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We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you! Presentation is loading. Please wait. Published byMargaret Wellford Modified about 1 year ago Right Triangles Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT BACKNEXT Click one of the buttons below or press the enter key © 2002 East Los Angeles College. All rights reserved. Consider the Right Triangle. EXIT BACKNEXT If we draw a vertical line from vertex C to a point D on our base, we form other right triangles. EXIT BACKNEXT We have now created three right triangles These triangles are all similar! EXIT BACKNEXT Recall that similar triangles have congruent (equal measure) corresponding angles. EXIT BACKNEXT We know the following EXIT BACKNEXT Similarly, EXIT BACKNEXT Our picture becomes... EXIT BACKNEXT Notice we can dissect this right triangle. We must rotate the first right triangle ¼ turn clockwise so the two triangles have the same alignment. EXIT BACKNEXT Since these triangles are similar, the following properties can be used. EXIT BACKNEXT It can be shown that the original right triangle ABC is similar to the smaller two right triangles. EXIT BACKNEXT If we separate the figure into three triangles and use the same alignment for all three we get... EXIT BACKNEXT Similar proportions can be created. CAD B A C D B C EXIT BACKNEXT Example 1) Determine the value for X EXIT BACKNEXT We really have EXIT BACKNEXT Example 2) EXIT BACKNEXT Example 3) EXIT BACKNEXT Example 4) EXIT BACKNEXT Summary EXIT BACKNEXT End of Right Triangles Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Us At: Our Website: EXIT BACKNEXT Similar Triangles I Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT BACKNEXT © 2002 East. A triangle with at least two sides congruent is called an Isosceles Triangle. bc a In this triangle, sides b and c are congruent. What is similar about these objects? What do we need to pay attention to when objects are rotated? The shapes below are examples of regular polygons. Look at the sides and angles of each shape. Octagon rectanglehexagon triangle The following shapes. Proving the Distance Formula. What is the distance between points A and B? We can use the Pythagorean Theorem to find the distance. Honors Geometry Section 4.1 Congruent Polygons. To name a polygon, start at any vertex and go around the figure, either clockwise or counterclockwise, Lesson 4 Contents Example 1Translate a Figure Example 2Find a Translation Matrix Example 3Dilation Example 4Reflection Example 5Rotation. G.9 Quadrilaterals Part 1 Parallelograms Modified by Lisa Palen. David Weeks, Mathsmadeeasy Graphs for GCSE Maths. MODULE I VOCABULARY PART II. FIGURE IT OUT! The first new term we will discuss is distance. Distance a measurement of the length of how far something. FRACTIONS Lets draw a rectangle, and divide it in two equal parts... Index FAQ Basic Functions Polynomials Exponential Functions Trigonometric Functions Trigonometric Identities The Number e. Solving 2-D Vectors Graphically Physics. Why? O You can and people have accurately represented a situation by drawing vectors to scale in order to recreate. CCSSM Stage 3 Companion Text Lesson 3-O. Warm-Up 1.Describe the translation that moves A(–3, 4) to A'(1, 3). 2.Describe the type of reflection that moves. Geometry. Geometry Part II Similar Triangles By Dick Gill, Julia Arnold and Marcia Tharp for Elementary Algebra Math 03 online. Transformational Geometry Math 314. Game Plan Distortions Distortions Orientations Orientations Parallel Path Parallel Path Translation Translation Rotation. Copyright © Cengage Learning. All rights reserved. 6.2 Law of Cosines. The Ambiguous Case of the Law of Sines. This is the SSA case of an oblique triangle. SSA means you are given two sides and the angle opposite one of them. Here we have an apple that is at a height h above the Earths surface. Trigonometric Functions Measuring Angles Areas of Sectors of Disks Definition of Basic Trigonometric Functions Trigonometric Identities. Lines of Symmetry pg. 5 (LT #1). Reflection Symmetry When a graph or a picture can be folded so that both sides will perfectly match. LINE of SYMMETRY: If you have not watched the PowerPoint on the unit circle you should watch it first. After youve watched that PowerPoint you are ready for this one. If. Tangent Lines Section 2.1. Secant Line A secant line is a line that connects two points on a graph. Notice the slopes of secant lines are different depending. 7-5 The SSA Condition and HL Congruence. The abbreviations SsA and HL stand for two other combinations of measures of corresponding sides and angles in. MODULE VI LET’S GO MEASURE A KITE! AREAS OF QUADRILATERALS We have already discussed how to find the area of certain parallelograms. Today, we are going. Vertex-Edge Graphs. The Basics ❖ A vertex-edge graph is a graph that includes edges and vertices. ❖A❖An edge is a side (shown by a straight line) ❖ V❖ Honors Geometry Section 8.4 The Side-Splitting Theorem. Learn to recognize, describe, and show transformations. Course Translations, Reflections, and Rotations. The Law of Cosines. Let's consider types of triangles with the three pieces of information shown below. SAS You may have a side, an angle, and then another. © 2016 SlidePlayer.com Inc. All rights reserved.

Study tips and Strategy Let me tell you first, in AIEEE, everything matters is marks whether you got the answer or not. No worry whether you used some logic or went through a long solution. The most thing that matter in this exam is your time. Time plays a very major role in securing a good rank in AIEEE. Most of the questions that are asked in AIEEE particularly physics and chemistry are very simple. You can consider AIEEE as 20-20 match, no matter how you solve, but take such a short that aims at your target. Preparing for AIEEE is almost similar as the preparation for JEE. Many IIT-JEE (Indian Institute of Technology - Joint Entrance Examination) toppers fail to do well in the AIEEE exam most of the time. What IIT-JEE demands is, building concepts and proper application of the concept, where as AIEEE demands speed and accuracy in the stipulated time, besides in-depth knowledge of the subject that JEE demands. Getting through the All India Engineering Entrance Examination (AIEEE) is not all about conceptual clarity and knowledge of the appropriate formulae and theories, as required in IIT-JEE. But here something you have to keep in mind and that is your problem solving speed. Examination Strategy :(How to prepare for AIEEE during the last precious months before exam) * First of all in AIEEE you don't need to have the deep knowledge of every topic as required in IIT-JEE. But it is the major requirement that you some how remember the formulas and some important properties. Try to maintain formula list for each chapters, so that at the end time, you just go through the formula list. This will really help you in the exams. * Buy some books with sample papers of AIEEE and some past year papers (you can get many of them in the internet also). * Solve free online tests under similar time constraints or downloade a few AIEEE papers and do them as timed tests. Do it sincerly. * Develop an habbit of judgeing the question .You should be able to judge the question whether you should go for it or not.This is very important thing.There will always few question mostly in mathematics section, which will be very tough.If you put your head in those question,you will waste your time. * Always try to estimate the time that the question will take to solve. If it is lenghty then, but you are confident that you can solve it, then use some special symbols for that question,leave those question and search for the easy questions(less time taking). And at last solve those special marked questions. and timed separately. * Always try to solve the question by some arguments. a) Chemistry . everything. due to lack of confidence. as most of the things are related to that book only. * Generally in the examination.. Examination Hour Strategy: * Attempt that paper only which you feel easy. Each part should be practiced. * Avoid negative marking . Try to solve the three parts of the paper in given time interval .. and focus instead on objective. Remember every second counts in the exam. But if you start other side. You can do these questions at the end * While solving any paper first of all divide all the AIEEE Model Papers.The above two points are very important because this is the time where you can test everything. so that whenever you sit for the main exam you should be prepared with every tools to attemp the question * Do not attempt lengthy questions at the start.. This is because your strengths and weaknesses in different subject areas vary. your strategy.. but the mathematics part is some what tough.. Ultimetly your main goal should be to solve the questions in less time counts. Try to argue your own answers.So.. because once you solve consecutively 5-10 questions. chemistry part is very easy. then you will acheive the rythm and confidence.50 Minutes c) Mathematics . then later you will get difficulties in easy problems also. you use for practice. always test some new ways of solving the questions. your tricks to solve the question. numerical questions to practice as many types of questions as you can. logics.70 Minutes If you follow this time frame in the actual exam. or putting some values for some parameter of the question.40 Minutes b) Physices . into the following three parts. * You should also go through the NCERT books.A 1 marks decrease in your score can make a difference of hundred rank. you will still be left with a comfortable cushion of 15 minutes. * Increase your accuracy and speed. then use some special symbols for that question.70 Minutes The extra 20 min.This is very important thing. but you are confident that you can solve it. * Develop an habbit of judgeing the question . * Always avoid negative marking. which ultimetly is the waste of time. that you have left.leave those question and search for the easy questions(less time taking).you will waste your time and you will not get any thing. that are left solve those questions.If you put your head in those question. if the front paper contains 8 questions then you move to your OMR answer sheet 8 times.40 Minutes b) Physices . And at last solve those special marked questions. * If you are stuck in some question. then without any delay switch to some other problem. So. later if you get time then solve that question .50 Minutes c) Mathematics . fill the bubbles correctly. and then tranfer your answer to the OMR sheet. So for example. * It is the common habit of the students to transfer the answers to the ovals as soon as it is solved. * Always try to estimate the time that the question will take to solve.You should be able to judge the question whether you should go for it or not.a) Chemistry . * Let me tell you that 1 mark difference can lead to huge rank difference. which will be very tough.There will always few question mostly in mathematics section. beacause of long solution or some other reason. If it is lenghty then. Your strategy should be like : solve the whole page (1 page).

Shear Force : A good example of shear force is seen with a simple scissors. The two handles put force in different directions on the pin that holds the two parts together. The force applied to the pin is called shear force. COMPRESSION. Consequently, what is the definition of shear stress? Shear stress, force tending to cause deformation of a material by slippage along a plane or planes parallel to the imposed stress. The resultant shear is of great importance in nature, being intimately related to the downslope movement of earth materials and to earthquakes. What is the definition of a shear force? Shearing forces are unaligned forces pushing one part of a body in one specific direction, and another part of the body in the opposite direction. William A. Nash defines shear force in terms of planes: “If a plane is passed through a body, a force acting along this plane is called a shear force or shearing force.” What are the units of shear strain? Measures of strain are often expressed in parts per million or microstrains. The true shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. What is a shear in weather? Wind shear refers to a change in wind speed or direction with height in the atmosphere. Wind shear can also refer to a rapid change in winds over a short horizontal distance experienced by aircraft, conditions that can cause a rapid change in lift, and thus the altitude, of the aircraft. What is the shear modulus? In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain: where = shear stress is the force which acts is the area on which the force acts = shear strain. What is an example of torsion? The twisting force acting on the object is known as torque, and the resulting stress is known as Shear stress. A common example of torsion in engineering is when a transmission drive shaft (such as in an automobile) receives a turning force from its power source (the engine). What is the difference between static and dynamic loads? A static load bearing is the weight applied without any build up of energy, and therefore is to remain motionless. Force, pressure, and gravity remain static or are applied gradually. A dynamic load bearing is measured by the application of rapid force or pressure to an object. What is point of Contraflexure in beams? In a bending beam, a point is known as a point of contraflexure if it is a location at which no bending occurs. In a bending moment diagram, it is the point at which the bending moment curve intersects with the zero line. In other words where the bending moment changes its sign from negative to positive or vice versa. What is a shearing injury? Diffuse axonal injury is the shearing (tearing) of the brain’s long connecting nerve fibers (axons) that happens when the brain is injured as it shifts and rotates inside the bony skull. DAI usually causes coma and injury to many different parts of the brain. What is an example of a compression? Compression crushed this can. Licensed from iStockPhoto. noun. The definition of compression is the action or state of being squished down or made smaller or more pressed together. When a pile of material is squished together and made smaller and more dense, this is an example of compression. What is meant by bending moment? A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. Beams can also have one end fixed and one end simply supported. What is static load? Dead loads are static forces that are relatively constant for an extended time. They can be in tension or compression. The term can refer to a laboratory test method or to the normal usage of a material or structure. Live loads are usually unstable or moving loads. What is the difference between tension and compression? Both are made up of a coil spring that is devised for elasticity and strength, but that is where their likeness ends. The main difference is that tension springs are meant to hold two things together while compression springs are designed to keep components from coming together. What is a compressive force? Compression Force is the application of power, pressure, or exertion against an object that causes it to become squeezed, squashed, or compacted. Objects routinely subjected to compression forces include columns, gaskets, disc brakes, and the components of fuel cells. What do you mean by bending stress? Bending stress is a more specific type of normal stress. When a beam experiences load like that shown in figure one the top fibers of the beam undergo a normal compressive stress. The stress at the horizontal plane of the neutral is zero. The bottom fibers of the beam undergo a normal tensile stress. What is the matrix for a shear? In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value. What is a shear failure? In engineering, shear strength is the strength of a material or component against the type of yield or structural failure where the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is parallel to the direction of the force. What is meant by shear wall? Shear wall is a structural member used to resist lateral forces i.e. parallel to the plane of the wall. For slender walls where the bending deformation is more, Shear wall resists the loads due to Cantilever Action. In other words, Shear walls are vertical elements of the horizontal force resisting system. What is meant by an axial force? If the load on a structure is applied through the center of gravity of its cross section, it is called an axial load or that would be force applied to the lengthwise centerline of an object. Axial force is the compression or tension force of the member. What is a shear moment? Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of a structural element such as a beam. What is shearing in engineering? Shearing, also known as die cutting, is a process which cuts stock without the formation of chips or the use of burning or melting. Strictly speaking, if the cutting blades are straight the process is called shearing; if the cutting blades are curved then they are shearing-type operations. What is the shear force in a beam? Shear force is the force in the beam acting perpendicular to its longitudinal (x) axis. For design purposes, the beam’s ability to resist shear force is more important than its ability to resist an axial force. Axial force is the force in the beam acting parallel to the longitudinal axis. What is torsion in engineering? Definition of torsion. 1 : the twisting or wrenching of a body by the exertion of forces tending to turn one end or part about a longitudinal axis while the other is held fast or turned in the opposite direction; also : the state of being twisted. 2 : the twisting of a bodily organ or part on its own axis.

However, this is not correct. The groups are different with regard to what is being studied. on follow-up testing and treatment. Example: In a t-test for a sample mean µ, with null hypothesis""µ = 0"and alternate hypothesis"µ > 0", we may talk about the Type II error relative to the general alternate this content You just assume this is the case in order to perform this test because we have to start from somewhere. Joint Statistical Papers. Before you even start the study you may do power calculations based on projections. A low number of false negatives is an indicator of the efficiency of spam filtering. check over here Similar considerations hold for setting confidence levels for confidence intervals. This value indicates that there is not strong evidence against the null hypothesis, as observed previously with the t-test. An alternative hypothesis may be one-sided or two-sided. Lubin, A., "The Interpretation of Significant Interaction", Educational and Psychological Measurement, Vol.21, No.4, (Winter 1961), pp.807–817. Is it dangerous to use default router admin passwords if only trusted users are allowed on the network? Badbox when using package todonotes and command missingfigure How to deal with being asked to smile more? Power Of The Test In the ideal world, we would be able to define a "perfectly" random sample, the most appropriate test and one definitive conclusion. We could decrease the value of alpha from 0.05 to 0.01, corresponding to a 99% level of confidence. Type 1 Error Example Instead, α is the probability of a Type I error given that the null hypothesis is true. If the significance level for the hypothesis test is .05, then use confidence level 95% for the confidence interval.) Type II Error Not rejecting the null hypothesis when in fact the Assuming each pair is independent, the null hypothesis follows the distribution B(n,1/2), where n is the number of pairs where some difference is observed. Matched Pairs In many experiments, one wishes to compare measurements from two populations. Type 1 Error Calculator Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. When comparing two means, concluding the means were different when in reality they were not different would be a Type I error; concluding the means were not different when in reality The Skeptic Encyclopedia of Pseudoscience 2 volume set. If this is the case, then the conclusion that physicians intend to spend less time with obese patients is in error. http://stats.stackexchange.com/questions/61638/what-is-the-relation-of-the-significance-level-alpha-to-the-type-1-error-alpha ABC-CLIO. Type 2 Error Definition Similar problems can occur with antitrojan or antispyware software. Probability Of Type 1 Error Please select a newsletter. Correct outcome True positive Convicted! news A threshold value can be varied to make the test more restrictive or more sensitive, with the more restrictive tests increasing the risk of rejecting true positives, and the more sensitive I edited my question accordingly. –what Jun 13 '13 at 10:00 You seem to be talking about the same thing both times; in some circumstances, you may see people A typeII error occurs when failing to detect an effect (adding fluoride to toothpaste protects against cavities) that is present. Probability Of Type 2 Error The Sign Test Another method of analysis for matched pairs data is a distribution-free test known as the sign test. A positive correct outcome occurs when convicting a guilty person. Why does removing Iceweasel nuke GNOME? http://u2commerce.com/type-1/type-ii-error-statistical-significance.html RETURN TO MAIN PAGE. Retrieved 10 January 2011. ^ a b Neyman, J.; Pearson, E.S. (1967) . "On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference, Part I". Type 3 Error Also from About.com: Verywell, The Balance & Lifewire Malware The term "false positive" is also used when antivirus software wrongly classifies an innocuous file as a virus. Based solely on this data our conclusion would be that there is at least a 95% chance on subsequent flips of the coin that heads will show up significantly more often In actuality the chance of the null hypothesis being true is not 3% like we calculated, but is actually 100%. It can be thought of as a false negative study result. Type 1 Error Psychology It does NOT imply a "meaningful" or "important" difference; that is for you to decide when considering the real-world relevance of your result. Inventory control An automated inventory control system that rejects high-quality goods of a consignment commits a typeI error, while a system that accepts low-quality goods commits a typeII error. A medical researcher wants to compare the effectiveness of two medications. They also cause women unneeded anxiety. check my blog The probability that this is a mistake -- that, in fact, the null hypothesis is true given the z-statistic -- is less than 0.01. Retrieved 2010-05-23. For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug. Download a free trial here. Please Share This Page with Friends:FacebookTwitterGoogleEmail 6 thoughts on “p-Value, Statistical Significance & Types of Error” Aliya says: December 3, 2015 at 5:54 am Thanks a lot. The company chooses a random sample of 100 individuals who have used the cream, and determines that the mean recovery time for these individuals was 28.5 days. In the test score example above, the P-value is 0.0082, so the probability of observing such a value by chance is less that 0.01, and the result is significant at the Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the

The effect of the relativity correction on the law of reflection from a moving mirror MetadataShow full item record This thesis utilizes the Lorentz transformation formulas from the special theory of relativity in dealing with the law of reflection from a moving mirror. It was found necessary in the proof to consider the source of the light ray and its final destination. For this reason the mirror was taken to be one side of a square. The light source was at one corner of the opposite side of the square and the light ray was so directed that it would be reflected to the other corner. For a physical treatment of the problem a cubical box would be necessary, but for the mathematical treatment the square was sufficient. In the first part of the thesis the square was moving away from the earth at a fixed angle and a constant velocity. The earth was considered to be the stationary system. If an observer was placed on the square he would note that the light would be reflected to the corner of the square. The problem was to investigate the effect of the relativity correction on the law of reflection from a moving mirror and as a subsidiary problem the effect of a moving or stationary ether upon reflection from a mirror at rest. A medium, called the ether, for the light to pass through was brought into the problem. It was assumed stationary with respect to the earth. The first formula derived was the law of reflection from a moving mirror using the methods of classical physics. This law was general in that it did not involve the source of light. It dealt entirely with the angles of incidence and reflection, and the velocity of the mirror with respect to the earth. Next, a classical treatment of the moving square was made. This was treated as a geometrical problem. It was assumed for this case and the next that the quantities that could be measured physically were the velocity of the square, the angle at which it was moving, and the angle of incidence from the point of view of an observer on the mirror. It was assumed in the construction that the reflected ray would hit the corner of the square. Thus the formula for the angle of reflection was derived. For this angle to be correct it would have to agree with the angle given by the law of reflection from a moving mirror. To show that the two angles were not identical a numerical example was worked out. The angles were widely divergent. Then the Lorentz transformation formulas were applied. Actually, the only formula needed was the one which showed how much a length would apparently be shortened in its direction of motion. This shortening is from the stationary observer's point of view. The effect of this apparent shortening was to alter the angles in the geometric construction. The equations for the angles of incidence and reflection were then formulated in terms of the quantities assumed to be known. This angle of reflection to be correct must agree with the angle from the law of reflection from a moving mirror for the same angle of incidence. The relativity correction was applied to the law of reflection. It was assumed that the two angles of reflection were identically equal. It was then proven that this assumption was correct. It should be noted that the normal that the stationary observer uses for calculating the angles of incidence and reflection is the normal from his point of view and not that of the observer on the square. The stationary ether had no effect upon the outcome of the problem. In the second part of the thesis the square was considered to be stationary and the ether moving with a constant velocity. The problem was to find out what effect this moving ether has upon the light being reflected to the corner of the square. This problem was divided into two cases which were assumed to be independent. They were (1) the effect of the ether upon the light and (2) the effect of the ether upon the mirror. In the first case the ether can change the velocity and the direction of the light ray. By taking a particular orientation of the square it was shown that if a transformation was applied that would lead to the correct results the velocity of the ether would have to reduce to zero. However, this gives a result that the observer has already obtained without considering a moving ether and transformations. In the second case the ether can distort the square. To get the correct results the transformation would have to bring it back to a square. Again the velocity of the ether would have to reduce to zero. The observer would have measured his apparatus as a square without taking into account any distortions. Thus in both cases a moving ether is artificial and meaningless. Therefore in both parts of the problem the ether whether moving or stationary adds nothing to the solution. It is well known that light reflects from a mirror in a square according to the formula i = r and the light is reflected to the corner of the square if the mirror is at rest relative to an observer. If this is true then light reflected from a mirror in a square in motion relative to an observer should lead the observer from his calculations to predict that the light will be reflected to the corner of the square. This paper shows that it is necessary to apply the relativity correction to the law of reflection from a moving mirror to have theory agree with facts. Thesis (M.A.)--Boston University RightsBased on investigation of the BU Libraries' staff, this work is free of known copyright restrictions

« ΠροηγούμενηΣυνέχεια » For let a plane XY intersect all the arms of the plane angles on the same side of the vertex at the points A, B, C, D, E: and let AB, BC, CD, DE, EA be the common sections of the plane XY with the planes of the several angles. Within the polygon ABCDE take any point o; and join O to each of the vertices of the polygon. Then since the _8 SAE, SAB, EAB form the trihedral angle A, .. the <* SAE, SAB are together greater than the 2 EAB; the Ľ SAE, SAB are together greater than the < OAE, OAB. Similarly, the 4* SBA, SBC are together greater than the 2* OBA, OBC: and so on, for each of the angular points of the polygon. Thus by addition, the sum of the base angles of the triangles whose vertices are at s, is greater than the sum of the base angles of the triangles whose vertices are at O. But these two systems of triangles are equal in number; .:. the sum of all the angles of the one system is equal to the sum of all the angles of the other. It follows that the sum of the vertical angles at S is less than the sum of the vertical angles at O. But the sum of the angles at O is four rt. angles ; .. the sum of the angles at S is less than four rt. angles. Q.E.D. NOTE. This proposition was not given in this form by Euclid, who established its truth only in the case of trihedral angles. The above demonstration, however, applies to all cases in which the polygon ABCDE is convex, but it must be observed that without this condition the proposition is not necessarily true. A solid angle is convex when it lies entirely on one side of each of the infinite planes which pass through its plane angles. If this is the case, the polygon ABCDE will have no re-entrant angle. And it is clear that it would not be possible to apply xi. 20 to a vertex at which a re-entrant angle existed. EXERCISES ON BOOK XI. 1. Equal straight lines drawn to a plane from a point without it have equal projections on that plane. 2. If S is the centre of the circle circumscribed about the triangle ABC, and if SP is drawn perpendicular to the plane of the triangle, shew that any point in SP is equidistant from the vertices of the triangle. 3. Find the locus of points in space equidistant from three given points. 4. From Example 2 deduce a practical method of drawing a perpendicular from a given point to a plane, having given ruler, compasses, and a straight rod longer than the required perpendicular. 5. Give a geometrical construction for drawing a straight line equally inclined to three straight lines which meet in a point, but are not in the same plane. 6. In a gauche quadrilateral (that is, a quadrilateral whose sides are not in the same plane) if the middle points of adjacent sides are joined, the figure thus formed is a parallelogram. 7. AB and AC are two straight lines intersecting at right angles, and from B a perpendicular BD is drawn to the plane in which they are: shew that AD is perpendicular to AC. 8. If two intersecting planes are cut by two parallel planes, the lines of section of the first pair with each of the second pair contain equal angles. 9. If a straight line is parallel to a plane, shew that any plane passing through the given straight line will intersect the given plane in a line of section which is parallel to the given line. 10. Two intersecting planes pass one through each of two parallel straight lines; shew that the common section of the planes is parallel to the given lines. 11. If a straight line is parallel to each of two intersecting planes, it is also parallel to the common section of the planes. 12. Through a given point in space draw a straight line to intersect each of two given straight lines which are not in the same plane. 13. If AB, BC, CD are straight lines not all in one plane, shew that a plane which passes through the middle point of each one of them is parallel both to AC and BD. 14. From a given point A a perpendicular AB is drawn to a plane XY; and a second perpendicular AE is drawn to a straight line CD in the plane XY: shew that EB is perpendicular to CD. 15. From a point A two perpendiculars AP, AQ are drawn one to each of two intersecting planes: shew that the common section of these planes is perpendicular to the plane of AP, AQ. 16. From A, a point in one of two given intersecting planes, AP is drawn perpendicular to the first plane, and AQ perpendicular to the second : if these perpendiculars meet the second plane at P and Q, shew that PQ is perpendicular to the common section of the two planes. 17. A, B, C, D are four points not in one plane, shew that the four angles of the gauche quadrilateral ABCD (see Ex. 6, p. 444) are together less than four right angles. 18. OA, OB, OC are three straight lines drawn from a given point O not in the same plane, and OX is another straight line within the solid angle formed by OA, OB, OC: shew that (i) the sum of the angles AOX, BOX, COX is greater than half the sum of the angles AOB, BOC, COA. (ii) the sum of the angles AOX, COX is less than the sum of the angles AOB, COB. (iii) the sum of the angles AOX, BOX, COX is less than the sum of the angles AOB, BOC, COA. 19. OA, OB, OC are three straight lines forming a solid angle at O, and OX bisects the plane angle AOB ; shew that the angle XOC is less than half the sum of the angles AOC, BOC. 20. If a point is equidistant from the angles of a right-angled triangle and not in the plane of the triangle, the line joining it with the middle point of the hypotenuse is perpendicular to the plane of the triangle. 21. The angle which a straight line makes with its projection on a plane is less than that which it makes with any other straight line which meets it in that plane. 22. Find a point in a given plane .such that the sum of its distances from two given points (not in the plane but on the same side of it) may be a minimum. 23. If two straight lines in one plane are equally inclined to another plane, they will be equally inclined to the common section of these planes. 24. PA, PB, PC are three concurrent straight lines, each of which is at right angles to the other two: PX, PY, PZ are perpendiculars drawn from P to BC, CA, AB respectively. Shew that XYZ is the pedal triangle of the triangle ABC. 25. PA, PB, PC are three concurrent straight lines, each of which is at right angles to the other two, and from P a perpendicular PO is drawn to the plane of ABC: shew that 0 is the orthocentre of the triangle ABC. THEOREMS AND EXAMPLES ON BOOK XI. DEFINITIONS. (i) Lines which are drawn on a plane, or through which a plane may be made to pass, are said to be co-planar. (ii) The projection of a line on a plane is the locus of the feet of perpendiculars drawn from all points in the given line to the plane. THEOREM 1. a straight line. The projection of a straight line on a plane is itself Let AB be the given st. line, and XY the given plane. It is required to shew that the locus of p is a st. line. XI. 7. .. the point p is in the common section of the planes Ab, XY; that is, p is in the st. line ab. But p is any point in the projection of AB, :: the projection of AB is the st. line ab. Q.E.D. THEOREM 2. Draw a perpendicular to each of two straight lines which are not in the same plane. Prove that this perpendicular is the shortest distance between the two lines. Let AB and CD be the two straight lines, not in the same plane. (i) It is required to draw a st. line perp. to each of them. Through E, any point in AB, draw EF parł to CD. Let XY be the plane which passes through AB, EF. From H, any point in CD, draw HK perp. to the plane XY. xi. 11. And through K, draw KQ par to EF, cutting AB at Q. Then KQ is also parl to CD; and CD, HK, KQ are in one plane. XI. 7. From Q, draw QP par to HK to meet CD at P. Then shall PQ be perp. to both AB and CD. For, since HK is perp. to the plane XY, and PQ is par to HK, Constr. :. PQ is perp. to the plane XY; XI. 8. :: PQ is perp. to AB, which meets it in that plane. XI. Def. 1. For a similar reason PQ is perp. to QK, :: PQ is also perp. to CD, which is parl to QK. (ii) It is required to shew that PQ is the least of all st. lines drawn from AB to CD. Take He, any other st. line drawn from AB to CD. Ex. 1, p. 429. :: HE is also greater than PQ.

ENGINEERING MATHEMATICS FORMULA PDF document issued by the Department of Engineering, but obviously reflects the particular Speigel, M.R., Mathematical Handbook of Formulas and Tables. math to math for advanced undergraduates in engineering, economics, physical sciences, and mathematics. The ebook contains hundreds of formulas, tables. Third edition as the Newnes Engineering Mathematics Pocket Book Fourth edition . formulae, definitions, tables and general information needed during. |Language:||English, Spanish, Japanese| |Genre:||Academic & Education| |ePub File Size:||21.73 MB| |PDF File Size:||8.44 MB| |Distribution:||Free* [*Regsitration Required]| Use differentiation and integration tables to supplement differentiation and integration techniques. Differentiation Formulas. 1. 2. 3. 4. 5. 6. 7. 8. 9. Solution of quadratic equations by formula . Basic Engineering Mathematics, 4th Edition introduces and then consolidates basic mathematical principles. Jan 19, Formula. Handbook including. Engineering. Formulae,. Mathematics,. Statistics and. Computer Algebra instruktsiya.info - pdf. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts. Ordinary differential equations[ edit ] Main articles: Ordinary differential equation and Linear differential equation An ordinary differential equation ODE is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable often denoted y , which, therefore, depends on x. Integration Formula Sheet - Chapter 7 Class 12 Formulas Thus x is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation , which may be with respect to more than one independent variable. Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and, in many cases, one may express their solutions in terms of integrals. Most ODEs that are encountered in physics are linear, and, therefore, most special functions may be defined as solutions of linear differential equations see Holonomic function. As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression , numerical methods are commonly used for solving differential equations on a computer. Partial differential equations[ edit ] Main article: Partial differential equation A partial differential equation PDE is a differential equation that contains unknown multivariable functions and their partial derivatives. This is in contrast to ordinary differential equations , which deal with functions of a single variable and their derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena in nature such as sound , heat , electrostatics , electrodynamics , fluid flow , elasticity , or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems , partial differential equations often model multidimensional systems. Follow ErForum by Email PDEs find their generalisation in stochastic partial differential equations. Non-linear differential equations[ edit ] Main article: Non-linear differential equations A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives the linearity or non-linearity in the arguments of the function are not considered here. There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory cf. Cancel reply. Please enter your comment! Please enter your name here. You have entered an incorrect email address! Get New Updates Email Alerts Enter your email address to subscribe this blog and receive notifications of new posts by email. Join With us. Engineering Mathematics Formula Sheet Today Updates. Statics and Dynamics By R. Hibbeler Book April Punmia, Ashok Kumar Jain, Arun April 8. April 7. Popular Files. January June Trending on EasyEngineering. A Systems Perspective By N April 1. Kraus Book Free Download December December December 1. Never Miss. Load more. Sponsored By.Visit website. Henry Yul De Aries. Numerical Methods: This can also be used as a review guide. The x- and y-intercepts of a graph. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity and the velocity depends on time. Exponential Functions. You should have learned the relevant theory before you apply these formulas. About the Author.

- Poisson's ratio Poisson's ratio ("ν"), named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to the applied load), divided by the relative extension strain, or axial strain (in the direction of the applied load). When a sample of material is stretched in one direction, it tends to contract (or rarely, expand) in the other two directions. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions. Poisson's ratio (ν) is a measure of this tendency. The Poisson's ratio of a stable material cannot be less than −1.0 nor greater than 0.5 due to the requirement that the shear modulusand bulk modulushave positive values. Most materials have between 0.0 and 0.5. Cork is close to 0.0, showing almost no Poisson contraction, most steels are around 0.3, and rubber is nearly incompressible and so has a Poisson ratio of nearly 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Some materials, mostly polymer foams, have a negative Poisson's ratio; if these auxetic materials are stretched in one direction, they become thicker in perpendicular directions. Assuming that the material is compressed along the axial direction: where: is the resulting Poisson's ratio,: is transverse strain (negative for axial tension, positive for axial compression): is axial strain (positive for axial tension, negative for axial compression). Cause of Poisson’s effect On the molecular level, Poisson’s effect is caused by slight movements between molecules and the stretching of molecular bonds within the material lattice to accommodate the stress. When the bonds elongate in the stress direction, they shorten in the other directions. This behavior multiplied millions of times throughout the material lattice is what drives the phenomenon. Generalized Hooke's law For an isotropic material, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axes in three dimensions. Thus it is possible to generalize Hooke's Lawinto three dimensions:: :where:, and are strain in the direction of , and axis: , and are stress in the direction of , and axis: is Young's modulus(the same in all directions: , and for isotropic materials): is Poisson's ratio (the same in all directions: , and for isotropic materials) The relative change of volume "ΔV"/"V" due to the stretch of the material can be calculated using a simplified formula (only for small deformations): where: is material volume: is material volume change: is original length, before stretch: is the change of length: If a rod with diameter (or width, or thickness) "d" and length "L" is subject to tension so that its length will change by "ΔL" then its diameter "d" will change by (the value is negative, because the diameter will decrease with increasing length): The above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used: where: is original diameter: is rod diameter change: is Poisson's ratio: is original length, before stretch: is the change of length. Orthotropic material, such as wood in which Poisson's ratio is different in each direction (x, y and z axis) the relation between Young's modulus and Poisson's ratio is described as follows: where: is a Young's modulusalong axis i: is a Poisson's ratio in plane jk Poisson's ratio values for different materials glasscomponent additions on Poisson's ratio of a specific base glass. [ [http://www.glassproperties.com/poisson_ratio/ Poisson's ratio calculation of glasses] ] ] Negative Poisson's ratio materials Some materials known as auxeticmaterials display a negative Poisson’s ratio. When subjected to strain in a longitudinal axis, the transverse strain in the material will actually be positive (i.e. it would increase in cross sectional area). For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for these bonds to stretch in the longitudinal direction, the hinges must ‘open’ in the transverse direction, effectively exhibiting a positive strain. [ [http://silver.neep.wisc.edu/~lakes/Poisson.html Negative Poisson's ratio ] ] Applications of Poisson's effect One area in which Poisson's effect has a considerable influence is in pressurized pipe flow. When the air or liquid inside a pipe is highly pressurized it exerts a uniform force on the inside of the pipe, resulting in a radial stress within the pipe material. Due to Poisson's effect, this radial stress will cause the pipe to slightly increase in diameter and decrease in length. The decrease in length, in particular, can have a noticeable effect upon the pipe joints, as the effect will accumulate for each section of pipe joined in series. A restrained joint may be pulled apart or otherwise prone to failure. [http://www.cpchem.com/hb/getdocanon.asp?doc=135&lib=CPC-Portal] Another area of application for Poisson's effect is in the realm of structural geology. Rocks, just as most materials, are subject to Poisson's effect while under stress and strain. In a geological timescale, excessive erosion or sedimentation of Earth's crust can either create or remove large vertical stresses upon the underlying rock. This rock will expand or contract in the vertical direction as a direct result of the applied stress, and it will also deform in the horizontal direction as a result of Poisson's effect. This change in strain in the horizontal direction can affect or form joints and dormant stresses in the rock. [http://www.geosc.psu.edu/~engelder/geosc465/lect18.rtf] Impulse excitation technique Coefficient of thermal expansion * [http://silver.neep.wisc.edu/~lakes/PoissonIntro.html Meaning of Poisson's ratio] * [http://silver.neep.wisc.edu/~lakes/Poisson.html Negative Poisson's ratio materials] * [http://home.um.edu.mt/auxetic More on negative Poisson's ratio materials (auxetic)] * [http://www.webelements.com/webelements/properties/text/definitions/poissons-ratio.html Poisson's ratio] Wikimedia Foundation. 2010. Look at other dictionaries: Poisson's ratio — n. [see POISSON DISTRIBUTION] Physics an elastic constant of a material equal to the ratio of contraction sideways to expansion lengthwise when the material is stretched … English World dictionary Poisson's ratio — Poisson s ratio. См. Коэффициент Пуассона. (Источник: «Металлы и сплавы. Справочник.» Под редакцией Ю.П. Солнцева; НПО Профессионал , НПО Мир и семья ; Санкт Петербург, 2003 г.) … Словарь металлургических терминов Poisson’s ratio — Puasono santykis statusas T sritis fizika atitikmenys: angl. Poisson number; Poisson’s ratio vok. Poisson Konstante, f; Poissonsche Konstante, f; Poissonsche Zahl, f rus. коэффициент поперечного сжатия, m; коэффициент Пуассона, m pranc.… … Fizikos terminų žodynas Poisson’s ratio — Puasono santykis statusas T sritis Standartizacija ir metrologija apibrėžtis Tempiamo arba gniuždomo bandinio skersinės ir išilginės santykinių deformacijų dalmens absoliučioji vertė. atitikmenys: angl. Poisson number; Poisson’s ratio vok.… … Penkiakalbis aiškinamasis metrologijos terminų žodynas Poisson’s ratio — Puasono koeficientas statusas T sritis fizika atitikmenys: angl. Poisson’s ratio vok. Poisson Konstante, f; Poissonscher Koeffizient, m rus. коэффициент Пуассона, m pranc. coefficient de Poisson, m; rapport de Poisson, m … Fizikos terminų žodynas poisson's ratio — noun also poisson ratio Usage: usually capitalized P Etymology: after S. Poisson : the ratio of transverse to longitudinal strain in a material under tension … Useful english dictionary Poisson’s ratio — Puasono koeficientas statusas T sritis Standartizacija ir metrologija apibrėžtis Tempiamų arba gniuždomų kūno sluoksnių skersinės ir išilginės deformacijų dalmens absoliučioji vertė. atitikmenys: angl. Poisson’s ratio vok. Poisson Konstante, f;… … Penkiakalbis aiškinamasis metrologijos terminų žodynas Poisson's ratio — Physics. the ratio, in an elastic body under longitudinal stress, of the transverse strain to the longitudinal strain. Also, Poisson ratio. [1925 30; see POISSON DISTRIBUTION] * * * … Universalium Poisson's ratio — noun Etymology: S. Poisson Date: 1886 the ratio of transverse to longitudinal strain in a material under tension … New Collegiate Dictionary Poisson's ratio — noun Of a material in tension or compression, the ratio of the strain in the direction of the applied load to the strain normal to the load. Abbreviated ν … Wiktionary

Microsoft Excel is a widely used office software which you must have tried or at least heard in life. It’s convenient, powerful and also complicated for some newbies since there seems to be too much to learn. They may end up doing repetitive work manually without knowing how to use Excel formulas to complete their tasks. It’s undoubtedly a huge waste of time and effort. In fact, you don’t need to spend much time on learning all the things about it. There are some useful Excel functions can greatly improve your working efficiency, and best of all, they are all easy to master. 1. SUM Fuction Comparing to calculate the total amount of a lot of data manually, Excel itself can do a better and quicker job to calculate the summation. It’s called SUM function. To make use of it: ① Select the cell you want to output the result of calculation, then click the icon of Insert Function above the table. ② Choose SUM in the popping out window and hit OK. ③ The formula will be filled in the cell and a window called Function Arguments will pop out. You can click the up arrow to select the range of data to calculate or type it manually. In my case I want to sum the data from B2 to D2 (including C2), so the complete formula should be “=SUM(B2:D2)”. ④ If you want to sum the data of B2 and D2, you should set the value of Number 1 and Number 2 as B2 and D2 separately, or input complete the formula as “=SUM(B2, D2)” directly. ⑤ Hit OK, Excel will work the total amount out immediately. ⑥ There’s one more small trick I want to share. If you want to apply the same formula in a range of cells, put your cursor at the right lower corner of the cell till it turns into a small black cross. Drag it to contain all the cells you want to apply the same formula. 2. Calculate Percentages It’s frequent at work to make sheets involving percentages of different departments in a whole company. Calculating percentages with a calculator surely won’t take a lot of your time. But if there’re piles of data to deal with in a short time, Excel functions can definitely do a quicker job. ① To calculate the percentage, you must know the total amount in advance. Like the table below, the first thing I should do is to sum the profits from F2 to F5 with the method of using SUM function. ② Then click G2, and input the formula “=F2/$F$6“. F6 refers to the value in the B22, which is also the total amount of profits. And the symbol “$” makes it a fixed value. It won’t change even when we apply this formula to other cells. ③ Press Enter key to work it out. Then with the trick I shared before, apply this formula to other cells in “Percentage” column. ④ If you want them to show as percentages instead of decimal fractions, just select all these cells and click the “％” icon in Home tab. The results will change into percentages immediately. 3. Evaluate the Average Value Evaluating the average value is also a common need while doing calculations in Excel. ① Select a cell as the place to work the average value out, Click the icon of Insert Function – Averages, and then make your choice the poping out window of Function Arguments. ② Or you can input the formula “=AVERAGE(number1:number2)” directly. Here I want to evaluate the average value of the data from B2 to B5. So the complete formula is “=AVERAGE(B2:B5)”. Press Enter key and you can see the result. 4. Merge the Content in Different Cells Sometimes you may need to merge the content from different cells in an existing table. If there’s a large amount of data for us to process in the table, Excel function can help you to save a lot of time and effort as well. Like this table below, I’m required to merge the content of Column A and Column B in the new created “Merge” column. ① The first step is to merge the content of the first row in Column A and Column B. Input the formula “=CONCATENATE(A2,B2)” in F2 and then press Enter key. The text in A2 and B2 will be combined and filled in the F2 right away. ② Put the cursor at the lower-right corner of F2, it will turn to a small black cross. Then press and hold the left mouse, drag it to select all the cells in “Merge” column. Thus you can apply the same formula to a whole column, and the content from two columns will be merged successfully in a moment. 5. VLOOKUP Function VLOOKUP can easily search through a column and find the corresponding data. For example, you can search for the score of a student in class quickly according to his or her name. It’s a very practical Excel function which can be massively used in work. The basic formula for VLOOKUP is: ① Assuming there’s a long list of record in the table below. Now I want to find the score of someone, the first step should be inserting the formula correctly. So I click the cell I want to output the search result, which is F2 in my case, and hit the small icon of Insert Function. ② Type “VLOOKUP” in the textbox of Search for a function and hit Go. Click VLOOKUP in the section of Select a function and click OK to insert the formula (you can directly type it in the cell, of course). ③ Then you can input the value of each factor of the formula. For example, I want find the score of student F in the table, so the Lookup_value should be “F“. And the values of score are listed in the third column, so the Col_index_num should be “3“. Then set the Range_lookup as “0” for an accurate result. ④ As for the Table_array, a convenient way is clicking the small arrow on the right of the textbox, then hold and drag the left mouse to select all the cells in the table. The range of table will be inserted to the textbox accordingly. ⑤ Click that small arrow again to get back to Function Arguments window. Now all the textboxes have been filled up. Click OK to close the window. VLOOKUP function has already found the data I’m looking for in F2. This is what I want to share for today, hope it can be a little inspiration to your work. In fact, except for the 5 useful Excel functions I mentioned above, there are many other practical functions and formulas can improve your efficiency of processing text and data. You can also give them a try in the future. Left the repetitive tasks to machine, invest your energy in some more creative work!

Calculate the tax disadvantage to organizing a U. S. business today, after passage of the Jobs and Growth Tax Relief Reconciliation Act of 2003, as a corporation versus a partnership under the following conditions. Assume that all earnings will be paid out as cash dividends. Operating income ( operating profit before taxes) will BrainMass Solutions Available for Instant Download How can organizations determine if they are structured in the most effective and efficient manner? What are some advantages and disadvantages of each: (a) team-based, (b) network-based, and (c) boundaryless organizations? You are working for an investment firm and have been asked to analyze and explain investments to your boss and clients who do not understand the various aspects of your job. You are working with the analysis of stocks, bonds and mutual funds. The following questions are examples of some of the analysis type problems that your bo 1. Find the median of the following set of numbers. 45, 21, 54, 22, 65, 61 Solution, explanation: 2. The following scores were recorded on a 100-point examination: 95, 75, 76, 86, 96, 71, 68, 81, 95, 76, 69, 82, 93, 88, 94 Find the mean and median final examination scores. Solution, explanation: II. Create an income statement that finds the gross profit and net income for ABC Computers for the year ending August 31, 2007 if the company had net sales of $28,625; cost of goods sold of $13,247; and operating expenses of $2,752. III. Complete a vertical analysis of the income statement in part two. IV. In 2 Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet av A local church is studying the amount of offerings in an envelope from their early Sunday mornings services. The church studied 500 envelopes and found the following: Less than $5 200 $5 to $9.99 100 $10 to $19.99 75 $20 to $49.99 75 $50 or more 50 What is the probabi Please see attached document. Sharifi Hospital basis its budgets on patient-visits. The hospital's static budget for October appears below: Budgeted number of patient visits 8,500 Budgeted variable overhead costs: Supplies (@$4.70 per patient-visit) $39,950 Laundry (@$7.80 per patient-visit) 66,300 Total v The following costs appear in Malgorzata Company's flexible budget at an activity level of 15,000 machine hours: Total Cost Indirect materials: $7,800 Factory rent: $18,000 What would be the flexible budget amounts at an activity level of 12,000 machine hours if indirect materials is a variable cost a Question 1 Jordan Company produced 150,000 floor lamps during the past calendar year. Jordan had 2,500 floor lamps in finished goods inventory at the beginning of the year. At the end of the year, there were 11,500 floor lamps in finished goods inventory. The lamps sell for $50 each. Jordan's accounting records provide the fol Pitkins Company collects 20% of a month's sales in the month of sale, 70% in the month following the sale, and 6% in the second month following sale. The remainder is uncollectible. Budgeted sales for the next four months are: January February March April Budgeted Sales $200,000 $300,000 $350,000 $250,000 The following information relates to Minorca Manufacturing Corporation for next quarter: January February March Expected sales (in units) 440,000 390,000 400,000 Desired ending finished goods inventory (in units) 28,000 30,000 35,000 How many units should Minorca plan on producing for the month of February? II. You are asked to make a depreciation schedule for a business asset. A depreciation schedule shows the remaining value of the asset at the end of each time period. Create a depreciation schedule for each of the following 1. A copy machine that costs $1,800, shipping $50 and installed for $125 depreciated using the stra Which of the following should not influence a firm's dividend policy decision? The firm's ability to accelerate or delay investment projects, A strong preference by most shareholders in the economy for current cash income versus capital gains, constraints imposed by the firm's bond indenture, the fact that much of the firm's Rooney Inc. recently completed a 3-for-2 stock split. Prior to the split, its stock price was $90 per share. The firm's total market value was unchanged by the split. What was the price of the company's stock following the stock split? The firm's target structure is consistent with which of the following? Maximum earnings per share (EPS), Minimum cost of debt (rd), Highest bond rating, Minimum cost of equity (rs), or Minimum weight average cost of capital (WACC). Millman Electronics will produce 60,000 stereos next year. Variable costs will equal 50% of sales, while fixed costs will total $120,000. At what price must each stereo be sold for the company to achieve an EBIT of $95,000? 1. The issuance price of a bond does not depend on the a. face value of the bond. b. riskiness of the bond. c. method used to amortize the bond discount or premium. d. effective interest rate. 2. Which of the following is true of a premium on bonds payable? a. It is a contra-stockholders' equity account. b. It Consider the following contingency table: Under 20 21-30 31-40 Male 12 12 17 Female 13 16 21 a. If one person is selected at random, what is the probability that person is Female? ______ b. If one person is selected at random, what is the probability that person is ei Swanson Inc. purchased $400,000 of Malone Corp. ten-year bonds with a stated interest rate of 8 percent payable quarterly. At the time the bonds were purchased, the market interest rate was 12 percent. Determine the amount of premium or discount on the purchase of the bonds. Swanson Inc. purchased $400,000 of Malone Corp. ten-year bonds with a stated interest rate of 8 percent payable quarterly. At the time the bonds were purchased, the market interest rate was 12 percent. Determine the amount of premium or discount on the purchase of the bonds. a. $92,442 premium b. $92,442 discount c. $81,1 On January 1, 2006, an investor paid $291,000 for bonds with a face amount of $300,000. The contract rate of interest is 8% while the current market rate of interest is 10%. Using the effective interest method, how much interest income is recognized by the investor in 2006 (assume annual interest payments and amortization)? Thirty flasks, 10 full, 10 half empty and 10 entirely empty, are to be divided among 3 sons so that flasks and content should be shared equally. How may this be done without pouring from one flask to another? You must evaluate a proposal to buy a new milling machine. The base price is $108,000, and shipping and installation costs would add another $12,500. You must evaluate a proposal to buy a new milling machine. The base price is $108,000, and shipping and installation costs would add another $12,500. The machine falls into the MACRS 3-year class, and it would be sold after 3 years for $65,000. The applicable depreciation rates are 33, 45, 15 and 7 percent. The machine would 1. Passenger comfort is influenced by the amount of pressurization in the airline cabin. Higher pressurization permits a closer-to-normal environment and a more relaxed flight. A study by an airline user group recorded the corresponding air pressure on 30 randomly chosen flights. The study revealed a mean equivalent pressure of Match the letter of the term to the definition of that term. Definitions 1. The average of the squared deviation scores from a distribution mean. ____ 2. Midpoint in the distribution of numbers. ____ 3. It has to do with the accuracy and precision of a measurement procedure. ____ 4. Examines if an observed causal rela Symbols Definitions 1. S (Uppercase Sigma) ____ a. Null hypothesis 2. m (Mu) ____ b. Summation 3. s (Lowercase Sigma) ____ c. Factorial 4. p (Pi) ____ d. Nonparametric hypothesis test 5. e (Epsilon) ____ e. Population standar Burger and more business is worth 250,000. it is expected to grow at 12% per year compounded annually for the next 5 years. Find the expected future value. If funds from the sale of the business today would be placed in an account yielding 6% compounded semiannually, what would be the minimum acceptable price for Burger and The earnings, dividends, and common stock price of Carpetto Technologies Inc. are expected to grow at 7 percent per year in the future. Carpetto's common stock sells for $23 per share, its last dividend was $2.00, and it will pay a dividend of $2.14 at the end of the current year. Assuming you have equal confidence in the outp The earnings, dividends, and common stock price of Carpetto Technologies Inc. are expected to grow at 7 percent per year in the future. The earnings, dividends, and common stock price of Carpetto Technologies Inc. are expected to grow at 7 percent per year in the future. Carpetto's common stock sells for $23 per share, its last dividend was $2.00, and it will pay a dividend of $2.14 at the end of the current year. Using the DCF approach, what is the cost of co Percy Motors has a target capital structure of 40 percent debt and 60 percent common equity, with no preferred stock. The yield to maturity on the company's outstanding bonds is 9 percent, and its tax rate is 40 percent. Percy's CFO estimates that the company's WACC is 9.96 percent. What is Percy's cost of common equity?

