynamics, but also the range of relevant observables, is not yet well understood. Energy dissipation from GW emission is not difficult to include in -body codes thanks to the development of the post-Newtonian (PN) formalism (e.g. 2014LRR....17....2B), and aspects of such corrections have therefore been studied. For example, using full -body simulations 2006MNRAS.371L..45K showed how large BHs can form in a GW capture run-away. Similarly, 2014ApJ...784...71S; 2017ApJ...840L..14S; 2017arXiv171107452S; 2018ApJ...853..140S performed isolated three-body scatterings which lead them to conclude that the GW captures forming during the interactions are likely to dominate the rate of eccentric BBH mergers forming in globular clusters (GCs) observable by the ‘Laser Interferometer Gravitational-Wave Observatory’ (LIGO). A monte-carlo (MC) approach for studying the evolution of GCs including scatterings up to binary-binary interactions with PN terms was recently presented by 2017arXiv171204937R, along with a similar study by 2017arXiv171206186S, who both confirmed that GW emission in the -body EOM is crucial for probing the population of eccentric BBH mergers forming in clusters (2017arXiv171107452S). Including tides is significantly more difficult than gravitational wave emission. This is due not only to our limited understanding of stellar structure and the mechanism(s) via which tides are excited and subsequently dissipated, but also because tidal effects are extremely time consuming to computationally evolve in an -body code. Some few-body studies have been done using full hydro dynamics (e.g. 2010MNRAS.402..105G), but doing large systematic studies are not yet possible due to computational limits. Other methods for studying the effects from dynamical tides include the impulsive approximation, where tidal energy and angular momentum losses are included by simply correcting the velocity vectors at pericenter ‘by hand’ every time two of the objects pass very close to each other (e.g. 2006MNRAS.372..467B). A similar approach was also used by Mardling:2001dl, and does indeed work. But, making such discontinuous corrections to the -body system often lead to poor performance and complicated decision making. Finally, other approaches include only solving for the evolution of a subset of the tidal modes, which can be done in both linear tidal theory (Mardling:1995hx) using the Press and Teukolsky (PT) approach (1977ApJ...213..183P), and non-linearly using the so-called affine model (1985MNRAS.212...23C; Luminet:1986cha; 1992ApJ...385..604K; 1992MNRAS.258..715K; 1993ApJS...88..205L; 1993ApJ...406L..63L; 1994ApJ...420..811L; 1994ApJ...423..344L; 1995ApJ...443..705L; 1995MNRAS.275..498D; 1996PThPh..96..901O; 2001ApJ...549..467I; 2003MNRAS.338..147I; 2017ApJ...846...36S); a model we will apply later in this paper. However, such prescriptions are still too computationally expensive for say parameter space studies and derivations of tidal capture cross sections (2017ApJ...846...36S); other strategies are therefore needed. In this paper we propose to include energy loss effects in few-body codes through a simple drag force term in the equations-of-motion. Many few-body codes have already been optimized to include drag forces, e.g., the 2.5 PN term that accounts for energy dissipation through the emission of GWs is no more than a simple drag force. Our ansatz in this paper is therefore to derive a general drag force that can be used to model any energy loss effects, and again, with tides as the main motivation. The only input our drag force model requires is an estimate for the amount of orbital energy lost if two of the objects undergo a near parabolic encounter. This has been calculated in several studies for tides (1977ApJ...213..183P; 1986ApJ...310..176L), and fitting formulae have also been provided to speed up these calculations (e.g. 1985AcA....35..401G; 1993A&A...280..174P). As illustrated in this paper, the use of such fitting formulae together with our proposed drag force model allows one to quickly evolve few-body systems with both energy losses from tides and GW emission. We note here that our model does not give any new insight into the two-body tidal problem, but it will be able to provide insight into how especially tides affect the evolution of chaotic few-body interactions. For that reason, our model has the same limitations as the two-body tidal problem, e.g., we are not able to predict what happens after a tidal capture; do the two objects merge or do they form a stable binary? However, what we are able to accurately probe and resolve the number of tidal captures forming in chaotic few-body interactions. We illustrate this in a few examples, by performing controlled two-body and three-body experiments with different tidal implementations, including our proposed drag force model. In the near future we plan to include this model into the MOCCA (MOnte Carlo Cluster simulAtor) code (Hypki2013; Giersz2013), which will allow us to perform systematic studies of how tidal energy losses in chaotic interactions could affect observables and feedback in to the underlying host cluster dynamics. These are key questions that have to be addressed, as new searches for transient phenomena will soon be monitoring the sky, including LSST (2009arXiv0912.0201L), JWST (2006SSRv..123..485G), and WFIRST (2013arXiv1305.5422S). The paper is organized as follows. In Section 2 we present our drag force model, and describe how to normalize it for different energy loss mechanisms. We especially discuss how to apply it for describing tidal energy losses, which is the main motivator for this paper. A short step-by-step description of how to implement the model in an -body code is also given. In Section 2 we numerically evolve a few two-body and three-body scattering experiments involving tidal and compact objects with the inclusion of our proposed drag force model. We especially compare our drag force results with other tidal prescriptions, including the impulsive and affine tidal approximations. Conclusions are given in Section LABEL:sec:conclusions. 