Fluid mechanics is the branch of physics that studies the behavior of fluids at rest and in motion. It deals with the forces that fluids exert on other objects and how fluids respond to those forces. The fluids studied can be either liquids or gases. This subject is important in many fields, including engineering, meteorology, oceanography, and aerospace. It is used in many industrial processes, such as internal combustion engines, turbine design, and many others. What is a fluid? Fluids are defined by their ability to flow and change shape, as opposed to solids, which maintain a fixed shape and volume. There are two main types of fluids: liquids and gases. Liquids, such as water and oil, are a type of fluid that has a definite volume but no definite shape, meaning they will take the shape of their container. Gases, such as air and natural gas, have no definite shape or volume, meaning they will expand to fill their container. Fluids can also be classified as incompressible or compressible. An incompressible fluid is a fluid that cannot be compressed, such as water. A compressible fluid is a fluid that can be compressed, such as a gas. The behavior of a fluid depends on its compressibility. For example, the speed of sound in a gas is affected by its compressibility. Fluids are also classified as Newtonian or non-Newtonian. A Newtonian fluid is a fluid that behaves in a predictable way, following the laws of physics, such as liquids like water and oil. Non-Newtonian fluids are fluids that do not behave in a predictable way, such as toothpaste and blood. They can exhibit different behavior under different flow conditions. In summary, understanding the properties and behavior of fluids is important in many industries and natural phenomena. The behavior of a fluid can be affected by its compressibility, viscosity and its ability to follow the laws of physics. Here are 10 most important equations in fluid mechanics: Newton's Law of Viscosity It states that the shear stress (the force per unit area) in a fluid is proportional to the rate of change of velocity of the fluid with respect to distance in the direction perpendicular to the direction of flow. Mathematically, it is represented as : τ = μ * (du/dy) where τ is the shear stress, μ is the viscosity of the fluid, and (du/dy) is the rate of change of velocity of the fluid in the direction perpendicular to the flow. The proportionality constant μ is known as the coefficient of viscosity. This law applies to fluids that behave in a predictable way, following the laws of physics, such as liquids like water and oil, which are known as Newtonian fluids. The viscosity of a fluid is a measure of its resistance to flow. Fluids with high viscosity are more resistant to flow, while fluids with low viscosity flow more easily. Pascal's law states that the pressure applied to an enclosed fluid is transmitted equally in all directions and at right angles to the container walls. Mathematically, it is represented as : P1 = P2 = P3 = .... = Pn where P1, P2, P3, ..., Pn are the pressures at different points within the fluid. This means that if an external force is applied to a fluid at one point, it will be transmitted throughout the entire fluid. This law is the basis of many hydraulic systems, such as brakes, lifts, and construction equipment. It is also used in many industrial processes such as oil and gas extraction. Hydrostatic law, also known as Pascal's law of fluids, states that the pressure exerted by a fluid at rest is transmitted equally in all directions and at right angles to the surface on which it acts. This means that if an external force is applied to a fluid at one point, it will be transmitted throughout the entire fluid. The pressure at a given point in a fluid at rest is determined by the weight of the fluid above it. Mathematically, it is represented as: P = ρgh where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid above the point in question. This law is widely used in fluid mechanics and many engineering applications such as pumps, valves, and pipelines. In fluid mechanics, the continuity equation is a fundamental principle that states that the mass flow rate of a fluid through any given cross-sectional area must remain constant. Mathematically, it is represented as: ∑(mass flow rate)in = ∑(mass flow rate)out A1V1 = A2V2 = A3V3 = .... = AnVn where A1, A2, A3, ..., An are the cross-sectional areas and V1, V2, V3, ..., Vn are the velocities of the fluid at different points within the fluid. This equation states that the mass of fluid flowing into any given point must be equal to the mass of fluid flowing out of that point. It is based on the principle of conservation of mass. It is a general statement of the conservation of mass, which applies to all systems, including fluids, gases and solids. The continuity equation is also related to other principles in fluid mechanics such as the conservation of mass, the equation of motion and the energy equation. It is used in many engineering applications such as fluid flow in pipes, pumps, and valves, and in the analysis of fluid systems. It is also used in the design and optimization of fluid systems. This is a fundamental principle in fluid dynamics that describes the relationship between the pressure, velocity, and height of a fluid as it flows through a pipe or over a surface. The equation states that "in an ideal incompressible fluid when the flow is steady and continuous, the sum of the pressure energy, kinetic energy and potential energy is constant at any point along a streamline". Bernoulli's equation is given by: P + 1/2 * ρu^2 + ρgh = constant where, P is the pressure of the fluid, ρ is the density of the fluid, u is the velocity of the fluid, g is the acceleration due to gravity, h is the height of the fluid above a reference point. **Applicable for incompressible, non-viscous, steady flow fluids**. This equation is important in the study of fluid flow, as it can be used to predict the behavior of fluids in a variety of situations, including the flow of air over an airplane wing, the flow of water through a pipe, and the flow of fluids in pumps and turbines. For example, in the case of an airplane wing, the air flowing over the top of the wing is moving faster than the air flowing underneath the wing, which results in a lower pressure above the wing and a higher pressure below the wing. This creates lift, which is what allows the airplane to fly. Navier Stokes Equation The Navier-Stokes equations are a set of partial differential equations that describe the motion of a fluid, including its velocity, pressure, and temperature. These equations are based on the conservation of mass, momentum, and energy. They take into account the effects of viscosity, which is the resistance of a fluid to flow, and turbulence, which is the chaotic, irregular motion of a fluid. The equations also include the effects of forces such as gravity and pressure gradients, and can be used to model fluid flow in a wide range of situations, including laminar flow, turbulent flow, and boundary layer flow. These equations can be represented in vector form as: ∂u/∂t + (u.∇)u = -1/ρ ∇p + μ/ρ ∇²u + g where, u is the velocity vector of the fluid t is time ρ is the density of the fluid p is the pressure of the fluid μ is the dynamic viscosity of the fluid f is any external forces acting on the fluid.

represents a Nakagami distribution with shape parameter μ and spread parameter ω. - NakagamiDistribution is also known as Nakagami- distribution. - The probability density for value is proportional to for , and is zero for . - NakagamiDistribution allows μ and ω to be any positive real numbers. - NakagamiDistribution allows ω to be a quantity of any unit dimension and μ to be a dimensionless quantity. » - NakagamiDistribution can be used with such functions as Mean, CDF, and RandomVariate. Background & Context - NakagamiDistribution[μ,ω] represents a continuous statistical distribution supported on the interval and parametrized by positive real numbers μ and ω (called a "shape parameter" and a "spread parameter", respectively), which together determine the overall behavior of its probability density function (PDF). Depending on the values of μ and ω, the PDF of a Nakagami distribution may have any of a number of shapes, including unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Nakagami distribution is sometimes referred to as the Nakagami -distribution or Nakagami -distribution. - The Nakagami distribution was first proposed in a 1960 article by Minoru Nakagami as a mathematical model for small-scale fading in long-distance high-frequency radio wave propagation. In the years since, many applications of the distribution have been wave related. In particular, the Nakagami distribution has been used to model phenomena related to medical ultrasound imaging, communications engineering, and meteorology. It has also been used in various other fields, including hydrology, multimedia, and seismology. - RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Nakagami distribution. Distributed[x,NakagamiDistribution[μ,ω]], written more concisely as xNakagamiDistribution[μ,ω], can be used to assert that a random variable x is distributed according to a Nakagami distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation. - The probability density and cumulative distribution functions for Nakagami distributions may be given using PDF[NakagamiDistribution[μ,ω],x] and CDF[NakagamiDistribution[μ,ω],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. - DistributionFitTest can be used to test if a given dataset is consistent with a Nakagami distribution, EstimatedDistribution to estimate a Nakagami parametric distribution from given data, and FindDistributionParameters to fit data to a Nakagami distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Nakagami distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Nakagami distribution. - TransformedDistribution can be used to represent a transformed Nakagami distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Nakagami distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Nakagami distributions. - NakagamiDistribution is related to a number of other distributions. Before the formulation of the Nakagami distribution, the RayleighDistribution and RiceDistribution were commonly used models for wave fading, and the three distributions are qualitatively very similar. In addition, NakagamiDistribution generalizes both RayleighDistribution and HalfNormalDistribution, in the sense that the CDF of NakagamiDistribution[1,2 σ^2] is precisely that of RayleighDistribution[σ], while the PDF of NakagamiDistribution[1/2, π/(2 θ^2)] is exactly that of HalfNormalDistribution[θ]. Moreover, NakagamiDistribution[μ,ω] has the same PDF as both GammaDistribution[μ,Sqrt[ω]/Sqrt[μ],2,0] and as the limit of RiceDistribution[μ,α,Sqrt[ω/2]] as α→0. NakagamiDistribution is also related to HoytDistribution, NormalDistribution, and LogNormalDistribution. Examplesopen allclose all Basic Examples (4) In the theory of fading channels, NakagamiDistribution is used to model fading amplitude for land-mobile and indoor-mobile multipath propagation and also in the presence of ionospheric scintillation. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise: Show that SNRdist is a GammaDistribution: Properties & Relations (7) RayleighDistribution is a special case of Nakagami distribution: HoytDistribution is related to Nakagami distribution: Wolfram Research (2010), NakagamiDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/NakagamiDistribution.html (updated 2016). Wolfram Language. 2010. "NakagamiDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/NakagamiDistribution.html. Wolfram Language. (2010). NakagamiDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NakagamiDistribution.html

How to solve the distance formula In this blog post, we will be discussing how to solve the distance formula. How can we solve the distance formula What is the difficulty in solving problems with formula or model thinking? In the simple example of throwing stones to find the distance, the difficulty is not to solve the quadratic equation of one variable, but that your thinking ability can transform the problem in this business scenario into a problem of solving the company. After completing this transformation, all that remains is the process of matching using your formula library. You can think about how much education at the current school stage teaches students how to transform their thinking ability? More school learning often relies on a large number of question type exercises and produces mechanical memory. For example, when you see the business scenario of stone throwing, you already know that it is converted into equation solving. This problem is not that you know how to analyze and convert, but that you have done similar problems. [answer] a. Analysis: calculate the growth of the average of each business income compared with the previous year. Since the options give percentages, the average growth rate is obtained. According to the formula and the options, item a is selected for this question. Calculate the growth of the average in the first half of 2017 compared with the same period of the previous year. The data in the options are all in units (yuan / set), so the average growth is obtained. According to the formula: In the school learning stage, the most impressive thinking framework is the formula and model, that is, all kinds of problems in the real world can be finally transformed or abstracted into specific formulas or models to solve. For example, when you throw a stone with a parabola, how far can the stone be thrown? Based on the knowledge of physics and mathematics, problems in reality will be transformed into solving a quadratic equation of one variable. The solution of the quadratic equation of one variable can be directly brought into the standard formula to get the answer. Various formula models themselves are the most basic thinking framework, because formula itself is a commonly used axiomatic post theorem obtained by human beings through induction and abstraction after studying various phenomena in the real world. And your actual process of solving the problem is only to apply the formula for deduction. Through the explanation of the above questions, I believe that we have a better grasp of solving such problems. We need to accurately find the total amount and the number of copies through the description of the question stem. When it comes to growth related problems, we need to determine the content of the examination according to whether there is a unit in the final request, and finally substitute it into the corresponding formula to solve it. I believe that through today's study, we can have a better understanding of the problem of average growth. We should practice more on weekdays so that practice makes perfect. Through the explanation of the above questions, I believe that we have a better grasp of solving such problems. We need to accurately find the total amount and the number of copies through the description of the question stem. When it comes to growth related problems, we need to determine the content of the examination according to whether there is a unit in the final request, and finally substitute it into the corresponding formula to solve it. I believe that through the introduction of public education in China, we can have a better understanding of the problem of average growth. We should practice more on weekdays so that practice makes perfect. If the formula related to the proportion of the current period is used, the stem usually gives the partial rate, that is, the molecular rate, and the value of the proportion. It is necessary to find the rate of the proportion before using the formula of the denominator rate to solve the denominator, that is, the overall rate. Basic formula of growth problem (Solving growth volume, growth rate, base period value, current period value), proportion (basic formula and modification of proportion, growth volume of proportion, judging the rise / fall of proportion), multiple, average (basic formula and modification of average, growth rate of average, judging the rise / fall of average), cross, etc. Solve your math tasks with our math solver It has helped me so much, especially since the lockdown because the corona virus. I can't always ask my teacher to explain the question so it's easier just to use the app. I love the fact that the app gives you the solution instead of just the answer. the app has really helped during this pandemic!! the app is a best App It helps me when I am confused It is based with kinds of Mathematical expression like Algebra. This app is best because it will help you It's a good app but it was offline now it is online if it is possible to make it usable offline, please!

Reinforced concrete retaining walls (RCRWs) are referred to as structures that withstand the pressure resulting from the difference in the levels caused by embankments, excavations, and/or natural processes. Such situations frequently occur in the construction of several structures, such as bridges, railways, and highways. Due to the frequent application of RCRWs in civil engineering projects, minimizing the construction cost of such structures is an issue of crucial importance. The satisfaction of both geotechnical and structural design constraints is a key component in the design of RCRWs. In most cases, primary dimensions are initially estimated based on reasonable assumptions and the experience of the designer. Then, in order to reach a cost-effective design while satisfying the design constraints, the design variables (particularly the wall dimensions) need to be revised by using a trial-and-error process, which makes it rather grueling. On the other hand, there is no guarantee that the final design will be the best possible one. To eliminate this problem, which can hinder the designer from reaching a cost-effective solution, and by considering the advances in computational technologies during the recent decades, it makes sense to express the design in the form of a formal optimization problem. The design optimization of RCRWs has received significant attention during the last two decades. Some of the pertinent works are briefly investigated herein. As a benchmark work, Saribas and Erbatur [1 ] used a nonlinear programming method and investigated the sensitivity of the optimum solutions to parameters such as backfill slope, surcharge load, internal friction angle of retained soil, and yield strength of reinforcing steel. The simulated annealing (SA) algorithm has been also applied to minimize the construction cost of RCRWs [2 ]. Camp and Akin [4 ] developed a procedure to design cantilever RCRWs using Big Bang–Big Crunch optimization. They captured the effects of surcharge load, backfill slope, and internal friction angle of the retained soil on the values of low-cost and low-weight designs with and without a base shear key. Khajehzadeh et al. [5 ] used the particle swarm optimization with passive congregation (PSOPC), claiming that the proposed algorithm was able to find an optimal solution better than the original PSO and nonlinear programming. In their work, the weight, cost, and CO2 emissions were chosen as the three objective functions to be minimized. Gandomi et al. [6 ] optimized RCRWs by using swarm intelligence techniques, such as accelerated particle swarm optimization (APSO), firefly algorithm (FA), and cuckoo search (CS). They concluded that the CS algorithm outperforms the other ones. They also investigated the sensitivity of the algorithms to surcharge load, base soil friction angle, and backfill slope with respect to the geometry and design parameters. Kaveh and his colleagues (e.g., [7 ]) optimized the RCRWs using nature-inspired optimization algorithms, including charged system search (CSS), ray optimization algorithm (RO), dolphin echolocation optimization (DEO), colliding bodies of optimization (CBO), vibrating particles system (VPS), enhanced colliding bodies of optimization (ECBO), and democratic particle swarm optimization (DPSO). Temur and Bekdas [11 ] employed the teaching–learning-based optimization (TLBO) algorithm to find the optimum design of cantilever RCRWs. They concluded that the minimum weight of the RCRWs decreases as the internal friction angle of the retained soil increases, and increases with the values of the surcharge load. Ukritchon et al. [12 ] presented a framework for finding the optimum design of RCRWs, considering the slope stability. Aydogdu [13 ] introduced a new version of a biogeography-based optimization (BBO) algorithm with levy light distribution (LFBBO) and, by using five examples, it was shown that this algorithm outperforms some other metaheuristic algorithms. In this work, the cost of the RCRWs was used as the criterion to find the optimum design. Nandha Kumar and Suribabu [14 ] adopted the differential evolution (DE) algorithm to solve the design optimization problem of RCRWs. The results of sensitivity analysis showed that width and thickness of the base slab and toe width increases as the height of stem increases. Gandomi et al. [15 ] studied the importance of different boundary constraint handling mechanisms on the performance of the interior search algorithm (ISA). Gandomi and Kashani [16 ] minimized the construction cost and weight of RCRWs analyzed by the pseudo-static method. They employed three evolutionary algorithms, DE, evolutionary strategy (ES), and BBO, and concluded that BBO outperforms the others in finding the optimum design of RCRWs. More recently, Mergos and Mantoglou [17 ] optimized concrete retaining walls by using the flower pollination algorithm, claiming that this method outperforms PSO and GA. By taking a look at the studies so far reported, it can be noticed that there has been no work done in assessing the effect of using different available methods of determining the bearing capacity on the optimum design of the RCRWs. The current study investigates this important issue. In order to model and design the RCRWs, a code is developed in MATLAB [18 ]. To reach a design with minimum construction cost, an optimization problem is defined and the construction cost is considered as the single objective function to be minimized. The design criteria, including both geotechnical and structural limitations, are considered as the optimization constraints. The wall geometrical dimensions and the amount of steel reinforcement are used as the design variables. The particle swarm optimization (PSO) [19 ] algorithm is used to find the optimum solution.

NCERT Solutions for Class 9 Maths Chapter 2 Polynomials (Ex 2.2) Exercise 2.2 Loading More Solutions... Vedantu’s Class 9 Maths Chapter 2 Exercise 2.2 includes the following problems from polynomial- Before discussing polynomials, it is necessary that you have a clear idea about variables and exponents. Consider the following equation - 2x + 3x = 10 Here we don't know at first what X is. When we solve the equation, we come to the conclusion that here X is 2. So, what X does here is it answers the ‘what’ of a question. In this equation, the question that is implied is - what should we multiply 2 and 3 with so that the addition of the product results in 10? Again when we are confronted with the following equation - 2x + 3x = 20 the result of X changes. So, the value of X varies the situation. This is what we call variable. This variable is generally denoted by x,y,z etc. Again, if you look at the above two examples, the values of 2 and 3 never change. In both situations, the values of 2 and 3 remain fixed. These are called constants. Now, look at this, Here the value of 2 increases exponentially ( 2*2*2). The power 3 is known as an exponent. If you look at the algebraic expression - 2x + 3x - written above, you will see there are two parts in it - 2x and 3x. These parts are called terms. Polynomial is a mathematical expression that has multiple terms and consists of constants, variables and non-negative or non-fractionated exponents. The polynomial will only involve addition, subtraction or multiplication. So 2x + 3x is a polynomial in the variable X. Again, 2³ + X³ + 3² is a polynomial. But X + 1∕X is not a polynomial since 1/x can be written as . As said earlier, a polynomial does not allow negative exponents. Expressions that have √X are not polynomials too since √X can be written as and we know that this is not accepted by polynomials. If you look at the expression 2x + 3x, the numbers before the variables are called coefficients. Here are a few other exercises that you will find in the NCERT Solutions for Class 9 Maths Chapter 2 PDF – In this exercise of Class 9 Maths Chapter 2, students will be introduced to the concept of polynomials. They will obtain an indent knowledge regarding polynomials in one variable and how to solve the associated sums through shortcut techniques. Here are some important questions for exams that are provided in the exercise. Question 1: Finding the polynomials among the expressions in one variable and stating reasons for the answers. Question 2: Finding the coefficients of x2 in the provided expressions. Question 3: Examples of a binomial of degree 35 and monomial of degree 100. Question 4: Determining the degree of polynomials. Question 5: Classifying linear, quadratic and cubic polynomial expressions. This chapter is a continuation of Exercise 2.2 Class 9 of NCERT maths book. Students will get in-depth knowledge about the Factor Theorem by solving all questions provided in this exercise step by step. Question 1: Finding the remainder when divided by expressions provided. Question 2: Determining remainders. Question 3: Checking if provided expressions are factors. Vedantu's Polynomials Class 9 solutions will help you understand all the topics included in the Polynomials chapter. NCERT guidelines specifically advise teachers to build the fundamental knowledge of the subjects in the students. Vedantu follows this NCERT advice thoroughly. That is why you will find not only solutions to the questions on polynomials but also the explanations on the logic behind our solutions. For example - in question number 2 of Ex 2.1 of our Polynomials Class 9 NCERT Solutions when we are asked to find the coefficients of given polynomials we showed how has an invisible 1 before it. So the coefficient of X² is 1. Our answers in Class 9th Maths NCERT Solutions Chapter 2 include every step. No step has been skipped. You will not struggle to understand how we reached the result from a given equation. So in question 1(i) of Ex 2.2, we showed how we arrived at the answer 3 by solving the equation step by step. This will not only help the students to understand the granularities in the chapter but will also help them to get good marks. Every Polynomial formula Class 9 has been used in our solution. The students will get to learn all the aspects of the chapter. Reading our PDF, a student can answer any question from this chapter with confidence. The Polynomials Class 9 PDF is written in a lucid language. Apart from the Maths expressions, we have used simple, easy to understand words so that students can understand the solutions. Even the math expressions are adequately explained wherever deemed fit. The Class 9 Maths Chapter 2 PDF has accurate solutions to every polynomials problem. The answers are written by expert teachers who know how to solve a Maths problem so that students understand the process and even the examiners get impressed by the detailed process. These teachers are well aware of NCERT guidelines and follow them to write their answers. There is no unnecessary information in our solutions. Vedantu’s Polynomial Class 9 PDF is neatly organised. The solutions are written in an uncluttered, easy to understand way. That is why in question 1 of Ex2.3, you will find that we have used red ink to indicate the cutting off of -2x² and +2x². This visually pleasurable way of reading the solutions will help the students to maintain their focus on the solutions. You can use our NCERT Solutions for Class 9 Maths Chapter 2 Exercise 2.2 to learn all the intricacies of polynomials. These solutions can provide you with everything that you need to excel in the chapter. The answers in our PDF are not written by part-time bloggers, but they are written by professional teachers who took care to make the solutions as helpful as possible. The Class 9 Maths Chapter 2 Exercise 2.2 Solutions are absolutely free of cost. You don’t need to pay a single penny to download our PDF. You can simply enter your email id and we will send the download link to your email. Vedantu is not motivated by profit; we are genuinely interested in making the learning process of students better. We believe every Indian student has the right to get a quality education. Our PDF on polynomials is exhaustive and can help you in finding solutions to all the questions asked in the textbook. Maths needs explanation - countless of times. You might want to ask questions about a particular solution in our PDF. So to help you, Vedantu holds online classes where you can ask questions directly to the teachers. This interaction is particularly necessary for understanding Maths chapters. The schedule of the classes is announced beforehand. We have helpful videos too which will help you understand the solutions in a much more easy and efficient way. Our online teachers are experts in their subjects and come from respectable institutions like IITs. These teachers know how to teach Maths to students in such a way that students do not get bored. The teachers are aided by simultaneous images, videos and slides. This visual way of teaching ensures easy retention in students. With our Master Class, you can experience a whole new learning experience. We are offering free seats in our master class for a limited period of time. These masterclasses can enhance your knowledge in ways you cannot fathom. We do not employ cheap tricks and tips. Our main focus is on building the basic knowledge of the students. 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Also known as the golden number, golden ratio, golden number, golden ratio. Remember those times in math class when you questioned how and when you would ever use math in the real world? I also. But now that I’m a designer, I really wish I had stayed up in Geometry and learned about the golden ratio. Well in my design career, I learned about the golden ratio and how to use it in design. When I studied it, I realized that this knowledge was one of the missing pieces of my design education. Who knows?! In this post, you will get a crash course on what the golden ratio is, why and how I use it in logo design, and how you can start applying it to your work. The golden ratio is a mathematical principle that can be found in nature, anatomy, color, and even sound waves. Due to its pleasant nature, it has been used in art, paintings, architecture, music, and design for thousands of years. Scientific studies have shown that we perceive things that contain the Golden Ratio as a beautiful, harmonious, and bordering perfection, even when we do not know it. I found that the more I incorporate the golden ratio into my design, the more satisfying the result is. Similar to using a grid in layout design, using Golden Ratio provides a framework for my layout decisions. Of course, you cannot superficially use the Golden Ratio and think that everything will always be perfect. Using these proportions in your design work will take time and practice. But the more you use it, the more you start to see the relationships between the elements of your design and begin to detect and correct areas that are inconsistent in your work and that just don’t feel right. Now to get technical. Let’s look at some key terms and concepts. Each builds on the previous one. To get to the Golden Ratio, we must first understand the Fibonacci number sequence. It can be found everywhere; from the number of petals, there are in a flower to the spirals in sunflower or pineapple to the pattern of keys in a piano. Once you get familiar with this sequence, you will start to see it everywhere. The sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21 … So let’s see how this works. Each number after the first two in the sequence is the sum of the 2 previous numbers. … and so. Now let’s put this in visual terms that designers can appreciate. If we square each number to identify the spatial area, our sequence will now look like this: And if you are an expert in puzzles, you will begin to see that the progression of the numbers forms a visual pattern: Crazy, right? You just wait. It gets even better. The golden ratio This is where the Golden Ratio comes in. The ratio of the numbers in the Fibonacci sequence tends towards the golden ratio. The ratio of two consecutive numbers in the sequence gets closer and closer to the golden ratio, 1.618. The golden ratio is the ratio of two quantities where the ratio of the small quantity (a) to the large quantity (b) is the ratio of the large quantity (b) to the set (a + b). The golden spiral The Fibonacci pattern is not only found in nature but there is another pattern within it, called the Fibonacci Spiral (also known as the Golden Spiral). By adding a circular arc to each square, we will produce a perfect spiral: This spiral may look quite familiar to you if you’ve ever examined a seashell or even seen a picture of a galaxy. Now that you know what the golden ratio is, let’s use it in the logo design. This is where the fun begins! Note: for the sake of simplicity, I use the term “golden ratio” when I’m talking about the golden rectangle, spiral, or ratio, as the terms are often used interchangeably. One of the ways I design logos is by taking all the main forms of the Golden Ratio so that the proportions are harmonious. Then I combined the shapes and used a grid to align all the parts. And finally, I refine the points, the shapes and the connections. Pro Tip: Don’t use more than one golden ratio when designing. If you need a smaller ratio, take it from the golden ratio you started with. Another way I use the Golden Ratio is to determine the height and width of the logo, as well as the proportions of the strokes. Pro Tip: Not all strokes can align with the Gold Ratio. In this example, the horizontal strokes must be slightly thinner than the vertical strokes to appear optically the same. Even the Greeks altered it a bit . Lastly, I use the golden ratio to help me decide the location, size, and length of key elements in a logo. So you want to use the golden ratio? Incorporating the golden ratio into your workflow will help you make better and faster design decisions. Wondering how you can use the golden ratio in your next logo to create a more harmonious and pleasing design? To really understand it, I would suggest studying first the logos of the Masters, like Paul Rand and Saul Bass, and observe the Golden Relationship in their work. The goal is to train your eye to see. Trace the logos and take note of what you discover. Did the logo use the golden ratio? Did the points within the Golden Ratio of various parts of the logo intersect? Not far from your studio, you will begin to notice the proportions, the relationships between the forms, and the alignment. When you learn to see, you will inevitably make better design decisions. Next, use the golden ratio as a way to correct the proportions once you have the logo design 90% complete. As you use the golden ratio to refine your work, you will begin to notice how you will incorporate it earlier in your design process. Even if you don’t rush. The goal is to use it as you would a guide and not force it. Final tip: use it as a guide, not as a rule. Your eyes have the last word.

In the GPS, the time variable becomes a coordinate time in the rotating frame of the earth, which is realized by applying appropriate corrections while performing synchronization processes. Synchronization is thus performed in the underlying inertial frame in which self-consistency can be achieved. With this understanding, I next need to describe the gravitational fields near the earth due to the earth’s mass itself. Assume for the moment that earth’s mass distribution is static, and that there exists a locally inertial, non-rotating, freely falling coordinate system with origin at the earth’s center of mass, and write an approximate solution of Einstein’s field equations in isotropic coordinates:1. The angle is the polar angle measured downward from the axis of rotational symmetry; is the Legendre polynomial of degree 2. In using Eq. (12), it is an adequate approximation to retain only terms of first order in the small quantity . Higher multipole moment contributions to Eq. (13) have a very small effect for relativity in GPS. One additional expression for the invariant interval is needed: the transformation of Eq. (12) to a rotating, ECEF coordinate system by means of transformations equivalent to Eqs. (3). The transformations for spherical polar coordinates are: The Earth’s geoid. In Eqs. (12) and (15), the rate of coordinate time is determined by atomic clocks at rest at infinity. The rate of GPS coordinate time, however, is closely related to International Atomic Time (TAI), which is a time scale computed by the BIPM in Paris on the basis of inputs from hundreds of primary time standards, hydrogen masers, and other clocks from all over the world. In producing this time scale, corrections are applied to reduce the elapsed proper times on the contributing clocks to earth’s geoid, a surface of constant effective gravitational equipotential at mean sea level in the ECEF. Universal Coordinated Time (UTC) is another time scale, which differs from TAI by a whole number of leap seconds. These leap seconds are inserted every so often into UTC so that UTC continues to correspond to time determined by earth’s rotation. Time standards organizations that contribute to TAI and UTC generally maintain their own time scales. For example, the time scale of the U.S. Naval Observatory, based on an ensemble of Hydrogen masers and Cs clocks, is denoted UTC(USNO). GPS time is steered so that, apart from the leap second differences, it stays within 100 ns UTC(USNO). Usually, this steering is so successful that the difference between GPS time and UTC(USNO) is less than about 40 ns. GPS equipment cannot tolerate leap seconds, as such sudden jumps in time would cause receivers to lose their lock on transmitted signals, and other undesirable transients would occur. To account for the fact that reference clocks for the GPS are not at infinity, I shall consider the rates of atomic clocks at rest on the earth’s geoid. These clocks move because of the earth’s spin; also, they are at varying distances from the earth’s center of mass since the earth is slightly oblate. In order to proceed one needs a model expression for the shape of this surface, and a value for the effective gravitational potential on this surface in the rotating frame. For this calculation, I use Eq. (15) in the ECEF. For a clock at rest on earth, Eq. (15) reduces to, a “Terrestrial Time” scale (TT) has been defined by adopting the value . Eq. (18) agrees with this definition to within the accuracy needed for the GPS. From Eq. (15), for clocks on the geoid,10. Note that these effects sum to about 10,000 times larger than the fractional frequency stability of a high-performance Cesium clock. The shape of the geoid in this model can be obtained by setting and solving Eq. (16) for in terms of . The first few terms in a power series in the variable can be expressed as Better models can be found in the literature of geophysics [18, 9, 15]. The next term in the multipole expansion of the earth’s gravity field is about a thousand times smaller than the contribution from ; although the actual shape of the geoid can differ from Eq. (20) by as much as 100 meters, the effects of such terms on timing in the GPS are small. Incorporating up to 20 higher zonal harmonics in the calculation affects the value of only in the sixth significant figure. Observers at rest on the geoid define the unit of time in terms of the proper rate of atomic clocks. In Eq. (19), is a constant. On the left side of Eq. (19), is the increment of proper time elapsed on a standard clock at rest, in terms of the elapsed coordinate time . Thus, the very useful result has emerged, that ideal clocks at rest on the geoid of the rotating earth all beat at the same rate. This is reasonable since the earth’s surface is a gravitational equipotential surface in the rotating frame. (It is true for the actual geoid whereas I have constructed a model.) Considering clocks at two different latitudes, the one further north will be closer to the earth’s center because of the flattening – it will therefore be more redshifted. However, it is also closer to the axis of rotation, and going more slowly, so it suffers less second-order Doppler shift. The earth’s oblateness gives rise to an important quadrupole correction. This combination of effects cancels exactly on the reference surface. Since all clocks at rest on the geoid beat at the same rate, it is advantageous to exploit this fact to redefine the rate of coordinate time. In Eq. (12) the rate of coordinate time is defined by standard clocks at rest at infinity. I want instead to define the rate of coordinate time by standard clocks at rest on the surface of the earth. Therefore, I shall define a new coordinate time by means of a constant rate change:10 (see Eq. (18)). When this time scale change is made, the metric of Eq. (15) in the earth-fixed rotating frame becomes This work is licensed under a Creative Commons License.

The Photodynamics of Moving Bodies Back to Contents In the quantum theory we associate with a photon two kinematic properties that correspond to two dynamic properties. We associate the photon=s energy with an angular frequency, and we associate the photon=s linear momentum with a wave number, Those kinematic properties come to us as the reciprocals of the elapsed time and the spanned distance that we associate with one radian of one cycle of the electromagnetic wave of which the photon is a curd (ah, yes, the cottage cheese theory of light). I now want to work out the equations that will transform those properties as one observer would measure them into the same properties as another observer, moving relative to the first along their common x-axis, would measure them. In doing so I will obtain something like a reciprocal Lorentz Transformation. As usual, we will use a series of imaginary experiments to work out the equations that we want. As we do with real experiments, we so design our imaginary experiments that they will produce events that will reveal the information we want. Our imaginary measurements of those events will have missing pieces whose mathematical form we infer from the requirement that they be consistent with the known laws of physics and of logic. We begin with Observer Stationary Stan, who has a body of mass M floating at the origin of a coordinate grid that he has etched into the phlogistonic emulsion embedded in the Šther filling the region in which he will conduct his experiment. We can use non-existent fantasy materials, such as phlogiston and Šther, in these experiments as aids to the imagination so long as they do not interfere with the action of the experiment: in this case we use the phlogistonic emulsion as a medium on which we can draw and make measurements without in any way changing the properties of the experimental body or of any photons involved in Stan=s experiments. Stan contrives to make his experimental body emit two identical photons in opposite directions in the x-y plane (I will ignore the z-direction in this essay for convenience, since whatever we infer for the y-direction applies also to the z-direction). Emission of those photons reduces the mass of the emitting body byΔM, so each photon carries energy Because photons possess no rest mass, they have the relationship E=pA c between their energies and their linear momenta. Thus each of Stan=s photons carries linear momentum in which the vector c designates the fact that the photon moves at the speed of light in a specific direction. Because the two photons go in opposite directions, their linear momenta cancel out and the linear momentum of the emitting body remains equal to zero. Stan has put three clocks into his setup. The clock on the emitting body stops at time t0 when the body emits the two photons. The two other clocks, one at y=a, x=b and the other at y=-a, x=-b, stop at time t1 when they absorb the photons. As the photons travel they etch a trace of their passage into the phlogistonic emulsion, thereby drawing straight lines that make an angleθ with the x-axis such that Now let=s turn our attention to Observer Mobile Monica. She moves at speed V in the negative direction parallel to Stan=s x-axis and she has contrived for her x=-axis to coincide with Stan= s x-axis for convenience. She has also established her y=-axis parallel to Stan= s y-axis. Thus, as Monica observes them, Stan, his coordinate grid, and his apparatus move at the speed V in her positive x= -direction. If Stan measures between two events the distances and duration x, y, and t, then Monica will measure between those same two events the distances and duration x=, y=, and t=, which she can relate to Stan=s measurements through the Lorentz Transformation: 1. In Monica=s frame Stan=s frame moves a distance Vt= between the two events, so Monica must add that distance to the dilated version of the distance that Stan measures between the events in the x-direction. Further, because Stan=s clocks tick off time that has dilated in Monica=s frame to fill more time ticked off Monica=s clocks, Monica must account for that dilation in her calculation; 2. Oriented perpendicular to the direction of relative motion, the y-direction doesn=t change for either observer; 3. As noted, Monica must account for the dilation of time elapsed on Stan=s clocks. If Stan=s clocks are separated from each other by some distance in the x-direction, Monica must add a temporal offset to the dilated time to compensate the fact that in her frame Stan=s fore clock appears pushed into the past relative to his aft clock; Monica also knows that in her frame Stan=s emitting body initially ponders a mass That fact implies that the photons carry away an energy that corresponds to but Monica wants a proof. Conservation of energy should give her a good enough proof, but she wants to augment it with a proof that involves a different conservation law. Imagine that a rod made of massless, transparent unobtainium extends in the negative y-direction from Stan=s emitting body. The body emits a burst of photons into the rod, they propagate to its far end, and there they condense into a body of massμ (the physical laws that process would violate are irrelevant to the subject of our imaginary experiment, so we can ignore them just as we ignore other irrelevant details, such as the color of the emitting body). Because of Newton=s third law of motion, the emitting body recoiled when it emitted the photons, so it moved a small distance y1 from the origin of Stan=s coordinate grid as the photons moved a distance y2 before condensing into a small body. No external forces acted on that little system, so it remains balanced on the origin of Stan=s coordinate grid; that is, the system=s center of mass remains on the grid=s origin. That fact means that if Stan puts a needle on his x-axis and pushes it against the unobtainium rod where it crosses the grid=s origin, the compound body will accelerate but it will not rotate. From the counterbalancing of the inertially instigated torques in the composite body Stan can then calculate Reality has the fundamental property that any event that exists for one observer necessarily exists for all observers (even if they don=t actually observe it). Thus Monica will also observe Stan=s composite body accelerating (though at a rate different from the one that Stan calculates from his measurements) and not rotating. That latter observation tells Monica that the composite body is balanced in her frame, that the inertially induced torques acting on the component bodies counterbalance each other, so she writes her own version of Equation 11 as Monica knows that y=2=y2 and y=1=y1. She also has Equation 9, so now she knows that the law of conservation of angular momentum necessitates that Equation 10 stand true to Reality. Referring back to Stan=s original experiment, Monica now knows for certain that the two photons each carry an amount of energy equal to By way of Equation 1 and Stan=s measurement of each photon=s angular frequency, Monica also knows that This puzzles Monica because she knows that frequency corresponds to the reciprocal of an elapsed time, so time dilation should make the frequencies of the photons in her frame smaller, not larger. For the moment she leaves it as a paradox that she will resolve later and turns her attention to the photons= linear momenta. In Stan=s frame the emitting body doesn=t move, so it suffers no change of linear momentum when it emits the photons. But in Monica=s frame the body initially carries momentum M=V, so when it emits the two photons it loses an amount of linear momentum equal toΔM=V. To satisfy the laws of conservation of linear momentum, the photons must have taken that momentum with them, so now Monica knows that each photon took to add to the linear momentum conferred upon it by its emission. She must add that amount to the x-component of the basic momentum of the photon as it appears in her frame. If she and Stan were to observe a photon propagating in the x-direction only and if, at some instant, they could take a picture of it by way of some Štherial camera, they would find that the photon spans the distance between two points. But that distance is shorter for Monica than it is for Stan, so the wave number of the photon and, therefore, its momentum must be correspondingly larger; that is, Monica has When she adds in the momentum from Equation 15, she gets She knows right away that the component of the momentum in the y-direction cannot differ between her frame and Stan=s, so she has And lastly Monica looks at the photon=s energy and associated angular frequency. She already has Equation 14, which tells her how the photon=s fundamental energy differs in her frame from its value in Stan=s frame. But in her frame another phenomenon comes into play. Whenever a body absorbs or emits light, that light exerts a force upon the body as it enacts the transfer of momentum, thereby obliging the body to exert the necessary equal and oppositely directed force upon the light. If the body is moving, then the latter force does work upon the light, changing its energy. For a single photon emitted in the direction of the body=s motion (or for the component of a photon=s momentum parallel to the direction of motion) the net work done on the photon corresponds to Adding that result to Equation 14 gives Monica In accordance with Equations 1 and 2, Monica divides Equations 17, 18, and 20 by Planck=s constant (S) and obtains the wave-parameter analogue of the Lorentz Transformation: With those equations she conducts a simple test. She calculates which gives her the wave-parameter analogue of the Minkowski metric. Now she can go back and resolve a minor paradox that she discerned earlier. She takes the case in which k=x=0, which represents a photon moving only in the y-direction in her frame, and calculates from Equation 21 that Substituting that into Equation 23 to calculate the corresponding frequency gives her which displays what she originally anticipated based on time dilation acting on the emitted photons. She now has the wave-parameter analogue of the Lorentz-Fitzgerald contraction. And finally Monica substitutes kx=kCosθ and ω/c=k into Equation 21 to get From her previous study she knows that Stan=s photons trace in her frame paths that make an angleθ= away from her x=-axis in accordance with Monica substitutes that into Equation 27, divides the equation by Cosθ+V/c, and multiplies the equation by 1+Cosθ(V/c) to get which is the reciprocal, to within a constant factor, of the relativistically Doppler shifted wavelength of the photon. Substituting kx=Cosθ(ω/c) into Equation 23 gives her which describes the relativistic Doppler shift of the photon=s frequency. Now Stan and Monica have a version of the Lorentz Transformation that they will find useful in the development of the quantum theory, among other things. Back to Contents

+-- William Hinshaw | 1808-1885 | | Jacob Madison Hinshaw ---+ B: c1840 | D: 1861 | +-- Rhody - c1806-? M: Almarinda Walker +-- William A. Hinshaw, c1860-c1866 +-- Jacob Madison Hinshaw, Jr., 1862-1941 |Jacob Madison Hinshaw     [ID 03549]||Click here to switch to Ahnentafel view:| (Jacob Madison Hinshaw)159    (Jacob Madison Hensie)162    (Jacob Madison Hinchey)162,a,45,b    (Madison J. Hinchy)c    (Matt Hinshaw)159    (Matt Hincher)162 Born about 1840, Tennessee.d,174,e,159 He married Almarinda Walker, Sep 22 1858, Hawkins County, Tennessee.45,159,162,174,a,b    (Almarind Walker)162,e    (Lorinda Walker)45,162,a,f,b    (Rinda Walker)g    (Elias Beal, Security).f    Almarinda, daughter of Gabriel Walker & Rebecca Jane Hinshaw, was born about 1843, Tennessee.162,g,h,i Jacob and family were shown in the 1860 census (Jul 13-16 1860), Greene County, Tennessee:e Jacob Madison Hinshaw died Nov 30 1861, Green County, Tennessee; age 22; buried Long Family Cemetery, near Bulls Gap, Tennessee.159 Jacob was hung for his Union loyalties.    Tennessee voted for secession on June 8, 1861, with residents of eastern Tennessee voting two-to-one against secession but losing the vote to the state's larger western population.    Rev. William Carter, who had been a delegate to an early 1861 convention of pro-Union loyalists at Greeneville, devised a plan to burn down the railroad bridge over Lick Creek, near Potterstown in Greene County.    Carter went to Washington and met with President Abraham Lincoln, General George McClellan and Secretary of War William Seward.    Lincoln gave his personal approval for the plan and assured Carter that the Union army would invade eastern Tennessee from Kentucky immediately after the bridge burning to protect the Union loyalists.159 Carter returned to Green County and, with the help of David Fry, assembled a party of 40 to 60 loyalist men.    The men gathered at the home of Jacob Harmon on the night of November 8.    In a corner of a large room was placed a small wooden table, over which was spread a United States flag.    Each man stepped forward, one at a time, and placed his left hand on the flag, raised his right hand, and took an oath to "do what was ordered of him that night and to never disclose what he had done.".159 After midnight, the raiders set out on horseback for the two-mile ride to the wooden railroad bridge across Lick Creek.    About 2:00 A.M. on November 9, they captured several Confederate guards inside a tent at one end of the bridge.    Some of the men set fire to the bridge, while others gave the guards a choice: swear loyalty to the United States or die on the spot (none of the guards chose to die that night).159 By the next day (November 10), Confederate authorities had tracked down five of the conspirators, including Jacob Hinshaw.    The promised invasion of Union soldiers never happened.159 On November 11, Confederate Colonel W.B. Wood sent a dispatch to General Samuel Cooper requesting instructions.    Colonel Daniel Leadbetter, Provisional Army, was assigned to the command of troops to rebuild the bridge.159 By November 25, time was running out for Jacob and his co-conspirators.    That day, J.C. Ramsey, district attorney for the Confederate District of Tennessee, sent a dispatch to Confederate War Secretary Benjamin: "The military authorities in command at this post have determined to try the bridge-burners and other men charged with treason by a court-martial.    What shall I do?    Answer.".    Benjamin sent an immediate and abrupt reply: "I am very glad to hear of the action of the military authorities and hope to hear they have hung every bridge-burner at the end of the burned bridge".159 On November 30, a dispatch from Colonel Leadbetter was sent from Greeneville, Tennessee: "Two insurgents have to-day been tried for bridge-burning, found guilty, and hanged.".    The two insurgents were Henry Fry and Jacob Madison "Matt" Hinshaw, who were both hanged from a large tree near the old railway depot in Greeneville.    Colonel Leadbetter ordered their bodies be left hanging on display for hours, as a warning to others.159 Jacob left his wife with 18-month old son William, and two months after Jacob was hanged Almarinda gave birth to another son, who was named after his late father Jacob.    After Jacob was hanged, his father William enlisted in the 2nd Illinois Light Artillery at age 53.159 In 1862, the U.S. Congress passed a special act that made Jacob and the other executed bridge-burners posthumous members of the Union Army.    They were enrolled in Company F of the 2nd Tennessee Infantry.    The act allowed the men's widows and their nearly 20 fatherless children to draw small pensions from the government that had abandoned the loyalist raiders it had promised to protect.    The act also allowed the men's graves, all of which are in Pottertown-area cemeteries, to be marked with official U.S. government markers.159 In 1865 Almaranda Hinshaw applied for a Civil War Pension as widow of Jacob M. Hinshaw, Company F, 2nd Tennessee Infantry.k    She also listed minor child James Hinshaw [sic].k Almarinda ("Rinda") remarried on Jun 15 1865 to William B. Jenkins.159,h,f Almarinda and family were shown in the 1870 census (Aug 11 1870), Hawkins County, Tennessee:h Photo: Artist's rendition of Jacob's hanging 159 See also - "The Pottertown Bridgeburners" narrative on the events leading to the hanging of Jacob Madison Hinshaw: http://www.rootsweb.com/~tngreene/potter.html See also - "Civil War Times", December 1997: http://www.britannica.com/bcom/magazine/article/0,5744,25823,00.html See also - Dave Mathews' study of the 2nd Tennessee Infantry: http://home.fuse.net/damathew 45. The Church Of Jesus Christ of the Latter Day Saints (LDS) International Genealogical Index (IGI) - Tennessee. 159. "Civil War Times", December, 1997, pages 48-54 & 86-87, Contributed by Jeff Hatfield (). 162. Letter from Donahue Bible, historian and author, discussing Hinshaws of eastern Tennessee. 174. Family history information contributed by James C. Henshaw. (a) "Early East Tennessee Marriages", Sistler, 1987. (b) "Tennessee Marriages, 1851-1900"; http://www.ancestry.com. (c) 1850 census, 9th Division, Greene County, Tennessee; roll 880, page 164, dwelling #387. (d) 1850 census, 9th Division, Greene County, Tennessee; roll 880, page 168, dwelling #443. (e) 1860 census, 23rd Civil District, Blue Spring P.O., Greene County, Tennessee; page 413A, line #36, dwelling #1405, family #1405. (f) "Tennessee Marriage Records - Volume 2, Hawkins County 1789-1865", by Prentiss Price; 1958, Clinchdale Press, Knoxville, Tennessee; microfilm #0469471 Item 1 in the LDS Family History Library. (g) Contribution from Andra Walker Clark () citing: records of genealogist Prentiss Price. (h) 1870 census, White Horn P.O., District 14, Hawkins County, Tennessee; roll M593-1535, page 183, line #36, dwelling #107, family #107. (i) 1880 census, District 14, Hawkins County, Tennessee; roll T9-1260, ED 75, page 168C, line #11, dwelling #147, family #147. (j) 1860 census, 23rd Civil District, Blue Spring P.O., Greene County, Tennessee; page 413A-B, line #39, dwelling #1406, family #1406. (k) National Archives and Records Administration. "General Index to Pension Files, 1861-1934": "Civil War Pension Index". [database] Provo, UT: Ancestry.com, 2000. (l) 1880 census, District 14, Hawkins County, Tennessee; roll T9-1260, ED 75, page 167B, line #46, dwelling #144, family #144. If you have additional information on this person, please share! If you would like to be automatically notified by email whenever an update is made affecting this page then enter your email address and click the "Enter" button below: How is this person related to other ancestors? To find out, enter the database ID number of another ancestor, and then click the "Enter" button below: Return to HFA Home Page This site uses spambot thwarting technology to hide email addresses from all known email harvesting programs used by spammers. [This page was computer generated]

Are you tired of searching for a simple and straightforward method to find the median? Look no further! In this article, we will demystify the process and provide you with a step-by-step guide on how to find the median with ease. Whether you’re a student struggling with math homework or simply curious about this statistical measure, our concise and jargon-free explanation will help you master the art of finding the median in no time. So, let’s dive in and unravel this mathematical mystery! Introduction to Finding the Median Finding the median is a fundamental statistical concept that allows us to understand the middle value of a set of data. Whether you’re analyzing a small sample or a large population, understanding how to find the median is crucial to gaining insights and drawing conclusions from your data. In this comprehensive guide, we will explore the concept of median, step-by-step instructions on finding it, techniques for dealing with outliers or missing values, practice exercises, and common mistakes to avoid. The Importance of Finding the Median The median is a measure of central tendency that is exceptionally useful in situations where the data may contain outliers or extreme values. Unlike the mean, which can be heavily influenced by outliers, the median provides a more robust representation of the „typical” value in a dataset. It is particularly handy in skewed distributions or when the data contains extreme values that do not reflect the overall trend. The Relevance of Median in Various Fields The concept of median finds its applications in a wide range of fields, such as economics, healthcare, finance, and social sciences. For example, in economics, the median household income is often used as a measure to understand the income distribution and inequality within a country. In healthcare, the median survival time can indicate the effectiveness of a treatment. The median price of a house can provide insights into the real estate market. Understanding how to find the median is, therefore, crucial for making informed decisions and drawing accurate conclusions from data. Understanding the Concept of Median To effectively find the median, it is essential to grasp the underlying concept. The median represents the middle value in a dataset, dividing it into two equal halves, with 50% of the values falling below and 50% above this central point. It is not affected by the actual values of the dataset but only by their order. In other words, the median is the value that separates the data into two halves, with an equal number of values on either side. Benefits of Using the Median Limitations of the Median Step-by-Step Guide to Finding the Median To find the median, follow these simple steps: 1. Arrange the dataset in ascending order from smallest to largest. 2. Calculate the total number of values (N) in the dataset. 3. Determine whether N is even or odd. 4. If N is odd, the median is the middle value in the ordered dataset. 5. If N is even, the median is the average of the two middle values. 6. The median is found! It is important to remember that the dataset must be sorted in ascending order for these steps to work correctly. Now, let’s dive into finding the median in more detail under both odd and even scenarios. Finding the Median in a Set of Data with an Odd Number of Values Finding the median in a dataset with an odd number of values is relatively straightforward. Let’s say we have a dataset with nine values. We follow the step-by-step approach discussed above. 1. Arrange the dataset in ascending order: 12, 15, 18, 19, 20, 26, 28, 31, 40. 2. Calculate the total number of values: N = 9. 3. Since N is odd, the median is the middle value. 4. The middle value is the fifth value in the ordered dataset, which is 20. Therefore, the median is 20. It’s important to note that in datasets with an odd number of values, there is always a single middle value. This value becomes the median, representing the center of the data distribution. Finding the Median in a Set of Data with an Even Number of Values Finding the median in a dataset with an even number of values requires one additional step compared to finding it in an odd-sized dataset. Let’s consider a dataset with 10 values. 1. Arrange the dataset in ascending order: 10, 12, 15, 18, 19, 21, 24, 30, 35, 40. 2. Calculate the total number of values: N = 10. 3. Since N is even, the median is the average of the two middle values. 4. The two middle values are the 5th and 6th values, which are 19 and 21. 5. Calculate the average of these two values: (19 + 21) / 2 = 20. 6. Therefore, the median is 20. By taking the average of the two middle values, we can find the median in a dataset with an even number of values. This technique ensures a representative value that falls between the two central points. Dealing with Missing or Outlier Values in Finding the Median While calculating the median, missing or outlier values can pose challenges. However, there are strategies to handle these situations and ensure accurate results. When dealing with missing values, the first step is to determine the reason for their absence. If it is a random or non-systematic issue, and the missing values are only a small fraction of the dataset, you may choose to exclude them. However, if the missing values are systematic or significant in number, they should be dealt with more carefully. One approach is to impute missing values based on other available information or statistical techniques such as regression, mean, or median imputation. By doing so, you can estimate the missing values and proceed with finding the median using the complete dataset. Outliers are extreme values that deviate significantly from the rest of the dataset. It’s important to assess whether these outliers are genuine, data entry errors, or a result of some other phenomenon. In some cases, outliers may need to be excluded if they do not represent the true nature of the data. If you decide to include the outliers, it is essential to acknowledge their impact on the median. Outliers can shift the median towards their side, especially if they are located close to or within the middle of the ordered dataset. It’s crucial to consider the context of the data and the potential influence of outliers when interpreting the median. Practice Exercises for Finding the Median To enhance your understanding and skill in finding the median, it’s crucial to practice with various datasets. Here are some exercises to help you master the concept: 1. Dataset 1: 2, 5, 8, 11, 15, 18, 20 2. Dataset 2: 12, 15, 20, 32, 32, 36, 41, 42 3. Dataset 3: 17, 21, 24, 28, 30, 34, 39, 50, 78 4. Dataset 4: 9, 12, 15, 15, 21, 21, 26, 27, 29 By practicing with different datasets, you will gain confidence in finding the median and become more comfortable with its interpretation. Common Mistakes to Avoid When Finding the Median While finding the median may seem straightforward, there are common mistakes that even experienced analysts can make. By being aware of these pitfalls, you can ensure accurate results and proper interpretation of your data. Mistake 1: Forgetting to Sort the Data One of the most common mistakes is failing to sort the dataset in ascending order before finding the median. Without proper ordering, you may obtain incorrect results, leading to misinterpretation and flawed analysis. Always double-check that your dataset is appropriately arranged before proceeding. Mistake 2: Mishandling Even-Sized Datasets In datasets with an even number of values, determining the median requires calculating the average of the two middle values. Neglecting this additional step or calculating the average incorrectly can lead to inaccurate results. Pay close attention to the steps when dealing with even-sized datasets. Mistake 3: Ignoring Missing or Outlier Values Failing to account for missing values or not considering the impact of outliers can significantly affect the median. Make sure to address missing values appropriately through imputation techniques and critically assess the influence of outliers on the central tendency measure. By avoiding these common mistakes, you can confidently find the median and utilize it effectively in your data analysis. The median is a powerful statistical tool that allows us to understand and interpret a dataset’s central tendency accurately. By following the step-by-step guidelines outlined in this comprehensive guide, you can confidently find the median in both odd and even-sized datasets. Additionally, understanding how to handle missing or outlier values ensures accurate results despite potential challenges. Through practice exercises, you can hone your skills and avoid common mistakes associated with finding the median. Whether you’re analyzing data for business, research, or personal purposes, the ability to find and interpret the median is a valuable asset in making informed decisions and drawing reliable conclusions.

This home is not currently listed for sale or rent on Zillow. A Zestimate® home valuation is Zillow's estimated market value. It is not an appraisal. Use it as a starting point to determine a home's value. Learn more A Rent Zestimate® is Zillow's estimated monthly rental price, computed using a proprietary formula. It is a starting point in determining the monthly rental price for a specific property. - Lot: 8,712 sqft - Single Family - Built in 1961 - Views: 3,472 all time views - Cooling: Central - Heating: Forced air - Last sold: Aug 2011 for $111,000 - Ceiling Fan - Fenced Yard - Flooring: Carpet, Hardwood, Tile - Parking: Garage - Attached - 2nd Bedroom Flooring \ Hardwood, 2nd Bedroom Level \ Main Level, 2nd Bedroom Window Treatments (Y\N) \ None, 3rd Bedroom Flooring \ Hardwood, 3rd Bedroom Level \ Main Level, 3rd Bedroom Window Treatments (Y\N) \ None, 4th Bedroom Level \ Not Applicable, Additional Rooms \ Recreation Room, Additional Sales Information \ None, Addtl Room 1 Flooring \ Carpet, Addtl Room 1 Level \ Basement, Addtl Room 1 Name \ Recreation Rm, Addtl Room 10 Level \ Not Applicable, Addtl Room 2 Level \ Not Applicable, Addtl Room 3 Level \ Not Applicable, Addtl Room 4 Level \ Not Applicable, Addtl Room 5 Level \ Not Applicable, Addtl Room 6 Level \ Not Applicable, Addtl Room 7 Level \ Not Applicable, Addtl Room 8 Level \ Not Applicable, Addtl Room 9 Level \ Not Applicable, Age \ 51-60 Years, Appliances \ Oven\Range, Avail Furnished (Y\N) \ No, Basement (Y\N) \ Full, Basement Bathrooms (Y\N) \ Yes, Basement \ Finished, Built Before 1978 (Y\N) \ Yes, Compensation paid on \ Net Sale Price, Dining Room Level \ Main Level, Dining Room \ Combined w\ LivRm, Disability Access and\or Equipped \ No, Driveway \ Concrete, Electricity \ Circuit Breakers, Equipment \ CO Detectors, Equipment \ Ceiling Fan, Equipment \ Fan-Attic Exhaust, Equipment \ Sump Pump, Equipment \ TV-Cable, Exterior Property Features \ Storms\Screens, Family Room Flooring \ Carpet, Family Room Level \ Basement, Family Room Window Treatments (Y\N) \ None, Fees\Approvals \ Credit Report, Fees\Approvals \ Letters of Reference, Fireplace Location \ Family Room, Fireplace Location \ Living Room, Fireplace Type\Details \ Gas Logs, Fireplace Type\Details \ Gas Starter, Fireplace Type\Details \ Wood Burning, Foundation \ Concrete, Franchisor Feed(y\n) \ No, Garage On-Site \ Yes, Garage Ownership \ Yes, Garage Type \ Attached, Heat\Fuel \ Forced Air, Heat\Fuel \ Gas, Interior Property Features \ 1st Floor Bedroom, Interior Property Features \ 1st Floor Full Bath, Interior Property Features \ Bar-Wet, Interior Property Features \ Hardwood Floors, Kitchen Flooring \ Ceramic Tile, Kitchen Level \ Main Level, Kitchen Type \ Galley, Kitchen Window Treatments (Y\N) \ None, Laundry Level \ Not Applicable, Lease Terms \ 1 Year Lease, Living Room Flooring \ Hardwood, Living Room Level \ Main Level, Living Room Window Treatments (Y\N) \ None, Lot Description \ Corner, Lot Description \ Fenced Yard, Management \ Self-Management, Master Bedroom Bath (Y\N) \ Full, Master Bedroom Flooring \ Hardwood, Master Bedroom Level \ Main Level, Master Bedroom Window Treatments (Y\N) \ None, Monthly Rent Incl \ None, Offered for Sale or Rent \ No, Other Information \ School Bus Service, Parking Type \ Garage, Pet Information \ Deposit Required, Pet Information \ Pet Count Limitation, Pets Allowed (Y\N) \ Yes, Property Address On IDX? (Y\N) \ Yes, Property Type \ Residential Rental, Roof Type \ Asphalt\Glass (Shingles), Sewer \ Sewer-Public, Short Term Lease Ok (Y\N) \ No, Special Compensation Info. \ None, Square Feet Source \ Assessor, Status \ Active, Type of Rental Property \ Detached, Water \ Public, Waterfront (Y\N) \ No - Garbage disposal - Range / Oven - Dining room - Exterior material: Brick, Vinyl - Roof type: Asphalt - Structure type: Ranch - Unit count: 1 - Floor size: 1,076 sqft - Heating: Gas - Laundry: In Unit - Parcel #: 0933176001 - Pets: Dogs - Zillow Home ID: 4661146 The Value Range is the high and low estimate market value for which Zillow values a home. The more information, the smaller the range, and the more accurate the Zestimate. See data coverage and accuracy table Don't agree with your home's Zestimate? Owners can edit their home facts to make the Zestimate more accurate. Plus, you can leave an opinion on your Zestimate value below. Just click "Owner Estimate". The Rent Range is the high and low estimate for which an apartment or home could rent. The more information we have, the smaller the range, and the more accurate the Rent Zestimate. See data coverage and accuracy table Popularity on Zillow Homes like this sold for $169-210K. Market guideZillow predicts 60174 home values will decrease 1% next year, compared to a 0.6% fall for Saint Charles as a whole. Among 60174 homes, this home is valued 16.3% less than the midpoint (median) home, but is valued 39.4% more per square foot. Foreclosures will be a factor impacting home values in the next several years. In 60174, the number of foreclosures waiting to be sold is 16.2% greater than in Saint Charles, and 888.6% higher than the national average. This higher local number may prevent 60174 home values from rising as quickly as other regions in Saint Charles. Learn more about forecast calculations or 60174 home values. 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Directed Paths on Hierarchical Lattices with Random Sign Weights We study sums of directed paths on a hierarchical lattice where each bond has either a positive or negative sign with a probability . Such path sums have been used to model interference effects by hopping electrons in the strongly localized regime. The advantage of hierarchical lattices is that they include path crossings, ignored by mean field approaches, while still permitting analytical treatment. Here, we perform a scaling analysis of the controversial “sign transition” using Monte Carlo sampling, and conclude that the transition exists and is second order. Furthermore, we make use of exact moment recursion relations to find that the moments always determine, uniquely, the probability distribution . We also derive, exactly, the moment behavior as a function of in the thermodynamic limit. Extrapolations () to obtain for odd and even moments yield a new signal for the transition that coincides with Monte Carlo simulations. Analysis of high moments yield interesting “solitonic” structures that propagate as a function of . Finally, we derive the exact probability distribution for path sums up to length for all sign probabilities. Sums of directed paths are present in numerous models of disordered systems. Polymer configurations in a disordered matrix, dynamics of interfaces grown by deposition and Feynman path sums for electron hopping between impurities[2, 3] are only a few of the relevant examples. In this paper, we focus on the latter example involving a model first introduced by Nguyen, Spivak and Shklovskii (NSS), for interference effects in the strongly localized regime. In the directed path sign model one studies the sum of all possible directed paths between two sites on a lattice. On each lattice bond, one places a random sign with probability . Each directed path evolved is then computed by multiplying the values of the bonds it crosses. Finally the sum of all paths is obtained. The proponents of the model obtained, numerically for small systems, that a second order transition occurred at between a phase with preferential sign (for the path sum ), and a phase with no preferential sign. NSS also offered appealing arguments based on the behavior of . Presumably, such a parameter grows exponentially above the transition while it goes to zero below . The physical relevance of this transition lies in the fact that it may signal the change between Aharonov-Bohm oscillations of period and those of in the context of hopping conduction. The NSS argument was later contended by Shapir and Wang arguing that correlations between paths implied that does not necessarily go to zero for any . Subsequently, Wang et al used an exact enumeration scheme to probe the transition for small lattices of maximum size . The work found no evidence of a transition above negative sign probability . Such conclusions were supported by Zhao et al on the basis of numerics, for large square lattices, where it was assured that the transition did not exist above in two dimensions. Nevertheless, the decay of the order parameter as a function of system size was found to be anomalously slow for finite (see also ). Therefrom, more recently Spivak, Feng and Zeng discussed numerical results that suggest a finite jump in the order parameter indicating a first order transition for the sign problem. The authors also imply that the moments increase faster than as indicating there is no unique relation between and the probability distribution . This is an important point since the moments, in such a case, may not contain information about the transition. Finally, in a recent paper by Nguyen and Gamietea, a renewed extensive study of the parameter proves that, at least according to such a parameter, no transition exists; only a strong crossover from logarithmic to exponential behavior is observed. Besides the numerical approaches, mean field type approximations by Obukhov point to a second order transition for dimensions . Furthermore, Derrida and Cook also took up the problem, analytically, using a sparse matrix approach. They generalized the model to random phases, which includes random signs as a special case. Their approach is mean field in nature and results in a phase diagram where the sign transition is of second order (see also). Nevertheless, mean field results may not apply to low dimension due to the importance of path crossings. Here we address the following issues: i) What is the order of the sign transition through scaling analysis of the order parameter proposed, ii) do moments of the path sums determine the probability distribution uniquely? and iii) what is the exact behavior of the parameter above and below the transition?. A new perspective will be gained by using hierarchical lattice: Such lattices, while still amenable to analytical manipulation, include crucial path correlation effects absent in mean field. The paper is organized as follows: Section II discusses the sign model and describes hierarchical lattices. In section III we perform detailed Monte Carlo simulations, close to the transition, for systems of up to size . A scaling analysis is performed for the order parameter to distinguish between first and second order transitions. In section IV we study the moments exactly, using moment recursion relations. We find that moments determine the distribution uniquely according to Carlemans theorem, and find possible indications of a phase transition from odd and even moment extrapolations to . In this section we also discuss the high moment behavior, unveiling interesting structures as a function of the sign probability . Subsequently, we probe the parameter exactly showing, for the first time, its unambiguous crossover between exponential and logarithmic behavior. In section V we obtain the exact probability distribution for lattice sizes and sample the distribution for up to as a function of . We end with the conclusions and a discussion of the mapping of the moments to an -body partition function in one dimension as a continuum model that might aid in explaining the curious high moment behavior. Ii The sign model Imagine two reference points on a lattice between which one would like to evolve all possible directed paths and compute a “partition function” where represents each individual contributing path. By directed it is meant that paths always propagate in the forward direction without loops or overhangs. The random medium in which these paths evolve can be represented by assigning local weights on the bonds or sites that are picked up by the paths as they wander to their final destination. Such a model has been used as a paradigm simulating, for example, a coarse-grained polymer or interface wandering in a random matrix with locally favorable energy minima. The model is interesting because it yields anomalous lateral wandering and energy exponents for the interface/polymer as compared to those generated by simple diffusion, signaling a new disorder induced universality class in dimensions. Another application, in an entirely different field, is in the context of Variable Range Hopping, a mechanism for conduction in insulators. In this context, one also needs to sum over Feynman paths to compute the transition probability, between impurities, of current bearing electrons. The Feynman paths, in this case, are directed because they are tunneling paths. Any elongation of the latter, in the form of loops or overhangs, is exponentially less probable. For further justification of the model we refer the reader to the review in reference. NSS studied such tunneling processes proposing a directed path model where the local weights are random signs. In such a model, the path is a product of the signs it picks up en route to the final site. Writing eq.(1) more explicitly where is a random sign according to the distribution . The probability in the NSS model emulates the relative abundance of levels above and below the Fermi energy. This model has been very successful in explaining qualitative and quantitative features of conduction in the strongly localized regime. In particular, intriguing interference effects producing a characteristic periodicity of magnetic field oscillations and changes in the localization length due to non-local effects[3, 17]. In spite of the seemingly different nature of disorder in the NSS model, replica arguments and numerics have shown that it belongs to the same universality class of directed polymers with positive weights[3, 18], at least for close to . We have taken up the sign model on hierarchical lattices as mentioned in the introduction. A hierarchical lattice is a recursive structure built by repeating a chosen motif. Depending on the latter motif, one can build integer dimensional objects emulating an Euclidean lattice. For this work we chose the Berker lattice or diamond. Such motif (see Fig. 1) has the parameter corresponding to the number of branches between the initial and final points. The lattice size is related to the recursion order as i.e. the number of bonds on any directed path between and . The number of bonds on the lattice (or mass) is given by , so that the effective dimension of the lattice is . In this work we will use except if otherwise stated. Qualitative features of critical behavior of many statistical models are correctly reproduced on such structures with no unphysical effects. In fact, mapping to hierarchical lattices is the basis of the Migdal-Kadanoff renormalization procedure, of frequent use in critical phenomena. As noted before, an important feature of hierarchical lattices over either Bethe lattices/mean field approaches is that path intersections are taken into account. Thus, we expect that the resulting simulations will be more faithful to low dimensional behavior. In fact, we will present, in section IV, further evidence of the adequacy of hierarchical lattices making contact with known recent results on the sign transition. Iii Sign phase transition in two dimensions In this section, we have undertaken Monte Carlo simulations on hierarchical lattices to check for scaling properties. Paradoxically, scaling has only been discussed once before in connection with the transition and it is a primary tool to assess its nature. It will be especially useful to clearly distinguish between first order and second order transitions. Hierarchical lattices were generated to or order ten. Averages were taken over twenty thousand realizations of disorder for a series of values between and . As the size of the system increases more detailed data was collected close to the transition regime . Figure 2 shows Monte Carlo data for the order parameter as a function of . A definite plateau at develops as increases for low , signaling a definite change in the order parameter (positively signed paths dominate). For the proposed order parameter we should expect the scaling form . Figure 3 shows a good collapse for the same data as the previous figure. As the order parameter is always between zero and one, we only need to find and the correlation length exponent . For the hypothetical transition we find the values and (). The latter exponent is very different from that of percolation on these lattices (; so the role of percolation if any, is not apparent. If the transition were first order the exponent would be the dimensionality of the system . The non trivial scaling found can also be seen by taking the derivative of the order parameter and plotting its maximum as a function of the system size. These criteria rule out a first order transition. We have also monitored the evolution of with size. The specific value of was found from the peak values of the derivative of the order parameter . The resulting values are plotted in Fig. 4 where, within error bars, the values of and are confirmed. Summarizing, scaling is very good around and does not correspond to the scaling of a first order transition. Furthermore, there is no sign of a discontinuity in the order parameter as suggested in ref.. We thus conclude that, on hierarchical lattices, the transition exists and is second order as mean field predicts. These conclusions are in agreement with work by Roux and Coniglio on hierarchical lattices. There, they analyzed the variable where is the fraction of positive (negative) paths arriving at site , and they suggest a clear positive phase. The order of the transition for hierarchical lattices is not analyzed in detail in their paper. Nevertheless, they noted undue emphasis of hierarchical lattices on the result, and the possible impact of this on the scaling properties of various quantities. We will come back to such observations, briefly, in section V. Iv Moment recursion relations A statistic we can probe exactly on hierarchical lattices are the moments of the probability distribution. This is possible because of recursion relations derived by Cook and Derrida and generalized to arbitrary moment and hierarchical order (system size) by Medina and Kardar. The recursion relation for is the following where is the moment number and is the hierarchical lattice order. This expression is readily generalized to other integer by changing the binomial factor to a multinomial and including the additional branches. Hence, one can emulate higher dimensional networks. The simple form of this recursion permits, given the local moments at order one, to compute moments to any given lattice size. Appropriate programming of the recursion relations, with arbitrary precision computations, is linear in time with lattice order. The behavior of the moments for the sign model is extremely rich as we shall see in the following. As found in ref. , after a few hierarchical orders, the values converge rapidly to a limiting form as a function of . Such limiting form is important because it also signals the convergence to a unique limiting distribution, at least, if moments do not grow faster than . The asymptotic form of the moments can be obtained for [13, 21], that is, moments grow exponentially with for . Nevertheless, for , lower moments grow slightly faster than exponential ( with ), gradually converging to exponential growth for larger moments. The latter implies, according to the condition that the moments determine the distribution uniquely. There are various forms of such a theorem, but the above is the strongest version due to Carleman. If one substitutes above, , -our asymptotic result- the criterion is satisfied. Even if grows slightly faster i.e. the above sum diverges because . Any faster growth would violate eq.(5), factorial growth being the borderline case. That the moments satisfy eq.(5) is one of our central results. In Fig. 5 we show a sequence of moments as a function of the moment number . The different curves, starting from below, represent hierarchical orders one thru nine (sizes thru ). One readily notes convergence to a definite law. The inset shows a comparison between the growth of and that of moments for the particular case of . The asymptotic behavior is already reached at , larger sizes falling on the same curve. For values close to , the moment sequence has a characteristic sawtooth shape, where even moments are at the crests and the odd at the troughs. Such structure is not a finite size effect. We have checked this for up to on the hierarchical lattice. As all the odd moments go to zero while the even remain finite as expected. On the other hand, as is reduced the sawtooth disappears, first for the higher moments and then for the lower. In this respect there appears to be a phase transition for each moment at different values of , in a way reminiscent of that discussed by Cook and Derrida (in their case as a function of ‘temperature’). The transition for the first two moments occurs close to which is close to that found from Monte Carlo simulations in the previous section. On this basis it is plausible that the disappearance of sawtooth shape is related to the transition. Figure 6 shows a set of curves for , and various values of , and up to . The last six orders of the hierarchical lattice collapse on the same curve indicating we have achieved asymptotics. For the highest value of one notes the sawtooth behavior, while it disappears for all moments below . Nevertheless, additional structure is observed at moments beyond for and , where a shoulder develops and moves towards larger values as increases, undeformed, in a solitonic manner. Although the analysis of these structures is out of the scope of this paper, it is interesting to analyze it in the light of a mapping to a one dimensional many body problem. In such a mapping the moment number corresponds to the number of particles interacting like charges on contact. Thus we speculate that the shoulders could be related to sudden changes in the character of the ground state as the particle number (moment number) increases. We will discuss this in more detail in the final section. For even smaller values the curve starts to resemble the well known limit given by eq.(4), and depicted as a flat line at in Fig. 6. From the figure one can graphically identify the value of as a function of using the relation . The quantity is a “free energy” that may reflect the sign transition. We have followed the value at intercept mentioned before as a function of below . When the moments “zigzag” there are two possible extrapolations, while below the assumed transition the moments lead to a single prediction of the free energy. The results are depicted in Fig. 7, where the curves merge around within the error of the extrapolation procedure. Such a value coincides with our Monte Carlo prediction. One can validate the relevance of hierarchical lattices by checking the exact computation of the variable with . Such a quantity has been discussed extensively in previous work[2, 4, 8, 24]. As mentioned before, was initially suggested as a candidate for a kind of order parameter that diverged exponentially above the transition and went to zero below. Observations by Shapir and Wang showed, nevertheless, that path correlations (crossings) invalidated the vanishing of the parameter for any value of . It has been argued that for small there is a crossover from exponential growth (for ) to logarithmic growth (for )[8, 24]. Shapir and Wang, on the other hand found a change from for to for . Yet, they observe that the former result is incorrect because partial overlaps of pairs of walks should be accounted for. Simulations on regular lattices to date can only do very poorly in proving the surmised logarithmic behavior below . Here we have computed to sizes for various values in a few CPU minutes. We have found a clear confirmation of logarithmic to exponential crossover as increases. Figures 8 and 9 show and its derivative as a function of respectively. The scales used permit rapid identification of the corresponding behavior. It should be noted that, on euclidean lattices, it is reported the behavior reported is , where but depends weakly on . On hierarchical lattices we can also demonstrate analytically that there is no transition in the variable . Following Cook and Derrida, eq.(3), for the first two moments, can be written for general as Now, after defining one can write a recursion relation for as It is simple to determine that has in general three fixed points, . For () a critical fixed point arises and exhibits a phase transition as NSS proposed. On the other hand, for there are only two trivial fixed points; is unstable and is stable, indicating that always diverges as found above. Values of close to one correspond to , while close to zero correspond to . Analyzing the behavior of the recursion for near the fixed point one can derive from Eq.IV that . The behavior close to , which should be logarithmic, is also verified (numerically) although we have not arrived at a simple closed expression. In summary, hierarchical lattices provide similar results to those expected on euclidean lattices thus seeming a good testing ground for the sign transition. As a final word; we have computed higher order cummulants of finding no features of special interest related to the transition. The only result worth mentioning is that for , where is the th cummulant of . In what follows we will take advantage of the special structure of hierarchical lattices to compute the full probability distribution for . V Probability distribution for Monte Carlo sampling of the distribution of is handicapped by the models distribution broadness. For such reasons Wang et al undertook an exact enumeration study to probe the NSS order parameter . Because of the high computer demand of exact enumeration, they could only access sizes of for all . Here we use a scheme, on hierarchical lattices, permitting access to exactly for all and sampling of the distribution for . The procedure is as follows: As a hierarchical lattice is built recursively following a chosen motif, one can write the following recursion relation for the probability distribution. where and denote contiguous elements on separate branches of the hierarchical lattice. , where is the sign probability discussed in previous sections. The number of possible outcomes for or number of different paths goes as ( for , and for ). This growth is extremely fast, although many values will be degenerate for any particular disorder realization. Note that while is easily accessible, going an order further, puts the calculation out of reach, no intermediate sizes being available on hierarchical lattices. For we have resorted to a coarse-graining procedure in the following manner: the exact results for involve terms which we cannot evolve exactly to the next order. Nevertheless, we can make a coarse-grained distribution by averaging occurrences in groups of to obtain different values. One can then go to up to by repeating this procedure. Beyond such a size, the coarse graining procedure does not incorporate sufficient detail to see anymore changes in the distribution, so within our resolution we have achieved its limit form. Fig. 10a shows the probability distribution for for significant values of . The probability distribution is astonishingly complex, even for small sizes, revealing rich interferences in the paths sums . Note that it would be hopeless to sample the distribution using Monte Carlo as there are sixty to one hundred and thirty orders of magnitude of probability. Fig. 10b shows different values for a sample as a function of using the coarse-graining procedure described above. As expected the distribution is symmetric for and gains asymmetry () as moves towards zero. Note that falls slower than exponential, on average, about the peak value. It is notable the speed with which the distribution appears symmetric beyond . This feature is understood in the ‘zigzag’ behavior of the moments, where odd moments are much smaller than even moments and their separation increases exponentially as is increased. Having the information of the exact distribution one is also able to obtain the exact order parameter introduced in section III. No qualitative differences were found with curves reported in figures 1 and 2 at least to sizes , so sampling of , involved in Monte Carlo, seems to be good enough to draw the conclusions about the transition (see section III). In Fig. 11 we have depicted the distribution for and without joining the points for the probability amplitude (as was done in Fig. 10). A fractal structure is apparent; The whole distribution, in the shape of an approximate triangle, is built from, scaled, identical triangular structures up to the resolution achieved by the coarse-graining procedure. Similar complexity is expected for the sign problem on Euclidean lattices. An interesting final point to make in this section is that, in view of the unique relation between distribution and moments (see section IV), it is possible to use known inversion formulas. In this way one could derive the limiting distribution exactly to any order desired. Vi Summary and Discussion We have provided evidence for the existence of a phase transition for the directed path sign model on hierarchical lattices. Nontrivial finite size scaling of the order parameter close to the transition, points to a second order phase transition as found from mean field type approaches. From numerical computations, the threshold on diamond hierarchical lattices is and the correlation length exponent is . The latter exponent is very different from that of percolation on the same lattice . The study of exact moment recursion relations for led us to the definitive conclusion that the moments uniquely determine the probability distribution, according to Carleman’s theorem. Using extrapolations of the derivative of integer moments () to , we were able to find a “free energy” . Such a free energy splits into two possible extrapolations (from even and odd moments) as one goes through the transition point by increasing . The latter transition point coincides with that found in Monte Carlo simulations of the sign transition. We have not completely interpreted this connection in the present paper. Furthermore, we studied the high moments of the partition function below the transition, and found a very interesting non-monotonic behavior including step structures that propagate on the moment number axis, as changes. Using the fact that moments can be computed exactly we studied the celebrated ratio proposed by NSS. We have shown, analytically that indeed in the ratio does not show a transition as suspected numerically[4, 8, 24] on regular lattices. Furthermore, we have shown that hierarchical lattices exhibit the same logarithmic to exponential crossover for surmised in references and . Finally, we studied a recursion relation for the full probability distribution for , finding an extremely complex structure even for systems as small as . Previous remarks by Roux and Coniglio of anomalous accumulation of probability at are confirmed. Nevertheless, their claim that the hierarchical lattice becomes essentially one dimensional for large , and thus, the probability distribution should approach a Gaussian is not borne out from our results. One obvious difference is that for a Gaussian all cumulants, larger than two, should be zero which is in disagreement with exact results of section IV. No evident signal of the transition, beyond that already inferred from the order parameter , is found from the full probability distribution. Medina and Kardar have studied the moments for the sign problem interpreting them as partition functions for -body one dimensional Hamiltonians with contact interactions. Most of the focus, however, has been on the low behavior that yields cummulants of . Nevertheless, it would be interesting to interpret the findings of this paper, regarding high moments, in the light of a many body-theory. A previous effort by Zhang focused on the Hartree-Fock approximation valid only for a large number of particles (higher moments). In Zhang’s approach the sign model was equivalent to finding the ground state of the many body Hamiltonian where is a charge that acts via contact interaction of the particle: for and for . Zhang’s approach yields . Our findings predict, from the relation , for large (see also ), where increases as . For lower the behavior is non-trivial and is certainly not represented a simple powerlaw of . Therefore, Zhangs results represent some kind of intermediate regime. A more detailed solution of eq. (9) might yield the ‘solitonic’ patterns reported here (see section IV) which are not well understood. As speculated in section IV, the ground state formed by particles with attractive and repulsive interactions might change, suddenly, at critical particle numbers generating discontinuities in the derivative of . More work is needed in this direction. The highly non-monotonic behavior displayed by the moments calls for caution regarding the regime of validity of moments dependencies on the moment number reported in the literature[3, 23]. Claims of a non-unique relation between moments and the probability distribution were based on expressions only valid in the limit, which is clearly unrelated to the constraints of Carlemans theorem. Obviously, the conclusions of this paper are only valid in the measure to which hierarchical lattices agree with continuum results. For a discussion of the latter point see reference. Acknowledgements.EM thanks G. Urbina and A. Hasmy for careful reading of the manuscript and R. Paredes for a valuable suggestion. EGA thanks R. Rangel for useful discussions. This work was supported by CONICIT through grant S1-97000368, a grant from the Research Fund of UNEXPO and the POLAR Foundation. - E. Medina, T. Hwa, M. Kardar, and Y.-C Zhang, Phys. Rev. A 39, 3053 (1989). - V. L. Nguyen, B. Z. Spivak, and B. I. Shklovskii, Pis’ma Zh. Eksp. Teor. Fiz/ 41, 35 (1985) [JETP Lett. 41, 42 (1986); Zh. Eksp. Teor. Fiz. 89, 11 (1985) [Sov. Phys.-JETP 62, 1021 (1985)]. - E. Medina, and M. Kardar, Phys. Rev. B 46, 9984 (1992). - Y. Shapir, and X. R. Wang, Europhys. Lett. 4, 1165 (1987). - X. R. Wang, Y. Shapir, E. Medina, and M. Kardar, Phys. Rev. B 42, 4559 (1990). - H. L. Zhao, B. Spivak, M. Gelfand, and S. Feng, Phys. Rev. B 44, 10760 (1991). - B. Spivak, S. Feng, F. Zeng, JETP Lett. 64, 312 (1996). - V. L. Nguyen, and A. D. Gamietea, Phys. Rev. B 53, 7932 (1996). - S. Obukhov, in Hopping Conduction in Semiconductors ref. p.338. - J. Cook, and B. Derrida, J. Stat. Phys. 61, 961 (1990). - In ref. Cook and Derrida add a temperature parameter. The limit corresponds to the pure sign model we treat here. - Y. Y. Goldschmidt, and T. Blum, J. Phys. I 2, 1607 (1992). - E. Medina, and M. Kardar, Jour. Stat. Phys. 71, 967 (1993). - M. Kardar, Y.-C. Zhang, Phys. Rev. Lett. 58, 2087 (1987). - B. I. Shklovskii, and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Berlin, 1984). - B. I. Shklovskii and B. Z. Spivak, in Hopping Conduction in Semiconductors, edited by M. Pollak and B. I. Shklovskii (North-Holland, Amsterdam, 1990). - F. P. Milliken, and Z. Ovadyahu, Phys. Rev. Lett. 65, 911 (1990). - S. Roux, and A. Coniglio, J. Phys. A 27, 5467 (1994). - A. N. Berker, and S. Ostlund, J. Phys. C 12, 4961 (1979). - J. Lee, and J. M. Kosterlitz, Phys. Rev. B 43, 3265 (1991). - J. Cook, and B. Derrida, J. Stat. Phys. 57, 89 (1989). - W. 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Presentation on theme: "Chemical Equations Chemical Reaction: Interaction between substances that results in one or more new substances being produced Example: hydrogen + oxygen."— Presentation transcript: Chemical Equations Chemical Reaction: Interaction between substances that results in one or more new substances being produced Example: hydrogen + oxygen water Reactants of a Reaction: Starting materials that undergo chemical change; written on the left side of the equation representing the reaction Products of a Reaction: Substances formed as a result of the reaction; written on the right side of the equation representing the reaction The arrow points towards the products formed by the reaction Individual products and reactants are separated by a plus sign Chemical Equation: A written statement using symbols and formulas to describe the changes that occur in a reaction Example: 2H 2 (g) + O 2 (g) 2H 2 O (l) Letter in parentheses indicates the state of the substance: gas (g), liquid (l), solid (s), dissolved in water (aq) If heat is required for the reaction to take place, the symbol is written over the reaction arrow Balanced Equation: Equation in which the number of atoms of each element in the reactants is the same as the number of atoms of that element in the products Law of Conservation of Mass: Atoms are neither created nor destroyed in chemical reactions Example: CaS + H 2 O CaO + H 2 S Reactants Products Is this equation balanced? Example: NO(g) + O 2 (g) NO 2 (g) Is this equation balanced? Adjust the coefficient of the reactants and products to balance the equation 2NO + O 2 2NO 2 Reactants Product Practice: SO 2 + O 2 SO 3 Is this equation balanced? Practice: H 2 + Cl 2 HCl Is this equation balanced? 2SO 2 + O 2 2SO 3 H 2 + Cl 2 2HCl Avogadro’s Number: The Mole 1 Dozen = 12 1 Gross = Mole = 6.02 x mole of atoms = x atoms 1 mole of molecules = x molecules The Mole and Avogadro’s Number Example: How many molecules are contained in 5.0 moles of carbon dioxide (CO 2 )? Step Identify the original quantity and the desired quantity. 5.0 mol of CO 2 original quantity ? molecules of CO 2 desired quantity x conversion factor = Step Set up and solve the problem. 5.0 molx 6.02 x molecules 1 mol = 3.0 x molecules CO 2 Unwanted unit cancels. Step Write out the conversion factors. 1 mol 6.02 x molecules or 6.02 x molecules 1 mol Choose this one to cancel mol. Molar Mass: the mass (in grams) of one mole a particular substance There is a unique relationship between molar mass and atomic weight: Molar mass (in grams) is always equal to the atomic weight of the atom! Examples: Atomic weight of carbon is 12.01amu Molar mass of carbon is g/mol Atomic weight of helium is 4.00 amu Molar mass of helium is 4.00 g/mol Mole (mol): The number of particles (atoms or molecules) in a sample of element or compound with a mass in grams equal to the atomic (or molecular) weight Example: Atomic weight of sodium = g of sodium contains 1 mole (6.02 x ) atoms in g of sodium Mass to Mole Conversions:Relating Grams to Moles Because molar mass relates the number of moles to the number of grams of a substance, molar mass can be used as a conversion factor. Example: How many moles are present in 100. g of aspirin (C 9 H 8 O 4, molar mass g/mol)? Step Identify the original quantity and the desired quantity g of aspirin original quantity ? mol of aspirin desired quantity x conversion factor = Step Write out the conversion factors g aspirin 1 mol g aspirin or Choose this one to cancel g aspirin. The conversion factor is the molar mass, and it can be written in two ways. Step Set up and solve the problem g aspirinx 1 mol g aspirin =0.555 mol aspirin Unwanted unit cancels. Calculations Using the Mole 1 mole represents the mass of a sample that contains Avogadro’s number of particles Example: Atomic wt. of Potassium (K)= 39 amu 1 mol K atoms = 6.02 x atoms = 39 g K Example: Atomic wt. of Sulfur (S) = 32 u 1 mol S atoms = 6.02 x atoms = 32 g S 1 mol S atoms = 6.02 x atoms 6.02x atoms = 32 g S 1 mol S atoms = 32 g S Practice: 1 mol S atoms = 6.02 x atoms 6.02x atoms = 32 g S 1 mol S atoms = 32 g S What is the mass in grams of 1 atom of Sulfur? 1 S atom= g 1 S atom x 32 g S/6.02 x atoms = g 1 S atom = 5.32 x g Practice: 1 mol S atoms = 6.02 x atoms 6.02x atoms = 32 g S 1 mol S atoms = 32 g S How many moles of Sulfur in 98.6 grams? 98.6g = moles Sulfur 98.6g x 1 mole/32g = moles sulfur = 3.07 moles sulfur Practice: 1 mol S atoms = 6.02 x atoms 6.02x atoms = 32 g S 1 mol S atoms = 32 g S How many atoms in this sample of 98.6g of S? 98.6g = atoms of S 98.6g x 6.02x10 23 atoms/32g =atoms of S = 1.85x10 24 atoms S Practice: 1 mol S atoms = 6.02 x atoms 6.02x atoms = 32 g S 1 mol S atoms = 32 g S What is the mass of 1 atom of Sulfur? 1 atom = g 1 atom x 32g/6.02x10 23 atoms = g = 5.33 x g S The mole concept applies to molecules, as well as to atoms Chemical formulas indicate relative quantities of atoms within a compound Example: H 2 O has 2 H atoms for every 1 Oxygen atom C 6 H 12 O 6 has how many atoms of each element? Formula Weight (F.W.): sum of all atomic weights of all atoms in a compound; expressed in amu A mole of a molecule will have a mass in grams equal to its formula weight Example: F.W. of CH 4 N 2 O (urea) = ? Atomic weights: C = 12 amu H = 1 amu N = 14 amu O = 16 amu F.W. CH 4 N 2 O = (1) + 2 (14) + 16 = 60 amu Molar mass of CH 4 N 2 O is 60 g/mol Practice: Formula weight of H 2 O = ? F.W. of H 2 O = atomic weight of H atoms + atomic weight of O atom Atomic weight of H =1amu, O =16 amu F.W. of H 2 O = Molar mass of H 2 O = 18 amu 18 g/mol More Practice: Prozac, C 17 H 18 F 3 NO, is a widely used antidepressant that inhibits the uptake of serotonin by the brain. What is the molar mass of Prozac? 309 g/mole 17C (12.0) + 18H (1.0) + 3F (19.0) + 1N (14.0) + 1 O (16.0) = = Chemical Equations and the Mole Stoichiometry: Study of mass relationships in chemical reactions; ratios of different molecules Example: CH 4 + 2O 2 CO 2 + 2H 2 O 10 CH O 2 10 CO H 2 O 6.02 x10 23 CH x10 23 O 2 6.02 x10 23 CO x10 23 H 2 O 1 mol CH mol O 2 1 mol CO mol H 2 O 16.0g CH g O 2 44.0g CO gH 2 O 6.02 x10 23 CH x10 23 O 2 6.02 x10 23 CO x10 23 H 2 O 1 mol CH mol O 2 1 mol CO mol H 2 O 16.0g CH g O 2 44.0g CO gH 2 O How many moles of O 2 would be required to react with 1.72 mol CH 4 ? 1.72 mol CH 4 = mol O mol CH 4 x 2 mol O 2 /1 mol CH 4 = 3.44 mol O 2 How many grams of H 2 O will be produced from 1.09 mol of CH 4 ? 2-part problem: first find moles of H 2 O then convert moles to grams Use this equation to obtain conversion factor for moles of CH 4 to moles of H 2 O: 1 mol CH mol O 2 1 mol CO mol H 2 O 1.09 mol CH 4 = mol H 2 O 1.09 mol CH 4 x 2 mol H 2 O/1 mol CH 4 = 2.18 mol H 2 O 2.18 mol H 2 0 = grams H 2 O 2.18 mol H 2 0 x 18.0g H 2 O/1 mol H 2 O = 39.2grams H 2 O Theoretical and Percent Yield Theoretical yield: the maximum amount of product that would be formed from a particular reaction in an ideal world Actual yield: the amount of product formed from a particular reaction in the real world (usually less than the theoretical yield) Percent yield: ratio of actual yield to theoretical yield, times 100 % % yield = actual yield/theoretical yield x 100% Example: theoretical yield = 39.2g water actual yield = 35.5 g water % yield = 35.5g/39.2g x 100% = 90.6 % yield Oxidation and Reduction Reactions An oxidation-reduction reaction involves the transfer of electrons from one reactant to another. In oxidation, electrons are lost Zn Zn e - (loss of electrons--LEO) In reduction, electrons are gained. Cu e - Cu (gain of electrons--GER) Identify each of the following as an oxidation or a reduction reaction: A. Sn Sn e- B. Fe e - Fe 2+ C. Cl 2 + 2e - 2Cl - Practice: Oxidation Reduction Zn + Cu 2+ Zn 2+ + Cu Zn acts as a reducing agent because it causes Cu 2+ to gain electrons and become reduced. Reducing agent: a compound that is oxidized while causing another compound to be reduced Oxidizing agent: a compound that is reduced while causing another compound to be oxidized Cu 2+ acts as an oxidizing agent because it causes Zn to lose electrons and become oxidized. oxidizedreduced Examples of Everyday Oxidation–Reduction Reactions: Iron Rusting 4 Fe(s) + 3 O 2 (g) 2 Fe 2 O 3 (s) Fe 3+ O 2– neutral Feneutral O Fe loses e – and is oxidized. O gains e – and is reduced. Inside an Alkaline Battery Zn + 2 MnO 2 ZnO + Mn 2 O 3 neutral ZnMn 4+ Zn 2+ Mn 3+ Zn loses e − and is oxidized. Mn 4+ gains e − and is reduced.