2 Drag Force Model In this section we describe and derive our proposed tidal drag force model, that in principle can be used to dynamically evolve chaotic few-body systems with the inclusion of any type of energy loss mechanism; however, our main motivation is how tidal effects impact the evolution. In short, our approach is to model orbital energy losses by introducing a drag force that acts against the relative motion between any pair of objects in the few-body system. For deriving the drag force, we assume that the largest energy loss occurs during close pairwise encounters, and that these encounters can be considered as isolated two-body systems during the period where most of the energy is lost (see Figure 1). This is an excellent assumption, as basically all of the relevant energy loss mechanisms depend steeply on the relative distance between the objects, implying that most of the energy loss in few-body systems do indeed take place during close pairwise encounters (e.g. 2017ApJ...846...36S). The amount of energy that is lost over a single close passage for an isolated two-body system has been studied extensively in the literature, both for GWs (Peters:1964bc; Hansen:1972il; 1977ApJ...216..610T) and tides (1977ApJ...213..183P; 1985AcA....35..401G; 1985AcA....35..119G). Following our assumption that every object pair in the few-body interaction can be treated as an isolated binary when modeling the energy loss, this allows us to derive the normalization of the drag force, a computation has to be done at each time step. The functional form of the drag force that controls how energy is lost over a given orbit has to be such that most of the loss takes place near the pericenter of the two considered objects. This emulates how the energy is actually lost in many mechanisms, and it naturally makes the assumption of pairwise two-body isolation a good approximation for the purpose of modeling energy losses. We note that with such a drag force the pairwise energy loss will be close to that found from the impulsive approach, where the individual velocity vectors are ‘corrected by hand’ at each pericenter passage (e.g. 2006MNRAS.372..467B). However, the loss of energy due to the drag force will happen continuously over the orbit, which makes our approach both more realistic and easy to implement in modern few-body codes. In fact, besides the implementation of the drag force into the few-body code described in 2017ApJ...846...36S that we will use in this paper, we have already successfully implemented it in the regularized code used in (e.g. 2016ApJ...831...61T) In the sections below we illustrate how a drag force with the properties described above can be constructed. We also discuss its limitations, and what can be improved. 2.1 Drag Force Functional Form We consider two objects on a Kepler orbit, bound or unbound, with an initial semi-major (SMA) and eccentricity . For this system we now consider a drag force with magnitude that acts against the orbital motion of the two objects, as further described and illustrated in Figure 1. In this picture, the two-body system will loose an amount of orbital energy per differential line element integrated along the orbit. Assuming that the change in orbital angular momentum per orbit is negligible, one can in all relevant cases approximate the total energy loss over one orbit, denoted by , by the following integral (see 1977ApJ...216..610T for a similar procedure applied to GW energy losses), where is the differential change in time, is the true anomaly, and for a bound orbit and for an unbound orbit (see Figure 1). The term is simply the relative velocity between the two interacting objects, denoted by , which can be written as, where is the total mass of the two interacting objects, referred to as and , and is their relative distance. The term can be derived from Kepler’s relation , from which it follows, To proceed we now have to chose a functional form for the drag force, . As described, the form should both be simple to implement in a few-body code, possibly similar to the 2.5 PN drag force that has been successfully implemented in many recent few-body codes, while ensuring that most of the energy loss happens at pericenter. A first proposed form could be a force that is , where is some power; however, in this case one finds that the integral in Equation \eqrefeq:intDE does not have an analytical solution for any , including . This ‘problem’ relates to the fact that the circumference of an ellipse cannot be written out in a closed form. This is why ‘elliptical’ integrals always have to be solved numerically. However, if we instead choose a force that is , then the integral in Equation \eqrefeq:intDE can be written out in closed form for any , which allows us to analytically estimate the drag force normalization, or coefficient. This leads to a very fast derivation of the drag force per time step. In this paper we therefore choose to work with the following drag force, where is a normalization factor that to leading order depends on the orbital parameters for the two-body system and the considered energy loss mechanism. Although this choice of drag force could seem arbitrary, we note that the 2.5 PN drag force is exactly of this type with . Therefore, a code that is optimized to run with PN terms, should have no problem in evolving the system with our proposed drag force. Below we illustrate how to estimate the drag force coefficient . 2.2 Drag Force Normalization Coefficient The coefficient of the drag force introduced in the above Equation \eqrefeq:DF_eq, can be estimated using Equation \eqrefeq:intDE assuming that is known a priori for the considered two-body system. After some algebraic manipulations we find from solving equation \eqrefeq:intDE with our proposed drag force from Equation \eqrefeq:DF_eq that, where is the solution to the following integral This factor can be written out in closed form for any value of . In this paper we will study the performance for two different values of , namely for and . For these two cases evaluates to,

Teachers are best off using Khan Academy's 7th Grade Math content to support their own instructional practices. Before assigning students a lesson on a specific topic, it would be good to check out some of the practice exercises and videos first. The lessons here can work well as a preview to a given unit in class -- students can build context for what they're about to learn. Alternatively, teachers can assign students specific content as part of the 7th Grade Mission. For whole-class use, some videos (like those covering the number system) will be very helpful to students. However, some of the more specific videos (like those in the probability lessons) offer some procedural examples but might not offer as much help to kids who are stuck. Whether using the 7th Grade Mission or the tutorials, some of the conversational vocabulary in the videos might go beyond what kids at this age are used to hearing and reading. By trying out the content and watching the accompanying videos beforehand, teachers can better prepare their own mini-lessons, using the site as a nice complement to classroom instruction. Full Disclosure: Khan Academy and Common Sense Education share funders; however, those relationships do not impact Common Sense Education's editorial independence and this learning rating. Key Standards Supported Expressions And Equations Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Ratios And Proportional Relationships Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Statistics And Probability Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. The Number System Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Apply properties of operations as strategies to add and subtract rational numbers. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real- world contexts.