( ) 1.How are you?A你好吗？ B再见 ( ) 2.Good morning. A下午好。 B早上好。 ( ) 3.How many girls? A有多少个女孩？ B有多少条狗？ ( ) 4.How many blue balls? A 有多少个蓝色的球？ B有多少个绿色的球？ ( ) 5.Point to a girl. A指向一个女孩。 B指向一个男孩。 ( ) 6.I'm Sam. A我是大明。 B我是男孩。 ( ) 7.My name is Panpan. A我的名字叫熊猫。B我的名字叫盼盼。 ( ) 8.Good afternoon. A下午好。 B中午好。 ( ) 9.It's green. A 它是绿色的。 B它是蓝色的。 ( ) 10.Stand up! A 起立！ B 请坐！ ( ) 11.I'm fine. A 我很好。 B我五岁了。 ( ) 12.What's your name? A你叫什么名字？ B这是什么？ ( ) 13.I'm a boy. A我有个球. B我是个男孩。 ( ) 14.Open the door. A打开门。 B打开窗户。 ( ) 15.It' s the ceiling. A 它是地面。 B他是天花板。 ( ) 16.Sit down! A 坐下。 B起立。 ( ) 17.This is my desk. A这是我的书桌。B这是我的座位。 ( ) 18.See you! A看你。 B再见！ ( ) 19.This is our teacher. A这是我的老师。B这是我们的老师。 ( ) 20.This is a cat. A这是一只猫。B那是一只猫。 ( ) 21.Pleased to meet you. A很高兴见到你。B再见。 ( ) 22.What's that ? A那是什么？ B这是什么？ ( ) 23.Is it a dog ? A 它是一条狗吗？B它是一条狗 。 ( ) 24.What's in the box ? A盒子里有什么？B是狐狸吗？ ( ) 25.Where's my pen ？A我的钢笔在哪里？B我的铅笔在哪里？ ( ) 26.A balloon is on the bed. A床上有一个气球。 B 床下有一个气球。 ( ) 27.That is a cat. A那是一只猫。 B这是一只猫。 ( ) 28.You 're eight. A 你八岁了。 B 我八岁了 。 ( ) 29.It ' s you. A 是你。 B是我。 ( ) 30.It 's my pen. A 它是我的钢笔。B 它是我们的钢笔。 ( )31.It 's a black cat. A它是只黑色的猫。B它是顶黑色的帽子。 ( )32.You 're welcome! A 别客气！ B 你是好样的。 ( )33.Where's the cat ？A 猫在哪里？ B 球猫在哪里。 ( )34.Happy birthday to you ! A 祝你生日快乐！B 给你生日蛋糕. ( )35.It 's an orange house. A 它是所橙色的房子。 B 它是个橙色的盒子。 ( )36.I ' m six ， too. A 我六岁了。 B 我也六岁了。 ( )37.How old are you ? A 你几岁了？ B 你好吗？ ( )38.It 's in my hat. A 它在我的帽子里。 B它是我的小猫。 ( )39.A doll is under the bed. A 床下面有个娃娃。B床下面有条狗。 ( )40.No, it isn't. A 不是。 B 是的。 ( )41.Thank you . A 谢谢。 B 不用谢。 ( )42.Yes, it is. A 是的。 B 不是。 ( )43.Come to the front ! A 到前面来。 B 回去。 ( )44.Go back to your seat ! A 到后面去。 B 回到你的座位上 。 ( )45.Good boy.A 好孩子。 B 再见。 ( )46.I 'm seven. A 我七岁了。 B 我八岁了。 ( )47.What 's this ?A 这是什么？ B 那是什么？ ( )48.A book for you.A 给你一本书。 B给你一管钢笔。 1.（ ）I have _____ egg. A. a B. an C. the 2.( ) She _____ a nice schoolbag . A. have B. has C. is 3. ( ) How many ______ do you have ? I have 3. A. pen B. a pen C. pens 4. ( )My friend Tom _______ painting. A. likes B. like C.is like 5 ( ) _______she ? she's my sister. A. What's B. Where C. Who' 6. ( ) ______ is the teacher . She's in the classroom. A. What's B. Where C. Where's 7.( ) ________is the pencil-case ? It's 32 yuan. A. How old B. How many C. How much 8.( ) What colour is the walkman ? A. It's nice B. It's yellow and blue. C. Yes, I like it. 9 ( )Lucy _______ science .-------Me too. A. likes B. like C. is like 10.Let _____ clean the teacher's desk. ------ok A.I B. my C. me 11.This is ______ new computer. A. my B. me C.I 12.How many fans can you see ? A. I have 6. B.I can see six. C.I can see 13.How much is the yo-yo ? A. I am 12. B. It's 12 yuan. C. 12 books 14.Where are you from ? A. Chinese book. B. It's Chinese C. I am from China. 15.Where is my English book ? A. It's in the desk. B. It's in the pencil-case C. It's blue 16.How old are you ? A. I'm fine, thank you . B. I'm Ok C. I'm ten 17.What's her name ? A. My name is Lily B. His name is Tom C. Her name is Lily 18.Who is he ? A. He's Zhangpeng B. she is a girl C. He's a boy 19.What's in your schoolbag ? A. It's a book. B. Many story-books C.I have 10. 20.Let's clean the classroom.______________ A. It's nice and clean B. All right C. You're right. 21.What's ten plus ten ? A. It's twenty B. It's twelve C. It's two. 22.Are they farmers ? A. Yes, she is B. No ,they aren't C. They are farmers 23.Is she in the classroom ? A. Yes ,she is B. Yes ,he is C. she is in the kitchen 24.Is this your book ? A. No, they aren't B. It's a English book C. yes ,it is 25.Is this your uncle ? A. No. she isn't B. Yes, she is C. Yes, he is 26.What would you like for dinner ? A. yes B. I'd like some vegetable and beef. C. Thank you 27. Can I have some chicken ? A. Sure, here you are B. Yes ,you can C.I have some noodles 28.Help yourself. A. Thank you B.NO C. Here you are 29.Can I help ? A. Yes, pass me a plate, please B. No C. sorry 29. What's he ? A. He is my father B. He's a doctor C. He's a man 30.What's your aunt ? A. She is a nurse B. He is a nurse C. She's a woma 31.Who's that man ? A. He is my uncle B. She's my sister C. He's a baseball player A. sorry B. come and meet family C. Merry Christmas 33．Who's that boy ? ________ John. A. She's B. He's C. It's 34 Do you like sports ? A. Yes ,she is B. No, I do C. Yes, I do . 35 May I have a look ? A. No, you can't B. Look at the book C. sure ,here you are 36 _________ your aunt ? She's a driver . A. Where's B. What's C. Who's 37 Look _______the photo. A. in B. to C. at 38 There ________many books on the sofa .Please put them away. A. is B. has C. are 39 There_______ a teacher in the classroom A. is B. has C. are 40 __________is she ? She's my mother A. How old B. How C. Who 41 A. draws B. drawers C. drawer. 42 A. I am sorry. B. Fine, and you? C. Haha. 43. I like reading. I want to be a . A. writer B. nurse C. policeman 44. I like drawing. I want to be a A. fisher B. dancer. C. painter 45. What is your father? He is a . A. tree B .dryer C. spaceman 46. What can your father do? a car. A. swim B. ride C. drive

1 edition of On the asymptotic distribution of weighted least squares estimators found in the catalog. On the asymptotic distribution of weighted least squares estimators by Laboratory of Actuarial Mathematics, University of Copenhagen in Copenhagen Written in English |The Physical Object| The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals made in the results of every single equation.. The most important application is in data best fit in the least-squares sense minimizes. This video describes the benefit of using Least Squares Estimators, as a method to estimate population parameters. Check out This video outlines the conditions which are required for Ordinary Least Squares estimators to be consistent, and behave 'normally' in the asymptotic limit. Check out . Downloadable! The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically normal. This Lecture Note deals with asymptotic properties, i.e. weak and strong consistency and asymptotic normality, of parameter estimators of nonlinear regression models and nonlinear structural equations under various assumptions on the distribution of the data. The estimation methods involved are nonlinear least squares estimation (NLLSE Cited by: finds the asymptotic distribution of least squares estimates, e: for the case % Î, by com-puting a first-order Taylor expansion of the gradient Ï f. This requires tedious computations, especially if we consider %». Using Theorem 1, and a technique similar to the one used by. Relativity and high energy physics. Going to the Seaside Box 38. PROFILE OF HIRED FARMWORKERS, 1998 ANNUAL AVERAGES... AGRICULTURAL ECONOMIC REPORT... U.S. DEPARTMENT OF AGRICULTURE Resolutions of the legislature of South Carolina, touching the controversy between New York and Virginia, and a late law of that state on the same subject. Georg Letham, physician and murderer Abstract. This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions. ized least squares and the weighted least squares estimators of Chapters 4 and 9, respectively. In this section I provide a brief introduction to two aspects of asymp totic theory: convergence in probability and convergence in distribution. I refer the reader to White. This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions on the design matrix and some moment conditions on the error by: on the use of auxiliary data and a formal derivation of the asymptotic properties of the underlying Weighted Least Squares estimator. Hellerstein & Imbens had introduced very broadly a GMM model, based on empirical likelihood estimators. The necessity for the spe-cific framework in this paper arises for the purposes of data set combination. "Asymptotic distribution of the weighted least squares estimator," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(2), pagesJune. Request PDF | Asymptotic distributions for weighted estimators of the offspring mean in a branching process | It is known that conditional least squares estimator (CLSE) of the offspring mean for. Asymptotic Least Squares Theory: Part I We have shown that the OLS estimator and related tests have good finite-sample prop-erties under the classical conditions. These conditions are, however, quite restrictive in practice, as discussed in Section It is therefore natural to ask the following Size: KB. This paper derives the limiting distributions of least squares averaging estimators for linear regression models in a local asymptotic framework. We show that the averaging estimators with fixed weights are asymptotically normal and then develop a plug-in averaging estimator that minimizes the sample analog of the asymptotic mean squared by: Weighted Least Squares in Simple Regression The weighted least squares estimates are then given as ^ 0 = yw ^ 1xw ^ 1 = P wi(xi xw)(yi yw) P wi(xi xw)2 where xw and yw are the weighted means xw = P wixi P wi yw = P wiyi P wi: Some algebra shows that the weighted least squares esti-mates. Chapter 5. Least Squares Estimation - Large-Sample Properties In Chapter 3, we assume ujx ˘ N(0;˙2) and study the conditional distribution of bgiven X. In general the distribution of ujx is unknown and even if it is known, the unconditional distribution of bis hard to derive since b = (X0X) 1X0y is a complicated function of fx ign i=1. In order to analyze the stochastic property of multilayered perceptrons or other learning machines, we deal with simpler models and derive the asymptotic distribution of the least-squares estimators of Cited by: 6. in (), then the least squares estimator is asymptotically normally distributed with covariance matrix. Asy. Var[b] = σ. Q −1. plim 1. X X Q −1. () For the most general case, asymptotic normality is much more difficult to establish because the sums in () are not necessarily sums of independent or even uncorrelated random Size: KB. Weighted least-squares with weights estimated by replication 3 7 These methods have been discussed in the literature for normally distributed errors. Bement & Williams () use (), and construct approximations, as m - oo, for the exact covariance matrix of the resulting weighted least-squares. In a heteroscedastic linear model, we establish the asymptotic normality of the iterative weighted least squares estimators with weights constructed by using the within-group residuals obtained. is a unit root. Similarly, the limiting distribution of the standardized (by T) least squares estimators of the CI vector will also be nonnormal. Despite this complica- tion, the asymptotic representations greatly simplify the task of approximating the distribution of the estimators using Monte Carlo techniques. Haruhiko Ogasawara, in Handbook of Statistics, Abstract. Asymptotic distributions of the least squares estimators in factor analysis and structural equation modeling are derived using the Edgeworth expansions up to order O(1/ n) under estimators dealt with in this chapter are those for unstandardized variables by normal theory generalized least squares, simple or scale. squares which is an modification of ordinary least squares which takes into account the in-equality of variance in the observations. Weighted least squares play an important role in the parameter estimation for generalized linear models. 2 Generalized and weighted least squares Generalized least squares Now we have the modelFile Size: KB. vals and hypothesis testing, since the distribution of the estimator is lacking. In this article, we develop an asymptotic analysis to derive the distribution of RandNLA sampling estimators for the least-squares problem. In particular, we derive the asymptotic distribution of a general sampling estimator with arbitrary sampling : Ping Ma, Xinlian Zhang, Xin Xing, Jingyi Ma, Michael W. Mahoney. depending on which weighted least square estimator we use. To get the asymptotic distribution of the structural parameters ^ we apply Theorem in Amemiya () and we get that Var(^) = N 1(0W 1) 0W 1 W 1(0W) 1 where = @˙[email protected]: Let’s also consider the properties of the listwise deletion WLS estimation. Asymptotic results are given for approximated weighted least squares estimators in nonlinear regression with independent but not necessarily identically distributed errors. Asymptotic results on nonlinear approximation of regression functions and weighted least squares: Series Statistics: No 1Cited by:. Asymptotic oracle properties of SCAD-penalized least squares estimators Huang, Jian and Xie, Huiliang, Asymptotics: Particles, Processes and Inverse Problems, Weak convergence of the empirical process of residuals in linear models with many parameters Chen, Gemai and and Lockhart, Richard A., Annals of Statistics, Cited by: The aim of this work is to investigate the asymptotic properties of weighted least squares (WLS) estimation for causal and invertible periodic autoregressive moving average (PARMA) models with uncorrelated but dependent errors. Under mild assumptions, it is shown that the WLS estimators of PARMA models are strongly consistent and asymptotically Cited by: The main goal of this paper is to study the asymptotic properties of least squares estimation for invertible and causal weak PARMA models. Four different LS estimators are considered: ordinary least squares (OLS), weighted least squares (WLS) for an arbitrary vector of weights, generalized.

Posts by Jannie Total # Posts: 40 A bowl contains 100 green and red m&ms. For every 3 green you have 7 red pieces of candy. How many green m&ms are in the bowl? a brand name has a 50% recognition rate assume the owner of the brand wants to verify that rate by beginning with a small sample of 5 randomly selected consumers the probability that exactly 4 of the 5 consumers recognize the brand name Find the angle, in degrees and minutes, that a straight line makes with the positive direction of the x axis if its gradient is 0.853 Find the mass of oxygen gas needed to react with 0.2 moles magnesium and the mass of the product? Working out please The product of two consecutive positive numbers is 42 Find the numbers Please write this problem in quadratic equation form please What mass of magnesium, Mg, contains the same number of atoms as 12g of Carbon, C Could you please show the method too Thanx :) Suppose you were to leave $100 to be equally divided between your great great grandchildren 100 years from now. Given that you can secure annual interest at 6% pa, what would this amount be? Find the compound interest, compounding monthly? Please show all working out :) Multiple-choice questions in a test are graded by adding 2 marks for each correct response and subtracting 1 mark for each incorrect response (including no response). Rory and Jenny answered all the multiple-choice questions, with Rory scoring 27 and Jenny scoring 42. Rory ... A company employs 4 times as many unskilled workers as it does skilled workers. Unskilled workers earn $650 a week, and skilled workers earn $850 a week. The weekly wage bill for these workers is $17 250. How many skilled and unskilled workers are there? Simultaneous equations Con has twice as much money as Fiona. If Fiona earns an extra $15, Con will only have one-and-a-half times her amount. How much money does each person have? Simultaneous equations A hall has 1325 seats. These are arranged in rows of 35 seats or 40 seats. The ratio of 35-seat rows to 40-seat rows is 3 : 4. How many rows are there? Simultaneous equation please The linear equation F=ac + b is used to change temperature from degrees Celsius (°C) to degrees Fahrenheit (°F). Given that 0°C = 32°F and 100°C = 212°F, calculate the constants a and b. A truck is loaded with two different types of crates. When 20 of crate A and 25 of crate B are loaded, the trucks 8 tonne capacity is reached. When 2 of crate A and 16 of crate B are loaded, the same capacity is reached. Calculate the mass of each crate. Simultaneous ... 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Access the full text. Sign up today, get DeepDyve free for 14 days. In this work the problem of characterization of Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal modulated by random fluctuations of amplitude and phase is investigated. It is assumed that the amplitude and phase of signal values at discrete time moments of observations are distorted by adding realizations of independent and identically distributed random variables. The obtained results of theoretical analysis of such distorted signal spectra show that only in the case of amplitude modulation the DFT spectrum of the modulated bounded signal can be similar to the original signal spectrum, although there occur random deviations. On the her hand, if phase modulation is present, then the DFT spectrum of the modulated bounded signal n only shows random deviations but also amplitudes of peaks existing in the original spectrum are diminished, and consequently similarity to the original signal spectrum can be significantly blurred. Keywords: Discrete Fourier Transform, signal spectrum, amplitude modulation, phase modulation, random fluctuations 1. INTRODUCTION The Discrete Fourier Transform (DFT) based periodogram is a widely used tool for analyzing time series that can be decomposed as a sum of monochromatic oscillations plus noise. Important applications of the periodogram include detection of hidden periodicities and estimation of unknown oscillation parameters (amplitude and frequency). For example, it is well known that very accurate frequency estimates of the sinusoidal components can be obtained from the local maxima of a periodogram (Walker 1971). If time series of the analyzed signal observation values at discrete time moments xt , t is available, then its corresponding DFT is computed as follows (Gasquet and Witomski 1999) 1 ~ (1) xk xt exp( i 2 kt / N ) , Nt 0 for k 0,1,..., . Frequently, the well-known Fast Fourier Transform procedures are used to perform the relevant calculations (Cooley and Tukey 1965), (Press et al. 1992), (Singleto969). Theoretical as well as numerical properties of the DFT are described in time series analysis textbooks (Blackledge 2003), (Bloomfield 2000), (Bremaud 2002), (Brillinger 1975), (Koopmans 1974). Certain statistical properties of spectrum estimation using the DFT are investigated in the work of (Foster 1996a,b) and some her aspects like periodogram smohing are considered in (Speed 1985). 144 The present work is a continuation of the author's previous publication (Popi ski 1997) on theoretical, statistical, as well as numerical properties of the DFT spectrum. It deals with the problem of applicability of such technique to spectrum estimation of signals which are subject to random amplitude and phase modulation. The proposed approach is justified by the fact that all signals usually referred to as "periodic" have some amplitude and phase variation from period to period. For example an active sonar system transmits a periodic pulse train to detect targets. The received pulses are n perfectly periodic due to random modulation of the pulses from scattering and attenuation (Hinich 2003). Also El Niño signals are recognized as amplitude and phase modulated (Allen and Robertso996). The assumed concept of random modulation modeling is described in section 2. In section 3 the theoretical results related to the modulated signal DFT spectrum are presented bh in the case of a noiseless signal and in the case of signal observations corrupted by random errors. 2. MODULATION MODELING Let us consider finite duration time series of complex-valued signal observations at discrete time moments Re[ ] i Im[ ] , t 0,1,..., . We assume that the analyzed signal is of deterministic character and comprises some regular oscillations. Such a signal can represent for example the monochromatic oscillation o0t A0 exp(i 0t i 0 ) , t with frequency 0 , constant amplitude A0 , and constant phase 0 . Now, let us assume that the signal values amplitudes and phases are distorted by random additive fluctuations at and t , t respectively, which are realizations of independent and identically distributed random variables. We also assume that amplitude distortions are independent of phase distortions. Hence, we deal with an amplitude and phase modulated signal satisfying the mathematical model rt (1 at ) exp(i t ) exp(i t ) at exp(i t ) , (2) where exp(i t ) and at exp(i t ) , t are realizations of independent and identically distributed random variables. Let the distribution of t be uniform on the interval , which gives immediately for t ( , ) ( t ~ U ( , ) ) with 0 1 sin( ) , m E exp(i t ) exp(iu )du 2 (3) 2 1 2 2 2 E exp(i t ) m E exp(i t ) m2 exp(iu ) du m 2 1 m 2 . s one can clearly see the mean value m satisfies 0 m 1 and is equal to zero only for , so that we also have 0 About the distribution of the real-valued random variable at we assume only that Ea at 0 and Ea at 2 (4) (if 0 there is only phase modulation of the signal), which further implies for a t ma Ea E (1 at ) exp(i t ) Ea (1 at ) E exp(i t ) m , (5) Ea E (1 at ) exp(i t ) ma Ea (1 at ) 2 E exp(i t ) 2 ma m2 , and obviously we have . For example, at can be uniformly distributed on the interval [ A, A] , where A 0 , and then A2 / 3 . 3. MODULATED SIGNAL SPECTRA It follows immediately from (2) that in order to analyze the DFT spectrum of our model signal we shall consider the case of signals rt zt , t modulated by series of independent and identically distributed complex-valued random variables zt . Indeed, the case of zt exp(i t ) corresponds to phase modulation, the case of zt (1 at ) exp(i t ) to phase and amplitude modulation, and the case of real-valued modulation series zt at is needed for amplitude modulation analysis, respectively. In all three mentioned cases the random variables zt have constant mean value and variance according to (3), (4) and (5). In order to compute the DFT of the modulated signals of the form rt zt , t we apply the well-known convolution formula (Gasquet and Witomski 1999) ~ rk k j 0 ~ z ok j ~j j k ~ zj (6) for k 0,1,..., . Hence, if we want to analyze the DFT of the modulated signal ~ , it is necessary to rk characterize the statistical properties of the random modulating series DFT ~ j , z j 0,1,..., . Such a characterization is given in the following lemma. Lemma 1. If a series zt , t its DFT satisfies represents realizations of uncorrelated complex-valued random variables with constant mean value E z zt E z ~0 z , E z ~k z and finite variance E z zt 0 for kl 0 , then 1,..., , and E z ( ~k z where the symbol Proof. kl 1 N denes the Kronecker delta. E z ~k )( ~l z z E z ~l ) z The first property Ez ~0 follows immediately from the assumptions of the lemma and the z definition of the DFT. In order to prove the equality E z ~k 0 for k 1,..., , observe that z 1 1 1 E z zt exp( i 2 kt / N ) E z zt exp( i 2 kt / N ) exp( i 2 kt / N ) N t 0 Nt 0 N t 0 1 1 exp( i 2 k ) 0 . exp( i 2 kt / N ) N t 0 exp( i 2 k / N ) Also from the definition of the DFT and from the assumed zero correlation of the random variables considered we have 1 E z ( ~k E z ~k )( ~l E z ~l ) z z z z E z ( zt E z zt ) exp( i 2 kt / N ) ( z s E z z s ) exp(i 2 ls / N ) N2 t 0 s 0 E z ~k z 1 N2 1 N E z ( zt t 0 s 0 kl Ez zt )( z s E z z s ) exp( i 2 ( kt ls ) / N ) 1 N2 t 0 exp( i 2 (k l )t / N ) l )t / N ) 1 exp( i 2 (k l )) 1 exp( i 2 ( k l ) / N ) 0 since t 0 exp( i 2 ( k for k l, which proves the second property. Now, as we know the mean values of the random variables ~j , j z according to the above lemma, we easily obtain from (6) Ez ~ rk and further since E z ( ~k E z ~k )( ~l z z z yields for k ~ ok for E z ~l ) z kl (7) / N the same formula Ez ~ rk 2 Ez ~ rk Ez j 0 ~ z ok j ( ~j E z ~j ) z ~ k (z j 2 z E z ~j ) z (8) where k 0 the 2 t 0 last equality N j 0 follows from ~ ok 2 j 2 j k the N k 0 N well-known property of ~ 2 ok , the DFT ~ ok / N (Gasquet and Witomski 1999). From the Schwartz inequality (Bremaud 2002) we also have the following estimate related to rk rl the covariance of the random variables ~ and ~ Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez (~ rk E z ~ )( ~ rk rl Ez ~ ) rl Ez ~ rk Ez ~ rk Ez ~ rl 2 Ez ~ rl (9) . N where B Let us remark that for bounded signals satisfying B , t 0 is a real number, we immediately obtain from (8) and (9), 2 2 2 2 2 zB zB Ez ~ Ez ~ rk rk B2 Ez (~ Ez ~ )(~ Ez ~ ) rk rk rl rl , . (10) N2 k 0 N N In such a case the random variables ~ , k representing the spectrum of the rk modulated signal, have equal variances decreasing asymptically to zero as N (which implies that their covariances also decrease asymptically to zero). Bounded signals are of course of primary interest in this work since we intend to investigate spectra of regular oscillations of stationary character, modulated by random amplitude and phase fluctuations. In view of the equalities in (3), (4), (5) and lemma 1, the equalities (7), (8) and inequality (9) hold in the case of phase modulation series zt exp(i t ) with m and z2 1 m 2 , as well as in the case of amplitude and phase modulation series zt (1 at ) exp(i t ) with m and m 2 . Thus, taking into account the equality (7) and the inequalities in (10), we see that if a bounded signal is phase or amplitude and phase modulated according to our model, then its DFT spectrum will have random character. The modulated signal DFT ~ ~ rk rk spectrum has mean values E z rk ok , k with random deviations ~ Ez ~ imposed, i.e. random phase or amplitude and phase modulation diminishes the amplitudes of the original bounded signal spectrum by the factor 0 m 1 and corrupts the spectrum by additive deviations of random character, which have zero mean and known variance. In consequence, any peaks present in the spectrum of the original bounded signal may be less distinguishable in the modulated signal spectrum. Distortion of the bounded signal spectrum values close to 0 (i.e. if phase in the case of random phase modulation is small for ~ E ~ 2 0 . In the her extreme case of fluctuations are small), since then m z 1 and E z rk z rk we have m z 0 , and then the modulated signal spectrum will have purely stochastic character without any frequencies distinguished, so that any peaks present in the original signal spectrum will be completely blurred. In accordance with our model (2), if an original signal is only amplitude modulated, then rt rat , where rat at , t so we have ~ ok ~ , k 0,1,..., . rk ~ rak Hence, the spectrum of the amplitude modulated signal differs from the original signal rak spectrum only by the additive distortion terms ~ , k 0,1,..., . Having in view the ssumptions in (4) and putting zt at with z we can apply again lemma 1, which further yields E z ~ ok , as well as equality (8) and inequality (9) for random variables rk ~ ~ ~ E ~ , k 0,1,..., . Consequently, if the original signal is bounded, then by (10) rak rk z rk the distortion terms will have equal variances tending to zero as N , so that they will also have asymptically zero covariances. This means that they will n distort completely the original bounded signal spectrum on which they are superimposed. Some small peaks present in the original signal spectrum can be smohed due to amplitude modulation but possibly larger ones will be still distinguishable. In Figure 1. the above described effects of amplitude and/or phase signal modulation on the DFT spectrum of a model signal (sum of 7 monochromatic complex harmonics with constant amplitudes and phases) is shown. The random variables representing amplitude and phase fluctuations were distributed according to uniform distribution at ~ U ( A, A) and , ) with parameters A 1.0 and / 2 , respectively. Assume now that the time series of the original signal values , t corrupted by random errors t , according to the model ~U( is where yt (11) t t, are realizations of uncorrelated complex-valued random variables having zero mean 0 and finite second moment E | |2 Let us see what happens if the corrupted signal values are submitted to random modulation ryt zt yt , t of the same kind as considered above. For a modulating series zt , t satisfying the assumptions of lemma 1, we obtain on the basis of the DFT convolution formula (Gasquet and Witomski 1999) ~ ryk k j 0 ~ ~ yk j z j ~ yN j k ~ zj (12) 148 for k which by assertion of the lemma yields E z ~yk ~k , r y E E z ~yk r E ~k y ~ E (ok ~ )m k z ~ ok . (13) frequency Fig. 1. Spectra of a model series representing sum of 7 monochromatic oscillations (N=2048): original signal (black), phase modulated signal ( / 2 , blue), amplitude modulated signal ( A 1.0 , green), amplitude and phase modulated signal ( / 2 , A 1.0 , brown). Furthermore, simple calculation shows that and analogously as in (8) we obtain N which by (11) together with the equalities E lemma 1 finally gives E E z ~yk r E E z ~yk r yt ( ~k y 0, E t s ~ 2 ok ) , st , s, t and E yt E ~k 2 2 t (14) N for k as well as E E z ( ~yk E E z ~yk )( ~yl E E z ~yl ) r r r r E E z ~yk r E E z ( ~yk r E E z ~yk )(~yl r r 2 E E z ~yl r E E z ~yl ) r (15) E E z ~yk r E E z ~yl r N 0,1,..., . 150 Hence, we see that the presence of errors corrupting the original signal values does n change the character of the DFT spectrum of the modulated series ryt zt yt , t 0,1,..., . Indeed, the formulae (13), (14), (15) are analogous to (7), (8), (9), respectively, except for addition of 2 2 2 the term 2 ( z2 ) / N E z zt / N which now occurs because of non-zero second moment of the errors. For bounded signals to (10) , namely, B, t we can easily obtain the inequalities analogous 2 B2 , (16) 2 2 E Ez (~yk r 0,1,..., . E Ez ~yk )(~yl r r E Ez ~yl ) r Thus, our earlier assertions concerning the character of the modulated signal spectrum hold also in the case of a bounded signal corrupted by uncorrelated random errors which have zero mean and finite second moment. This means that only in the case of amplitude modulation the DFT spectrum of the modulated corrupted signal can resemble the one of the corrupted signal itself. 4. SUMMARY The properties of the DFT spectrum examined in this work are helpful for understanding the possible changes such a spectrum undergoes in the case of random amplitude and phase modulation of an original signal. Our modulation model includes additive distortions of stochastic nature in the amplitudes and phases of original signal values at observation moments. For bounded signals of deterministic character (like a sum of monochromatic oscillations with constant amplitudes and phases) it is proved that occurrence of random phase modulation of the signal can completely change the character of its DFT spectrum. Namely, the phase modulated signal spectrum can show purely stochastic character without any frequencies distinguished, like the spectrum of white noise. In the case of random amplitude modulation of such a signal it is shown that the modulated signal spectrum can still resemble the spectrum of the original signal, although small peaks can be significantly smohed. Similar conclusions can be deduced also in the case of a deterministic signal which is corrupted at the moments of observation by uncorrelated random errors with zero mean and finite second moment. It is worth remarking that our conclusions agree with observations of Ni and Huo (2007), concerning the importance of phase and amplitude information in signal and image reconstruction. The concept of phase randomization used for obtaining multivariate surrogate time series (Mammen and Nandi 2008) with distribution similar to observed series is also related to the subject considered here. Hinich (2003) used similar approach to amplitude modulation modeling, as the one applied in this work, to derive statistics for detecting randomly modulated pulses in noise. The Singular Spectrum Analysis (SSA) method has an important property, first ned by Vautard and Ghil (1989), that it may be used directly to identify modulated oscillations in the presence of noise. Allen and Robertson (1996) proposed a generalization of the "Monte Carlo SSA" algorithm which allows for objective testing for the presence of modulated oscillations at low 151 signal-to-noise ratios in multivariate data. They demonstrated the application of the test to the analysis of interannual variability in tropical Pacific sea-surface temperatures. Acknowledgements. This research work was supported by the Polish Ministry of Science and Higher Education through the grant No. N N526 160136 under leadership of Dr Tomasz Niedzielski at the Space Research Centre of Polish Academy of Sciences. REFERENCES Allen M.R., Robertson A.W. (1996) Distinguishing Modulated Oscillations from Coloured Noise in Multivariate Datasets, Climate Dynamics, Vol. 12, No. 11, 775784. Blackledge J.M. (2003) Digital Signal Processing, Horwood Publishing, Chichester, West Sussex, England. Bloomfield P. (2000) Fourier Analysis of Time Series: An Introduction, Wiley, New York. Bremaud P. (2002) Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis, Springer Verlag Inc., New York. Brillinger D.R. (1975) Time Series Data Analysis and Theory, Holt, Rinehart and Winston Inc., New York. Cooley J.W. and Tukey J.W. (1965) An Algorithm for the Machine Calculation of Complex Fourier Series, Mathematics of Computation, Vol. 19, 297-301. Foster G. (1996a) Time Series by Projection I: Statistical Properties of Fourier Analysis, The Astronomical Journal, Vol. 111, No. 1, 541-554. Foster G. (1996b) Time Series by Projection II: Tensor Methods for Time Series Analysis, The Astronomical Journal, Vol. 111, No. 1, 555-566. Gasquet C., Witomski P. (1999) Fourier Analysis and Applications Filtering, Numerical Computation, Wavelets, Springer Verlag Inc., New York. Hinich M.J. (2003) Detecting Randomly Modulated Pulses in Noise, Signal Processing, Vol. 83, Issue 6, 1349-1352. Koopmans L.H. (1974) Spectral Analysis of Time Series, Academic Press, New York. Mammen E. and Nandi S. (2008) Some Theoretical Properties of Phase-Randomized Multivariate Surrogates, Statistics, Vol. 42, No. 3, 195-205. Ni X. and Huo X. (2007) Statistical Interpretation of the Importance of Phase Information in Signal and Image Reconstruction, Statistics and Probability Letters, Vol. 77, Issue 4, 447-454. Popi ski W. (1997) On Consistency of Discrete Fourier Analysis of Noisy Time Series, Artificial Satellites Journal of Planetary Geodesy, Vol. 32, No. 3, 131-142. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T. (1992) Numerical Recipes The Art of Scientific Computing, Cambridge University Press, Cambridge. Singleton R.C. (1969) An Algorithm for Computing the Mixed Radix Fast Fourier Transform, IEEE Transactions on Audio and Electroacoustics, Vol. AU-17, No. 2, 93-103. Speed T.P. (1985) Some Practical and Statistical Aspects of Filtering and Spectrum Estimation, In Price J. F. (Editor), Fourier Techniques and Applications, Plenum Press, New York, 101-118. 152 Vautard R., Ghil M. (1989) Singular Spectrum Analysis in Nonlinear Dynamics with Applications to Paleoclimatic Time Series, Physica D, Vol. 35, 395424. Walker A.M. (1971) On the Estimation of a Harmonic Component in a Time Series with Stationary Independent Residuals, Biometrika, Vol. 58, No. 1, 2136. Received: 2010-12-11, Reviewed: 2011-02-04, Accepted: 2011-02-07. Artificial Satellites – de Gruyter Published: Jan 1, 2010 Access the full text. Sign up today, get DeepDyve free for 14 days.

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? 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? Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make? Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar". A few extra challenges set by some young NRICH members. The clues for this Sudoku are the product of the numbers in adjacent squares. How many solutions can you find to this sum? Each of the different letters stands for a different number. Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? How many different symmetrical shapes can you make by shading triangles or squares? 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What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Different combinations of the weights available allow you to make different totals. Which totals can you make? Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up? 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? 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? This task encourages you to investigate the number of edging pieces and panes in different sized windows. The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails. Bellringers have a special way to write down the patterns they ring. 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He starts off with a lucid dialogue of differential types and speedy strikes to the elemental theorems of calculus and Stokes’ theorem. the result's real arithmetic, either in spirit and content material, and an exhilarating selection for an honors or graduate direction or certainly for any mathematician wanting a refreshingly casual and versatile reintroduction to the topic. For these kinds of strength readers, the writer has made the technique paintings within the most sensible culture of artistic mathematics. This reasonable softcover reprint of the 1994 variation offers the varied set of subject matters from which complicated calculus classes are created in attractive unifying generalization. the writer emphasizes using differential kinds in linear algebra, implicit differentiation in larger dimensions utilizing the calculus of differential varieties, and the strategy of Lagrange multipliers in a normal yet easy-to-use formula. There are copious workouts to aid advisor the reader in checking out knowing. The chapters will be learn in virtually any order, together with starting with the ultimate bankruptcy that comprises many of the extra conventional issues of complicated calculus classes. moreover, it really is perfect for a path on vector research from the differential varieties element of view. The specialist mathematician will locate the following a pleasant instance of mathematical literature; the scholar lucky adequate to have undergone this ebook can have an organization clutch of the character of contemporary arithmetic and an excellent framework to proceed to extra complicated stories. Diagnostic checking is a vital step within the modeling method. yet whereas the literature on diagnostic exams is sort of vast and plenty of texts on time sequence modeling can be found, it nonetheless continues to be tricky to discover a publication that accurately covers equipment for acting diagnostic exams. Diagnostic assessments in Time sequence is helping to fill that hole. A revered advent to biostatistics, completely up to date and revised the 1st variation of Biostatistics: a strategy for the overall healthiness Sciences has served pros and scholars alike as a number one source for studying tips to observe statistical easy methods to the biomedical sciences. This considerably revised moment variation brings the booklet into the twenty-first century for today’s aspiring and training clinical scientist. - Nicolas Chuquet, Renaissance Mathematician - One-Dimensional Linear Singular Integral Equations: Volume II General Theory and Applications - Chaos and Fractals: A Computer Graphical Journey - Probability Theory: A Comprehensive Course (Universitext) (2nd Edition) - Fluctuations of Lévy Processes with Applications: Introductory Lectures (2nd Edition) (Universitext) Additional resources for International Symposium on Mathematical and Computational Biology 6. P. Flory, Principles of polymer chemistry (Cornell University Press, London, 1953). 7. K. Huang, “CSAW: Dynamical model of protein folding”, arXiv:condmat/0601244 v1 12 Jan 2006. 8. K. Huang, Biophys. Rev. , 2, 139 (2007). 9. K. 16,17. 10. B. Li, N. D. Sokal, J. Stat. Phys. 80, 661 (1995). 11. T. Kennedy, J. Stat. Phys. 106, 407 (2002). 12. M. Matsumoto, S. Saito, and I. Ohmine, Nature, 416, 409 (2002). 13. A. Suenga et. al. Chem. Asian J. 2, 591 (2007). 14. Z. Lei (unpublished). 15. Z. Lei and K. Submitted to Angewandte Chemie. 45. S. McAllister, P. J. A. Floudas, J. Rabinowitz, and H. Rabitz. Enhancing molecular discovery using descriptor-free rearrangement clustering techniques and sparse data sets. in prep. May 12, 2008 16:17 Proceedings Trim Size: 9in x 6in Wu THE SOLUTION OF THE DISTANCE GEOMETRY PROBLEM IN PROTEIN MODELING VIA GEOMETRIC BUILDUP DI WU Department of Mathematics, Western Kentucky University, USA ZHIJUN WU Department of Mathematics, Iowa State University, USA YAXIANG YUAN Institute of Computational Mathematics, Chinese Academy of Science, China A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. 5. Z. D. K. Gifford, N. M. S. Jaakola. K-ary clustering with optimal leaf ordering for gene expression data. , 19(9):1070–1078, 2003. 6. A. W. F. Edwards and L. L. Cavalli-Sforza. A method for cluster analysis. Biometrics, 21:362–375, 1965. 7. J. H. Wolfe. Pattern clustering by multivariate mixture analysis. Multivariate Behavioral Research, 5:329–350, 1970. 8. A. K. Jain and J. Mao. Artificial neural networks: a tutorial. IEEE Computer, 29:31–44, 1996. 9. R . W. Klein and R. C. Dubes. Experiments in projection and clustering by simulated annealing.

When it comes to money The only thing that is always important is not the amount of money. What matters is when you earn the money or when you give. So let's think about it or a little more Let's assume that we live in a world where When you put money in the bank, the bank tells you 10% risk-free interest rate guarantee. This is high by historical standards, but it makes it easier for us to calculate. So suppose you always leave the bank You get 10% risk-free interest. Now, with that in mind, I'm putting the scenarios aside and which of you do most of them Think what you want. So I could give you $ 100 right now. This is the first choice. $ 100 to you immediately instead of giving it to you in 1 year I could pay $ 109 and in the next 2 years this is the 3rd choice, to you I'd like to pay $ 120, and if that's your choice, someone is coming towards you on the street. I can lend you $ 100 now, or $ 109 (laughs) Borrow $ 109 in 1 year, $ 120 in 2 years and in your mind You can get 10% risk-free interest. Given that you do not need money immediately. We assume you will save money. Because you do not have to pay immediately. Which of the following is most desirable? Which of these do you want to get more of? If you have the full cost or the full amount of money If you want to know, you will say "Hey, look. $ 120, that's the most money." "I'll take it because that's the biggest figure." But probably in your mind "Well, I'll take it later, maybe am i missing something? " You will be right. If you took the money quickly, you would You would lose the opportunity to earn 10% risk-free income. If you compare them directly if you wanted, the process would be like this, "Well, let's see. If I take the first option, I'll get $ 100." If you put it in a bank, How much will it increase based on a 10% risk-free percentage? After 1 year, 10% of $ 100 is $ 10. So you get a $ 10 profit. So, after 1 year in your bank your deposit will be $ 110. So, just by doing a little work we really saw that the bank was $ 100 with a 10% risk-free return 1 year later we turned it into $ 110. which is better than getting $ 109 after 1 year. According to this scenario, or given such a situation or, alternatively, to do so you would like more. 1 year later you are ahead with $ 1. What about 2 years later? If you choose this, $ 100 is $ 110 after 1 year will be, then 10% of $ 110 is $ 11. You will want to add $ 11 to it, so it will be $ 121. Thus, he once again put $ 100 in the bank with risk-free interest, It is better to choose to earn 10% per annum. It will be $ 121. And this is for any of you From the guarantee that he will pay $ 121 in 2 years is a better situation. Once again, you are ahead with $ 1. So this idea is not just about the amount, but when you get that, that's the idea called the time value of money. Time value of money. Or another way of not thinking about it is to think about what the value of this money is over time. According to some expected interest rates and in doing so the money is equal to the future you can compare the amount of money. Now, another way to understand the value of time way or another of the approximate time value of money The concept is the idea of current value, current value. Maybe I'm talking about current and future value. Current and future value. Thus, based on these assumptions, this 10% assumption, if someone asks you, “After 2 years What is the current value of $ 121? " They are essentially asking you What is the current value? CD means current value. What is the current value of $ 121 after 2 years? That's $ 121 with no risk for the next 2 years what type of money or how much money to earn is equivalent to asking what is required. We know that. If you put $ 100 in the bank, After 2 years, you will get $ 121 with 10% risk-free income. The current value here The current value of $ 121 is $ 100. Or a way to think about current and future value if someone asks you what the future holds? So what is the future value of $ 100 in 1 year? So in 1 year. If the bank charges you 10% if guaranteed, the future value is $ 110. After 2 years, the future value of 2 years is $ 121. So, considering these, you let me report a more interesting problem. So let's say ….. We simplify the calculations during this period We will estimate a 10% risk-free interest rate. Suppose someone tells us in 1 year Will give $ 65 and we ask ourselves, "What is the current value of this?" So what is its current value. Remember, the current value is just the amount of money you ask asks, that is, if you bank it If you put it at a risk-free interest rate, Will it be equal to $ 65? Which of these 2 is equal to you? You will say, "Well, look. What is the amount of money?" Let's call it X. No matter what the amount of money, if I If I increase it by 10%, it really is X + 10% X + … let's write. + 10% xX … Let me clarify this way. X + 10% X should be equal to $ 65. If I receive 10% of this amount during the year it should be equal to $ 65. It's the same with 1X or we can say that 1X + 10% 0.10X is equal to 65 is the same as, or add these 2. 1.10X = 65, if you have a real current value here if you want to solve you just need to divide both sides by 1.10. We come to the conclusion that X is equal … So let's solve it. This will be clearer. So divide both sides by 1.0 and really 0 doesn't matter. We do not worry about accuracy, because it is exactly 10%. So it will be … they will be reduced and X will be equal to … Let's calculate with a calculator, X will be equal to 65:11, that is, $ 59.09, let's round it up. So X = 59.09 which is $ 65 1 is the current value of the year. or another way of thinking about it, if the 1-year future value of $ 59.09 If you want to know what will happen, for 10% of income You will receive $ 65..

Progression Of Geometry With Grades Kindergarten: In kindergarten, children learn to identify and describe two- and three-dimensional shapes. They are able to identify the geometric shapes and names regardless of their orientation. They analyze and compare shapes of different sizes and attributes. They also learn to compose simple shapes to form larger shapes. Grade 1: In this grade, children develop skills to reason with shapes and the attributes. They understand basic geometric shapes definition. They analyze shapes from their defining attributes, like number of sides. They compose two dimensional and three-dimensional shapes to make larger composite shapes. They learn to partition rectangles and circle into equal shares. They learn the vocabulary related to partitionings like quarters, fourths and halves. Grade 2: The kids learn to identify and draw shapes according to the given attributes. They learn to draw shapes as per the specified number of faces or geometry angles. They are able to identify cubes, pentagons, hexagons, quadrilateral, and triangles. They also learn to partition a rectangle into same-size squares. They then count these squares to determine the size of the rectangle. The kids learn to partition rectangles and circles into two to four equal shares. They describe the partitioned shapes using the words like half of, third of, thirds, halves. They understand the wholes as four fourths, three-thirds, and two halves. Lesson 56 Proving Triangle Congruence By Asa And Aas Question 1.Can the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use.Answer: Question 2.In the diagram, \ \, \ \, and \ \ . Prove ABC DEF.Answer: Question 3.In the diagram, S U and \\ . Prove that RST VYTAnswer: Geometry Word Problems Involving Perimeter Example:A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 morethan the equal sides, what is the length of the third side? Let x = length of the equal sides Sketch the figure Step 2: Write out the formula for perimeter of triangle. P = sum of the three sides Step 3: Plug in the values from the question and from the sketch. 50 = x + x + x+ 5 Be careful! The question requires the length of the third side. The length of third side = 15 + 5 = 20 Answer: The length of third side is 20. Recommended Reading: Abiotic Science Definition Adding And Populating A Geometry Column PostGIS stores a tables spatial information in geometry columns. Usually these columns have the word geom floating around in their name. If you select * on a table with a geometry column, the geom column will look like a long series of nonsense. Frequently, long/lat values get stored in columns in a database or spreadsheet. To make these into a geometry column that PostGIS understands, we: Lesson 51 Angles Of Triangles Draw an obtuse isosceles triangle and an acute scalene triangle.Answer:The figures of an obtuse isosceles triangle and an acute triangle are as follows: Question 2.ABC has vertices A, B, and C, Classify the triangle by its sides. Then determine whether it is a right triangle.Answer:A , B , and C and the triangle is ABCWe know that,To find whether the given triangle is a right-angled triangle or not,We have to prove,AC² = AB² + BC²Where,AC is the distance between A and C pointsAB is the distance between A and B pointsBC is the distance between B and C pointsWe know that,The distance between 2 points = ² + ²Now,Let the given points be considered as A, B, and CSo,AB = ² + ² = 3² + 3²= 9 + 9 = 18BC = ² + ²= ² + 0²AC = ² + ²= ² + 3² The measure of each acute angle is 90°, 64°, and 26° Don’t Miss: How Many Physics Questions Are On The Mcat Make An Angles Display OK, so its not an activity for the kids exactly, but lets face it: Angles can be pretty tricky when you are trying to learn all the different types and terms. Having a classroom display to help your students remember the different terms and angles is a must! We have a range of posters that are perfect to display in your classroom. Here are some of our fantastic angle posters available to download today! How To Uninstall Geometry Dash From Your Pc If youre not enjoying Geometry Dash, you can find more PC games in the Free Alternative section below. The easiest way to uninstall Geometry Dash is through the Android emulator. Ill use Bluestacks as an example, but the process is similar for most Android emulators. Read Also: Who Is Paris Jackson Mother The 18th And 19th Centuries It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, “Which one of these theories is correct for our physical space?” The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense distances. With the development of relativity theory in physics, this question became vastly more complicated. Introduction of mathematical rigor Analysis situs, or topology Whos The Ideal Client For Our Geometry Course Completion Service We serve all sorts of students, but, for the most part, clients fall into one of the categories below: - Students who want the assurance of a guaranteed high grade in order to protect their GPA and increase their chances of being admitted to the school of their choice, e.g. selective college admissions - Students who do not want to be bothered with specific projects and assignments that are tedious. Geometry courses in particular are notorious for this - Students who are currently going through a tough family/personal situation and no longer have the time and energy to devote to their current Geometry course - Students who simply want to direct their focus to more pressing subjects and interests Recommended Reading: Hawkes Learning Systems Business Statistics Answers The Nine Chapters On The Mathematical Art Areas for the By the beginning of the 9th century, the “Islamic Golden Age” flourished, the establishment of the House of Wisdom in Baghdad marking a separate tradition of science in the medieval Islamic world, building not only Hellenistic but also on Indian sources. Although the Islamic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy, and were responsible for the development of algebraic geometry. Al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today. In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof. Making Points Lines Line Segments And Rays Finally, students will get creative and use their coloring supplies to make points, lines, line segments, and rays with the letters of their name. Using dots and arrows, students will draw and label at least one point, line, line segment, and ray in their name. Check out the picture below for an example of how this might look: Here are a few ideas to help you make this geometry activity a success: - Consider using three separate pieces of graph paper, one for each activity, to prevent too many labels from cluttering up student name pictures. - You may instead find it helpful to do each geometry activity using a different marker or colored pencil. Use one color for classifying the angles, another color for measuring the angles, and a third color for drawing points, lines, line segments, and rays. - Give students extra practice by having them do this activity with their first name, middle name, and last name, or using their own name and siblings’ names. Looking for more fun geometry activities to practice with your elementary-aged students? Check out the interactive geometry lessons on iKnowIt.com today! Geometry And Shapes For Kids: Activities That Captivate Learn and build with the geometry and shapes for kids. Tons of fun math activities included and a FREE pattern block symmetry activity! Were you searching or googling to find engaging, hands-on geometry and shapes for kids? Are you looking for lessons and activity ideas that are fun and build a strong understanding of shapes and geometry? Then you have come to the right place! Exercise 53 Proving Triangle Congruence By Sas vocabulary and core concept check Question 1.What is an included angle?Answer: Question 2.COMPLETE THE SENTENCEIf two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then __________ .Answer: Monitoring progress and Modeling with Mathematics In Exercises 3-8, name the included an1e between the pair of sides given. Question 3.\ and \Answer: \ and \Answer: \ and \Answer: \ and \Answer: \ and \Answer: \ and \Answer: In Exercises 9-14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem . Explain. Given \ \, \ \Prove PQT RSTAnswer: In Exercises 19-22, use the given information to name two triangles that are congruent. Explain your reasoning. Question 19.SRT URT, and R is the center of the circle.Answer: ABCD is a square with four congruent sides and four congruent angles.Answer: RSTUV is a regular pentagon.Answer: Question 22.\ \, \ \, and M and L are centers of circles.Answer: CONSTRUCTIONIn Exercises 23 and 24, construct a triangle that is congruent to ABC using the SAS Congruence Theorem . Describe and correct the error in finding the value of x.Answer: HOW DO YOU SEE IT?What additional information do you need to prove that ABC DBC?Answer: Question 27.PROOFThe Navajo rug is made of isosceles triangles. You know B D. Use the SAS Congruence Theorem Answer: You May Like: Equilateral Geometry Definition Proving Triangle Congruence By Sas Work with a partner. Use dynamic geometry software.a. Construct circles with radii of 2 units and 3 units centered at the origin. Construct a 40° angle with its vertex at the origin. Label the vertex A.Answer: b. Locate the point where one ray of the angle intersects the smaller circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw ABC.Answer: c. Find BC, mB, and mC.Answer: d. Repeat parts – several times. redrawing the angle indifferent positions. Keep track of your results by copying and completing the table below. What can you conclude?USING TOOLS STRATEGICALLYTo be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data.Answer: Communicate Your Answer Question 2.What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent?Answer: How would you prove your conclusion in Exploration 1?Answer: We’ve Done Geometry Work For A Wide Range Of Topics Such As: - Reasoning and proofs: inductive reasoning, classification, & properties of congruence - Lines, segments, & rays: Parallel lines, perpendicular lines - Triangles and congruence: SAS, AAS, SSA, Pythagorean Theorem, Law of Sines, Law of Cosines. Mid-segments, perpendicular bisectors, angle bisectors, medians, & altitudes - Polygons & Quadrilaterals: parallelograms, rhombi, rectangles, squares, kites, & trapezoids - Circles: tangent lines, arcs, inscribed angles, chords, secants, & graphs - Perimeter & area: of triangles, quadrilaterals, polygons, prisms, & circles - Surface area & volume of polygons, circles, prisms, etc - Geometric solids : prisms, cylinders, pyramids, cubes, cones, & spheres - Transformations, congruence, and similarity: rotations, reflections, & dilations, & many more! - Geometric constructions: constructing bisectors constructing regular polygons inscribed in circles constructing circumcircles and incircles constructing a line tangent to a circle Don’t Miss: Afda Mean Median Mode Range Practice Answer Key Congruent Triangles Cumulative Assessment Question 1.Your friend claims that the Exterior Angle Theorem can be used to prove the Triangle Sum Theorem . Is your friend correct? Explain your reasoning.Answer: Question 2.Use the steps in the construction to explain how you know that the line through point P is parallel to line m.Answer: The coordinate plane shows JKL and XYZa. Write a composition of transformations that maps JKL to XYZAnswer: b. Is the composition a congruence transformation? If so, identify all congruent corresponding parts.Answer: Question 4.The directed line segment RS is shown. Point Q is located along \ so that the ratio of RQ to QS is 2 to 3. What are the coordinates of point Q? Q Updating A Geometry Column With Existing Lat/longs ST_MakePoint takes the x and y values youre using to make the point. For us, longitude is our x, and latitude is our y. The function requires these numbers to be in the type double precision. If yours arent already you can cast them in the statement as we do below. St_SetSRID sets the spatial reference system for the point we make. It takes the point and the EPSG code your setting it to. This EPSG code needs to be the same as what you set your geom column to when you added it. Don’t Miss: Who Is Paris Jackson Mom Big Ideas Math Geometry Answers Chapter 5 Congruent Triangles If you are looking across the web for better preparation resources regarding the Big Ideas Math Geometry Ch 5 Congruent Triangles then this is the one stop destination for all your needs. For better understanding of the concepts we have compiled all the Big Ideas Math Geometry Answers Ch 5 Congruent Triangles in a simple and easy to understand language. Boost your math skills and have a deeper understanding of concepts taking help from the Congruent Triangles BIM Book Geometry Solution Key. Assigning Numbers To Letters In Your Name Don’t Miss: Who Is Khloe Kardashian’s Biological Father Hexagons And Other Shapes There are all sorts of shapes to be found in flora. Conifers are in the shape of cones , dew drops that coalesce on the leaves of plants are spheres, and salt granules are perfect cubes. But the most common shape youll find in nature, and the one that most astounds mathematicians, is the hexagon. These six-sided shapes are everywhere! Beehives, insect eyes, and snowflakes are all made up of hexagons. And while free-floating bubbles are often spheres, when a bunch of bubbles mash up against one another, they turn into hexagons! What shapes can you find around your house or yard? How To Download Geometry Dash For Android For Free Read Also: Hawkes Learning College Algebra Answers Lines Rays And Angles This fourth grade geometry lesson teaches the definitions for a line, ray, angle, acute angle, right angle, and obtuse angle. We also study how the size of the angle is ONLY determined by how much it has “opened” as compared to the whole circle. The lesson contains many varied exercises for students. When two points are connected with a straight line, we get a line segment. We call this line segment AB or line segmentAB. The sides of a triangleare line segments. A line has no beginning point or end point. Imagine it continuing indefinitely in both directions.We can illustrate that by little arrows on both ends. We can name a line using two points on it. This is line EF or line. Or, we can name a line using a lowercase letter: this is line s. |A ray starts out at a point and continues off to infinity. We can showthat by drawing anarrow at one end of the ray. Think of the sun’s rays:they start at the sun and go on indefinitely. We can name a ray using its starting point and one other point that ison the ray: this is ray QP or ray. Or, we canname a ray using a lowercase letter: this is ray r. What is an angle? Many people think that an angle is some kind ofslanted line. But in geometry an angle is made up of two rays thathave the same beginning point. That point is called the vertex and the two rays are called the sidesof the angle. To name an angle, we use three points, listing the vertex in the middle.This is angle DEF or DEF. We can use the symbol for angle.

Can you use the numbers on the dice to reach your end of the number line before your partner beats you? Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it. In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first? 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? Exactly 195 digits have been used to number the pages in a book. How many pages does the book have? Place six toy ladybirds into the box so that there are two ladybirds in every column and every row. Can you put the numbers 1 to 8 into the circles so that the four calculations are correct? Use the information about Sally and her brother to find out how many children there are in the Brown family. Find all the numbers that can be made by adding the dots on two dice. Can you hang weights in the right place to make the equaliser An old game but lots of arithmetic! Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it. Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions. Have a go at this game which involves throwing two dice and adding their totals. Where should you place your counters to be more likely to win? Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd. Use the number weights to find different ways of balancing the equaliser. A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids. Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you. In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins? If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced? If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag? Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals? Find your way through the grid starting at 2 and following these operations. What number do you end on? Ben has five coins in his pocket. How much money might he have? 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. Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes? Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only? 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? 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? 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? Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 A group of children are using measuring cylinders but they lose the labels. Can you help relabel them? Using the statements, can you work out how many of each type of rabbit there are in these pens? 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? Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15? Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. 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! Use these head, body and leg pieces to make Robot Monsters which are different heights. Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done? A game for 2 or more players. Practise your addition and subtraction with the aid of a game board and some dried peas! 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? Can you substitute numbers for the letters in these sums? Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total? 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? Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon? You have 5 darts and your target score is 44. How many different ways could you score 44?

graph. ◇ Distinguish between linear and non-linear relationships and between relationships To show something in a two-variable graph, Variables. The relationship between two variables is graphed by drawing two axes perpendicular to. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The graph shows a relationship between two variables, x and y. A positive relationship or direct relationship is a relationship between two variables that Draw a curve that shows that the number of tickets sold at Disneyland tickets and . A positive or direct relationship is one in which the two variables (we will . or drawn properly on both axes, meaning that the distance between units has to be on the plane of the graph in order to show relationships between two variables. They assist learners in grasping fairly quickly key economic relationships. Years of statistical analysis have gone into the small graph you can examine to learn about key forces and trends in the economy. Further, they help your instructor to present data in a way which is small-scale or economical, and establish a relationship, frequently historical, between variables in a certain kind of relationship. They permit learners and instructors to establish quickly the peaks and valleys in data, to establish a trend line, and to discuss the impact of historical events such as policies on the data that we wish to analyze. Types of Graphs in Economics There are various kinds of graphs used in business and economics that illustrate data. These include pie charts segments are displayed as portions, usually percentages, of a circlescatter diagrams points are connected to establish a trendbar graphs results for each year can be displayed as an upward or downward barand cross section graphs segments of data can be displayed horizontally. You will deal with some of these in economics, but you will be dealing principally with graphs of the following variety. Certain graphs display data on one variable over a certain period of time. For example, we may want to know how the inflation rate has varied in the Canadian economy from We would choose an appropriate scale for the rate of inflation on the y vertical axis; and on the x horizontal axis show the ten years from to with on the left, and on the right. We would notice right away a trend. The trend in the inflation rate data is a decline, actually from a high of 5. We would see that there has been some increase in the inflation rate since its absolute low inbut not anything like the high. And, if we did such graphs for each of the decades in Canada sincewe would see that the s were a unique decade in terms of inflation. No decade, except the s, shows any resemblance to the s. We can then discuss the trends meaningfully, since we have ideas about the data over a major period of time. We can link the data with historical events such as government anti-inflation policies, and try to establish some connections. Other graphs are used to present a relationship between two variables, or in some instances, among more than two variables. Consider the relationship between price of a good or service and quantity demanded. The two variables move in opposite directions, and therefore demonstrate a negative or indirect relationship. Aggregate demand, the relationship between the total quantity of goods and services demanded in the entire economy, and the price level, also exhibits this inverse or negative relationship. If the price level based on the prices of a given base year rises, real GDP shrinks; while if the price level falls, real GDP increases. Further, the supply curve for many goods and services exhibits a positive or direct relationship. The supply curve shows that when prices are high, producers or service providers are prepared to provide more goods or services to the market; and when prices are low, service providers and producers are interested in providing fewer goods or services to the market. The aggregate expenditure, or supply, curve for the entire Canadian economy the sum of consumption, investment, government expenditure and the calculation of exports minus imports also shows this positive or direct relationship. Construction of a Graph You will at times be asked to construct a graph, most likely on tests and exams. You should always give close attention to creating an origin, the point 0, at which the axes start. Label the axes or number lines properly, so that the reader knows what you are trying to measure. Most of the graphs used in economics have, a horizontal number line or x-axis, with negative numbers on the left of the point of origin or 0, and positive numbers on the right of the origin. Figure 2 presents a typical horizontal number line or x-axis. In economics graphs, you will also find a vertical number line or y-axis. Here numbers above the point of origin 0 will have a positive value; while numbers below 0 will have a negative value. Figure 3 demonstrates a typical vertical number line or y-axis. When constructing a graph, be careful in developing your scale, the difference between the numbers on the axes, and the relative numbers on each axis. The scale needs to be graduated or drawn properly on both axes, meaning that the distance between units has to be identical on both, though the numbers represented on the lines may vary. You may want to use single digits, for example, on the y-axis, while using hundreds of billions on the x-axis. Using a misleading scale by squeezing or stretching the scale unfairly, rather than creating identical distances for spaces along the axes, and using a successive series of numbers will create an erroneous impression of relationship for your reader. If you are asked to construct graphs, and to show a knowledge of graphing by choosing variables yourself, choose carefully what you decide to study. Here is a good example of a difficulty to avoid. Could you, for example, show a graphical relationship between good looks and high intelligence? I don't think so. You often see pictures representing numerical information. These pictures may take the form of graphs that show how a particular variable has changed over time, or charts that show values of a particular variable at a single point in time. We will close our introduction to graphs by looking at both ways of conveying information. Time-Series Graphs One of the most common types of graphs used in economics is called a time-series graph. A time-series graph A graph that shows how the value of a particular variable or variables has changed over some period of time. One of the variables in a time-series graph is time itself. Time is typically placed on the horizontal axis in time-series graphs. The other axis can represent any variable whose value changes over time. The grid with which these values are plotted is given in Panel b. Time-series graphs are often presented with the vertical axis scaled over a certain range. The result is the same as introducing a break in the vertical axis, as we did in Figure These points are then plotted in Panel b. To draw a time-series graph, we connect these points, as in Panel c. The values for the U. The points plotted are then connected with a line in Panel c. Scaling the Vertical Axis in Time-Series Graphs The scaling of the vertical axis in time-series graphs can give very different views of economic data. We can make a variable appear to change a great deal, or almost not at all, depending on how we scale the axis. For that reason, it is important to note carefully how the vertical axis in a time-series graph is scaled. Consider, for example, the issue of whether an increase or decrease in income tax rates has a significant effect on federal government revenues. This became a big issue inwhen President Clinton proposed an increase in income tax rates. The measure was intended to boost federal revenues. Higher tax rates, they said, would cause some people to scale back their income-earning efforts and thus produce only a small gain—or even a loss—in revenues. Op-ed essays in The Wall Street Journal, for example, often showed a graph very much like that presented in Panel a of Figure It shows federal revenues as a percentage of gross domestic product GDPa measure of total income in the economy, since Various tax reductions and increases were enacted during that period, but Panel a appears to show they had little effect on federal revenues relative to total income. Her alternative version of these events does, indeed, suggest that federal receipts have tended to rise and fall with changes in tax policy, as shown in Panel b of Figure Which version is correct? Both graphs show the same data. It is certainly true that federal revenues, relative to economic activity, have been remarkably stable over the past several decades, as emphasized by the scaling in Panel a. And a small change in the federal share translates into a large amount of tax revenue. It is easy to be misled by time-series graphs. Large changes can be made to appear trivial and trivial changes to appear large through an artful scaling of the axes. The best advice for a careful consumer of graphical information is to note carefully the range of values shown and then to decide whether the changes are really significant. Testing Hypotheses with Time-Series Graphs John Maynard Keynes, one of the most famous economists ever, proposed in a hypothesis about total spending for consumer goods in the economy. He suggested that this spending was positively related to the income households receive. One way to test such a hypothesis is to draw a time-series graph of both variables to see whether they do, in fact, tend to move together. Annual values of consumption and disposable income are plotted for the period — Notice that both variables have tended to move quite closely together. This is consistent with the hypothesis that the two are directly related. Department of Commerce The fact that two variables tend to move together in a time series does not by itself prove that there is a systematic relationship between the two. Notice the steep decline in the index beginning in October, not unlike the steep decline in October Did the mystery variable contribute to the crash? It would be useful, and certainly profitable, to be able to predict such declines. The mystery variable and stock prices appear to move closely together. Was the plunge in the mystery variable in October responsible for the stock crash? The mystery value is monthly average temperatures in San Juan, Puerto Rico. Attributing the stock crash in to the weather in San Juan would be an example of the fallacy of false cause. Notice that Figure The left-hand axis shows values of temperature; the right-hand axis shows values for the Dow Jones Industrial Average. Two axes are used here because the two variables, San Juan temperature and the Dow Jones Industrial Average, are scaled in different units. Descriptive Charts We can use a table to show data. HERI conducts a survey of first-year college students throughout the United States and asks what their intended academic majors are. In the groupings given, economics is included among the social sciences. All three panels present the same information. Panel a is an example of a table, Panel b is an example of a pie chart, and Panel c is an example of a horizontal bar chart. Percentages shown are for broad academic areas, each of which includes several majors. Panels b and c of Figure Panel b is an example of a pie chart; Panel c gives the data in a bar chart. The bars in this chart are horizontal; they may also be drawn as vertical. Either type of graph may be used to provide a picture of numeric information. Key Takeaways A time-series graph shows changes in a variable over time; one axis is always measured in units of time. One use of time-series graphs is to plot the movement of two or more variables together to see if they tend to move together or not. The fact that two variables move together does not prove that changes in one of the variables cause changes in the other. Values of a variable may be illustrated using a table, a pie chart, or a bar chart. The table in Panel a shows a measure of the inflation rate, the percentage change in the average level of prices below. Panels b and c provide blank grids. We have already labeled the axes on the grids in Panels b and c. It is up to you to plot the data in Panel a on the grids in Panels b and c. Connect the points you have marked in the grid using straight lines between the points. What relationship do you observe? Has the inflation rate generally increased or decreased? What can you say about the trend of inflation over the course of the s? Here are the time-series graphs, Panels b and cfor the information in Panel a. The first thing you should notice is that both graphs show that the inflation rate generally declined throughout the s with the exception ofwhen it increased. The generally downward direction of the curve suggests that the trend of inflation was downward. Notice that in this case we do not say negative, since in this instance it is not the slope of the line that matters. Rather, inflation itself is still positive as indicated by the fact that all the points are above the origin but is declining. Finally, comparing Panels b and c suggests that the general downward trend in the inflation rate is emphasized less in Panel b than in Panel c. This impression would be emphasized even more if the numbers on the vertical axis were increased in Panel b from 20 to Just as in Figure Decide whether each proposition below demonstrates a positive or negative relationship, and decide which graph you would expect to illustrate each proposition. In each statement, identify which variable is the independent variable and thus goes on the horizontal axis, and which variable is the dependent variable and goes on the vertical axis. An increase in the poverty rate causes an increase in the crime rate. As the income received by households rises, they purchase fewer beans. As the income received by households rises, they spend more on home entertainment equipment. How can we estimate the slope of a nonlinear curve? After all, the slope of such a curve changes as we travel along it. We can deal with this problem in two ways. One is to consider two points on the curve and to compute the slope between those two points. Another is to compute the slope of the curve at a single point. When we compute the slope of a curve between two points, we are really computing the slope of a straight line drawn between those two points. Nonlinear Relationships and Graphs without Numbers They are the slopes of the dashed-line segments shown. These dashed segments lie close to the curve, but they clearly are not on the curve. After all, the dashed segments are straight lines. When we compute the slope of a nonlinear curve between two points, we are computing the slope of a straight line between those two points. Here the lines whose slopes are computed are the dashed lines between the pairs of points. Every point on a nonlinear curve has a different slope. To get a precise measure of the slope of such a curve, we need to consider its slope at a single point. To do that, we draw a line tangent to the curve at that point. A tangent line A straight line that touches, but does not intersect, a nonlinear curve at only one point. The slope of a tangent line equals the slope of the curve at the point at which the tangent line touches the curve. Consider point D in Panel a of Figure We have drawn a tangent line that just touches the curve showing bread production at this point. It passes through points labeled M and N. The vertical change between these points equals loaves of bread; the horizontal change equals two bakers. The slope of our bread production curve at point D equals the slope of the line tangent to the curve at this point. In Panel bwe have sketched lines tangent to the curve for loaves of bread produced at points B, D, and F. Notice that these tangent lines get successively flatter, suggesting again that the slope of the curve is falling as we travel up and to the right along it. In Panel athe slope of the tangent line is computed for us: Generally, we will not have the information to compute slopes of tangent lines. We will use them as in Panel bto observe what happens to the slope of a nonlinear curve as we travel along it. We see here that the slope falls the tangent lines become flatter as the number of bakers rises. Notice that we have not been given the information we need to compute the slopes of the tangent lines that touch the curve for loaves of bread produced at points B and F. In this text, we will not have occasion to compute the slopes of tangent lines. Either they will be given or we will use them as we did here—to see what is happening to the slopes of nonlinear curves. In the case of our curve for loaves of bread produced, the fact that the slope of the curve falls as we increase the number of bakers suggests a phenomenon that plays a central role in both microeconomic and macroeconomic analysis. As we add workers in this case bakersoutput in this case loaves of bread rises, but by smaller and smaller amounts.

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Chapter 2: MOTION AND SPEED Section 1DESCRIBING MOTION Motion occurs when an object changes its position. To know whether the position of something has changed, you need a reference point. A reference point helps you determine how far an object has moved. An important part of describing the motion of an object is to describe how far it has moved, which is distance. The SI unit of length or distance is the meter (m). 1 meter = 100 centimeters Sometimes you may want to know not only your distance, but also your direction from a reference point. Displacement is the distance and direction of an objects change in position from a reference point. DISTANCE VS. DISPLACEMENT What is speed? Speed is the distance an object travels per unit of time. Any change over time is called a rate. Speed is the rate at which distance is traveled. CALCULATING SPEED Speed = distance time If s = speed, d = distance, and t = time, this relationship can be written as: s = d t Suppose you ran 2 km in 10 minutes. Your speed or rate of change of position, would be: s = d = 2 km t 10 min 0.2 km/min = CONSTANT SPEED If an object is in motion and neither slows down nor speeds up, the object is traveling at a constant speed. (Ex. Car traveling on a freewayCRUISE CONTROL) CHANGING SPEED Much of the time, the speeds you experience are not constant. (Ex. Riding a bicycle for 5 km) CHANGING SPEED AVERAGE SPEED Describes speed of motion when speed is changing. AVERAGE SPEED is the total distance traveled divided by the total time of travel. For the bicycle trip, the total distance traveled was 5 km and the total time was 15 min. or .25 h. The AVERAGE SPEED was: s = d = 5 km = t 0.25 h 20 km/h INSTANTANEOUS SPEED INSTANTANEOUS SPEED is the speed at a given point in time. (Ex. CARS SPEEDOMETER) VELOCITY VELOCITY includes the speed of an object and the direction of its motion. Ex. HURRICANE traveling at a speed of 60 km/h; located 100 km east of your location Velocity VELOCITY IS SPEED WITH DIRECTION! VELOCITY SPEED same DIRECTION different (VELOCITY = DIFFERENT) VELOCITY SPEED constant DIRECTION changing (VELOCITY = CHANGING) VELOCITY SPEED constant DIRECTION changing (VELOCITY = CHANGING) SPEED UNITS REMEMBER VELOCITY includes the speed and direction of an object; Therefore, a change in velocity can be either a change in how fast something is moving or a change in the direction it is moving. CHAPTER 2: MOTION AND SPEED Section 2: ACCELERATION ACCELERATION is a change in velocity. Acceleration occurs when an object changes its speed, its direction, or both. When you think of acceleration, you probably think of something speeding up (positive acceleration); However, an object that is slowing down also is accelerating (negative acceleration). In both cases, acceleration occurs, because its speed is changing. Calculating ACCELERATION Remember Acceleration is the rate of change in velocity. The change in velocity or speed is divided by the length of the time interval over which the change occurred. Acceleration = change in velocity time How is the change in velocity calculated? Always subtract the initial velocity(the velocity at the beginning of the time interval) from the final velocity(the velocity at the end of the time interval). Change in velocity = final vel. initial vel. Change in velocity = vf vi a = (vf vi) = t s (units) m/s UNITS The SI unit for velocity is meters/second (m/s), and the SI unit for time is seconds (s). So, the unit for acceleration is meters/second/second. This unit is written as m/s 2 and is read meters per second squared. CALCULATING POSITIVE ACCELERATION Suppose a jet airliner starts at rest at the end of a runway and reaches a speed of 80 m/s in 20 s. Because it started from rest, its initial speed was zero. Its acceleration can be calculated as follows: a = (vf vi) = (80m/s-0m/s)= 4 m/s2 t 20s CALCULATING NEGATIVE ACCELERATION Now imagine a skateboarder is moving at a speed of 3 m/s and comes to a stop in 2 s. The final speed is zero and the initial speed was 3 m/s. The skateboarders acceleration is calculated as follows: a = (vf vi) = (0m/s-3m/s)= -1.5 m/s2 t 2s ACCELERATION Will always be positive if an object is speeding up Will always be negative if an object slowing down Chapter 2: MOTION AND SPEED Section 3MOTION AND FORCES What is a force? A force is a push or a pull that one body exerts on another. A force can cause the motion of an object to change. OBVIOUS VS. NOT SO OBVIOUS Some forces are obviousthe force applied to a soccer ball as it is kicked into the goal Some forces are not so obviousthe force of the floor being exerted on your feet OR gravity pulling down on your body BALANCED FORCES When two or more forces act on an object at the same time, the forces combine to form the net force. What is the net force acting on this box? The net force on the box is zero, because the two forces cancel each other. Forces on an object that are equal in size and opposite in direction are called balanced forces. UNBALANCED FORCES When two students are pushing with unequal forces in opposite directions. A net force occurs in the direction of the larger force. UNBALANCED FORCES The students are pushing on the box in the same direction. The net force is formed by adding the two forces together. IT IS IMPORTANT TO REMEMBER Students often assume that NO MOTION = NO FORCE (not true), but an objects lack of motion is because the forces acting on it are balanced. NO MOTION = BALANCED FORCES MOTION = UNBALANCED FORCES What is inertia? Inertia is the tendency of an object to resist any change in motion. (NEWTONS 1st LAWThe Law of Inertia) QUESTION: Would a bowling ball or a table tennis ball have a greater inertia? Why? RememberMass is the amount of matter in an object, and a bowling ball has more mass than a table-tennis ball. The INERTIA of an object is related to its MASS. The greater the mass of an object, the greater its inertia. MASS = INERTIA British Scientist Sir Isaac Newton (1642-1727) was able to describe the effects of forces on the motion of objects. These rules are known as Newtons Laws of Motion. According to Newtons first law of motion, an object moving at a constant velocity keeps moving at that velocity unless a net force acts on it (Part ICar-CC). Also, if an object is at rest, it stays at rest, unless a net force acts on it (Part IISoccer ball). SHORT VERSIONNewtons 1st Law An object will resist any change in motion. What happens in a car crash? This can be explained by the law of inertia When a car traveling about 50 km/h collides head-on with something solid, the car crumples, slows down, and stops within appproximately 0.1s. A passenger without a seatbelt Will continue to move forward at the same speed that the car was traveling Within 0.02 s after the car stops, unbelted passengers slam into the steering wheel, dashboard, etc. They are traveling at the cars original speed of Growing Pressure to expand the Definition of Violence… We recommend that workplace violence be defined, not only as physical violence but also as psychological violence such as: bullying, mobbing, teasing, ridicule or any other act or words that could psychologically... 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Acid stimulation will be excluded from SB 318/HB 191 because it... Short History of Assessment By Jeanne Pfeifer Major Changes in Assessment Norm Referenced Tests Criterion Referenced Tests Authentic Assessment Norm Referenced Norm Referenced Based on "the Bell Curve" Use standardized tests Comparing students to students Want to create a spread... Unseen Poem: Essay. Choose one of the two unseen poems on the following slides, and write a 3 or 4 paragraph analytical essay based on it. Spend 5 minutes reading the poem again and again, annotating on your third reading. Ready to download the document? Go ahead and hit continue!

|FHWA > Bridge > Tunnels > Technical Manual for Design and Construction of Road Tunnels - Civil Elements| Technical Manual for Design and Construction of Road Tunnels - Civil Elements Appendix C - Cut-and-Cover Tunnel Design Example The purpose of this design example is to provide guidance to the application of the AASHTO LRFD Bridge Design Specifications when designing concrete cut and cover box tunnel structures. Reference is made to the AASHTO LRFD specifications throughout the design example. Specific references to sections are denoted by the letter "S" preceeding the specification article. 1. Tunnel Section Geometry and Materials The tunnel is a reinforced concrete double-chamber box structure. It is located entirely below grade and is built using cut and cover construction. Because the water table is located above the tunnel, hydrostatic soil pressures surround the structure. Figure 1 shows the internal dimensions for one of the openings. These dimensions serve as the starting point for the structural dimensions shown in Figure 2. 1.1 Tunnel Section Dimensions Figure 2 shows the geometry of the underground cut and cover box cross-section. 1.2 Material Properties 2. Computer Model of Tunnel The analysis of the tunnel subjected to applied loads and the design of the structural components are performed using a model generated by general purpose structural analysis computer software. Concrete walls and slabs are modeled as a rigid frame, composed of groups of members that are interconnected by a series of joints (see Section 4.0 Analysis Model Input and Section 5.0 Analysis Model Diagram). All joints are located along the centroids of the structural components. Members are modeled as one foot wide segments in the longitudinal direction of the tunnel to represent a one-foot-wide "slice" of the structure. AASHTO LRFD factored loads and load combinations are applied to the members and joints as required. The structure is analyzed to determine member forces and reactions, which will be used to design individual structural components of the tunnel. 2.1 Model Supports Universal restraints are applied in the Y-translation and X-rotation degrees of freedom to all members. Spring supports located at joints spaced at 1'-0" on center are used to model soil conditions below the bottom slab of the tunnel. Springs with a K constant equal to 2600 k/ft are used, applied only in the downward Z direction. The spring support reaction will account for the earth reaction load. 3. Load Determination The tunnel is located completely below grade and is subjected to loading on all sides. The self weight load of the concrete structure is applied vertically downward as component dead load. Vehicular live loads and vertical earth pressure are applied in the vertical downward direction to the top slab. Buoyancy forces are applied vertically upward to the bottom slab. Lateral forces from live load, soil overburden, horizontal earth pressure, and hydrostatic pressure are applied to the exterior walls. Load designations are referenced from LRFD Section 3.3.2 (see Figure 3). 3.1 Total Dead Loads Dead loads are represented by the weight of all components of the tunnel structure and the vertical earth pressure due to the dead load of earth fill. Concrete dead load (per foot length) (DC) Vertical earth pressure (EV) 3.2 Live Load Live load represents wheel loading from an HS-20 design vehicle. It is assumed that the wheels act as point loads at the surface and are distributed downward in both directions through the soil to the top slab of the tunnel. The load distribution is referenced from LRFD Section 184.108.40.206.6. Figure 4 shows the distribution of the wheel loads to the top slab. Wheel Loads (LL) Live Load Surcharge (LS) Lateral earth pressure is typically represented by the equation: σ = k0γn The following lateral pressures are applied to the exterior walls of the tunnel (see Figure 5): Calculate the lateral earth pressures: EH1 = ko(γs × nsurch) = 0.080 ksf EH2 = ko(γs × ns + γsat × nsat) = 0.494 ksf EH3 = ko(γs × ns + γsat × nsat) = 1.314 ksf EH4 = kw(γw × nw) = 1.825 ksf 3.4 Buoyancy Load WA Area of water displaced, A A = B × H = 1855.125 sq. ft. Buoyancy = A × γw = 115.76 klf (along tunnel) OK WA = Buoyancy/B = 1.513 klf 3.5 Load Factors and Combinations Loads are applied to a model using AASHTO LRFD load combinations, referenced from LRFD Table 3.4.1-1. The loads, factors, and combinations for the applicable design limit states are given in Table 1. Table 1: Load Factors and Load Combinations 4. Analysis Model Input 4.1 Joint Coordinates The cross section of the tunnel model lies in the X-Z global plane. Each joint is assigned X and Z coordinates to locate its position in the model. See Section 5.0 and Figure 6 for a diagram of the model. 4.2 Member Definition Members are defined by a beginning joint and an end joint, Ji and Jj , respectively, where i and j represent joint numbers. All members are composed of concrete and represent a one foot wide "slice" of the tunnel section. 5. Analysis Model Diagram The computer model represents a one foot wide slice of the cross-section of the tunnel. Members are connected by series of joints at their endpoints to form a frame, and are located along the centroids of the walls and roof and floor slabs. Joints in the 100 series and 200 series represent the floor and roof slabs respectively. Joints in the 300 and 500 series represent the exterior walls, while the 400 series represents the interior wall. The bottom diagram of Figure 5 shows all joints in the structure, while the top diagram shows only the joints at the intersections of slabs and walls. Joints 302, 402, and 502 at the base of the exterior walls and joints 305, 405, and 505 at the top of the exterior walls are included to determine shear at the face of the top and bottom slabs. 6. Application of Lateral Loads (EH) Lateral pressures EH1 through EH4 from Section 3.3 are applied to the members of the model as shown below. See Figure 7 for the horizontal earth pressure and hydrostatic pressure load distributions. 6.1 Exterior Wall Loads Due to Horizontal Earth Pressure EH3 Calculate pressure at top of wall: ko(γs×ns + γsat×nsat) = = 0.494 ksf Pressure at base of wall = 1.314 ksf (see calculation in Sec. 3.3) Calculate interval increment for loading all exterior wall members: The two tables below show the lateral earth pressure values (ksf) at the beginning and end of each member of the exterior walls: 6.2 Exterior Wall Loads Due to Hydrostatic Pressure EH4 Calculate pressure at top of wall: kw(γw×nw) = = 0.312 ksf Pressure at base of wall = 1.825 ksf (see calcs. in Sec. 3.3) Calculate interval increment for loading all exterior wall members: Δ = = 0.303 ksf The two tables below show the lateral hydrostatic pressure values (ksf) at the beginning and end of each member of the exterior walls: Figure 7 shows the load distribution along the exterior walls (members 301 to 305 and 501 to 505) for horizontal earth pressure (EH3) and hydrostatic pressure (EH4). 7. Structural Design Calculations - General Information 7.1 Concrete Design Properties Modulus of elasticity of steel, Es = 29000 ksiYield strength of steel reinforcement, fy = 60 ksi Compressive strength of concrete, f'c = 4 ksi 7.2 Resistance Factors Resistance factors for the strength limit state using conventional concrete construction are referenced from AASHTO LRFD Section 220.127.116.11. Flexure Φ = 0.90 (Φ) varies from 0.75 to 0.9 (0.75 is conserverative) Shear Φ = 0.90 Compression Φ = 0.7 since no spirals or ties 8. Interior Wall Design 8.1 Factored Axial Resistance (S18.104.22.168) For members with tie reinforcement using LRFD eq. (22.214.171.124-3): Pn = 0.80 [0.85×fc×(Ag - Ast) + fy×Ast] Ast = 1.76 in2 (#6 at 6", ea. face) Ag = 144.00 in2 Where Ag= 12×12 in2 (assuming wall thickness = 1 foot) Pn = 471.37 kip Factored axial resistance of reinforced concrete using LRFD eq. (126.96.36.199-1): Pr = ΦPn Φ = 0.9 for flexure Pr = factored axial resistance Pn = nominal axial resistance Pu = factored applied axial force Pr = 424.24 kip Check Pu < Pr Pu = from computer model output = 78.00 kip < Pr OK 9. Top Slab Design 9.1 Slenderness Check (S188.8.131.52) k × (lu/r) = 33.55 34 - 12 (M,1/M2) = 30.38 Where M1 and M2 are smaller and larger end moments From analysis output where M1 = 77 kip-ft P1 = 28.4 kip M2 = 255 kip-ft P2 = 28.4 kip Consider slenderness since k × (lu / r) is greater than 34 - 12 (M1/M2) Calculate EI using LRFD eq. (184.108.40.206-1 and 220.127.116.11-2): Mno = 215.00 kip-ft M2 = 255.00 kip-ft Note: Mno does not include effects of vertical live load surcharge βd = Mno/M2 = 0.84 Approximate Method (LRFD 18.104.22.168.2) The effects of deflection on force effects on beam-columns and arches which meet the provisions of the LRFD specifications may be approximated by the Moment Magnification method described below. For steel/concrete composite columns, the Euler buckling load, Pe, shall be determined as specified in article 22.214.171.124 of LRFD. For all other cases, Pe shall be taken as: (LRFD eq. 126.96.36.199.2b-5) E = modulus of elasticity (ksi) I = moment of inertia about axis under consideration (in4)k = effective length factor as specified in LRFD 188.8.131.52 lu = unsupported length of a compression member (in) Pe = 2626.67 kips Moment Magnification (LRFD 184.108.40.206.2b) (The components for sidesway will be neglected. Bracing moment will not include lateral force influence. Live load surcharge is excluded also.) The factored moments may be increased to reflect effects of deformations as follows: LRFD eq. (220.127.116.11.2b-1): Mc = δb × M2b + δs × M2s Mu = 215.00 kip-ft MuLAT = -35.08 kip-ft LRFD eq. (18.104.22.168.2b-3) For members braced against sidesway and without transverse loads between supports, Cm: Cm = 0.6 + 0.4 (M1/M2) LRFD eq. (22.214.171.124.2b-6) Cm = 0.72 M1= smaller end moment M2= larger end moment Pu = factored axial load (kip) = 28.4 kips Φ = resistance factor for axial compression Pe = Euler buckling load (kip) δb = 1 M2b = moment on compression member due to factored gravity loads that result in no appreciable sidesway calculated by conventional first-order elastic frame analysis; always positive (kip-ft) M2b = 179.92 kip-ft Mc = 179.92 kip-ft Factored flexural resistance (LRFD 126.96.36.199.1) The factored resistance Mr shall be taken as: Mr = ΦMn Φ = resistance factor = 0.9 Mn = nominal resistance (kip-in) The nominal flexural resistance may be taken as: (LRFD eq. 188.8.131.52.2-1) Do not consider compression steel for calculating Mn. As = area of nonprestressed tension reinforcement (in2) fy = specified yield strength of reinforcing bars (ksi) ds = distance from extreme compression fiber to centroid of nonprestressed tensile reinforcement (in2) a = depth of equivalent stress block (in) = β1 × c β1 = stress block factor specified in Section 184.108.40.206 of LRFD c = distance from the extreme compression fiber to the neutral axis LRFD eq. (220.127.116.11.2-4) As= 2.0 in2 fy= 60.0 ksi fc= 4.0 ksi b= 12.0 in c = 3.46 in a = 2.94 in Mn = 3153.53 kip-in = 262.79 kip-ft ΦMn = 236.51 kip-ft OK (≥ Mc) Mr = 236.51 kip-ft Mr > Mu Create interaction diagram Assume ρmin = 1.0% Asmin = 3.6 in2 Asprov (total) = 4.00 in2 choose #9 at 6" Es = 29000 ksi β1 = 0.85 Yt = 15 in 0.85 × f'c = 3.4 ksi Ag' in2 = 360 in2 As = A's = 2.0 in2 At zero moment point using LRFD eq. (18.104.22.168-2) Po = 0.85 × f'c × (Ag - Ast) + Ast × fy = 1450 kip ΦPo = 1015 kip At balance point calculate Prb and Mrb cb = 16.65 in ab = β1× cb = 14.15 in = 70 ksi f's > fy; set f's = fy Acomp = c × b = 199.8 in2 y' = a / 2 = 7.07625 in ΦPb = Φ [0.85 × f'c × b × ab × As' × f's - As × fy] = 485 kip ΦMb = 7442 kip-in = 620 kip-ft At zero 'axial load' point (conservatively ignore compressive reinforcing) = 2.9 in ΦMo = 2838.2 kip-in = 237 kip-ft At intermediate points Note Φ may decrease from 0.90 to 0.75 as a increases from 0.0 to ab. Use 0.75 to be conservative. Acomp = a × 12 in2 ΦPn = Φ (Acomp - A's) × 0.85 × f'c + As' × f's - As × fy kips

The purpose of the present paper is to investigate some strong differential subordination and superordination implications involving the Komatu integral operator which are obtained by considering suitable classes of admissible functions. The sandwich-type theorems for these operators are also considered. MSC: 30C80, 30C45, 30A20. Keywords:strong differential subordination; strong differential superordination; univalent function; convex function; Komatu integral operator Let and be members of ℋ. The function is said to be subordinate to , or is said to be superordinate to , if there exists a function analytic in , with and , and such that . In such a case, we write or . If the function F is univalent in , then if and only if and (cf.). Following Komatu , we introduce the integral operator defined by Moreover, from (1.2), it follows that In particular, the operator is closely related to the multiplier transformation studied earlier by Flett . Various interesting properties of the operator have been studied by Jung et al. and Liu . Let be analytic in and let be analytic and univalent in . Then the function is said to be strongly subordinate to , or is said to be strongly superordinate to , written as , if for , as the function of z is subordinate to . We note that if and only if and . then is called a solution of the strong differential subordination. The univalent function is called a dominant of the solutions of the strong differential subordination, or more simply a dominant, if for all satisfying (1.4). A dominant that satisfies for all dominants of (1.4) is said to be the best dominant. Recently, Oros introduced the following strong differential superordinations as the dual concept of strong differential subordinations. then is called a solution of the strong differential superordination. An analytic function is called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, if for all satisfying (1.5). A univalent subordinant that satisfies for all subordinants of (1.5) is said to be the best subordinant. Definition 1.4 () Definition 1.5 () For the above two classes of admissible functions, Oros and Oros proved the following theorems. Theorem 1.1 () Theorem 1.2 () In the present paper, making use of the differential subordination and superordination results of Oros and Oros [8,9], we determine certain classes of admissible functions and obtain some subordination and superordination implications of multivalent functions associated with the Komatu integral operator defined by (1.1). Additionally, new differential sandwich-type theorems are obtained. We remark in passing that some interesting developments on differential subordination and superordination for various operators in connection with the Komatu integral operator were obtained by Ali et al.[11-14] and Cho et al.. 2 Subordination results From (2.2) with the relation (1.3), we get Further computations show that Using equations (2.2), (2.3) and (2.4), from (2.6), we obtain Hence, (2.1) becomes which evidently completes the proof of Theorem 2.1. □ Proof The proof is similar to that of [, Theorem 2.3d] and so is omitted. □ The next theorem yields the best dominant of the differential subordination (2.7). andqis the best dominant. Proof Following the same arguments as in [, Theorem 2.3e], we deduce that q is a dominant from Theorem 2.2 and Theorem 2.4. Since q satisfies (2.9), it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence, q is the best dominant. □ 3 Superordination and sandwich-type results The dual problem of differential subordination, that is, differential superordination of the Komatu integral operator defined by (1.1), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition. Proof From (2.7) and (3.1), we have which evidently completes the proof of Theorem 3.1. □ If is a simply connected domain, then for some conformal mapping h of onto Ω. In this case, the class is written as . Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1. Theorem 3.1 and Theorem 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for certain ϕ. andqis the best subordinant. Proof The proof is similar to that of Theorem 2.5 and so is omitted. □ Combining Theorem 2.2 and Theorem 3.2, we obtain the following sandwich-type theorem. The author declares that they have no competing interests. The author worked on the results and he read and approved the final manuscript. Dedicated to Professor Hari M Srivastava. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619). Flett, TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl.. 38, 746–765 (1972). Publisher Full Text Jung, IB, Kim, YC, Srivastava, HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl.. 176, 138–147 (1993). Publisher Full Text Liu, JL: A linear operator and strongly starlike functions. J. Math. Soc. Jpn.. 54, 975–981 (2002). Publisher Full Text Antonino, JA: Strong differential subordination to Briot-Bouquet differential equations. J. Differ. Equ.. 114, 101–105 (1994). Publisher Full Text Miller, SS, Mocanu, PT: Subordinants of differential superordinations. Complex Var. Theory Appl.. 48, 815–826 (2003). Publisher Full Text Ali, RM, Ravichandran, V, Seenivasagan, N: Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator. J. Franklin Inst.. 347, 1762–1781 (2010). Publisher Full Text Cho, NE, Kwon, OS, Srivastava, HM: Strong differential subordination and superordination for multivalently meromorphic functions involving the Liu-Srivastava operator. Integral Transforms Spec. Funct.. 21, 589–601 (2010). Publisher Full Text

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Price elasticity is the ratio between the percentage change in the quantity demanded (qd) or supplied (qs) and the corresponding percent change in price the price elasticity of demand is the percentage change in the quantity demanded of a good or service divided by the percentage change in the price. Combination of the route own price elasticity with cross price elasticities when all national routes have prices which vary identically thus, the less elastic result is consistent with thus, the less elastic result is consistent with. Page 1 of 34 chapter four elasticity we have seen in chapter three how a change in the price of the good results in change in quantity demanded of that good in the opposite direction (movement. The elasticity of demand (ed), also referred to as the price elasticity of demand, measures how responsive demand is to changes in a price of a given good more precisely, it is the percent change in quantity demanded relative to a one percent change in price, holding all else constant (ceteris paribus) demand of goods can be. Price elasticity elasticity, in layman terms can be defined as the ability of an object to stretch or transform in shape, and return to its original form. Price elasticity of demand and supply how sensitive are things to change in price. 07062017 the price elasticity of demand is simply a number it is not a monetary value what the number tells you is a 1 percent decrease in price causes a 167 percent increase in quantity demanded. 17032017 price/demand elasticity where the good has only a single source or a very limited number of sources is typically low external situations may create rapid changes in the price elasticity of demand for almost any product with low elasticity. Price elasticity of demand depends on the price difference between regular and fair trade coffee as well as the share of fair trade coffee demand compared to total coffee consumption the linear regression model is based mainly on functional utility, and shows. Calculation of price elasticity of demand suppose that price of a commodity falls down from rs10 to rs9 per unit and due to this, quantity demanded of the commodity increased from 100 units to 120 units. 13042017 calculating the price elasticity of demand you may be asked the question given the following data, calculate the price elasticity of demand when the price changes from $900 to $1000. Price elasticity in the case of block rates and joint consumption in section four, we present in section four, we present preliminary estimates of the price elasticity of demand for natural gas, electricity, and water in. Cross-price elasticity of demand measures the effect of the change in one good's price on the quantity demanded of the other good (defined as: the ratio of the percent change in the quantity demanded of one good to the percent change in the price of the other. 29062018 it's human nature if the price of a product goes up, consumers buy less of it if the price goes down, consumers buy more in economic terms, that's called price elasticity. 26112003 breaking down 'price elasticity of demand' if the quantity demanded of a product exhibits a large change in response to its price change, it is termed elastic, that is, quantity stretched far from its prior point. Unitary elasticity means that a given percentage change in price produces the same percentage change in quantity thus, at unit elasticity, total revenues will remain constant when price changes thus, at unit elasticity, total revenues will remain constant when price changes. 12072018 it says that the quantity demanded of a product is a function of five factors: price, income of the buyer, the price of related goods, the tastes of the consumer, and any expectation the consumer has of future supply, prices, etc. Abstract: the case is about price elasticity of demand in oligopoly market due to sudden change in its price in this case there are 6 more. Price elasticity of demand, also called the elasticity of demand, refers to the degree of responsiveness in demand quantity with respect to price. Elasticity of demand and supply and price changes – a quick summary elasticity determines how much a shift changes quantity versus price if d increases and s is perfectly inelastic, then price rises and quantity doesn't change. For our examples of price elasticity of demand, we will use the price elasticity of demand formula 1 widget inc decides to reduce the price of its product, widget 10 from $100 to $75 the company predicts that the sales of widget 10 will increase from 10,000 units a month to 20,000 units a month. Video: price elasticity of demand in microeconomics discover the definition and formula for price elasticity of demand see some real-world examples of how it is calculated, and find out what it means for demand of a good to be inelastic or elastic. Price elasticity, also known as price elasticity of demand, measures the change in demand for a good or service, given a price change a good is said to be elastic if a change in price creates a greater change in demand. 30112003 elasticity of supply works similarly if a change in price results in a big change in the amount supplied, the supply curve appears flatter and is considered elastic. When the price of a good changes, consumers' demand for that good changes we can understand these changes by graphing supply and demand curves and analyzing their properties we can understand these changes by graphing supply and demand curves and analyzing their properties. 16072018 price elasticity measures the responsiveness of the quantity demanded or supplied of a good to a change in its price it is computed as the percentage change in quantity demanded—or supplied—divided by the percentage change in price. Elasticity chapter summary the elasticity of demand measures the responsiveness of demand to changes in a factor that affects demand elasticities can be estimated for price, income, prices of related products, and advertising expenditures the own-price elasticity is the ratio of the percentage change in quantity demanded to the percentage change in price. Price elasticity of demand (ped) price elasticity of demand and its determinants price elasticity of demand: measures the responsiveness of quantity demanded to a change in price, along a given demand curve. Lecture notes on elasticity of substitution ted bergstrom, ucsb economics 210a march 3, 2011 today’s featured guest is \the elasticity of substitution.

Neural networks is the archival journal of the world's three oldest neural modeling societies: the international neural network society (inns), the. 2 artificial neural networksan artificial neural network , is a biologically inspired computational model formed from hundreds of single units, artificial neurons, connected with coefficients (weights) which constitute the neural structure. In this article, we develop a machine learning technique called deep learning (artificial neural network) by using tensorflow. Computers organized like your brain: that's what artificial neural networks are, and that's why they can solve problems other computers can't. In machine learning and cognitive science, artificial neural networks (anns) are a family of statistical learning models inspired by biological neural networks (the central nervous systems of animals, in particular the brain) and are used to estimate or approximate functions that can depend on a large number of inputs and are generally unknown. Artificial neural networks are computational models which work similar to the functioning of a human nervous system there are several kinds of artificial neural networks these type of networks are implemented based on the mathematical operations and a set of parameters required to determine the . Artificial neural networks (ann) the power of neuron comes from its collective behavior in a network where all neurons are interconnected the network starts evolving : neurons continuously evaluate their output by looking at their inputs, calculating the weighted sum and comparing to a threshold to decide if they should fire. Artificial neural networks are behind a lot of big advances -- a lot of big advances how can one tech . An artificial neural network is an interconnected group of nodes, akin to the vast network of neurons in a brain here, each circular node represents an . Artificial neural networks (ann) are the foundations of artificial intelligence (ai), solving problems that would be nearly impossible by human or statistical standards. Artificial neural networks for beginners carlos gershenson [email protected]ssexacuk 1 introduction the scope of this teaching package is to make a brief induction to artificial neural. Artificial intelligence neural networks - learning artificial intelligence in simple and easy steps using this beginner's tutorial containing basic knowledge of artificial intelligence overview, intelligence, research areas of ai, agents and environments, popular search algorithms, fuzzy logic systems, natural language processing, expert systems, robotics, neural networks, ai issues, ai . Artificial neural network software apply concepts adapted from biological neural networks, artificial intelligence and machine learning and is used to simulate, research, develop artificial neural network neural network simulators are software applications that are used to simulate the behavior of . Today, it's more common to use other models of artificial neurons - in this book, and in much modern work on neural networks, the main neuron model used is one called the sigmoid neuron we'll get to sigmoid neurons shortly. Artificial neural network market expected to reach more than moderate cagr growth forecast period 2018-2023, artificial neural network market categorizes by application type, component and end-user |artificial neural network industry. Neural networks for machine learning from university of toronto learn about artificial neural networks and how they're being used for machine learning, as applied to speech and object recognition, image segmentation, modeling language and human . Applied deep learning - part 1: artificial neural networks overview welcome to the applied deep learning tutorial series we will do a detailed analysis of several deep learning techniques starting with artificial neural networks (ann), in particular feedforward neural networks. Artificial neural networks (or ann) are at the very heart of the ai revolution that is shaping every aspect of society and technology but the anns that we have been able to handle so far are . Neural networks and deep learning currently provide some of the most reliable image recognition, speech recognition, and natural language processing solutions available. Artificial neural networks - application edited by: chi leung patrick hui isbn 978-953-307-188-6, published 2011-04-11. Join barton poulson for an in-depth discussion in this video artificial neural networks, part of data science foundations: fundamentals. Introduction to artificial neural netw orks • what is an artificial neural netw ork the network is provided with a correct answer (output) for every. Deep learning and artificial intelligence are quite buzz words now, aren’t they however, this field is not quite as new as the majority of people thinks we as humans were always interested in the way we think and the structure of our brain. Artificial neural networks (ann) are one of the commonly applied machine learning algorithm this article explains the working behind ann. All artificial neural networks are constructed from this basic building block - the processing element or the artificial neuron it is variety and the fundamental differences in these building blocks which partially cause the implementing of neural networks to be an art. “deep learning,” the machine-learning technique behind the best-performing artificial-intelligence systems of the past decade, is really a revival of the 70-year-old concept of neural networks. Introduction to neural networks, advantages and applications artificial neural network(ann) uses the processing of the brain as a basis to develop algorithms that can be used to model complex patterns and prediction problems. Artificial neural networks (anns) are computational models inspired by the human brain they are comprised of a large number of connected nodes, each of which performs a simple mathematical operation. Neural networks tutorial – a pathway to deep learning march 18, 2017 andy deep learning , neural networks 29 chances are, if you are searching for a tutorial on artificial neural networks (ann) you already have some idea of what they are, and what they are capable of doing. A basic introduction to neural networks what is a neural network the simplest definition of a neural network, more properly referred to as an 'artificial' neural network (ann), is provided by the inventor of one of the first neurocomputers, dr robert hecht-nielsen.

Energy of a Regular Black Hole We use Einstein, Landau-Lifshitz, Papapetrou and Weinberg energy-momentum complexes to evaluate energy distribution of a regular black hole. It is shown that for a regular black hole, these energy-momentum complexes give the same energy distribution. This supports Cooperstock hypothesis and also Aguirregabbiria et al. conclusions. Further, we evaluate energy distribution using Mller’s prescription. This does not exactly coincide with ELLPW energy expression but, at large distances, they become same. PACS: 04.20.Dw, 04.30.Bw Key Words: Energy, Regular Black Hole The definition of energy-momentum has always been a focus of many investigations in General Relativity (GR). This, together with conservation laws, has a crucial role in any physical theory. However, there is still no accepted definition of energy-momentum, and generally speaking, of conserved quantities associated with the gravitational field. The main difficulty is with the expression which defines the gravitational field energy part. Einstein used principle of equivalence and conservation laws of energy-momentum to formulate the covariant field equations. He formulated the energy-momentum conservation law in the form where is the stress energy density of matter. He identified as representing the stress energy density of gravitation. He noted that was not a tensor, but concluded that the above equations hold good in all coordinate systems since they were directly obtained from the principle of GR. The choice of a non-tensorial quantity to describe the gravitational field energy immediately attracted some criticism. The problems associated with Einstein’s pseudo-tensor resulted in many alternative definitions of energy, momentum and angular momentum being proposed for a general relativistic system. These include Landau-Lifshitz, Tolman, Papapetrou, Bergmann, Weinberg who had suggested different expressions for the energy-momentum distribution. The main problem with these definitions is that they are coordinate dependent. One can have meaningful results only when calculations are performed in Cartesian coordinates. This restriction of coordinate dependent motivated some other physicists like Mller, Komar and Penrose who constructed coordinate independent definitions of energy-momentum complex. Mller claimed that his expression gives the same values for the total energy and momentum as the Einstein’s energy-momentum complex for a closed system. However, Mller’s energy-momentum complex was subjected to some criticism. Komar’s prescription, though not restricted to the use of Cartesian coordinates, is not applicable to non-static spacetimes. Penrose pointed out that quasi-local masses are conceptually very important. However, different definitions of quasi-local masses do not give agreed results for the Reissner-Nordstrom and Kerr metrics and that the Penrose definition could not succeed to deal with the Kerr. These inadequacies of quasi-local definitions have been discussed in a series of papers. Thus each of these energy-momentum complex has its own drawback. As a result these ideas of the energy-momentum complex were severally criticized. Virbhadra revived the interest in this approach by showing that different energy-momentum complexes can give the same energy-momentum. Since then lot of work on evaluating the energy-momentum distributions of different spacetimes have been carried out by different authors. In a recent paper, Virbhadhra used the energy-momentum complexes of Einstein, Landau-Lifshitz, Papapetrou and Weinberg (ELLPW) to investigate whether or not they can give the same energy distribution for the most general non-static spherically symmetric metric. It was a great surprise that contrary to previous results of many asymptotically flat spacetimes and asymptotically non-flat spacetimes, he found that these definitions disagree. He observed that Einstein’s energy-momentum complex provides a consistent result for the Schwarzschild metric whether one calculates in Kerr-Schild Cartesian coordinates or Schwarzschild Cartesian coordinates. The prescriptions of Landau-Lifshitz, Papapetrou and Weinberg furnish the same result as in the Einstein prescription if the calculations are carried out in Schwarzschild Cartesian coordinates. Thus the prescriptions of Landau-Lifshitz, Papapetrou and Weinberg do not give a consistent result. On the basis of these and some other facts, Virbhadra concluded that the Einstein method seems to be the best among all known (including quasi-local mass definitions) for energy distribution in a spacetime. Recently, Lessner pointed out that the Mller’s energy-momentum prescription is a powerful concept of energy and momentum in GR. It has been shown recently that ELLPW energy-momentum complexes coincide for any Kerr-Schild class metric when one uses Kerr-Schild Cartesian coordinates. In this paper we use ELLPW and Mller energy-momentum complexes to obtain the energy distribution of a regular black hole which is represented by a Bardeen’s model. It is shown that ELLPW energy-momentum complexes give the same and acceptable results for a given space-time. Our results agree with Virbhadra’s conclusion that the Einstein’s energy-momentum complex is still the best tool for obtaining energy distribution in a given spacetime. This also supports Cooperstock’s hypothesis (that energy and momentum in a curved space-time are confined to the the regions of non-vanishing energy-momentum of matter and the non-gravitational field). The paper has been organised as follows. In the next section, we shall describe the regular black holes. In Secs. 3 and 4, we evaluate energy distribution using ELLPW and Mller’s prescriptions respectively. Finally, we shall discuss the results. 2 Regular Black Holes In 1968, Bardeen constructed a well-known model called Bardeen’s model. This model represents a regular black hole obeying the weak energy condition, and it was powerful in shaping the direction of research on the existence or avoidance of singularities. The model uses the Reissner-Nordstrm spacetime as inspiration. The metric expressed in standard spherical coordinates is given by the line element of the form where Bardeen replaced the usual Reissner-Nordstrm function When in Bardeen’s model, there is an event horizon. There are values of such that the region contains trapped surfaces. The spacetime obeys the null convergence, yet it contains no physical singularities. It is to be noticed that if we take charge , the above metric reduces to the Schwarzschild metric. 3 Energy of the Regular Black Hole In this section we shall use ELLPW energy-momentum complexes to evaluate the energy distribution of the regular black hole. To this end, we shall follow the procedure developed by Virbhadra. The basic requirement of the procedure is to bring the metric in the form of Kerr-Schild class and then transform the resulting metric in Kerr-Schild Cartesian coordinates. The metrics of the Kerr-Schild class are written in the following form where is the Minkowski spacetime, is the scalar field and is a null, geodesic and shear free vector field in the Minkowski metric. These can be expressed as where . It is to be noticed that, for the Kerr-Schild class metric, the vector field remains null, geodesic and shear free with the metric . Thus it follows from the above equation that Now we bring the metric given by Eq.(2) in Kerr-Schild class by using the following coordinate transformation which implies that This metric turns out to be static case of the Kerr-Schild class as given by Aguirregabiria et al.. In order to have meaningful results in the prescriptions of ELLPW, it is necessary to transform the metric in Kerr-Schild Cartesian coordinates. Let us now transform the metric in Kerr-Schild Cartesian coordinates by using The corresponding metric in these coordinates will become This is the Kerr-Schild class metric with and . We use the procedure of Aguirregabiria et al. to calculate energy distribution of the regular black hole in the ELLPW prescriptions. It turns out that we get the same energy in these prescriptions which is given as When we replace the value of from Eq.(4), it follows that the energy distribution of the regular black hole is which can be written as follows If we take the charge or at large distances, it reduces to the energy of the Schwarzscild metric given by 4 Energy Distribution in Mller’s Prescription Now we shall use Mller’s energy-momentum complex to evaluate energy of the regular black hole. This is the beauty of Mller’s method that it is independent of coordinates and consequently we can perform computations in spherical polar coordinates. The energy-momentum complex of Mller is given by The energy-momentum complex satisfies the local conservation laws The locally conserved energy-momentum complex contains contributions from the matter, non-gravitational and gravitational fields. and are the energy and momentum (energy current) density components respectively. The energy and momentum components are given as where is the energy and represent the momentum components. Using Gauss’s theorem, the energy expression E can be written as where is the outward unit normal vector over an infintesimal surface element . The only required component of , to evaluate energy of the Bardeen’s model, is given by Substituting this value of in Eq.(19), we have the following energy distribution in Mller’s prescription which can be written as We see that the energy expression for ELLPW and Mller’s prescriptions coincide at large distances. They are exactly the same for the Schwarzschild metric. There are two types of energy-momentum complexes in the literature. The first type depends on the coordinates and the other type is independent of coordinate. However, it has been shown by many authors that the first type give more meaning results. The debate on the localization of energy-momentum is also an interesting and a controversial problem. According to Misner et al, energy can only be localized for spherical systems. However, Cooperstock and Sarracino suggested that if energy can be localized in spherical systems then it can be localized in any spacetimes. The energy-momentum complexes are non-tensorial under general coordinate transformations and hence are restricted to Cartesian coordinates only. In their recent work Virbhadra and his collaborators have shown that different energy-momentum complexes can provide meaningful results. In this paper, we have evaluated energy of the regular black hole using prescriptions of ELLPW. It is worth noting that the energy turns out to be same in the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and Weinberg. It is clear that the definitions of ELPPW support the Cooperstock hypothesis for the regular black hole. We have also calculated this quantity using Mller energy-momentum complex. This is not exactly the same as evaluated by using ELLPW prescriptions. However, it can be seen from Eqs.(14) and (23) that, at large distances, these give the same result and reduces to the energy of Schwarzscild spacetime. We plot the energy distributions of ELLPW () and Mller () in the figures 1 and 2 respectively. I would like to thank Ministry of Science and Technology (MOST), Pakistan for providing postdoctoral fellowship at University of Aberdeen, UK. 1. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Addison-Wesley Press, Reading, MA, 1962)2nd ed. 2. R.C. Tolman, Relativity, Thermodynamics and Cosmology (Oxford Univ. Press, 1934)227. 3. A. Papapetrou, Proc. R. Irish. Acad. A52, 11 (1948). 4. P.G. Bergmann and R. Thompson, Phys. Rev. 89, 400 (1953). 5. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972). 6. C. Mller, Ann. Phys. (NY) 4, 347 (1958). 7. C. Mller, Ann. Phys. (NY) 12, 118 (1961). 8. A. Komar, Phys. Rev. 113, 934 (1959). 9. R. Penrose, Proc. Roy. Soc. London A381, 53 (1982). 10. D. Kovacs, Gen. Relatv. and Grav. 17, 927 (1985); J. Novotny, Gen. Relatv. and Grav. 19, 1043 (1987). 11. G. Bergqvist, Class. Quantum Grav. 9, 1753 (1992). 12. D.H. Bernstein and K.P. Tod, Phys. Rev. D49, 2808 (1994). 13. K.S. Virbhadra, Phys. Rev. D60, 104041 (1999). 14. K.S. Virbhadra, Phys. Rev. D41, 1081 (1990); D42, 1066 (1990); D42, 2919 (1990) and references therein. 15. S.S. Xulu, Int. J. Mod. Phys. A15, 2979 (2000); Mod. Phys. Lett. A15, 1511 (2000) and references therein. 16. I.C. Yang and I. Radinschi, Mod. Phys. Lett. A17, 1159 (2002). 17. M. Sharif, Int. J. of Mod. Phys. A17, 1175 (2002). 18. M. Sharif, Int. J. of Mod. Phys. A18, (2003); Erratum A, (2003); gr-qc/0310018. 19. G. Lessner, Gen. Relativ. Grav. 28, 527 (1996). 20. J. Bardeen, Proc. GR5 (Tiflis, USSR, 1968). 21. A. Borde, Phys. Rev. D 50, 3692 (1994); Phys. Rev. D 55, 7615 (1997). 22. C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (W.H. Freeman, New York, 1973)603. 23. F.I. Cooperstock and R.S. Sarracino, J. Phys. A.: Math. Gen. 11, 877 (1978).

Dilatons in curved backgrounds by the Poisson–Lie transformation Transformations between group coordinates of three–dimensional conformal –models in the flat background and their flat, i.e. Riemannian coordinates enable to find general dilaton fields for three–dimensional flat –models. By the Poisson–Lie transformation we can get dilatons for the dual –models in a curved background. Unfortunately, in some cases the dilatons depend on inadmissible auxiliary variables so the procedure is not universal. The cases where the procedure gives proper and nontrivial dilatons in curved backgrounds are investigated and results given. In the paper we have investigated conformally invariant three–dimensional –models on solvable Lie groups that were Poisson–Lie T–dual or plural to –models in the flat background. Several of them were nontrivial in the sense that they lived in a curved background and had nonvanishing torsion. In some cases we were not able to find the dilaton fields by the plurality procedure given in because necessary conditions for application of Poisson–Lie transformation were not satisfied for the constant dilaton. Recently we have found explicit forms of transformations between the group coordinates of the flat –models and their flat coordinates, i.e. we expressed the Riemannian coordinates of the flat metric in parameters of its solvable isometry subgroups . This enables us to write down the general form of the dilaton field satisfying the vanishing equations for the flat model in terms of the group coordinates and consequently the dilaton fields on the dual or plural nontrivial models. To set our notation let us very briefly review the construction of the Poisson–Lie T–plural –models by means of Drinfel’d doubles (For more detailed description see , , , ). The Lagrangian of dualizable –models can be written in terms of right–invariant fields on a Lie group that is a subgroup of the Drinfel’d double as are components of right–invariant forms (vielbeins) and are local coordinates of . and are submatrices of the adjoint representation of the group on the Lie algebra of the Drinfel’d double 111 t denotes transposition. The fact that for a Drinfel’d double several decompositions of its Lie algebra into Manin triples ) may exist leads to the notion of Poisson–Lie T–plurality . Namely, let be generators of Lie subalgebras of the Manin triple associated with the Lagrangian (1) and are generators of some other Manin triple in the same Drinfel’d double related by the transformation matrix as The transformed model is then given by the Lagrangian of the form (1) but with replaced by and is calculated by (3) from the adjoint representation of the group generated by . Note that for we get the dual model with , corresponding to the interchange so that the duality transformation is a special case of the plurality transformation (5) – (7). 2 Poisson–Lie transformation of dilatons In quantum theory the duality or plurality transformation must be supplemented by a correction that comes from integrating out the fields on the dual group in path integral formulation. In some cases it can be absorbed at the 1-loop level into the transformation of the dilaton field satisfying the so called vanishing equations where the covariant derivatives , Ricci tensor and Gauss curvature are calculated from the metric that is also used for lowering and raising indices, and the torsion is Unfortunately, the right-hand side of the formula (14) may depend on the coordinates of the auxiliary group . That’s why the transformation of the dilaton field cannot be applied in general but only if the following theorem holds The dilaton (14) for the model defined on the group exists if and only if where is extended as a left–invariant vector field on and For applications it is much easier to check a weaker necessary condition. A necessary condition for the existence of the dilaton (14) for the model defined on the group is where is the unit of the Drinfel’d double . For parametrization of in the form where is the submatrix in (5). The condition (18) could not be satisfied for some of the –models with constant dilaton field so that we were not able to find the transformed dilaton that satisfy the vanishing equations. The possibility to find the general dilaton fields for the flat models offers a possibility to overcome this obstacle and obtain more general dilatons in curved backgrounds. 3 Dilatons of –models on solvable three-dimensional groups All models investigated in the following admit nonsymmetric tensor but their torsions vanish so without loss of generality we shall deal with models having . 3.1 General dilatons in flat backgrounds It follows from the construction of classical dualizable models that they are given by decompositions (Manin triples) and matrices . Most of the flat and torsionless models found in can be formulated on the the Drinfel’d doubles with semiabelian decompositions where 1 is the three–dimensional abelian algebra. From the form of the vanishing equations (8–10) it is easy to see that the general form of their dilaton fields is where are coordinates that bring the flat metric to a constant form (see ) and are real constants satisfying By the Poisson–Lie transformation of (19) we can get dilatons for the dual –models but, as mentioned before, only if the necessary conditions are satisfied. Due to (15) and (19) the condition (18) reads Moreover, the matrix vanishes for and the flat coordinates can be chosen to satisfy . The condition (21) then simplifies to 3.2 Dilatons for –models dual to The first –model in the curved background we are going to investigate is given by the metric where and are constants. This metric has nonvanishing Ricci tensor but its Gauss curvature is zero. It belongs to the –model corresponding to the decomposition of the (for notation see ) and . On the other hand, it can be obtained by the Poisson–Lie transformation (6), (7) from the metric where are constants. The latter metric is flat and corresponds to the decomposition of the and . where relations between the constants are In fact, the metric (23) is the most general that can be obtained by the Poisson–Lie transformation from a flat metric corresponding to the decomposition of the . These are the coordinates that bring the flat metric to its constant form . where the coefficients satisfy the equation (20) that in this case reads However, this is not yet the final form of the dilaton field because it is expressed in terms of the coordinates of the –model given by (24) and it must be transformed to the coordinates of the –model given by (23). The transformation formulas between these coordinates follow from two different decompositions of elements of the Drinfel’d double , namely from the relation where are generators corresponding to the decomposition of the Drinfel’d double and are generators of the decomposition . They can be related by (25). Coordinates in terms of are then expressed as We have checked that the vanishing equations for and given by (23) are satisfied. Note that the condition is more strict than the necessary condition (22) that implies only. It means that the necessary condition (16) is not sufficient for the Poisson–Lie transformation of the dilaton. 3.3 Sigma models dual to A bit more complicated –model is given by the metric , where where and are constants. Again, this metric has nonvanishing Ricci tensor and its Gauss curvature is zero. It belongs to the –model corresponding to the decomposition of the and . Besides that it can be obtained by the Poisson–Lie transformation (6), (7) from the metric where are constants. This metric is flat and corresponds to the decomposition of the . where the relations between the constants are where the coefficients satisfy the equation (29). To get the final form of the dilaton field we must transform it to the coordinates . The transformation formulas follow from decompositions of elements of the Drinfel’d double , namely from the relation (30) where are generators corresponding to the decomposition and , related by (5) and (35), correspond to the decomposition . Coordinates in terms of are then expressed as In order that the dilaton does not depend on the coordinate we must set and the general form of the dilaton obtained by the Poisson–Lie transformation for the metric (3.3) is again (32). The vanishing equations are satisfied. Let us mention in the end that there are still other models with curved backgrounds dual to the flat ones, namely those corresponding to the Manin triples of the Drinfel’d doubles DD15 and DD19. Unfortunately, in these cases all coordinates depend on the so that only may be inserted into (14) giving results published in . We have investigated the possibilities to apply the Poisson–Lie transformation to the general solution of the vanishing equations for the flat metric. We have obtained dilaton fields for the metrics (23) and (3.3) having a nontrivial Ricci tensor. They are the most general dilatons that can be obtained by the Poisson–Lie transformation from the general dilatons (28), (37) of the dual flat metrics (24) and (34). An interesting but yet unsolved question is whether the dilaton (32) is the general solution of the vanishing equations for the curved backgrounds (23) and (3.3). This work was supported by the project of the Grant Agency of the Czech Republic No. 202/06/1480 and by the research plan LC527 15397/2005–31 of the Ministry of Education of the Czech Republic. Useful comments of Libor Šnobl are gratefully acknowledged. - L. Hlavatý and L. Šnobl, Poisson–Lie T–plurality of three–dimensional conformally invariant sigma models II : Nondiagonal metrics and dilaton puzzle, J. High En. Phys. 04:10 (2004) 045, [hep-th/0408126]. - R. von Unge, Poisson–Lie T–plurality, J. High En. Phys. 02:07 (2002) 014, [hep-th/00205245]. - L. Hlavatý and M.Turek, Flat coordinates and dilaton fields for three–dimensional conformal sigma models [hep-th/0512082]. - C. Klimčík and P. Ševera, Dual non–Abelian duality and the Drinfeld double, Phys. Lett. B 351 (1995) 455, [hep-th/9502122]. - C. Klimčík, Poisson-Lie T-duality, Nucl. Phys. B (Proc. Suppl.) 46 (1996) 116, [hep-th/9509095]. - L. Šnobl and L. Hlavatý, Classification of 6-dimensional real Drinfel’d doubles, Int.J.Mod.Phys. A17 (2002) 4043 [math.QA/0202210].

In the crypto community, an infamous bullish model on Bitcoin has buzzed in recent months, predicting a substantial appreciation in BTC-USD after the forthcoming halving, expected in May this year. This model is referred to as the Stock to Flow (SF or S2F) model, which was initially introduced by Plan B, an anonymous quantitative institutional investor from the Netherlands. The logic behind the model makes sense to us in so far as it treats Bitcoin as a mere commodity (producing no income) and aims at establishing a relationship between the market value of Bitcoin and its scarcity of Bitcoin, proxied by a stock-to flow ratio. Once this relationship is quantified, the price of Bitcoin (BTC-USD) can be inferred. At first glance, this logic seems to work and offers consistent results with other commodities like gold as silver. Source: Plan B The author uses logarithmic values instead of static values to show an actual linear relationship between the two variables. Fitting a model to non-linear data would result in meaningful prediction errors. But contrary to other traditional commodities, the stock-to-flow ratio is predictable because Bitcoin was designed with a fixed supply and predictable inflation schedule. To wit from PlanB: Supply of bitcoin is fixed. New bitcoins are created in every new block. Blocks are created every 10 minutes (on average), when a miner finds the hash that satisfies the PoW required for a valid block. The first transaction in each block, called the coinbase, contains the block reward for the miner that found the block. The block reward consists of the fees that people pay for transactions in that block and the newly created coins (called subsidy). The subsidy started at 50 bitcoins, and is halved every 210,000 blocks (about 4 years). That’s why ‘halvings’ are very important for bitcoins money supply and SF. Halvings also cause the supply growth rate (in bitcoin context usually called ‘monetary inflation’) to be stepped and not smooth. Modeling Bitcoin’s Value with Scarcity, March 2019 Based on PlanB’s model, the near-term outlook for Bitcoin could be extremely exciting considering that the forthcoming halving is scheduled in May 2020: Based on PlanB’s model, the predicted market value of Bitcoin after the May 2020 halving will be $1 trillion, translating into BTC-USD at $55,000, representing a stellar gain from its current price level. Source: PlanB, digitalik In this note, we wish to discuss the veracity of PlanB’s model and propose a more robust statistical approach to tackle the original model’s limitations. Unfortunately, our conclusions invalidate PlanB’s model because we do not find any significant short-term and long-term dynamics of the relationship between the market value of Bitcoin and the stock-to-flow ratio. Therefore, we cannot conclude that a substantial appreciation in BTC-USD will materialize after the May 2020 halving. The model equation, which can be estimated with Ordinary Least Squares (OLS), can be written as follows: Bitcoin=market value of Bitcoin; SF=Stock-to-flow ratio While the assumption of a linear relationship between the independent variable (NYSE:SF) and the dependent variable (Bitcoin) is respected, other assumptions also need to be verified to make sure that the results of the linear regression model are valid and interpretable. To be specific, the Gauss-Markov Theorem states that the OLS estimator gives the best linear unbiased estimator (BLUE) of the regression coefficients as long as: - The OLS residuals have a mean of zero - The OLS residuals are uncorrelated (no autocorrelation) - The OLS residuals have constant variance (homoscedasticity) Let’s check whether these three assumptions are verified by using PlanB’s data. We will use raw data from PlanB’s article, available here. The dataset consists of monthly data of the market value of Bitcoin (Bitcoin) and the stock-to-flow ratio (SF) from December 2009 to February 2019. Here are the key statistics of our dataset: Let’s kick off by double-checking the linear relationship between: There is indeed a clear linear relationship between the two variables. We can proceed. Let’s now fit our linear regression model using OLS. Here is the summary of our results: At first glance, the OLS estimators are statistically significant and the coefficient of determination (R-squared) is very high (94%), suggesting that the model fits very well with the data. R-squared is the % of variance explained by the model. However, we need to be careful and don’t put the cart before the horse. Let’s make sure that our three assumptions about the OLS residuals mentioned above are verified before making any straightforward conclusions. Do the OLS residuals have a mean of zero? We find that the OLS residuals have a mean of -1.63552854942978e-14, confirming that the expectation of our OLS residuals is zero. The first assumption is validated. Do the OLS residuals have constant variance (homoscedasticity)? Let’s plot the OLS residuals to detect potential homoscedasticity, that is, a non-constant variance. The OLS residuals seem to exhibit no clear pattern while the pink line is pretty stable. At first glance, it seems to us that the OLS residuals are homoscedastic. To confirm this, let’s run the Goldfeld-Quandt test, whereby the null hypothesis assumes that the OLS residuals are homoscedastic. Test conclusion: Considering that the p-value is greater than 0.1, we fail to reject the null hypothesis of homoscedasticity, thereby confirming our initial intuition. The second assumption is validated. Do the OLS residuals are uncorrelated (no autocorrelation)? To detect the presence of autocorrelation, we can start by plotting residual autocorrelation. The ACF suggests the presence of autocorrelation. To confirm this, let’s run the Durbin-Watson test, whereby the null hypothesis assumes that errors are uncorrelated. Test conclusion: The Durbin-Watson test confirms the presence of positive autocorrelation in the OLS residuals. Consequently, the assumption of non-serial correlation cannot be verified. We will discuss the implications of it later in the note. Is the model stable? To investigate this, we apply a Recursive Least Square (RLS) filter. Conclusion: The CUSUM statistic moves slightly outside the 10% significance bands, leading us to reject the null hypothesis of the stability of the model at the 10% level. Due to the autocorrelation in the OLS residuals, we cannot make statistically meaningful conclusions. The OLS estimates of the standard errors are likely to be smaller than their true values. As a result, the r-ratios cannot be properly interpretable because they are likely to look more significant than they really are. Because the t-ratios are not interpretable, we cannot know whether the estimators of our linear regression model: are statistically different from zero. The same applies to R-squared. In this regard, the super-high R-squared of 95% in the model above is likely to be misleading. We need to find a way to know whether we are dealing with a spurious regression (showing an apparent strong relationship between unrelated variables) or a genuine relationship. A genuine relationship between two non-stationary variables is possible if we have a stationary equilibrium relationship between these two variables, meaning there exists a linear combination between these two variables that is stationary. In this case, these variables are cointegrated. In our context ln(Bitcoin) and ln((SF)) ar two non-stationary variables integrated of order 1: Test conclusion: In both cases, the null hypothesis of a unit root can be rejected at the 5% level. We say than ln(Bitcoin) and ln((SF)) are cointegrated if there is a linear combination between ln(Bitcoin) and ln((SF)) such as: If these two variables are cointegrated, any deviation from equilibrium will only be temporary, thereby resulting in a genuine relationship. According to Engle and Granger (1987), if a set of variables are cointegrated, then there exists a valid error correction representation of the data. In the section below, we will follow the two-step procedure recommended by Engle and Granger in Co-Integration and Error Correction: Representation, Estimation, and Testing (1987) to develop an error correction model and study the true relationship between the value of Bitcoin and the stock-to-flow ratio. Step 1: Estimate our equilibrium equation and perform a test for cointegration Our long-term equation is written as follows: A test for cointegration consists of testing whether OLS residuals (û) are stationary. To do so, we run an Augmented Dickey-Fuller (ADF) unit root test on the OLS residuals. Here are the results: Test conclusion: The null hypothesis of a unit root can be rejected at the 1% level. Therefore, the OLS residuals do not have a unit root and are stationary. Because the cointegration is verified, we know that our OLS estimators are super-consistent. That said, the serial correlation in the residuals affects their efficiency. Consequently, traditional diagnostic tests need to be overlooked. Step 2: Estimate the Error Correction Model (ECM) Our ECM equation can be written as follows: Because this equation has only I(0) variables, there is no longer autocorrelation in the OLS residuals and as such, traditional diagnostic tests are appropriate. This ECM describes the short-term and long-term dynamics of the relationship between the market value of Bitcoin and the stock-to-flow ratio. The adjustment coefficient, α, which should vary between -1 and 0, measures the speed of the adjustment toward the long-term equilibrium. Should α be positive, the impulse response would be explosive, forcing us to reconsider our model. To determine the lag lengths (p and q) for our ECM, we will run two VAR regressions using each variable as an endogenous variable. We will use Akaike’s Information Criteria (NYSE:AIC) to select the optimal lag length. We initially selected a max lag length of 10. Here are our results: Lag selection: Based on Akaike’s Information Criteria (AIC), we deduce that p = q =2. Let’s now fit our linear regression model using OLS. Here is the summary of our results: Interpretations of our results: As we noted above, traditional diagnostic testing is now appropriate because the model does not suffer from autocorrelation in the OLS residuals. The Durbin-Watson statistic is very close to 2. The OLS estimator on the lagged OLS residuals is statistically significant. α = -0.15. This means that only 15% of the disequilibrium will dissipate in the next period, meaning that adjustment is rather sluggish. The OLS estimators on: - the constant (const) are statistically significant. This means that if the market value of Bitcoin at t-1 changes ceteris paribus by 1% percentage point, the market value of Bitcoin at t changes by 0.25 percentage point. The estimators of the short-run and the long-run elasticities of the market value of Bitcoin with respect to the stock-to-flow ratio are not statistically significant. We, therefore, cannot assert that the market value of Bitcoin is significantly elastic with respect to the stock-to-flow ratio, both in the short and long run. R-squared is decent at 21%, although it has substantially deflated from the R-squared of 95% resulting from PlanB’s model. Stability of the model Let’s finally check the stability of our new model. Again, we apply a Recursive Least Square (RLS) filter. Conclusion: The CUSUM statistic stays within the 10% significance bands, leading us to fail to reject the null hypothesis of the stability of the model (i.e., there is no structural break) at the 10% level. Therefore, it is safe to assert that the new model is stable over time, unlike the original model from PlanB. In our view, PlanB’s infamous model is misleading because the statistical results cannot be properly interpretable. This even prompted Vitalik Buterin, Etherum (ETH) co-founder, to tweet about it, criticizing the approach of being “post-hoc rationalized bullshit”. In our view, PlanB’s model emanates from a good intuition. This is why we have revisited the original model to account for the autocorrelation issue and attempt to investigate both short-term and long-term dynamics of the relationship between the market value of Bitcoin and the stock-to-flow ratio. Unfortunately, our model cannot assert the presence of significant elasticities of the market value of Bitcoin with respect to the stock-to-flow ratio, both in the short and long run. Our conclusions are as follows: - Our new model explains relatively well the market value of Bitcoin, with an R-squared of 21%, though to a much lesser extent than PlanB’s. - Our model cannot confirm the presence of significant short-term and long-term dynamics of the relationship between the market value of Bitcoin and the stock-to-flow ratio, invalidating PlanB’s conclusions. - The adjustment toward the long-term equilibrium is rather sluggish, suggesting that BTC/USD can remain away from its fair-value for a significant amount of time. Did you like this? Click the “Follow” button at the top of the article to receive notifications. Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article. Additional disclosure: Our research has not been prepared in accordance with the legal requirements designed to promote the independence of investment research. Therefore, this material cannot be considered as investment research, a research recommendation, nor a personal recommendation or advice, for regulatory purposes.

In the 21st century economy, students in mathematics may be studying cryptography and coding theory, cellular automata for modeling in the life sciences, graphs and networks with applications to computing, or the mathematics of finance, to name only a few of the possible applications. In traditional areas such as teaching, there is projected to be a great demand for secondary teachers of Mathematics in the United States over the next decade. Iowa State University offers a variety of undergraduate degrees related to mathematics. Consider the options, and discuss them with your advisor, to find the program of study that is right for you. A summary is offered at the bottom. This is the traditional degree program which offers training suitable for students planning to work in mathematics and computation for industry or government, or to continue their studies in graduate school. Majors normally spend the first two years obtaining a grounding in calculus and differential equations. At the junior and senior levels the department offers more than 25 undergraduate courses, including an introduction to combinatorics, abstract algebra, partial differential equations, complex variables, and mathematics of fractals. In addition, there are other courses at the graduate level which are open to qualified undergraduates. → Learn more This degree option is designed for students who want to major in mathematics with a clear specialization in some area of application. Certain courses in the area of application are counted towards the mathematics major. This also facilitates double majors or major/minor combinations. The major in mathematics + application area is not recommended for students who plan to continue with graduate school in mathematics. It can be very appropriate for students who plan to pursue a graduate degree in the application area. In consultation with a mathematics faculty advisor, the student prepares a program of studies tailor-made to his/her future plans or career needs. A number of programs have already been designed and pre-approved. Deviations from the programs may be proposed, and must be approved by the Mathematics Department Undergraduate Committee. → Learn more This degree will prepare you for teaching mathematics at the middle and high school level. The Mathematics Department and the Curriculum & Instruction Department share responsibility for this program. → Learn more The path to becoming an actuary begins with a college degree, often in mathematics or business, followed by the first actuary exam. The employer will typically pay for on-the-job training and further exams. For more details, see http://www.beanactuary.org/. Math + Actuarial Science is one of the options in the Mathematics + Application Area program of study, where the application electives and further recommended courses have been selected to help you prepare for the actuarial exams. → Learn more Bioinformatics and Computational Biology (BCBio) is an interdisciplinary science at the interface of the biological, informational and computational sciences. The program includes required courses from many different disciplines. Undergraduate study in the BCBio major is jointly administered by the Department of Computer Science, the Department of Genetics, Development, and Cell Biology, and the Department of Mathematics. → Learn more Minor in Mathematics A minor in Mathematics requires the following seven mathematics courses. The six credits of 300 level math classes must be taken at Iowa State University. - Math 165, 166, 265 - Calculus I, II, III - Math 201 - Introduction to Proofs - One of the following: - Math 317 - Theory of Linear Algebra - Math 407 - Applied Linear Algebra - One of the following: - Math 301 - Abstract Algebra I - Math 304 - Combinatorics - Math 314 - Graph Theory - Math 350 - Number Theory - Math 421 - Logic for Mathematics and Computer Science - Math 435 - Geometry I - Math 436 - Geometry II - One of the following: - Math 266 - Elementary Differential Equations - Math 267 - Elementary Differential Equations and Laplace Transforms - Math 331 - Topology - Math 341 - Introduction to the Theory of Probability and Statistics I - Math 365 - Complex Variables with Applications - Math 373 - Introduction to Scientific Computing - Math 414 - Analysis I The programs of study listed above can be grouped into three categories: Light Mathematics Course Load A minor in mathematics requires 7 math courses. Standard Course Load A major in mathematics requires 14 math courses, plus general requirements. A major in mathematics + application area requires between 9 and 11 math courses, plus 3 to 5 courses in the application area, for a total of 14 courses (plus general requirements). Higher Course Load The major in math with secondary teaching certification is essentially a double major. It requires a full math major (with minor variations), plus additional courses in curriculum and instruction. The degree in bioinformatics and computational biology requires more courses than most other majors (26 courses, in several departments, plus general requirements).

Three-Pion Interferometry of Relativistic Nuclear Collisions Three-pion interferometry is investigated for new information on the space-time structure of the pion source created in ultra-relativistic heavy-ion collisions. The two- and three-pion correlations are numerically computed for incoherent source functions based on the Bjorken hydrodynamical model, over a wide range of the kinematic variables. New information provided by three-pion interferometry, different from that provided by two-pion interferometry, should appear in the phases of the Fourier transform of the source function. Variables are identified that would be sensitive to the phases and suitable for observation. For a positive, chaotic source function, however, a variation of the three-pion phase is found to be difficult to extract from experiments. Effects of asymmetry of the source function are also examined. pacs:PACS number(s): 25.75 Gz Two-pion interferometry has been regarded as an important means of obtaining the space-time structure of dynamics involved in relativistic heavy-ion collisions. Over the years, extensive studies of two-pion interferometry have been carried out theoretically and experimentally to investigate how much information the correlations of two emitted pions can provide. As experiments become refined, measurements of three-pion correlations should become feasible, hopefully providing new information. The first result of such measurements has been recently reported from the CERN NA44 experiment. In the last few years, some theoretical investigations have been made regarding three-pion interferometry[2, 4, 5, 6]. Though the treatments of three-pion interferometry and the issues involved are similar to those regarding two-pion interferometry, three-pion interferometry is technically far more complicated than the two-pion case due to the involvement of an additional momentum and also to new aspects of the particle correlations. Consequently, theoretical work so far has been on limited aspects of the interferometry, focusing on a few kinematic variables, such as the sum of three relative invariant momenta, over small ranges of their values. In the coming years, especially when RHIC becomes in operation, we expect that experiments on three-pion interferometry will become more detailed and will be made over larger ranges of various kinematic variables. We report here an investigation of three-pion interferometry over a wide range of kinematic variables. Our major objective is to clarify how much new information we can extract from three-pion interferometry, regarding the space-time structure of the pion source created in ultra-relativistic heavy-ion collisions. This work is similar to that of Heinz and Zhang in its objective, but it differs in scope. We numerically calculate the two- and three-pion correlations for the same source functions and compare the two types of correlations. The calculations are carried out with various model source functions that are based on the Bjorken hydrodynamical model. Throughout this work, we assume pion emission from the source to be completely chaotic. We also neglect possible final-state interactions (including the Coulomb interactions) between the emitted pions and the source. The new aspect of the correlations that three-pion interferometry can provide is the phase of the source function’s Fourier transform. The information content of the phase differs from that of the magnitude of the Fourier transform. Since two-pion interferometry can provide only the magnitude, we hope that three-pion interferometry can provide new information. To this end, we carefully identify the variables that are sensitive to the phase and suitable for observation. Since the phase is expected to be greatly affected by asymmetry of the source function, we also examine that issue. In Sec. II, we summarize the formulation of both the two- and the three-pion interferometries, mainly to define the correlation functions used. In Sec. III, we describe the various source functions that are used in the calculation, based on the Bjorken hydrodynamical model. The choice of optimal variables in the three-pion correlations is given in Sec. IV, and various numerical results are described in Sec. V. Discussion and conclusions are presented in Sec. VI. The two- and three-particle correlation functions that we discuss are reasonably well-known, but for clarity we sketch the formalism of, in our case, bosons, and we define the correlation functions and various variables that appear in this work. The field operator of emitted bosons, such as pions and kaons, obeys the Klein-Gordon equation, where is the mass of the particle, and the source current. The incoming and outgoing states of are specified in terms of the creation and annihilation operators and for these states. They are related to as where is on-shell and . We define an important quantity in this work, a source function, , in terms of : where denotes the quantum ensemble average over various incoming states. The Fourier transform of is defined as where is the amplitude and the phase. For a real , we have and , and thus . Boson spectra and correlation functions will be expressed in terms of or , and is written in the small-momentum expansion as where , and . Since is written in terms of odd moments of , an asymmetric (about ) source causes a strong, nonlinear dependence of on . In this work, we examine the of simple forms as discussed in Sec. III. The one-particle spectrum is given by Higher-order correlations of are assumed to satisfy Gaussian reduction, such as The two-particle spectrum is then written as where for the pair of the -th and -th momenta, we define their average and relative momenta as respectively. The two-particle correlation function is then expressed as Hereafter, we denote for and for . Equation (11) shows that the two-particle correlation function is independent of . Similarly, the three-particle spectrum and correlation are given by From the preceding discussion of , we have for a real source function. When , Eq. (6) yields is thus expected to vary prominently when the source is asymmetric (in ). Equation (13) shows that the three-particle correlation function depends on , which is absent in the two-particle correlation. The rest of the three-particle correlation is expressed in terms of ’s and can thus be determined from the two-particle correlations. The three-particle correlation is a function of three momenta, , and . For convenience, we introduce the average total momentum, Since ’s satisfy the identity, we have three independent momentum variables, which we will take to be , , and . Among various choices of three variables, another convenient choice is a set of ’s. For completeness, we show the relations between the first set and ’s: Iii Source function We apply the Bjorken hydrodynamical model to describe the evolution of the hot region created in heavy-ion collisions. Based on the Cooper-Frye spectrum, one writes the source function as where , , and denote the coordinate variables defined as , and , respectively; and are the momentum variables defined as and , respectively. Note that we specify the momentum variables for a momentum pair, and , by the subscript , such as , and those for three momenta with no subscript such as . We set the -axis to be the beam direction, and the -axis parallel to , or perpendicular to the beam axis. is a four-velocity of flow, and is in the Bjorken hydrodynamical model. The profile function determines the source shape in the - space. The shape along the transverse direction is taken to be of a Gaussian form. is the freeze-out temperature, and is the measure of freeze-out hypersurface. We assume that freeze-out occurs on the hypersurface where is constant. In this work, we examine the following five forms of the profile function. Simple (box-type) profile: Iv Optimal variables The three-particle correlation function depends on three momenta, which have nine components. As noted in Sec. II, we choose the three momenta to be , , and . In this work, we focus on the dependence of the correlation functions on relative momenta of the emitted bosons by fixing the value of . This leaves the two relative momenta to be the remaining variables. In order to identify new information in the correlation function, we should choose the variables that could provide the most rapid variation of . Figure 1 illustrates the variation of as a function of and with all other components of the relative momenta set to be zero. Here, we use the Heinz profile with the parameters of fm, fm, , and MeV. Figure 1 shows that is unity along the - and -axes and also along the line of , or the -axis (because of Eq. (16)). Figure 1 also shows that varies most prominently along the lines of , , and . Because of symmetry, however, it is sufficient to examine the variation around one of the lines, such as the line of . The reason why the line of provides a prominent variation of is seen as follows. When and , we have where the (weak) dependence on in is not explicitly shown. Setting and , we find where . When and , vanishes at , or , yielding the minimum. It follows then that the optimum choice is . With this constraint, the variables are now reduced to a single relative momentum. The source functions under consideration (as listed in Sec. III) are almost symmetric about the beam and transverse axes, but they can be asymmetric about the time-axis. Which component of should we choose so as to describe most effectively the asymmetric property of the source functions? is an obvious choice, but we find that or is the most convenient variable. From Eq. (10), we have , which gives . Thus as a function of or reflects time-axis asymmetry. Furthermore, for finite , our source functions are not simply a function of without symmetry along the longitudinal direction. Because of this, may be the variable more suitable for identifying the time-axis asymmetry of the source function. In the following, we will examine both and . is parallel to , while is perpendicular to it. Following the practice in the literature on two-pion interferometry, we will also denote and as and , respectively. Note that when the dependence of on or is examined, the other components of are fixed in our calculation. V Numerical results Figures 2, 4, 6, 8, and 10 illustrate the dependence of and on in the Heinz, Simple, Gaussian, Theta, and Exponential profile functions, respectively. Each figure is shown for , and 2. In the figures, we set , 140 MeV, 140 MeV, , and fm for all profile functions, and also fm for the Heinz, Theta, and Exponential profile functions. These parameter values reasonably satisfy the recent CERN-SPS experiment on Pb + Pb at 158 GeV/A. Note that describes the two-particle correlation since is normalized to be unity at , and that is not observed in the experiment. Figures 3, 5, 7, 9, and 11 show and (half of) its coefficient in the three-particle correlation function, , as functions of ( ). Each figure is shown for and 2 and 140 MeV with the parameter values of , , , and as in Figs. 2, 4, 6, 8, and 10. In Figs. 3, 5, 7, 9, and 11, we see that for all profile functions, at , and becomes more prominent as increases. tends to be smaller for the asymmetric profile functions (such as the Simple and Exponential profile functions) than for the symmetric ones, though the difference between them is not substantial. tends to deviate from zero more slowly than the phases of ’s, ’s. For example, we see in Figs. 4 and 5 that when reaches around 150 MeV, we still have . The reason for this is general because is defined as and is constrained by the identity, . In fact, the small-momentum expansion of Eq. (6) applied to yields that the term linear in vanishes, as discussed in Sec. II. Furthermore, when starts to deviate from zero, its coefficient in the three-particle correlation function, , tends to become small. In fact, we see in the figures that gets halved for all profile functions before decreases to 0.9, and even nearly vanishes when decreases further. We expect that in actual experiments it will be difficult to identify with a value much smaller than unity. This is the major finding of this work. We find the same difficulty when we choose as the independent variable. Figure 12 shows and as functions of for the Heinz profile for 140, 200, and 300 MeV, with fm, , , fm, and fm. Figure 13 also shows and as functions of for the three values of with 1.5 and the same values of , , , , and . We see in Fig. 13 that when starts to deviate from unity, its coefficient, , becomes small quite rapidly, as in the previous cases of being the independent variable. Note that decreases more quickly as a function of than of , as seen in Figs. 13 and 3. The quick reduction occurs because of the factor , when the transverse momentum, , is used as the variable. fm is chosen to reproduce experimental results, but we find that even if a much smaller value of is used, becomes quite small when is off unity. The difficulty also remains at different values of . Figures 14 and 15 show , , , and as functions of for (6.5, 0.65), (4,8, 2.8), and (3.2, 3.2) fm, with MeV, MeV, , and . These parameter sets yield the recently measured value of the longitudinal size, 3.5 fm. Figure 15 shows that depends on rather strongly. But the variation of takes place where is small, and will be difficult to observe. Vi Discussions and summary We have investigated the three-particle correlation function for chaotic source, using various profile functions of the source in the Bjorken hydrodynamical model. In all profile functions we have examined, the coefficient of in the three-particle correlation function, , decreases quite rapidly, as decreases from unity. Extraction of is thus difficult when decreases from unity. This result is in agreement with Heinz and Zhang, who previously made a similar investigation for small relative momenta. In order to clarify why is difficult to observe, we first identify the portion of the source function that most strongly influences the phase, . appears in the Fourier transform of as . As noted previously, is an odd function of , and for a real source function. The imaginary part of , , thus represents the behavior of relevant to the present discussion and is an odd function of . The portion of that most strongly affects is thus its odd-function part, , in with and . Consider that and vary over distances of typical scale, and , respectively. Here, and correspond to the spread widths of and the distance to maximum of , respectively, as Fig. 16 illustrates. and also vary over the scales of and , respectively. As increases, increases. The region where is appreciable is . The amplitude becomes small in the region where is larger than . Thus, if , the variation of is not difficult to observe. This is, however, untenable as long as is positive everywhere. Consider in some region: we then have in that region, and furthermore, . The last inequality contradicts the desired inequality. In this discussion, we have assumed the source function, , to be positive everywhere, as done in common practice. How realistic is this assumption? If is a statistical phase-space distribution, should not be negative everywhere. Generally speaking, however, can be locally negative, provided its integrals over or be positive. This property is similar to, for example, that of the Wigner function. Furthermore, that is locally negative suggests that it would be associated with some dynamics involving quantum correlation. This aspect of the interferometry is currently under investigation. Measurement of the three-pion correlations from the CERN NA44 experiment has been recently reported for the total relative momentum of up to about 300 MeV, and the extraction of up to about 60 MeV. This total relative momentum is too small to observe the variation of that is examined in this work. Taken as a constant, is found to be about 0.2 and is much smaller than the value of unity for a chaotic source, as discussed in this work. The reason for the small value is unknown, but it may be because of partial coherence of the source, or even because of a more exotic reason. In summary, we have investigated three-particle interferometry from chaotic sources using the Bjorken hydrodynamical model. The optimal variables are identified suitable for extracting new information through the phase of the source function’s Fourier transform. The three-particle correlations are calculated over a wide range of the kinematic variables. A variation of the three-pion phase is found to be difficult to observe experimentally because its coefficient, as it appears in the three-particle correlation function, becomes small in that region. This is the case if the source function is positive everywhere as conventionally assumed, and suggests the interesting possibility of the source function being locally negative. Acknowledgements.We acknowledge M. C. Chu for his valuable contribution at the initial stage of this work. This research is partially supported by the U.S. National Science Foundation under grants PHY88-17296 and PHY90-13248 at Caltech, and the U.S. Department of Energy under grant DE-FG03-87ER40347 at CSUN. - A. Sakaguchi et al. (NA44 collaboration), Nucl. Phys A638, 103c (1998); J. Schmidt-Sørensen et al. (NA44 collaboration), ibid., 471c (1998); H. Bøggild et al. (NA44 Collaboration), Phys. Lett. B455, 77 (1999). - U. Heinz and Q. H. Zhang, Phys. Rev. C56, 426 (1997). - U. Heinz, Nucl. Phys. A610, 264c (1996). - J. G. Cramer and K. Kadija, Phys. Rev. C53, 908 (1996). - H. Heiselberg and A. P. Vischer, Phys. Rev. C55, 874 (1997). - T. J. Humanic, Phys. Rev. C60, 014901 (1999). - R. Ganz et al. (NA49 collaboration), nucl-ex/9808006. - We follow the conventions of J. D. Bjorken and S. D. Drell, as in their book, Relativistic Fields (McGraw Hill, N. Y., 1964): Four-vectors are denoted in regular letters and three vectors in bold letters, and the square of the four-momentum of a physical particle is the square of its mass. - J. D. Bjorken, Phys. Rev. D17, 140 (1983). - H. Nakamura and R. Seki, Caltech Kellogg preprint (1999).

A legendary letter was sealed more than three centuries ago. What was in it was a sort of puzzle, a challenge so simple that any elementary student could understand. However, it is such a complex problem to prove that the greatest minds in history have failed trying. Between the lines of that letter, one said the following: That is the whole problem. The whole numbers are those with which we learned to count: the one, the two, the three ... Without frills. A prime number is all that greater than one and you can only divide it between yourself and the unit without decimals: two, three, five, seven, eleven ... That and adding is the only thing you need to know To understand the problem, there is no more. This is the conjecture that Christian Goldbach proposed to Leonhard Euler in 1742, the unbeatable beast that has defeated every great mind that has dared to face her. However, I am sure that a small part of you has been tempted to test it. So why not try? Let's try the eight. We can do this by adding three and five, so it is fulfilled. And the fourteenth? It would be three plus eleven, and twenty one is two and nineteen. I assure you that you can try with everyone you want, because you will not find a single case in which the conjecture is not fulfilled. And we know this because some machines have tried before you. Computers have checked the first four trillion numbers, a four followed by eighteen zeros. It might seem that this is enough to prove that the conjecture is true, after all, four trillion numbers supporting it are many numbers. However, mathematicians do not get that. You have to be sure that there is not a lost number out there that is not met, because, as huge as it may seem, four trillion are left at nothing compared to the infinite ocean of figures out there. Mathematicians need to be totally and absolutely sure, everything else is pure opinion and only with opinions do not do mathematics. So, there are mostly two ways to be satisfied. The first is the simplest, the one we all try when we are taught the problem: find a counterexample, take the opposite, find a case that is not met so that we can reject the conjecture. That is what we have tried, but as we have seen, it does not seem very fruitful ... although, wait. And the two? It is an integer and even number, but it is not the sum of two cousins because it can only be formed by adding two ones, and the number one is not a cousin. For the 2 Goldbach is not fulfilled! Is it possible that we have resolved Goldbach's conjecture? I'm afraid not, because this apparent lack of confidence is a historical curiosity. In the 18th century, the 1 was still considered a prime number, so Goldbach could express the two as "one plus one" and remain so calm. In fact, excluding one from the list of prime numbers had to update the conjecture. The modern version would be something like this:That every integer even greater than two is the sum of two prime numbers seems to me to be a completely true theorem, but I can't prove it“So let's forget number two and worry about the rest. As we have seen, a counterexample has not yet been found, and although by definition there are still infinite numbers to prove, mathematicians suspect that no matter how much they search they will not find a single case in which Goldbach is not fulfilled. They argue that, as the numbers grow, the ways in which they can express themselves also increase. For example: ten can be done by adding five to itself or three to seven. One hundred, for example, can be constructed in six different ways, and from what we have seen, the number of possible combinations skyrockets as we study larger numbers. In fact, there is a very visual way to see it: Goldbach's comet. It is a graph whose horizontal axis represents the number to study and in the vertical we find how many different forms we can decompose. The key is that the comet does not stop growing and it becomes difficult to think that at some point it will plummet by marking zero on the vertical axis, indicating an even number that cannot be constructed by adding two cousins. However, this is still not enough evidence. It is only a suspicion, but nothing assures us that there is an exception among all the numbers that we cannot check by hand, so we have to change our strategy. Luckily there are other ways to prove a conjecture: finding a general argument applicable to all numbers and showing that, if Goldbach were false, mathematics would have to contradict themselves. Something like looking for a nonsense, showing that we are facing something impossible. A classic example of this strategy is Euclid's theorem, which posed the existence of infinity of prime numbers. His way of demonstrating it was as follows: Let's take a few prime numbers and multiply them together, for example: 2x3x7. The result is never a prime number, because it is divisible among all the previous ones giving, in this case, 42. Now we must add a 1 and bingo! We have already achieved a prime number, 43, because adding 1 stops being divisible among the cousins we have used to build it. However, this is not always the case, for example: 3x5 + 1 gives 16, which is not a cousin. And here is the final trick, because, if the result is not a cousin and at the same time it is not divisible among the cousins we have used to build it, it means that its divisor is a new prime number, in this case the two: 2x2x2x2. Following these steps we can build as many prime numbers as we want, demonstrating that there are infinity of them. This type of reasoning is what mathematicians look for, ways to prove or deny conjectures beyond any doubt avoiding logical inconsistencies. However, these approaches have also failed to resolve Goldbach's conjecture. Nothing has turned out, and although from time to time someone appears saying they have tried it, they are always false alarms that show nothing. However, you may have heard that some years ago a mathematician resolved Goldbach's blissful conjecture, and it is "true," except that it was Goldbach's weak conjecture, something quite different. In 2013 Harald Andrés Helfgott managed to prove Goldbach's weak conjecture, which reads as follows: “Any odd integer greater than 5 can be expressed as the sum of three prime numbers”For example: 15 is the sum of 3, 5 and 7. Helfgott's strategy was somewhat intermediate to the two we have proposed. On the one hand, the computers had already calculated that the weak conjecture was fulfilled for all odd integers between 5 and 8875x10 ^ 30 (representing approximately 9 followed by thirty zeros). On the other hand, some great experts in number theory had shown that, from a sufficiently large number, all of the following had to fulfill the conjecture. The demonstration of the latter is very long for this article, but the important thing is to be clear that this "large enough number" has been reduced with better evidence over the years and that when Helfgot faced the conjecture, he was in 2x10 ^ 1346 (a 2 followed by one thousand three hundred forty-six zeros). That means we already knew that the largest and smallest numbers met the conjecture, but between them there was a huge chasm and their mission was to reduce it. He could have looked for better computing methods so that computers were able to calculate even more numbers by brute force. However, 8875x10 ^ 30 was already something huge (to give you an idea, it is estimated that there are 1x10 ^ 80 protons in the universe) and as much as they improved, the machines could not reach 2x10 ^ 1346. So Helfgot decided to face it in another way and try to reduce that number "large enough" to make it smaller than 8875x10 ^ 30 so that the conjecture was demonstrated for all integers through one or another method. And he succeeded, he managed to lower it to overlap it with computer calculations. Goldbach's weak guess was true. Meanwhile, the strong conjecture remains undefeated and has not advanced much in recent decades. A drought that is largely due to its complexity, but to which many other things may have contributed. Not a few professionals believe that focusing their research on the Goldbach conjecture is almost a condemnation of failure. Many of those who work in it do it timidly, in their free time and without being able to devote the time they deserve. This perception is so popular that it is treated even in one of the most famous mathematical novels: Uncle Petros and the Goldbach conjecture, from Apostolos Doxiadis. At this step the question is obvious: Will Goldbach's conjecture ever be resolved? Will it be true and will be consolidated as a theorem? Or in an unexpected turn of events will it have to be ruled out? It only remains to wait while Goldbach's old statement echoes in our ears, like a siren song that makes anyone who agrees to dance with her run aground against the cliffs. DON'T KEEP IT UP - The conjecture has not yet been proven. It is true that in 2013 Helfgot demonstrated Goldbach's weak conjecture, but when we refer to "Goldbach's conjecture" we are talking about the strong one. - The Fundamental Theorem of Arithmetic implies that 1 cannot be a cousin by saying that: any integer greater than 1 can be written as a single product of prime numbers. If we accept 1 as a cousin there would be infinite ways to break down each number (10 would be 2x5, but also 2x5x1, 2x5x1x1 and so on). There are more reasons, but this is possibly the easiest to understand. - Christian Goldbach, "Letter to Leonhard Euler" (Moscow, June 7, 1742) - Leonard Eugene Dickson “History of the Theory of Numbers, Volume I: Divisibility and Primality” Dover Books on Mathematics (2005) - Harald Andrés Helfgott “Major arcs for Goldbach’s problem” arXiv (2013) - Tomás Oliveira e Silva, Siegfried Herzog & amp; Silvio Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4x10 ^ 18” Mathematics of Computation 83: 2033-2060 (2013)

Definition:. Preliminary Geometric Design of an Airport in Palestine. Advantages of the airport:. The importance of this research reflects the need for Palestinian airport for: 1) Investment sector. 2) Civilization. 3) Politically advantages 4) Necessity for the PNA. 5) Tourism. Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. Definition: Preliminary Geometric Design of an Airport in Palestine. Advantages of the airport: • The importance of this research reflects the need for Palestinian airport for: • 1) Investment sector. • 2) Civilization. • 3) Politically advantages • 4) Necessity for the PNA. • 5) Tourism Airport site selection • the FAA (Federal Aviation Administration) recommends a minimum site selection analysis that includes the following factors: • Operational capability • Ground access • Development costs • Environmental consequences • Socioeconomic factors • Consistency with area-wide planning site selection according ICAO and FAA codes Al-Buqaiaa site : • The selected site away from any population communities to avoid sound pollution. • The selected site does not affect the population areas growth. • According to the wind rose extracted from the wind movement in the proposed site showed slight wind speed with (N/W) to avoid any cross wind in the runway direction. The selected site discreption: • ± 10m on mean sea level. • Intermediate of the west bank. • Sandy soil. • The weather is moderate and suitable (ambient temp, humidity) . AIRPORT SITE COORDINATION • 1 X:187470 Y:129505 • 2 X:189446 Y:129446 • 3 X:186898 Y:125544 • 4 X:188882 Y:125262 Passengers trends during 20 years of airport operation (Million passengers)/year. Passenger classification / year: • regular daily passengers (Dep/Arr) • High season passengers. • Transit passengers • Omra pilgrims passengers • Haj pilgrims passengers • Christians pilgrims passengers R.W Considerations The following factors should be considered in locating and orienting a runway: • Wind • Airspace availability • Environmental factors (noise, air and water quality) • Obstructions to navigation • visibility • Wildlife hazards Runway Orientation provided Wind rose by ministry of transpiration Runway orientation • The orientation of Runway will be 170o-350o (17-35) Runway. Which is the orientation that satisfies 95% coverage (crosswinds below a critical value) considering yearly wind conditions, with respect to the topography of the airport selected site. Number of Runways • In this project one primary runway has been designed because the following reasons: • 1- Max capacity for one primary runway is 40-50 operation/hour that means; if the average capacity of airplanes for the fleet mix is 125 passenger/operation. Then the hour capacity for the runway is: 45* 125 = 5625 pax/hour As it cleared before the demand for the airport after 20 years will be about 14 million/year, so the peak-hour-flow is: Average monthly passengers =14000000*0.08417 = 1178380 pax Average daily passengers=1178380 * 0.03226 =38014 pax Peak-day-flow=1.26*38014 = 47898 pax Peak-hour-flow= 0.0917* 47898= 4392 Pax/Hour < 5625 pax/hour So one primary runway is sufficient to cover the demand for 20 years coming. • 2- The nature of topography for the selected site VS the provided wind rose general orientation, make a conflicting to construct parallel runway, because of non-satisfaction for FAR requirements. Runway Length Justification • Airplanes today operate in a variety of different environments and available field lengths. However, the suitability of those runway lengths is governed by the existing and forecast fleet mix, critical aircraft operational requirements, and the following variables: • Airport elevation above mean sea level • Mean maximum temperature • Wind velocity • Aircraft operating weights • Takeoff and landing flap settings • Effective runway gradient • Runway surface conditions (dry, wet, contaminated, etc.) • Operational use. • Presence of obstructions within the vicinity of the approach and departure path. • Locally imposed noise abatement restrictions and/or other prohibitions Runway Length Justification • Boeing 747-400 designate as the critical aircraft for determining the primary runway length requirements of this project. • Accordance with FAA guidance, the critical individual aircraft takeoff and landing operating weights for the B747-400, B737-300 and B747-700 were obtained from Boeing’s Airplane Characteristics for Airport Planning manuals associated with these specific aircraft (Table B-2) Takeoff Length Requirements • To accurately determine takeoff length requirements, the takeoff chart for the B747-400 with dry runway, zero wind, and zero effective runway gradient conditions within the airport’s mean daily maximum temperature of the hottest month at the airport was used. Landing Length Requirements • Landing length requirements were determined by obtaining the landing chart for the B747-400 with the highest flap setting (30 degrees), zero wind, and zero effective runway gradients. Runway length design (Sample calculations) • Data: • Airplane Boeing 747-400 Mean daily maximum temperature of hottest month at the airport 34.4 C • Airport elevation 0 (on MSL) • Maximum design landing weight (see table B-2)574,000 pounds (260,362 KG) • Maximum design takeoff weight (see table B-2)875,000 pounds (396,894 KG) • Maximum difference in runway centerline elevations 10 feet Runway length design (Sample calculations) • Proceed horizontally to the length axis to read 3352.8m. Interpolation is allowed for this design parameter for (Takeoff length requirements 100% Useful Load) • – Adjust for non-zero effective runway gradient • 11,000 + (10 x 10) = 11,000+ 100 = 11,100 feet (3383 m) • (5) The takeoff length requirement is 11,100 feet (3383 m) • (6) Step 5 – Adjust for temperature: • Because it its 0 on S.L T1 =59° F • L2= (0.005*(94-59)*11,100) +11,100= 12931.5 feet Takeoff length requirements 100% Useful Load • Where T1 is standard temperature • L2 is adjusted length of Runway • For Takeoff length requirements 95% Useful Load: • – Adjust for temperature: • Because it its 0 on S.L T1 =59° F • L2= (0.005*(94-59)*10,171) +10,171=11951 feet • Adjust for non-zero effective runway gradient=12051 feet OR(3700 m) Takeoff length • Selection of Exit: • As mentioned previously, the critical individual aircraft is B747-400 which has touch-down speed is 141 Knots to 166 Knots. • Assume that the touch-down speed=150 Knots and Dth approximate=1000 ft. and as explained before in the text the suitable exit speed for 30o high speed exit is 60 mil/hr. • So: • Vth=150*1.687=253 ft/s • Ve= 60*1.466=87.98 ft/s • And a=3.3 ft/s2 • Then.. • Dc=(253)^2-(87.96)^2/(2*3.3)= 8526 ft • D=Dth+Dc= 1000+8526=9526 ft (From the edge of the runway). • And equal about 2900m from the beginning of the runway. Selection of Exit: • To be sure that the airport will serve all categories, and in comfortable way, the exit location for category (C) should be determined then. • Category (C) touch-down speed= 121 knots • Assume • Dth=700 ft • Ve= 40 mil/hr for 300 exit flap and.. a=5 ft/sec2 • Vth=121*1.687=204.13 ft/sec • Ve=40*1.466=518.64 ft/sec • And a=5 ft/sec^2 • Dc=(204.13)^2-(58.64)^2/ (2*5)= 3823 ft • Dth=700 ft • D=3823+700=4523 ft which is equal about 1378m from the beginning of the runway. Selection of Exit • Because of the runway is symmetry, the separation between exits will be like shown in figure: Imaginary Surfaces summary • Primary= aligned (longitudinally) with each runwayand extends 200 ft. from each runway end • Approach= longitudinally centered with the run way and extends beyond the primary surface • Horizontal= horizontal plane 150 ft. above the established airport elevation. Constructed by swinging arcs around the end of the primary surface • Conical= 20:1 slope surface extending beyond the horizontal surface • Transitional= constructed to join approach and horizontal or approach and transitional surfaces Imaginary Surfaces summary A = Utility runways B = Runway larger than utility C = Visibility minimums > 3/4 of a mile D = Visibility minimums =< 3/4 of a mile Airport Reference Code • From the manufacturer airplane criteria see appendix A: • The speed Approach is 154 knots. So that the speed is 141 knots or more but less than 166 knots, there for the Aircraft Approach Category is (D). • Wingspan is 213 ft and tail height is 64.3 ft which they are within the flowing limits 171 - <214 and 60 - <66 in order. There for the Airplane Design Group (ADG) is (V) • So that the Airport Reference Code is (D-V) Runway components design • Obstacle Free Zone (OFZ) • Runway Blast Pad • Runway Protection Zone (RPZ): • Runway Safety Area (RSA) • Shoulder • Taxiway Safety Area (TSA) RPZ ARea RPZ Area = 49.978 Acer = 198207 m2 Gradients • Surface gradient standards: Aircraft approach categories C & D: • The longitudinal and transverse gradient standards for runways and stop ways are as follows and as illustrated in following figure • The max longitudinal grade is ± 1.5 %; however, longitudinal grades may not exceed ±0.8% in the first and last quarter of the runway length. It is desirable to keep longitudinal grades to a minimum. • The max allowable grade change is ±1.5%. use longitudinal grade changes only when absolutely necessary. • Vertical curves for longitudinal grade changes are parabolic. The length of the vertical curve is minimum of 1000 ft (300m) for each 1 % change. • The minimum allowable distance between the points of intersection of vertical curves is 1000 ft (300m) multiplied by the sum of the grade changes ( in percent ) associated with the two vertical. Taxiway Dimensional standards • Taxiway and Taxi lane Object Free Area (OFA) • Taxiway Shoulders • Taxiway safety area (TSA) Major Terminal Components • Apron. • Connector. • Main Terminal Building. • Airport Access System. Alternative Terminal Building :Concepts • Simple Terminal Concept. • Linear Concept • Pier Concept. • Satellite Concept. Estimating Number of gates: • To estimate number of gates, a lot of data is needed, for example: • Peak hour passengers • Equivalent airplane factor • Fleet mix not available • Expected destinations at the same time. Peak hour passengers: • Average monthly passengers =3440000*0.08417 = 289545 pax • Average daily passengers=289545 * 0.03226 =9341 pax • Peak-day-flow=1.26*9341 = 11769 pax • Peak-hour-flow= 0.0917* 11769= 1079. Pax That means the capacity of airplane of 1 equivalent factor = about 125-135 pax. • So that number of demand gates =1079/135 = 8 gates • In this project it seen recommended to add 4 gates ,for emergency case • And 4 gates for ceremonial usage. • To be the Total estimated gates are ( 16 )

- Time-frequency representation A time-frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time-frequency analysismeans analysis of a TFR. TFRs are often complex-valued fields over time and frequency, where the modulus of the field represents "energy density" (the concentration of the root mean squareover time and frequency) or amplitude, and the argument of the field represents phase. A signal, as a function of time, may be considered as a representation with perfect "time resolution".In contrast, the magnitude of the Fourier transform(FT) of the signal may be considered as a representation with perfect "spectral resolution" but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in time, different events occur in the signal. TFRs provide a bridge between these two representations in that they provide "some" temporal information and "some" spectral information simultaneously. Thus, TFRs are useful for the analysis of signals containing multiple time-varying frequencies. Formulation of TFRs One form of TFR can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such formulations are known as quadraticTFRs because the representation is quadratic in the signal. This formulation was first described by Eugene Wignerin 1932 in the context of quantum mechanicsand, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the Wigner-Ville distribution. Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms whenever multiple frequencies are superimposed. This was partly addressed by the development of the Choi-Williams distributionin 1989 but most recent applications of TFRs have turned to linear methods. The cross-terms which plague quadratic TFRs may be evaded by comparing the signal with a different function. Such representations are known as linear TFRs because the representation is linear in the signal. The "windowed Fourier transform" (also known as the short-time Fourier transform) localises the signal by modulating it with a window function, before performing the Fourier transform to obtain the frequency content of the signal in the region of the window. Wavelet transforms, in particular the continuous wavelet transform, expand the signal in terms of wavelet functions which are localised in both time and frequency. Thus the wavelet transform of a signal may be represented in terms of both time and frequency. Before 1991, the notions of time, frequency and amplitude used to generate a TFR from a wavelet transform were derived intuitively. In 1991, Nathalie Delpratgave the first quantitative derivation of these relationships, based upon a stationary phase approximation. Continuous wavelet transform * [http://tfd.sourceforge.net/ DiscreteTFDs — software for computing time-frequency distributions] * [http://tftb.nongnu.org/ TFTB — Time-Frequency ToolBox] Wikimedia Foundation. 2010. Look at other dictionaries: Time-frequency analysis — is a body of techniques for characterizing and manipulating signals whose component frequencies vary in time, such as transient signals.Whereas the technique of the Fourier transform can be used to obtain the frequency spectrum of a signal whose… … Wikipedia Bilinear time–frequency distribution — Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub field field of signal analysis and signal processing called time–frequency signal processing, and, in the statistical analysis of time series data.… … Wikipedia Time series — Time series: random data plus trend, with best fit line and different smoothings In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at … Wikipedia Frequency domain — In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time. Speaking non technically, a time … Wikipedia Representation theory of finite groups — In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of… … Wikipedia Time from NPL — Map showing the location of the Anthorn VLF transmitter within Cumbria … Wikipedia Frequency spectrum — Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. A source of light can have many colors mixed together and in different amounts (intensities). A rainbow, or… … Wikipedia Frequency comb — A frequency comb is the graphic representation of the spectrum of a mode locked laser. An octave spanning comb can be used for mapping radio frequencies into the optical frequency range or it can be used to steer a piezoelectric mirror within a… … Wikipedia Time to digital converter — In electronic instrumentation and signal processing, a time to digital converter (abbreviated TDC) is a device for converting a signal of sporadic pulses into a digital representation of their time indices. In other words, a TDC outputs the time… … Wikipedia Time — This article is about the measurement. For the magazine, see Time (magazine). For other uses, see Time (disambiguation). The flow of sand in an hourglass can be used to keep track of elapsed time. It also concretely represents the present as… … Wikipedia

Integrability of the Pairing Hamiltonian Departamento de Física Comisión Nacional de Energía Atómica (CNEA), Ave. del Libertador 8250, 1429 Buenos Aires, Argentina. Centro Brasileiro de Pesquisas Físicas Rua Xavier Sigaud 150, CEP 22290-180, RJ, Rio de Janeiro, Brazil PACS: 21.69n 21.60Jz 74.20Fg 05.45+b 03.65-w 46.90+s Keywords: Pairing Interaction, Integrability, Time dependent Hartree Fock, Constants of the motion, Poincare section. We show that a many-body Hamiltonian that corresponds to a system of fermions interacting through a pairing force is an integrable problem , i.e. it has as many constants of the motion as degrees of freedom. At the classical level this implies that the Time-dependent Hartree-Fock- Bogoliubov dynamics is integrable and at the quantum level that there are conserved operators of two-body character which reduce to the number operators when the pairing strength vanishes. We display these operators explicitly and study in detail the three-level example. Two kinds of simple models are commonly used in nuclear physics for displaying the essential properties of the nuclear interaction, the particle-hole and the particle-particle models. The simplest one, the Lipkin model, consists of two levels of equal degeneracy and fermions interacting through a particle-hole force. It has been extensively used to test various approximation schemes and its classical version, provided by the Time-Dependent Hartree -Fock (TDHF) approach, is integrable. When this model is extended to three or more levels [2, 3, 4] one finds that the TDHF approximation yields a classical problem which is non integrable, displays various degrees of chaotic behavior and has an intricate and interesting phase-space structure . In principle the analogous situation for the particle-particle interaction can be thought to behave in a similar way. A two-level model with a pairing force is integrable [6, 7] and one would expect that the extension to three or more levels would yield non-integrable TDHF-Bogoliubov (TDHFB) dynamics. In this paper we report the fact that this is not so and that the problem of a pairing force acting in a restricted shell model space with single -particle levels turns out to be integrable both classically and quantum mechanically. We display explicitly the constants of the motion involved and we study their properties, their group structure and their classical limits. The outline of the paper is as follows. In Section 2 we review the pairing model, in both its quantum and classical aspects. The classical limit is obtained by a large scale degeneracy argument that leads to the TDHFB equations of motion. Section 3 is devoted to the search for new constants of the motion, i.e. new operators commuting with the Hamiltonian. These constants of the motion, which are not unique, turn out to be non trivial two body operators involving the coupling constant and the single particle energies. A set of commuting operators is thus constructed that renders the problem integrable. In Section 4 we treat the three- level case and show explicitly the consequences of this integrability both at the quantum and classical levels. The last Section is devoted to conclusions and final remarks. 2 The model The pairing force is a very general interacting mechanism that has an ubiquitous role in the quantum many body problem. In electron systems it leads to the superconducting mechanism and in nuclear physics to the collectivity associated to the pairing degree of freedom. In this latter case it provides a simplified description of the short range part of the nuclear interaction . A schematic model that incorporates this basic mechanism can be defined by interacting fermions that can occupy different single-particle shells of degeneracy and single-particle energies . The fermions interact via a monopole pairing force. The Hamiltonian of such a system is The are the usual fermion operators obeying anticommutation relations which create or annihilate a fermion on the i-th shell which has degeneracy . The operators count pairs of particles in each shell by . The operators conform an algebra whose Casimir operator is The full dynamics of the system occurs in the group space of . The classical limit of the model is obtained from the TDHFB approximation when the degeneracy of each level goes to infinity . is the semiclassical parameter analogous to in the usual semiclassical treatments. One way of obtaining this limit is through the time dependent variational principle implemented through coherent states that are constructed from the vacuum (or minimal weight) state , characterized by The coherent state in this representation is The equations of motion obtained through the time-dependent variational principle with this state are equivalent to the TDHFB equations. To obtain them we use the variational principle appropriate for non-normalized states with an action defined as (we set ) A detailed derivation is given in . The variables are not canonical but the transformation yields canonical variables satisfying . The finite range of these variables is a consequence of the Pauli exclusion principle between correlated fermion pairs. In terms of the variational equations become ordinary hamiltonian equations in complex form, where is the classical hamiltonian associated to the problem. In the same way for any quantum operator its classical limit in this representation is, Therefore the operators and have their classical analogues, , which can be written in terms of the variables This last set of operators obey the classical Poisson bracket relations . The classical pairing hamiltonian can then be expressed, in analogy with the quantum one, directly in terms of the generators, Energy conservation is guaranteed by the time-dependent variational principle so that the motion occurs in the -dimensional manifold defined by . There is a further constant of the motion linked to the conservation of the total number of pairs The TDHFB equations are then classical hamiltonian equations in a phase space of variables. The conservation of and implies that the case is classically integrable, a fact that was exploited in to compute energy levels and transition matrix elements semiclassically. 3 Search for additional constants of the motion The proof of the integrability for requires the existence of independent, well defined , global functions (constants of the motion), whose Poisson brackets with each other and with the hamiltonian vanish. Following Hietarinta a quantum mechanical hamiltonian of degrees of freedom is defined to be quantum integrable if there are independent, well defined, global operators (quantum invariants) which commute with each other and with the hamiltonian. Then the energy spectrum of a quantum integrable hamiltonian system is naturally labeled by quantum numbers , which are the eigenvalues of the corresponding quantum invariants. Likewise the stationary states are simultaneous eigenfunctions of the corresponding operators. For the cases and the integrability is trivial. and (or ) provide the commuting operators and and (or ) the corresponding classical conserved quantities. But in the case of no new obvious quantum invariants are present and we could expect the system to be non integrable and therefore display generically regions of chaotic behavior. However we now show how to construct non trivial operators, independent of the hamiltonian and the total number of pairs and commuting with them, which make the problem integrable. For this purpose let us construct the more general two-body operator which is hermitian and conserves the total number of pairs with and , and require it to commute with the hamiltonian The following relations among its coefficients are sufficient for the commutativity but they do not determine completely the coefficients. Thus several solutions can be found, and one should further check that the operator constructed is independent of and . For example if we choose we then would have is the Casimir of the algebra in the ith-level. In this case is not useful because it is not independent of the conserved magnitudes that we already know. However another choice leads to a set of operators In the non interacting limit () these operators become the natural set of commuting operators . A straightforward but tedious calculation shows that for any value of and . On the other hand, it is easy to see that and can be written in terms of the as Therefore we have constructed a set of commuting operators which also commute with the pairing hamiltonian and are number conserving. They extend to the fully interacting case the trivial properties of the number operators of the system. Consideration of these operators then demonstrate the integrability of the quantum problem. The simultaneous eigenvalue equations gives the eigenvalues of the hamiltonian as It should be noticed however that the actual solution is by no means simplified by this knowledge. The operators are two body operators as complicated in principle as the hamiltonian itself and they cannot be used (except by diagonalizing them) to separate the hamiltonian into invariant subspaces. however, as we will see, the simple fact of their existence has very drastic implications on the structure of both eigenvalues and eigenfunctions. In the classical limit the associated operators for the set written in terms of the classical operators are Taking into account (1), (15), (25) and (31) we can see that the classical operators , and have the same structure (in terms of the algebra generators ) as their quantum analogs, except for additive constants. It is then easy to see that Analogously to the quantum case, the mean number of pairs and the energy are The functions are surfaces on the dimensional phase space. The trajectories lie in the intersection of these surfaces, which are dimensional tori labeled by the constants . Chaotic motion therefore cannot occur in this system. 4 Manifestations of integrability in the three-level case In this section we restrict our analysis to the three-level case () and equal degeneracy and show the consequences of the integrable behavior, both in the quantum and in the classical solutions. As in the previous section we will start with the quantum treatment. Although the problem has three degrees of freedom we have already seen that the total number of pairs in the system is a conserved magnitude. We can then use this fact to reduce explicitly the dimensionality to two degrees of freedom and therefore we only need to display another quantum invariant operator to have an integrable quantum problem. We then choose it as which is a linear combination of the operators defined in the previous section, The quantum integrability shows up clearly when we display a grid of the simultaneous eigenvalues of and . This is done in Fig. 1 for the case where the total number of pairs is , which corresponds to a set of levels. We find that the eigenvalues lie on a regular grid that includes all eigenvalues. The fact that this grid is not parallel and equally spaced reflects the fact that and are not action variables that quantize at integer spaced values. Of course a point transformation to a set and exists if and are independent. But we do not construct this variables explicitly. It is clear however that the grid we obtain is a smooth deformation of a regular one. This proves that the two operators ( and ) are independent and that their simultaneous eigenvalues form a set of good quantum numbers. For the classical analysis we introduce the non-interacting action-angle coordinates, where is the mean number of pairs in the level and is now a continuous classical variable. In these variables the classical hamiltonian is, where we have taken as the energy reference. The analysis is best performed by explicit elimination of the conserved quantity . We introduce the canonical transformation, We have scaled the variables so that they are all in the same range and adopted the mid-shell value, i.e. . The problem is reduced to a space of two degrees of freedom whose effective hamiltonian is, In the above coordinates the classical version of is It can be explicitly verified that . We have also tested in the numerical integration of the equations of motion that both and are constants; and use this fact to control the numerical accuracy of the solutions. Integrability is also apparent in Poincaré sections, as Fig. 2 shows. Only separatrices (no chaotic layers) are observed for any value of the coupling constant. The salient feature of this figure is the appearance of forbidden regions, quite a common occurrence in spin systems. Our analysis shows that the simple pairing force is integrable both at the quantum and at the classical levels. The reason why this simple feature has escaped attention can be found in the fact that the pairing force has been used almost exclusively in the context of quantum many-body physics and for energies close to the ground state. Thus there are no significant consequences for ground state or RPA modes, which are in any case integrable. However the structure of the highly excited states and eigenfunctions should show this consequence. For example, the fluctuation properties of the eigenstates will follow a Poisson rather than a GOE statistics , while eigenfuntions will show the traces of conserved quantities and will not be ergodic. There are very few models of interacting particles that are integrable and the fact that this simple model belongs to this class comes as a surprise. We have reported some consequences in the TDHFB dynamics and on the spectrum. Other features should follow naturally, for example the statistics of levels should be very different than the particle-hole case. Also the addition of a small perturbation to this hamiltonian should follow the KAM perturbation scheme and not a fluctuating spectrum characteristic of the perturbation of chaotic systems. From the technical side it is quite useful to have at one’s disposal a model which takes into account important features of the nucleon-nucleon interaction and which remains integrable for arbitrary values of its strength. Thus perturbation expansions can have a steadying point which need not be the non interacting fermion system. We acknowledge stimulating discussions with Prof. O. Bohigas. This work was supported in part by CONICET Pid 3233/92. A.M.F.R. gratefully acknowledges scholarships from Comisión Nacional de Energía Atómica (Argentina) and from CLAF-CNPq (Brasil). - H.J. Lipkin, N. Meshkov and A.J. Glick, Nucl. Phys. 62 (1965) 188,199,211. - R.D. Williams and S.E. Koonin, Nucl.Phys.A391 ,72 (1982) - D.C. Meredith, S.E. Koonin, and M.R. Zirnbauer, Phys. Rev. A37, 3499 (198 8) - P. Lebœuf and M. Saraceno, Phys. Rev. A41, 4614 (1990) - J. Högaasen-Feldman, Nucl. Phys 28 (1961) 258. - M.C. Cambiaggio, G.G. Dussel and M. Saraceno, Nucl. Phys A415 (1984) 70-92. - R. Broglia and B. Sorensen, Nucl. Phys A110 (1968) 241. - A. Bohr and B.R. Mottelson, Nuclear Structure Vol I, (Benjamin New York 1969) - P. Kramer, M. Saraceno , Geometry of the time dependent variational principle in quantum mechanics, Lecture notes in physics, vol 140 (Springer,Berlin 1981). - A. Perelomov, Generalized coherent states and applications (Springer Berlin 1986). - A. K. Kerman and S.E. Koonin, Ann of Phys. 100 (1976) 332. - J. Hietarinta, J. Math Phys 25 (1984) 1833. - N. Srivastava and G. Müller, Z. Phys. B 81 (1990) 137. - O. Bohigas, in Chaos and Quantum Physics, Eds. M.-J. Giannoni, A. Voros and J. Zinn-Justin, (Elsevier Science, New York, 1990). - F. Calogero, J. Math Phys 12 (1971) 419; M.A. Olshanetsky and A.M. Perelomov Phys Rep 94 (1983) 313. - V.I. Arnol’d, Russ. Math. Survey 18 (1963) 9, 85. FIG. 1. Simultaneous eigenvalues of the Hamiltonian () and for and . The variables have been scaled dividing them by . FIG. 2. Poincaré surface of section for the three-level pairing hamiltonian . The mid-shell value has been taken and the section is performed in the case (see text).

عنوان مقاله [English] Petrological and geological studies are of great importance in natural resources management. The fractal technique is used as an instrument to achieve accurate results in a shorter time. Geological maps are very useful in natural resources management, industry, especially refineries, and mine exploration. Due to the large scale of available geological maps, small scale geological maps should be provided in details. The fractal dimension, as a measure of surface roughness over a variety of scales, can be used to model the dissipation of erosive products due to climatic elements and fluvial transport. Nowadays, by using new fractal technique and terrestrial survey, more accurate results on geological units can be obtained in a shorter time. This research aims to compare the performance of two techniques of quantitative parameters non-dimensionalization in geomorphology such as drainage network density index, and fractal dimension, to separate geological units in the Taft watershed, Yazd province. Materials and Methods Taft watershed, as the study area, located in the Yazd province, that is situated between 53° 43' 38'' to 54° 14' 54'' E. longitude and 31° 33' 22'' to 31° 49' 06'' N. latitude. There is high diversity of geological and lithological units, including gd (Granodiorite), K^(t-1) (Taft lime), and K^S (conglomerate and sandstone) in this watershed. Three geological units selected in the study area. In each geological unit, , 10 plots of 2 km×2 km (samples), and 10 plots of 2 km×2 km (tests) were selected respectively inside and outside of the study area for analysis. To identify and distinguish three studied geological units, drainage network was drawn in each geological units through geological map of the Iranian Geological Survey and satellite images of the Google Earth and field observation. Afterwards, using Fractalyse and ArcGIS softwares, their fractal dimension and density were calculated. Output results of the Fractalyse software is some numbers that one of them indicates the fractal dimension of those lines. Fractal dimension number is between one and two. The area and network length of each plot were calculated by ArcGIS 10.2 software. Then, the drainage network density of each plot was calculated by equation 1. Drainage Network Density= Drainage Network Length (km)/ Plot area (km2) (1) Efficiency of the two dimensionless indices of drainage network density and fractal in separating geological units were compared by two methods: In each geological unit, the calculated numbers of the samples and tests should be averaged individually. Then, equation 2 is used to calculate the validation of each geological units. Validity= sample/test (2) Sample: Average fractal dimension of drain networks for sample plots. Test: The mean fractal dimension of drainage networks for test plots. Sample: Average density of drainage networks for sample plots. Test: The average density of the drainage networks for test plots. B) Comparing the sample and test by using QQ diagram, the line equation, the coefficient of determination and the angel of deviation: In drainage network fractal dimension, QQ diagram is plotted between samples and tests' fractal dimension. Line equation, coefficient of determination and angel of deviation were calculated. Moreover, the QQ graphs were plotted and calculations carried out on the drainage network density. Results and discussion Results of the first comparing method (validation) in both techniques are very good and similar. In second comparing method (i.e. QQ graph, angle of deviation, and coefficient of determination), coefficient of determination in the drainage network density in K^S, gd and K^(t-1)geological units were 0.99, 0.93 and 0.94, respectively, which are more than drainage network fractal dimension. The standard deviation in drainage network fractal dimension in K^S and gd and K^(t-1)geological units are +17.05, - 1.48, and +8.37 respectively, these values in the drainage network density are much lower (i.e. better). The angle of deviation in drainage network density of K^S, gd and K^(t-1) geological units are + 0.21, -2.4, and +0.22, respectively. According to the results, the drainage network density index is better than the drainage network fractal dimension in identifying and separating of the studied geology and lithological units of K^S, gd and K^(t-1) in the region. The results of the accuracy assessment of both techniques are very good and similar to each other. Therefore, in this comparison, both techniques of drainage network density index and drainage network fractal dimension have high efficiency in identifying and separating of the geological studies unit. However, The drainage network density technique is the best technique of quantitative parameters non-dimensionalization in geomorphological studies in identifying and separating of geological units in the Taft watershed, Yazd. Keywords: QQ Diagram, Fractal Technique, Drainage Network Density, Yazd.

- How many electrons can 3p hold? - How many electrons are in each shell? - What element has most electrons? - What values of L are possible for n 3? - How many 2p electrons are in neon? - Which element has 3 electrons in the n 4 energy level? - How many orbitals are there in n 3? - How many p orbitals are there in n 3? - What is the only metalloid in Period 3? - How many electrons are in a 2p sublevel? - What element has a 4s sublevel with 2 electrons? - Are zn2+ and Ni Isoelectronic? - What is 1s 2s 2p 3s 3p? - Does the 3p sublevel have 3 electrons? - What is the L quantum number? - Which element has the electron configuration of 1s 2 2s 2 2p 6 3s 2 3p 4? - Why is 3d higher energy than 4s? - How many electrons can n 3 hold? - How many p orbitals are there in N 1? - What element has 3 energy levels and 7 valence electrons? - Which atom has three 2p electrons in its ground state? How many electrons can 3p hold? six electronsThe 2p, 3p, 4p, etc., can each hold six electrons because they each have three orbitals, that can hold two electrons each (3*2=6). The 3d, 4d etc., can each hold ten electrons, because they each have five orbitals, and each orbital can hold two electrons (5*2=10). First, we look at the n=1 shell (the first shell).. How many electrons are in each shell? Each shell can contain only a fixed number of electrons: The first shell can hold up to two electrons, the second shell can hold up to eight (2 + 6) electrons, the third shell can hold up to 18 (2 + 6 + 10) and so on. The general formula is that the nth shell can in principle hold up to 2(n2) electrons. What element has most electrons? HydrogenElectronsAtomic NumberElementEnergy Levels or “shells”M1Hydrogen (H)2Helium (He)3Lithium (Li)37 more rows What values of L are possible for n 3? Because n=3, the possible values of l = 0, 1, 2, which indicates the shapes of each subshell. How many 2p electrons are in neon? six electronsNeon is the tenth element with a total of 10 electrons. In writing the electron configuration for neon the first two electrons will go in the 1s orbital. Since 1s can only hold two electrons the next 2 electrons for Ne go in the 2s orbital. The remaining six electrons will go in the 2p orbital. Which element has 3 electrons in the n 4 energy level? Lithium has 3 electrons; 2 of the 3 electrons occupy the s sublevel in principal energy level 1. The 3rd electron must go in the next available sublevel, 2s. How many orbitals are there in n 3? nine orbitalsThere are nine orbitals in the n = 3 shell. There is one orbital in the 3s subshell and three orbitals in the 3p subshell. The n = 3 shell, however, also includes 3d orbitals. The five different orientations of orbitals in the 3d subshell are shown in the figure below. How many p orbitals are there in n 3? Orbitals and Electron Capacity of the First Four Principle Energy LevelsPrinciple energy level (n)Type of sublevelMaximum number of electrons (2n2)2p83s18p7 more rows What is the only metalloid in Period 3? SiliconOne metalloid of period 3 is Silicon. How many electrons are in a 2p sublevel? 6 electronsThere can be two electrons in one orbital maximum. The s sublevel has just one orbital, so can contain 2 electrons max. The p sublevel has 3 orbitals, so can contain 6 electrons max. The d sublevel has 5 orbitals, so can contain 10 electrons max. What element has a 4s sublevel with 2 electrons? calciumEven though the 4s sublevel is filled, the last electron went into that sublevel, making it a member of the s-block. It has 2 valence electrons. The element is calcium, a metal. Are zn2+ and Ni Isoelectronic? So, sure, they’re isoelectronic, but not because of having the same electronic structure; instead, it’s purely due to how many electrons they have, not where they are. … They have the same number of valence electrons, but they aren’t identical compounds. What is 1s 2s 2p 3s 3p? Physicists and chemists use a standard notation to indicate the electron configurations of atoms and molecules. For atoms, the notation consists of a sequence of atomic subshell labels (e.g. for phosphorus the sequence 1s, 2s, 2p, 3s, 3p) with the number of electrons assigned to each subshell placed as a superscript. Does the 3p sublevel have 3 electrons? phosphorus has three electrons in its 3p sub level. neon has its highest energy level completely filled. What is the L quantum number? RulesNameSymbolValue examplesPrincipal quantum numbernn = 1, 2, 3, …Azimuthal quantum number (angular momentum)ℓfor n = 3: ℓ = 0, 1, 2 (s, p, d)Magnetic quantum number (projection of angular momentum)mℓfor ℓ = 2: mℓ = −2, −1, 0, 1, 2Spin quantum numbermsfor an electron s = 12, so ms = −12, +12 Which element has the electron configuration of 1s 2 2s 2 2p 6 3s 2 3p 4? Electron ConfigurationsABSodium1s2 2s2 2p6 3s1Magnesium1s2 2s2 2p6 3s2Aluminum1s2 2s2 2p6 3s2 3p1Sulfur1s2 2s2 2p6 3s2 3p416 more rows Why is 3d higher energy than 4s? We say that the 4s orbitals have a lower energy than the 3d, and so the 4s orbitals are filled first. … The electrons lost first will come from the highest energy level, furthest from the influence of the nucleus. So the 4s orbital must have a higher energy than the 3d orbitals. How many electrons can n 3 hold? Questions and AnswersEnergy Level (Principal Quantum Number)Shell LetterElectron Capacity2L83M184N325O502 more rows How many p orbitals are there in N 1? There are n2 orbitals for each energy level. For n = 1, there is 12 or one orbital. For n = 2, there are 22 or four orbitals. For n = 3 there are nine orbitals, for n = 4 there are 16 orbitals, for n = 5 there are 52 = 25 orbitals, and so on. What element has 3 energy levels and 7 valence electrons? ElementElement NumberNumber of Electrons in each LevelBeryllium42Boron53Carbon64Nitrogen7551 more rows Which atom has three 2p electrons in its ground state? nitrogenSimilarly, for nitrogen in its ground state, Hund’s rule requires that the three 2p electrons singly occupy each of the three 2p orbitals. This is the only way that all three electrons can have the same spin.

Work supported in part by US Department of Energy cont Download Pdf - The PPT/PDF document "Work supported in part by US Department ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement. Presentation on theme: "Work supported in part by US Department of Energy cont"— Presentation transcript: Page 1 Work supported in part by US Department of Energy contract DE-AC02-76SF00515 Measurement of the Decelerating Wake in a Plasm a Wakefield Accelerator I. Blumenfeld , C.E. Clayton , F.J. Decker , M.J. Hogan , C. Huang , R. Ischebeck , R.H. Iverson , C. Joshi , T. Katsouleas , N. Kirby , W. Lu K.A. Marsh , W.B. Mori , P. Muggli , E. Oz , R.H. Siemann , D.R. Walz and M. Zhou Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, USA University of California, Los Angeles, California 90095, USA University of Southern California, Los Angeles, California 90089, USA Abstract. Recent experiments at SLAC have shown that high gradient acceleration of electrons is achievable in meter scale plasmas. Results from these experiments show that the wakefield is sen- sitive to parameters in the electron beam which drives it. In the experiment the bunch lengths were varied systematically at constant charge. The effort to extract a measurement of the decelerating wake from the maximum energy loss of the electron beam is discussed. Keywords: PWFA, decelerating wake measurement PACS: 52.40.Mj, 52.70.Nc, 41.75.Lx, 29.20.Ej INTRODUCTION Plasma wakefield accelerators (PWFAs) have shown great potential as a mechanism for acceleration of particles to high energy. Experiments have demonstrated both high gradients and long propagation distances[1, 2]. However, for a future collider design a deeper understanding of the experimental relationship between the wake and the electron beam is necessary. EXPERIMENTAL SETUP In the SLAC experiment E167, a 42 GeV beam with 1 10 10 electrons, 60 and was injected into a confined neutral lithium vapor of density 10 17 per cm with a length of 85cm, full width at half max (FWHM). The en- ergy spectrum of the incoming beam was measured by recording the incoherent x-ray synchrotron radiation emitted by the beam in a soft vertical chicane placed in a region of a high horizontal dispersion. This energy spectrum could then be matched in the 2D tracking code LiTrack to recover the temporal profile of the beam[1, 4]. Bunch lengths were found to be 10 50 . With a transverse spot size of 10 this beam was dense enough to ionize the lithium and drive a non-linear wake[1, 6]. The initial energy spread had a full width of 4%. The measureable effect of the interaction between the beam and the plasma is the large broadening of the energy spectrum as the bunch length and the plasma wavelength SLAC-PUB-13390 September 2008 Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA Page 2 Energy 50 100 150 200 250 300 350 20 40 60 80 50 100 150 200 250 300 350 x 10 pixel # counts a) b) Region 1 Region Region 3 Region 4 FIGURE 1. n example of a) and image, and b) derived energy profile, from the energy spectrometer diagnostic. Four distinct regions can be seen. Region 1 contains the accelerated charge, region 2 the beam head which has too low a current to ionize, region 3 the beam core driving the wake, and region 4 the lowest energy charge. Region 4 exhibits a peak followed by a lower energy rolloff. are on the same order, leading to the beam sampling all phases of the wake. The particle energies were measured after exiting the plasma using a magnetic spectrometer and Cerenkov radiation. The electrons were dispersed in energy by a dipole magnet centered 2.18m from the plasma exit with 12 05 kG . The electrons then passed through an air gap where their Cerenkov radiation was imaged at two locations from the magnet center, 0.85m, for low energies, and 1.85m, for high energies. Figure 1 shows an example of the beam as it appears after the energy spectrometer at the first imaging location. The entire beam can be seen on the image with four distinct regions. Region 1, where the accelerated charge sits, and Region 4, where the maximum decelerated charge is, are of most interest. ENERGY MEASUREMENTS Measurement of the accelerating gradient in a PWFA driven by short electron bunches is already well established. The essential problem is identifying the highest energy electrons on the diagnostic images with confidence. There are several strategies, which vary experiment to experiment, for achieving this, from charge contours to two screen methods[2, 7]. The remaining open question, to be addressed here, is how to relate a low energy measurement to the maximum decelerating wake. In the profile shown in Fig. 1 there is a peak in Region 4 where the lowest energy charge sits, followed by a long rolloff of lower energy particles. Understanding this feature will be the key to translating a particle energy measurement into a wake measurement. In reality the raw image is a convolution of various beam attributes, some due to the nature of the incoming beam and others due to effects in the plasma. The three effects Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA Page 3 that dominate the interpretation of a low energy measurement are the y-spot size, the peak in the wakefield and radiative losses due to oscillation of beam particles in the plasma. The camera resolution is not large enough to give a contribution. These three effects all contribute to the profile that appears when the image is summed. The strategy for estimating how these affect the measurement will be to reconstruct a theoretical profile, convolving all these effects with some reasonable assumptions, and look at its qualitative and quantitative attributes. The calculated function is d vs. , where N is the number of particles and y is the dispersion direction. This reconstruction will give a guide as to where to choose the low energy point and how to translate it to the wake. Spot Size In the absence of dispersion, the electron beam has a finite spot size at the diagnostic location in the dispersion direction y. This contribution to the measured profile is pot ps (1) where is the beam spot size in y. As the beam is dispersed in y, cannot be directly measured. As an upper limit for this effect, it is reasonable to use the beam spot in x. Since the y-emittance is an order of magnitude smaller, the y-spot is likely smaller. Measured x spot sizes were mm RMS width. Peak in the Wakefield The shape of the wakefield has an effect on the measurement as at the peak deceler- ating field , d 0, where ct is the beam coordinate. In order to quantify this effect the shape of the decelerating field is taken to be a parabola around this point with no variation as the beam propagates, (2) where A defines the sharpness of the field peak. The energy of a beam particle interacting with this field is GL , where is the original energy and L is the interaction length. Using this expression, and inverting from above, it is possible to derive the number of particles per energy slice due to the shape of the wake, L AL 3) where and d is the longitudinal density of the electron beam which is assumed to be constant around the wakefield peak. Then, using the expression for the dispersion 4) Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA Page 4 where s the dispersion and is the position of a particle of infinite energy. The contribution to d can be calculated, assuming the energy spread imprinted by the plasma is much larger than the initial spread, yielding akepeak 5) This relationship shows that the energy spectrum peaks around the minimum energy. Radiative Losses Through Propagation in the Plasma The energy radiated off as a particle oscillates in an ion-column is 12 (6) where is the particle energy, s the maximum displacement from the propagation axis as the particle oscillates, is the relativistic factor of the particle, and is the plasma wavenumber, a relationship which has been experimentally verified[9, 10]. This expression is for a single particle and dependent only on the maximum radial displacement from the propagation axis , and not on or independantly. To extend this to a beam distribution the quantity d is needed. Using appropriate transformations of the phase space density for a symmetric beam matched into the ion column and integrating over all angles, (7) Using this, it is now possible to determine the effect of radiative losses on the deceler- ating wake measurement. First, from Eq. 6 an upper limit can be set on the total energy lost to radiation while propagating in the plasma. The relativistic factor of the beam and the plasma density are set as constant. Since the power radiated is proportional to , the actual loss will be less as the particles are also losing energy to the wake as they propagate. The propagation distance is taken to be 90cm as, while the FWHM of the confined plasma was 85cm, deceleration continues past the half density point. This yields the maximum energy lost to radiation, , as 12 (8) In the absence of radiation each longitudinal slice will be at the same energy, as the longitudinal wake has no transverse variation for a symmetric drive beam. Therefore each longitudinal slice has an expression analagous to Eq. 7, which can be used in conjunction with the derivative of Eq. 8 to obtain long long 12 long c kp (9) Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA Page 5 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.2 0.4 0.6 0.8 1.2 dispersion (m) Intensity (arb) 30 GeV maximum decelerating wake Corresponding reconstructed image FIGURE 2. reconstructed profile convolving the dominant effects. The solid red curve shows how a single particle with initial energy of 42 GeV sitting in a 30 GeV decelerating wake would be measured on the diagnostic due to the 2 pixel resolution limit of the optical system. The dotted black curve shows how this maximum decelerating wake actually appears taking into account that it is sampled by a beam with finite transverse and longitudinal size. where long is the number of particles in each longitidinal beam slice. Inverting Eq. 8 yields long long ae (10) where 12 mc kp . This shows that each longitudinal slice develops a tail of lower energy particles as the particles away from the axis of propagation radiate. To calculate the effect on the energy spectrum, the departure from the nominal energy of each longitudinal slice by an off-axis particle, , is taken to be small. Its position can then be calculated through a linearization of the dispersion expression, Eq. 4, 11) where is the position of the measured particle and . Then, utilizing Eq. 10, ad ong ye (12) RESULTS AND CONCLUSIONS The separate contributions of the three dominant effects to a low energy measurement have been calculated. Now these effects must be convolved. Figure 2 shows an example of this convolution, where the maximum decelerating gradient was taken to be 30 GeV/m. The peak of the solid red curve shows how a particle influenced by only this field would appear on the diagnostic, while the dotted black curve shows what actually will be measured. The peak of the dotted black curve is therefore the measurement that should be identified with the wake and corresponds to the peak in Region 4 of Fig. 1. Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA Page 6 The peak in the dotted black curve is shifted slightly off that in the solid red curve. This shift can be corrected for by making a series of such reconstructions for all possible maximum decelerating wakes. This allows a one-to-one mapping from the peak in the low energy part of the measured profile to the real decelerating wake. The results of this calculation show that the corrections to the effective gradient, calculated using the plasma length and energy at this peak, are small, on the order of 1-2%. Thus, in order to map the minimum particle energy to the maximum decelerating wake, the location of the peak in the low energy part of the spectrum is the desired measurement. There is a small correction that can be made to account for the shifting of this peak. The particles in the long tail have lower energy due to radiative effects and not the wake by itself, and therefore should be ignored. Using this method, decelerating gradients of 20 35 GeV were measured. These will be combined with accelerating gradient measurements to yield an experimental limit on the transformer ratio. ACKNOWLEDGMENTS This work was supported by US Department of Energy contracts DE-AC02-76SF00515, DE-FG02-93ER40745, DE-FG03-92ER40727, DE-FG52-06NA26195, DE-FC02- 07ER41500, DE-FG02-03ER54721, DE-FG02-92ER40727, and NSF grant NSF-Phy- 0321345. The authors also would like to thank Melissa Berry for her assistance with the energy spectrometer design. REFERENCES 1. M. J. Hogan, et al., Phys. Rev. Let. 95 , 054802 (2005). 2. I. Blumenfeld, et al., Nature 445 , 741–744 (2007). 3. K. Bane, SLAC-PUB-11035, Tech. rep., Stanford Linear Accelerator Center (2005). 4. C. Barnes, Longitudinal Phase Space Measurements and Application to Beam Plasma Physics , Ph.D. thesis, Stanford University, Stanford, CA 94305 (2006), available as SLAC Report 799. 5. I. Blumenfeld, et al., to be submitted (2008). 6. C. L. O’Connell, et al., PRSTAB , 101301 (2006). 7. R. Ischebeck, et al., “Energy Measurement in a Plasma Wakefield Accelerator,” in PAC’07 , 2007, p. 4168. 8. E. Esarey, et al., Phys. Rev. E 65 , 056505 (2002). 9. S. Wang, et al., Phys. Rev. Let. 88 , 135004 (2002). 10. D. Johnson, et al., Phys. Rev. Let. 97 , 175003 (2006). 11. K. A. Marsh, et al., “Beam Matching to a Plasma Wake Field Accelerator using a Ramped Density Profile at the Plasma Boundary,” in PAC’05 , 2005, p. 2702. 12. J. B. Rosenzweig, et al., Phys. Rev. A 44 , 6189 (1991). Presented at the Advanced Accelerator Concepts Workshop 2008, 7/27/2008 to 8/2/2008, Santa Cruz, CA, USA

If dense matrices are to be handled in connection with solving systems of linear algebraic equations by gaussian elimination, then pivoting either partial pivoting or complete pivoting is used in an attempt to preserve the numerical stability of the computational process see golub and van. Gauss elimination method matlab program code with c. C program for gauss elimination method code with c. After outlining the method, we will give some examples. The task below is a case in which partial pivoting is required. Gaussian elimination is usually carried out using matrices. As i have mentioned above, there are several methods to solve a system of equations using matrix analysis. Elimination process begins, compute the factor a 2 1 pivot 3. A symmetric positive definite system should be solved by computing its cholesky factor algorithm 3. Though the method of solution is based on additionelimination, trying to do actual addition tends to get very messy, so there is a systematized method for solving the threeormorevariables systems. For a large system which can be solved by gauss elimination see engineering example 1 on page 62. Hello every body, i am trying to solve an nxn system equations by gaussian elimination method using matlab, for example the system below. The c program for gauss elimination method reduces the system to an upper triangular matrix from which the unknowns are derived by the use of backward substitution method. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. Grcar g aussian elimination is universallyknown as the method for solving simultaneous linear equations. Gaussian elimination technique by matlab matlab answers. Now, lets analyze numerically the above program code of gauss elimination in matlab using the same system of linear equations. The entries a ik which are \eliminated and become zero are used to store and save. Origins method illustrated in chapter eight of a chinese text, the nine chapters on the mathematical art,thatwas written roughly two thousand years ago. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to. How to solve linear systems using gaussian elimination. Gaussian elimination lu factorization qr factorization wz factorization 2 iterative methods generate sequence of approximations that converge in the limit to the solution jacobi iteration gaussseidal iteration sor method successive overrelaxation vasilije perovi. How to use gaussian elimination to solve systems of equations. A system of linear equations and the resulting matrix are shown. And one of these methods is the gaussian elimination method. The goals of gaussian elimination are to make the upperleft corner element a 1, use elementary row operations to get 0s in all positions underneath that first 1, get 1s. The following code performs gauss elimination on a given matrix and reduces it to upper triangular matrix in echelon form. For inputs afterwards, you give the rows of the matrix oneby one. Solving linear equations with gaussian elimination martin thoma. We will never get a wrong solution, such that checking nonsingularity by computing the determinant is not required. Gauss elimination and gauss jordan methods using matlab code. Pdf we present a method by which the breakdown of the interval gaussian elimination caused by division of an interval containing zero can be avoided. The gaussian elimination algorithm, modified to include partial pivoting, is for i 1, 2, n1 % iterate over columns. Lecture 08 system of equations gauss elimination, pivoting. Multiplechoice test gaussian elimination simultaneous linear. Rediscovered in europe by isaac newton england and michel rolle france gauss called the method eliminiationem vulgarem common elimination. It moves down the diagonal of the matrix from one pivot row to the next as the iterations go on. Gaussian elimination with partial pivoting youtube. Each equation becomes a row and each variable becomes a column. Gaussian elimination is the most basic n umerical metho d for solving a dense linear system of equations ax b. In this post i am sharing with you, several versions of codes, which essentially perform gauss elimination on a given matrix and reduce the matrix to the echelon form. So, this method is somewhat superior to the gauss jordan method. The basic idea of gaussian elimination is the factorization of a as the product lu of a lower triangular matrix l with ones on its diagonal and an upper triangular matrix u, the diagonal entries of which are called the pivot elements. Ive found a few sources which are saying different things about what is. Gaussianelimination september 7, 2017 1 gaussian elimination this julia notebook allows us to interactively visualize the process of gaussian elimination. The method of practical choice for the linear system problem ax b is gaussian elimination with partial pivoting section 3. Gaussian elimination with pivoting method file exchange. In this question, we use gaussian elimination to solve a system of linear equations using partial pivoting and backwards substitution. There are man y v ariations on ho w to organize the computations, but tak en as a whole gaussian elimination is probably one of the most widely kno wn n umerical algorithms. Gaussian elimination lecture 10 matrix algebra for. Gaussian elimination is probably the best method for solving systems of equations if you dont have a graphing calculator or computer program to help you. I have some trouble with understanding the difference between partial and complete pivoting in gauss elimination. Recall that the process ofgaussian eliminationinvolves subtracting rows to turn a matrix a into an upper triangular matrix u. Except for certain special cases, gaussian elimination is still \state of the art. An additional column is added for the right hand side. Code without partial pivoting and backsubstitution. Though the method of solution is based on addition elimination, trying to do actual addition tends to get very messy, so there is a systematized method for solving the threeormorevariables systems. Multiplechoice test gaussian elimination simultaneous. Using gaussian elimination with pivoting on the matrix produces which implies that therefore the cubic model is figure 10. Gauss elimination method with partial pivoting the. With the gaussseidel method, we use the new values as soon as they are known. F or decades, scien tists ha v e solv ed problems of ev er. A system of linear equations can be placed into matrix form. Giorgio semenza, in studies in computational mathematics, 2006. First of all, i have to pick up the augmented matrix. The previous example will be redone using matrices. This method is called gaussian elimination with the equations ending up. Gaussian elimination revisited consider solving the linear. I solving a matrix equation,which is the same as expressing a given vector as a linear combination of other given vectors, which is the same as solving a system of. Complete pivoting an overview sciencedirect topics. A diagonal b identity c lower triangular d upper triangular. This implementation isnaivebecause it never reorders the rows. The gaussian elimination algorithm, modified to include partial pivoting, is for i 1, 2, n1. Pdf interval gaussian elimination with pivot tightening. Gaussian elimination september 7, 2017 1 gaussian elimination this julia notebook allows us to interactively visualize the process of gaussian elimination. With the gauss seidel method, we use the new values as soon as they are known. When we use substitution to solve an m n system, we. To avoid this problem, pivoting is performed by selecting. Find gaussian elimination course notes, answered questions, and gaussian elimination tutors 247. Ive found a few sources which are saying different things about what is allowed in each pivoting. This additionally gives us an algorithm for rank and therefore for testing linear dependence. So, we are to solve the following system of linear equation by using gauss elimination row reduction method. Solving simultaneous linear equations using lu decomposition. Comparison of numerical efficiencies of gaussian elimination and gaussjordan elimination methods for the solutions of linear simultaneous equations, department of mathematics m. It is the number by which row j is multiplied before adding it to row i, in order to eliminate the unknown x j from the ith equation. Since we normalize with the pivot element, if it is zero, we have a problem. Eliminate the first term in row 2, then move to the next column and gauss it. It will obviously fail if a zero pivot is encountered. In this section we will reconsider the gaussian elimination approach. Gaussian elimination is summarized by the following three steps. If the optional step argument is supplied, only performs step steps of gaussian elimination. The gaussian elimination algorithm with or without scaled partial pivoting will fail for a singular matrix division by zero. The first step is to write the coefficients of the unknowns in a matrix. To solve such systems, there are direct methods and iterative methods n nnn n. Naive gauss elimination in general, the last equation should reduce to. This function solves a linear system axb using the gaussian elimination method with pivoting. For every new column in a gaussian elimination process, we 1st perform a partial pivot to ensure a nonzero value in the diagonal element before zeroing the values below. Given a matrix a, performs gaussian elimination to convert a into an uppertriangular matrix u. Applications of the gauss seidel method example 3 an application to probability figure 10. Chapter 06 gaussian elimination method introduction to. Introduction to numerical analysis for engineers systems of linear equations mathews cramers rule gaussian elimination 3. Apr 30, 2017 in this question, we use gaussian elimination to solve a system of linear equations using partial pivoting and backwards substitution. Pivoting, partial or complete, can be done in gauss elimination method. If the b matrix is a matrix, the result will be the solve function apply to all dimensions. Can i get the matlab gui implementation of gauss elimination. Apr 19, 2020 as i have mentioned above, there are several methods to solve a system of equations using matrix analysis. The gaussseidel method main idea of gaussseidel with the jacobi method, the values of obtained in the th iteration remain unchanged until the entire th iteration has been calculated. Since here i have four equations with four variables, i will use the gaussian elimination method in 4. Gaussian elimination with partial pivoting applies row switching to normal gaussian. View gaussian elimination research papers on academia. Forward elimination an overview sciencedirect topics. This method reduces the effort in finding the solutions by eliminating the need to explicitly write the variables at each step. Gaussian elimination algorithm no pivoting given the matrix equation ax b where a is an n n matrix, the following pseudocode describes an algorithm that will solve for the vector x assuming that none of the a kk values are zero when used for division. Nonsingularity is implicitly verified by a successful execution of the algorithm. The function accept the a matrix and the b vector or matrix. Course hero has thousands of gaussian elimination study resources to help you. The goals of gaussian elimination are to make the upperleft corner element a 1, use elementary row operations to. Gaussian elimination we list the basic steps of gaussian elimination, a method to solve a system of linear equations. Gaussian elimination, also known as row reduction, is an algorithm in linear algebra for solving a system of linear equations. The reduction of a matrix a to its row echelon form may necessitate row interchanges as the example shows. Uses i finding a basis for the span of given vectors. Motivation partial pivoting scaled partial pivoting gaussian elimination with partial pivoting meeting a small pivot element the last example shows how dif. Applications of the gaussseidel method example 3 an application to probability figure 10. The gauss seidel method main idea of gauss seidel with the jacobi method, the values of obtained in the th iteration remain unchanged until the entire th iteration has been calculated. Gauss elimination and gauss jordan methods using matlab.113 564 878 1412 285 191 1022 613 1152 855 1166 1208 616 1070 1531 403 372 989 456 1428 986 1072 392 793 857 408 1290 207 911 1033 518 715 1106 1259 1525 876 701 636 1463 343 1367 460 1350 542 1096

Magnetothermodynamics of BPS baby skyrmions The magnetothermodynamics of skyrmion type matter described by the gauged BPS baby Skyrme model at zero temperature is investigated. We prove that the BPS property of the model is preserved also for boundary conditions corresponding to an asymptotically constant magnetic field. The BPS bound and the corresponding BPS equations saturating the bound are found. Further, we show that one may introduce pressure in the gauged model by a redefinition of the superpotential. Interestingly, this is related to non-extremal type solutions in the so-called fake supersymmetry method. Finally, we compute the equation of state of magnetized BSP baby skyrmions inserted into an external constant magnetic field and under external pressure , i.e., , where is the ”volume” (area) occupied by the skyrmions. We show that the BPS baby skyrmions form a ferromagnetic medium. The Skyrme model skyrme is considered one of the best candidates for an effective model of low energy QCD. Using results from the large expansion, it is known that the proper degrees of freedom in this limit are mesons, while baryons emerge as collective excitations, i.e., solitons called skyrmions, with an identification between baryon number and topological charge. To get phenomenologically precise relations between solitons and baryons (nuclei), one has to perform the standard semiclassical quantization of the spin and isospin degrees of freedom, as well as include the electromagnetic interaction, which obviously contributes to the masses of particles. Fortunately, although the Skyrme model has not yet been derived from the underlying microscopic quantum field theory, its coupling to the electromagnetic field is completely determined by the symmetries and anomalies of QCD wit1 . The resulting gauged Skyrme model is rather difficult to analyse, and the electromagnetic properties of nucleons as well as atomic nuclei, although very important, could not yet be extracted in the full nonlinear Skyrme-Maxwell description. The electric part of the energy of the nuclei is typically approximated by the Coulomb energy Coul , where the back reaction of the Maxwell field on the Skyrme matter field is not taken into account. Let us remark that some first numerical results for the Skyrme model minimally coupled to the electromagnetic field (but without the anomalous or Wess-Zumino-Witten term contribution) have been found in GSk . Further, very recently some knotted soliton solutions have been obtained for the restriction of the minimally gauged Skyrme model i.e., the gauged Faddeev-Skyrme-Niemi model, however within the toroidal ansatz which limits the solutions to the charge sectors shnH . As has been mentioned already, a precise derivation of the Skyrme model (or in fact any effective low energy model) from QCD is one of the most urgent, however still unsolved, tasks in modern theoretical physics. The lack of a systematic derivation means that the precise form of the Skyrme type action is not known. The usual assumption (based on a perturbative approach) restricts the model to three terms: the sigma model part (Dirichlet energy), the Skyrme term (obligatory to avoid the Derrick arguments for the non-existence of static solutions) and a potential (providing a mass for the perturbative pionic excitations). It is, however, one of the main problems of the usual Skyrme model that it leads to unphysical binding energies, which are in strong disagreement with the experimental data. The underlying reason for this is that the usual Skyrme model is not a BPS theory, i.e., the energies of skyrmions are not linearly related to their topological charges. As atomic nuclei seem to be close to BPS objects (the masses are almost linear in the baryon charge with a deviation, at most), the corresponding effective model should be a (near) BPS one. There exist two quite different realizations of this concept. The first proposal is based on the observation that the inclusion of infinitely many vector mesons (Kaluza-Klein modes) can bring the original Skyrme model towards the Yang-Mills action Sut1 , hidden . In the second proposal, the crucial observation is that within all Skyrme type Lagrangians (i.e., with no additional fields) there exists a special one with the BPS property. It has a rather simple form and consists of two mutually balancing terms: a derivative part (the baryon (topological) current squared) and a potential BPS-Sk . Moreover, this model possesses the volume preserving diffeomorphism symmetry, which allows to interpret it as a field theoretical description of the liquid droplet model. In addition, the static energy-momentum tensor of the model is the energy-momentum tensor of a perfect fluid, further strengthening the case for this interpretation. As a consequence, there are infinitely many solitonic solutions saturating a topological bound, which leads to a linear energy - topological charge relation. Therefore, the classical binding energies are zero. Further, finite binding energies have been recently derived by taking into account the semiclassical quantization of the spin-isospin degrees of freedom, the Coulomb interaction as well as the isospin breaking potential. The obtained values are in very good agreement with the nuclear data and the semi-empirical (Weizsäcker) formula, especially for higher nuclei BPS nucl , Marl . This result allows to consider the BPS Skyrme model as a serious candidate for a lowest order approximation of the correct effective model of QCD at low energies, especially for the bulk quantities. In addition to the binding energies, there are many properties of nuclei and nuclear matter which should be understood within the framework of the (near) BPS Skyrme model. It is another advantage of this model that, due to its generalized integrability and BPS nature (solvability), many relevant questions can be answered in an analytical manner. One of the most important ones is related to the thermodynamic and magnetic properties of nuclei and nuclear matter. In particular, an understanding of how BPS skyrmions respond to an external magnetic field and to pressure would provide us with the corresponding equation of state, which is required for the analysis of nuclear matter in various conditions, from heavy nuclei to neutron stars. Unfortunately, even the BPS Skyrme model is quite complicated after the minimal coupling. To overcome the computational difficulties and learn something about the electromagnetic properties of BPS Skyrme type solitons, one can analyze lower-dimensional analogs, as has been done successfully already in many occasions. In fact, there exists a dimensional version of the Skyrme model, usually referred to as the baby Skyrme model, which supports solitonic solutions (baby skyrmions) old -shnir (for the gauged version see GPS , schr1 ). This field theory also possesses its BPS limit, whose Lagrangian consists of the (2+1) dimensional version of the Skyrme term and a potential GP -Sp1 . Moreover, there is again a gauged version of this model, the so-called gauged BPS baby Skyrme model, which has been analyzed recently in the case of an asymptotically vanishing magnetic field BPS-g . It is the aim of the present paper to further investigate baby skyrmions in the gauged BPS baby Skyrme model from the perspective of the equation of state for BPS baby skyrmion matter. In particular, we will focus on the issue of how the energy and volume of the solitons change if they are put in an asymptotically constant magnetic field and exposed to external pressure. The paper is organized as follows. In section II we give a general overview on the gauged BPS Skyrme model. We prove the existence of a topological bound for the regularized energy in the case of a non-vanishing but constant asymptotic magnetic field. The BPS equations saturating the bound are presented. In section III we solve the system for the so-called old baby potential, both numerically and analytically in the weak coupling limit. We find the equation of state and related quantities (magnetic compression, magnetization, susceptibility) and prove a ferromagnetic behavior of the BPS baby skyrmion matter. Then, in section IV we introduce pressure and derive the pressure-modified BPS equations. Section V is devoted to the analysis of the equation of state with nonzero pressure and external magnetic field, again for the old baby potential. In section VI we present a toy model for which the equation of state can be obtained analytically for any value of the electromagnetic coupling constant. Finally, we discuss our results. Ii The BPS baby Skyrme model in a constant magnetic field ii.1 The gauged BPS baby Skyrme model Here we briefly summarize the properties of the BPS Skyrme model coupled minimally with the electromagnetic gauge field. The model is defined by the following Lagrange density BPS-g Without loss of generality we assume that the constant vector and the potential is a function of the third component of the unit vector field . The pertinent field equations are and the inhomogeneous Maxwell equation is The full energy functional is Further, we assume and the standard axially symmetric static ansatz which leads to an identically vanishing electric field and to the magnetic field . Note, that positive (topological charge) corresponds to a negative magnetic field ( is always negative as we will see below), while baby anti-skyrmions (negative ) would lead to a positive magnetic field. Moreover, we are interested in topologically nontrivial matter field (unit vector field) configurations, which requires the appropriate boundary conditions. then provides the topological charge (winding number) of . The field equations can be rewritten as where now and . It is also convenient to introduce the new variable which allows to rewrite the equations as the following system of autonomous second order equations Further, introducing a new target space variable this may be further simplified to where now and . It has been previously found that the model preserves many properties of the original ungauged version GP , restr-bS , Sp1 . First of all, there is a BPS bound which can be saturated by the corresponding BPS configurations. The important assumption in the proof was the boundary condition for the magnetic field that it asymptotically vanishes. Then, the energy is bounded from the below by where the inequality is saturated for the pertinent BPS solutions. Here is the topological charge (winding number) and is the average value of the derivative of the superpotential (see below) over the target space manifold. The resulting BPS baby skyrmions may be of the compacton type with the magnetic field completely confined inside the compact baby skyrmions. Further, the flux is not quantized (except in the large limit). One interesting conjecture, verified in many particular examples, was the absence of gauged BPS baby skyrmions for potentials with more than one vacuum. This strongly differs from the ungauged case where such topological solitons do exist. Secondly, the model is integrable in the sense of generalized integrability gen-int (no conditions for the gauge field) which means that there are infinitely many conservation laws (genuine conservation laws, which are not related to the gauge transformations). Moreover, the static energy functional possesses the area preserving diffeomorphisms as its symmetry group. Therefore, the moduli space of BPS solutions is infinite-dimensional. This also means that our assumed ansatz does not restrict the form of the solutions. One may use the base space area preserving diffeomorphisms to construct solutions with arbitrary (not axially symmetrical) shapes. ii.2 Constant asymptotical magnetic field The problem we want to solve next is how the external constant magnetic field modifies the BPS gauged baby skyrmions originally obtained in BPS-g . Obviously, the field equations remain unchanged but the boundary conditions are different. Now, where the last condition leads to an asymptotically constant magnetic field . Here, can be finite (compactons - for example in the case of the old baby Skyrme potential) or infinite. As the zero boundary conditions played a crucial role for the proof of the existence of the BPS bound, as well as for its saturation by solutions of the BPS equations, it is not obvious whether all these properties survive after the change of the boundary conditions. Here we restrict ourselves to . The corresponding analysis for negative topological charge is straightforward and requires the interchange of to . ii.3 The BPS bound for constant asymptotical magnetic field Here we would like to derive a BPS bound in the case of an asymptotically constant magnetic field. This requires some important improvements in the original derivation. Consider the following non-negative integral where and are (at the moment arbitrary) functions of the field variable . Further, where is a constant equal to the asymptotic value of the magnetic field. Further, the ”superpotential” is a function of the field variable which depends on the potential (see Eq. (II.30)), as we shall see in a moment. The last terms in (II.22) can be written as as the first part vanishes at the compacton boundary where by definition. Then where the superpotential equation relating the potential and the superpotential reads which differs from the expression found in BPS-g for zero asymptotic magnetic field by the term linear in (and in ). By construction, , which leads to . Let us remark that this new superpotential equation can be brought to the form of the original superpotential equation by the following redefinition However, now the boundary conditions for the superpotential are changed. It is convenient to define a regularized energy where we subtract the infinite contribution from the asymptotically constant magnetic field Obviously, the inequality is saturated if which are the BPS equations in the case of a constant asymptotic magnetic field. For the shifted superpotential we get the usual form of the BPS equations It remains to be shown that the solutions of these equations obey the full second order equations of motion, The Maxwell equation follows in the same way as in the case since the derivative of (II.36) does not depend on the value of . Further, from the superpotential equation we get And then we follow the same derivation as in the case. Namely, rewriting the first equation of motion as and using the above formulas we get which is the same as for . The remaining steps: using the covariant derivative definition, use and the definition of , do not depend on . That ends the proof. Finally, let us observe that in the axially symmetric ansatz the BPS equations read or for the shifted superpotential ii.4 The regularized flux Another important quantity is the flux of the magnetic field As the magnetic field extends to infinity the flux will also take an infinite value. However, for compactons, which is the case discussed in the paper, the magnetic field outside the solitons is exactly equal to the external field. Due to that we are rather interested in the value of the flux integrated over the area of the solitons, which is equivalent (up to an additive constant) to the following definition of the regularized flux where the axially symmetric configuration has been assumed. Then, using the definition of the magnetic field and the behavior at the boundary we find It is also possible to prove that this value depends only on the model (coupling constants and the form of the potential) but not on the local behavior of a particular solution. Dividing one BPS equation by the other we find where the constant can be computed from the boundary values of the fields at , Therefore, we get and, specifically at where, by definition, , where the constants depend on the model (potential), It is clear that once or . This behavior is confirmed by numerical results. For the regularized flux we then get where is the ”volume” (area) of the compacton. We use the word ”volume” and the letter to maintain close contact with the standard thermodynamic notation. We already showed that the first part, , may be expressed as a target space integral and, therefore, does not depend on the specific solution . In other words, it is one and the same thermodynamic function for all equilibrium configurations (BPS solutions). In a next step, let us demonstrate that also the ”volume” (and, consequently, the full regularized flux) is a thermodynamic function, i.e., a given function of for all BPS solutions. The BPS equation (II.46) may be re-expressed like where we used (II.58) in the second step. Integrating both sides over their respective ranges leads to and to the regularized flux which, indeed, is a thermodynamic function, as announced. ii.5 The magnetization The thermodynamic magnetization is defined as minus the change of the thermodynamic energy of a sample (in our case, the skyrmion) under a variation of the external magnetic field. Here, the electromagnetic part of the thermodynamic energy must be calculated from the difference of the electromagnetic fields with and without the sample, which precisely corresponds to our definition of the regularized energy, i.e., We use the BPS bound (II.34) for the energy and express the average value of over the target space like where we treated as a function of in the last two terms, which we shall continue to do, i.e., in what follows. The magnetization then is and, obviously, is a thermodynamic function (i.e., the same function of for all equilibrium configurations). In standard thermodynamics there is a simple relation between the magnetization and the difference between full and external magnetic flux in the sample. In our conventions, this relation reads We shall see that this relation continues to hold in our model, although the proof is not trivial and makes use of the BPS nature of the model, specifically of the superpotential equation. Using the variable instead of , the superpotential equation may be re-expressed like To express the first derivative , it is useful to introduce a first order (infinitesimal) shift about a given value , then the magnetization at is and the thermodynamic relation (II.68) becomes The superpotential equation at zeroth order in is and serves to determine for a given , potential and given coupling constants. The first order superpotential equation is (remember that does not depend on ) and serves to determine for a given . Indeed, introducing a new variable the above equation becomes and may be easily solved via the method of the variation of the integration constant, leading to or, in terms of the variable In particular, for we find and was used. From this last result, the thermodynamic relation (II.72) follows immediately. Iii Constant magnetic field and the old baby potential iii.1 Numerical computations The system introduced above significantly simplifies in the case of the old baby Skyrme potential Then the field equations can be integrated to The corresponding energy integral is Effectively, the problem depends on two coupling constants. The dependence on the topological charge can be included into a redefinition of the base space coordinate while a particular value of just fixes the energy scale. So, let us choose and treat and as parameters (now dimensionless) defining different theories. Moreover, the external magnetic field is another free parameter. As in the case we expand the functions at the boundary In the numerical computations we assumed (the results for and are very similar) and then looked for a few different and scanned for a wide range of . Examples of gauged BPS baby skyrmions are plotted in Fig. 1 for different values of the external magnetic field. The electromagnetic coupling constant is . At this point it is useful to remember that the gauged baby BPS skyrmions without external magnetic field have a magnetic field which is everywhere negative (for positive bayon number ) and a negative magnetization proportional to the baryon number BPS-g . In other words, these gauged skyrmions show a ferromagnetic behaviour. For a negative external field we therefore expect that the negative magnetic field will become stronger (i.e., more negative). As the gauge potential for negative magnetic field is restricted to the interval , as follows easily from eq. (II.58), the stronger (more negative) magnetic field is achieved by shrinking the size of the skyrmion. Concretely, for strong negative we approach a singular configuration: the skyrmion profile gets flatter and flatter inside (approximately constant charge density) with a rapid but smooth approach to the vacuum at the boundary whereas has a more and more linear dependence on tending to . In the limit where the size of the compacton goes to 0 as and the solutions approach the step function and a linear function for and , respectively. The approach to the limiting step function solution is faster for higher values of the electromagnetic coupling constant . For high positive values of , the magnetic field changes sign everywhere, and the resulting gauge potential is a simple monotonously increasing function from 0 to . For a positive but sufficiently small , however, the phenomenon of magnetic flux inversion occurs. That is to say, the magnetic field is negative in a ball (because the magnetic field without exterenal field is more negative in the core region), becomes zero at and positive in the shell (because must hold at the compacton boundary). The corresponding gauge potential is, therefore, a decreasing function in the ball close to the center but an increasing function in the shell. Finally, the value of the gauge potential at the compacton boundary determines the total magnetic flux inside the compacton. Specifically, the total magnetic flux inside the compacton may become zero, in constrast to the regularized flux or magnetization, which is always negative for positive baryon number. The baby skyrmion profile is a simple monotously decreasing function for all values of . We show an example of the magnetic flux inversion in Fig. 2. In Fig. 3 and Fig. 4 we show how the compacton size and the compacton energy, respectively, depend on the external magnetic field. iii.2 Non-dynamical constant magnetic field Although the system can be reduced to BPS first order equations it is still too complicated to find analytical solutions. However, one may consider a simplified case where the magnetic field is treated as an external field . That is to say, we do not consider the back reaction of the system on the magnetic field in the vicinity of the BPS baby skyrmion. It has been found, after comparison with the numerical results, that this approximation works quite well and provides an exact description in the small electrodynamical coupling constant limit . iii.2.1 Equation of state and As the magnetic field is only a non-dynamical external field, we may reduce the system to one equation where the magnetic field plays the role of a ”deformed metric” in which baby skyrmions exist. (In fact, curved metrics may arise in some gravitational context grav , which points to another possible application of the BPS skyrmions.) Hence, The resulting equation can be analytically solved for the old baby potential with the boundary conditions where can be finite (compacton) or infinite (usual soliton). However, infinite is excluded by the asymptotic behavior of equation (III.10). Indeed, for large we get that which contradicts the boundary value for at infinity. The final solution is is an equation fixing the size of the compacton. It provides an approximate but exact relation between the two-dimensional ”volume” and the external magnetic field The validity of this approximation is restricted by the following condition which follows from the equation of motion for the magnetic field when the approximated (non-back reaction) solution is inserted. For small magnetic field we may use which agrees with the size of the non-gauged case. For large magnetic field we can use . Thus, i.e., the size of the solution grows linearly with the magnetic field. Next, we consider the energy where . Hence, we find the relation between the total energy and the external magnetic field, however, in an implicit way Equation (III.15) and the last expression are the main results of this section since they provide exact formulas for the and relations in the BPS gauged baby model. iii.2.2 Magnetic compressibility For small magnetic field and the last expression can be computed using the L’Hospital formula In order to find at vanishing we differentiate (III.14) Now, assuming we find that i.e., We plot the numerical results for and for general coupling (i.e., with the backreaction taken into account) in Fig. 5. Then the energy is which agrees with the non-gauged case. On the other hand, for large value of the magnetic field we find that Another consequence of (III.27) is that the magnetic compressibility is finite It is quite interesting that the magnetic compressibility very weakly depends on the electromagnetic coupling constant for a wide range of . In fact, for , see Fig. 6. Hence, the non-backreaction approximation works especially well for the magnetic compressibility. Moreover, we can also obtain the magnetic compressibility for large magnetic field. Now, Hence, asymptotically the magnetic compressibility tends to zero. iii.2.3 Magnetization and ferromagnetic medium Another interesting quantity is the magnetization at vanishing external field, Then,

Teaching your child about money reinforces the basic number facts of the base 10 number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. At first, for younger children, you can count the number of each denomination you have on hand. And, as you move up the currency you can show how each is related to the others. Another important concept to teach here grouping similar coins and bills. One way to reinforce this grouping, is to count more than $2 worth of pennies, nickels and/or dimes. Let your child count until half of the coins have been counted, then distract him/her to read a few pages of his/her favorite book, better if it relates to counting. After that task has been completed, return your child to continue counting the coins. He/she will probably start over, and here is where you step in to show your child how to group the coins in separate piles; doing so makes it easy to return to the task and later to actually find the value of all coins. Notice how I related each denomination to pennies. As your child masters these equivalences he/she will make relationships between the others, like 2 nickels makes a dime. Here we go.... start with a bag of 100 pennies. Use real currency here, you are not doing your child any favors with fake money. Have your child start counting the pennies, as I mentioned earlier. Now interrupt and return. Suggest to you child to place these pennies into separate piles with the same number of pennies in each. 10 is a number familiar to probably every child and is the number to use. Your child already counts 1, 2, 3, ..., 9, 10 and starts over, 11, 12, 13, ..., 19, 20 etc. Interrupt your child again, then return. Point out it is much easier to continue then to start all over again. 10 groups of 10 = 100 = 1 dollar Your child has now separated the pennies into 10 piles of 10. Count these, 10, 20, 30, 40, 50, 60 , 70, 80, 90, 100. Point out if all the pennies are placed in a pile, you wouldn't know there were 100 pennies. We've grouped the pennies into equal piles of 10. Point this out, it is most important. Move the piles next to each other and place a dollar piece next to it (or a dollar bill if you have no dollar piece.) Make the equivalence between the pennies and the dollar. 100 pennies is 1 dollar. As k your child which would be easier to carry around, a bag of 100 pennies or a single dollar coin (or bill). This is still too abstract! What's a penny? What's a dollar? Ok, a "cent" is a "penny." So, in words, if a toy costs 1 dollar and 35 cents, then ask your child how many pennies it would take to buy that toy? Help your child make the connection between the pennies already counted and the price of this toy. Not enough pennies. Now place the dollar piece there and ask your child if there is enough. How many of the pennies would be required with the dollar? How many pennies are left? 10 dimes = 1 dollar Once again, have your child divide the pennies into 10 piles of 10. Ask your child to verify he/she has 100 pennies. Repetition is most important throughout. Now, explain with 10 dimes handy that each dime is the same as each pile of pennies by placing a dime by each pile. Ask your child if the pennies added up to a dollar. Hopefully with the answer yes, then, ask how many dimes add up to a dollar. Help your child arrive to the answer 10. So, help your child make the connection that 10 dimes represents 100 pennies which represents a dollar. Now with the pennies, dimes and dollar, ask your child to pay for the toy mentioned above. Any answer as long as it's correct is fine. For example, use the dollar and 35 pennies. But help your child connect the dimes, the dollar and the pennies to pay for the toy with the dollar, 3 dimes and 5 pennies. 50 cent piece (half dollar) Here we go again, have your child divide the pennies into 10 columns of 10 pennies apiece. Reinforce that 100 pennies in 10 columns of 10 pennies apiece is the same as 1 dollar. Now separate the first 5 columns from the last five columns. Place a 50 cent piece above each group of 5 columns. Explain to your child that each 50 cent piece is 50 pennies. So 2 half dollars is 100 pennies is 1 dollar. Now, ask your child to place the dimes back into the picture, one above each column. then ask, how many dimes are in the half dollar. This time have your child separate the pennies into piles of 5 each. When done, ask how many piles there are. then ask if this makes sense? 10 piles of 10 is 20 piles of 5. this can be hard to grasp, if so, have your child separate into columns of ten, then carefully pull the bottom five pennies from each column a bit below the top five. Now count the groups of 5. Have your child place a nickel by each 5 pennies. And say with your child "5 pennies is a nickel" for each nickel. So 20 nickels represents 100 pennies which is 1 dollar. Recall the half dollar exercise. How many nickels are in the each of the two groups? 10 nickels is a half dollar. Have your child group the pennies in columns of 10 each. Now count the pennies starting from 1 down one column then the next. When your child reaches 25, group those pennies together, then start counting again, 1 to 25. You should have 4 groups of 25 pennies, 2 and 1/2 columns each. Now have your child place a quarter next to each group. So, each quarter is 25 pennies, and 4 quarters are 100 pennies. Now is a good time to have your child relate dimes and nickels to quarters, quarters to half dollars, etc. Ask your child with all of these choices how he/she would pay for that 1 dollar and 35 cent toy. Explore all possibilities. Finally, have your child play play bank teller. Ask your child to convert one denomination to the other. More advance, ask your child to make change for a purchase of some pretend item. Oh, one final note, its fine to tell your child that $1.35 means 1 dollar and 35 cents. That is, the number to the left of the decimal point (dot) is the number of dollars and the number to the right of the dot is the number of pennies. Have fun! And don't try to do this all in one sitting! This takes time!

First, for an excellent explanation of the person you have to be to understand all this, please visit this blog The Inevitability Principle of Blanched Artichokes is the principle that every person will eat blanched artichokes either in this life or the next. And when I say the next life, every thinking person will realize that if you have not eaten blanched artichokes in this life, you have eaten them in the life before this one, by simple logic. The effects of eating blanched artichokes stretch back over your life previous in the case that you did not happen to eat them in that life, and if you did you end up with effects^2 being applied to your life. This is due to the fact that the squared tangent of effects plus the number of lifes concurrently lived by one person (that is, one) is equal to the inverse of the cosine squared of said effects. If r is equal to effects, it is simple to see that the effects of eating blanched artichokes are related to both time and pressure by the elementary equation of pressure times volume being equal to n times r times temperature and that energy is equal to negative R of h being divided by a number, n, when n is 1, 2, 3, 8, 42, or 1789387458301. Because energy is also equal to mass times the speed of light squared, it is easily proved that r is related to the speed of light. One could also prove this by calculating the average warp speed of an antimatter particle and dividing it by the numbers of oscillations per second of the nearest black hole, multiplying by pi times 3, squaring the result, taking the r root of it, and calculating the average sine function of the answer, then adding 2.99792458 times ten to the eighth power, the result of this equation being equal to the speed of light, or c. As you can see from these elementary proofs, the inevitability principle of blanched artichokes has great impact upon our daily existence, although this is not generally known, and was not known to the thinking men until recently. However, before I explain some of the many intricacies of this principle, let me give you an example. Because I know that you are currently under the effects of blanched artichokes, I can calculate what you ate for breakfast based upon the time of sunrise of the 12th of June in AD 38, the most prominent menu item in McDonalds in the year 2011, your average weight in 1984, and the angle of refraction of the cornea of your left eye. You may find that difficult to believe, but if I were to know all these things, I could calculate your breakfast. For instance, since I know that the sunrise of the 12th of June in AD 38 was precisely 6:04:52.4783932, the most prominent menu item in McDonalds in 2011 was the 13-pack of Chicken McNuggets made with 99% real white meat, my average weight in 1973 was approximately -0 pounds, and the angle of refraction of the cornea of my left eye is exactly 92 degrees, 43 minutes, and 75 seconds, by an ingenious process I determine that I had organic plain yogurt with Polaner strawberry jelly mixed in. And, as that was exactly what I had for breakfast, I have thus demonstrated the inevitability principle of blanched artichokes. The inevitability principle of blanched artichokes is an extremely useful principle because it can be used to determine the past based on past, present, and future events. A practical example of this would be that tomorrow I calculated that this paper would be finished. It can be used to calclulate the success of projects before they complete. As well, the extent of the inevitability principle of blanched artichokes is far from being determined as we are just beginning to explore it and our knowledge grows at an exponential rate based on the equation of knowledge being equal to the inverse of cosine squared cofunction plus the hypontenuse of a squared right rhombic triangle divided by the fifth root of the extent of the spacetime fabric plus the time we spend investingating the inevitability principle of artichokes squared. Due to our lack of knowledge about the inevitability principle of blanched artichokes, thinking men often have conversations with other nonthinkers concerning seemingly benign topics that, with the proper knowledge, are used to expand our knowlege of the inevitability property of blanched artichokes. Allow me to finish with another, more complex, example. This evening I had a conversation concerning a memory program, and the person with whom I was having the conversation said, "Mrs. DeStumpf's room". This led me to deduce that Hilary Clinton was not president in 2034. Allow me to explain. I earlier proved r in relation to c, which sets the stage for calculating this. First, I must know the following information: the mass of a proton, 1.6726231 times ten to the negative twenty-seventh power, the exact time of the arrival of an email notifiying me that the Office Live Workspace has started, 13:28:54.3278, the number of pens in my pen box, eight and one marker, the eqilibrium constant of the cell of the reaction Zn(s) + 2Ag+(aq) -> Zn2+(aq) + 2Ag(s), being equal to 1.56V, the Number of Hilary Clinton, 45, the CD Set number of Microsoft Publisher 2000, X04-84021-M, and the my lottery card pick of the day from the Irish Lottery, 2-3-12-19-32-37. The calculation is as follows: because r is related to c, we can calculate that c squared times m (in this case, 1.6726231 times 13:28:54.3278) is equal to e, which is equal to r of h divided by n, in this case being 8. From this we calculate h and insert that into another equation calculating l as h divided by m times v squared plus the square root of X04-84021-M to the r power, resulting in l which we can use to calculate list A as being equal to 1.56 times negative 45 plus A of 0. These equations result in the quantumn numbers of n being equal to 8, l being equal to 5, and m being equal to 13, which is not possible. Therefore Hilary Clinton wasn't president in 2034.

Wednesday, June 15, 2011 Super-Virial Proto-Grounding L. Edgar Otto June 15, 2011 I present here a rather simple description of the ground of being as physics, perhaps too simple as in the computations to be misleading or confusing. But it is a more relaxed view with greater ways to see the degrees of freedom and nature of symmetries. Viriality is the what we call those spaces that consider duplications or halving of things for stable orbits that essentially establishes three space and the ways that forces and structures appear in the more relaxed physics of the world. It is the basis of our ideas of super symmetry, that is of a greater generalization of how the world works than just the low dimensions of three and four space. This is not to say that these dimensions do not have an essential role, but so do the higher dimensions. Thus there are analogs where they apply, and regardless of scale or what may insure a finite mass for some dynamic object, which are as simple as multiples of integers we regard as powers. At first I thought I could map the double quaternion explanation of Rowlands to my quasic grid- but this is not simply the case. Nor can we accept but one form of the intelligible logic of things where a different form may conflict or be irrelevant to our comprehensive and thus exclusive system. The appeal to some such logic may break down even if it does not appear to do so- and where we may hold logic itself as only designed as intelligible the question of its general validity, of thought itself, can just as well be a reassurance of the freedom to explore by intellect our reality. Some ideas we have heretofore questioned, in light of these expanded notions seem to have a solid basis after all- at least to explain why some idea worked but was not justified save only by explanations just as unclear or even bizarre. If we have a quasic grid, with many states that we can interpret as fractal like, we can ask, as with computer drawing, what is the grid that underlies it all? That grid I have called the epsilon-delta honeycomb. It is there that we can map the 32 x 32 elements of Rowlands and Dirac's algebra. The order is strait foreward and linear as if the square roots, being equal of the quasic grid- that is all the Cartesian and ultimately Euclidean concepts we extend to ideas like holographics. Of course we can just as well impose or suggest such ideas in either design or division of the space for the order is distributed over a plane in the quasic grid even though from some perspective the factors contain clear and equal binary information. It is clear, in many cases, that in physics, the essential viriality that doubles some things as they reflect the difference in potential and kinetic energy, or say the phase of spin of some objects, leads to some mistakes in the physical calculations- Consider the bending of light in a gravitational field by Einstein and the test of it at eclipse by Eddington to make this correction. It is moreover a core property of numbers that in the end justifies Eddington's Plus one Idea. Number theorists would do well to take advantage of the view presented here so as to double such values of corresponding orders which we can say is a match between these views of spaces- ones that may tell us something of number theory itself in terms of numbers as well of the general notion of dimensions and measure. Poet that I have come to accept if not fancy myself to be I am at a little loss of clear terms here. So let us from one side of the mirror of notions imagine the two divisions of general space as the HF (holographic) and FX or (fractal) for now but that only a part of the picture (I mean one has to imagine a world of mirrors but there are more than two sides on which to be on a side of something). I suspect it turns our that the HF x FX related or divided into the general quasic field amounts to 2^2n +1 which is what we should expect applying this to a advanced numerical view of physics. But this is not general enough and does not take advantage of all that is so far known in number theory. We can imagine terms like Ordinal and Anti-Ordinal, Cardinal and Counter Cardinal to apply as well to the general archetypal concepts here. While we say that even in Maxwell there can be residual neutral flow we can also say there is residual neutral fixed singularities or points of rest. Yet is not established that such singularity complexes or where they apply to a field idea has to be necessarily dependent on the general grand or great-grand ideas of unification theories. It may be that this too is a quasic relaxation as a relation in the general reality with its consequent restrictions and new freedoms for complexity and even transcendence of that now real. So since there is no set epsilon-delta honeycomb all is a general grid of quasic space without the fixed quasicity. That is one can think of hierarchies or not or one can reject them. The same for ideas of universe and multiverse. So we can imagine a fixed complex of singularities when the quasic span around it expands in depth or we can imagine the internal states of such a singularity complex growing in a details and definition. Or in some cases the physical object may become its inverse just as surely as our notions may be limited or expand in the depth and span of our theoretical and perceptive views. A singularity may be a given or arise or it may be ultimately independent of the general states, intermittently a "creative" space. An associahedron of simplexes seemingly limiting dimensions and as dynamic kinetically is like a rocket ship thru space and appears to self sustain in its motion. Of course the purely more fractal like spaces, more like falling and less like superficial surface quantum theory, can be dynamically seen as well. Our notions and the universe may have conflicting dynamics as well as where they meet in concepts. Thus: And as formal as I now present the work done (the other work has applications hopefully soon to be on a domain somewhere and that can use these ideas in general as sets of color notations for the combined general idea of spaces and connectivity) I suggest this simple relationship of integers of which I do invite exploration by others due to the snow blind simplicity that may occur in simple clear understanding. 16^2 or 256 is a natural addition in combining the HF and FX spaces into one unified space, (or rather recombining them). To this we add its virial doubling or 512. Thus we get 768 (that is twice the rotations and inversions of a tesseract) BTW we can consider Galois and the equations of the fifth degree of icosahedra and the like and apply them to either space alone but this does not ground the total physics in ways that justifies the lack of or existence of hierarchical ideas. Not to say these theories do not have a place or use as part of the bigger picture. Now we add to this 256 not just the virial doubling but binary powers of it to get these analogs to virialty: 256 + 1024 = 1280 256 + 4096 = 4352 256 + 16384 = 16640 and so on... which makes the sequence when summed or divided 2 5 17 65 257 and so on. Now 768 + 1280 = 2048 which is a binary number in the low viriality levels. We of course can divide numbers by the group numbers and keep such things in mind also, for example 1152 + 128 = 1280, 1152 being the group of the 24 cell in 4D. * * * Philosophic Musings just before this Post: An essence or substance cannot be deduced as existing reality by assumption it a given or emptiness, nor a necessity by chiral or other distinguished contrasting mechanisms, where it can be a "gray" blend unclear as 1 or 0. Yet, at least by experience and observation we can make the case so assert partial substance as a totality in relation to the obvious being or something neutral as comprehensive over reality. The physical difference between exist or not exist may be ambiguous (as if a higher uncertainty or as problematic in an intelligible way or not.) and still be generally absolutely neutral as a vacuum or field, independent of the reality as a fact or persistence both pragmatically and non-necessary- thus a working resolution of paradoxes. To the existent some essence does not exist (including the virtual multiverse- things that may exist may be unified higher or be independent in diffuse unity and freedom unto transcending of known systems. BTW we can map the epsilon delta notations on the grid as powers of 5 (thus in groups such powers relate to more general spaces and polyhedral or group structures) As a single quasic order (much like certain fractal fillings of a plane. For example: 1,1 ; 1,4 ; 4,1 ; 4,4 ; 1,7 ; 3,6 ; 4,7 ; would map to 0 ;5; 10; 15; 20; 25; 30; ... In the Condenser Mechanisms the color shift may interchange 0 and 1, that is a wild card * can equal 1 or 0 in all its meanings such as exist or not, possible or impossible... This may also explain beyond the color shifts and ideas of creation and annihilation in quantum theory hierarchies of light and knowing (in some interpretations) a much clearer grounding of this experience of consciousness. As far as the use of color goes, the combining of primary colors in say the hex notation is intelligible in the 3D;4D grids but its use as a labeling has a relaxed way of applying the same symbols if we keep it in mind where distinctions are meant. Footnote: Of course computer graphics have a new depth of applications of numbers and are thus intelligible- it strikes me that my habit of utilizing the 80 x 80 gif range as a limit to drawing avatars on the philosophychatforum was useful in the greater understanding of such numerical and digital relations in what at first may appear as non-linear and so diffused levels of mathematical explorations. * * * Footnote: It would appear (in the reply to Ulla's p-adic question here) that some answers are there to expanded the grounding of Pitkanen's and other ideas including the relation to things in braid space formalism, and especially spinor theory, have anticipated his post just after mine. Of course it would have been better to talk directly instead of interpreters pro and against between us. So may all the theoreticians come closer to the general breakthroughs needed for this enquiry. I continue to post because of the frayed ends and stray statements needed for closure and a conclusion. * * *

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cd = (d0 * c0 - d1*c1)/(d0 - d1) = (951 x 5.750 - 950 x 5.722)/(951 - 950) TRd = (T0*c0 - T1*c1)/cd = (2728.083 x 5.750 - 2717.094 x 5.722)/cd COMPARISON MODEL Loglikelihood H0 Value -154318.940 H0 Scaling Correction Factor 5.722 for MLR H1 Value -151645.790 H1 Scaling Correction Factor 2.887 for MLR ... Chi-Square Test of Model Fit Value 2717.094* Degrees of Freedom 950 P-Value 0.0000 Scaling Correction Factor 1.968 for MLR NESTED MODEL: Loglikelihood H0 Value -154323.099 H0 Scaling Correction Factor 5.750 for MLR H1 Value -151645.790 H1 Scaling Correction Factor 2.887 for MLR ... Chi-Square Test of Model Fit Value 2728.083* Degrees of Freedom 951 P-Value 0.0000 Scaling Correction Factor 1.963 for MLR Thanks for your confirmation. Let me ask a follow-up question. TRd was found to be 4.304958... Since the material says, "For MLM and MLR the products T0*c0 and T1*c1 are the same as the corresponding ML chi-square values," am I supposed to use 3.841 as critical value to determine whether the calculated S-B scaled chi-square difference is significant at the level of .05 or not? That is, is the difference (4.304958...) significant since TRd > 3.841? I forgot asking another question. The equality constraint was imposed on a single parameter (which measures the effect of child maltreatment on violent offenses) for two ethnic groups, whites and Asian Americans. In the comparison model, the coefficient was found to be .031 (SE = .027) for whites, whereas it was .654 (SE = 1.430). As you can see, both coefficients are not significant, although the S-B scaled chi-square difference is larger than 3.841. As I supposed to say the coefficient is significantly different between whites and Asian Americans even though the coefficient was found to be not significant in each ethnic group? 2nd post: Each coefficient being significantly different from zero or not is not the same as testing that they are the same. Typically, if you use the independent-sample z test of equality using your SEs, you get the same thing as the chi2. Step 1 on the Mplus website (http://www.statmodel.com/chidiff.shtml) for Difference Testing Using the Loglikelihood is: 1. Estimate the nested and comparison models using MLR. The printout gives loglikelihood values L0 and L1 for the H0 and H1 models, respectively, as well as scaling correction factors c0 and c1 for the H0 and H1 models, respectively. Does this refer to H0 and H1 values given for the SAME model (i.e., in the same output file); or for DIFFERENT models (estimated in separate runs, with separate output files)? I ask because while I have seen BOTH H0 and H1 values in some output files, I only see H0 for in a model I estimated using a NBI dependent variable, as seen below. There is no H1 value offered. Can I still utilize the steps on the website to compare the fit of this model with that of another nested model, using the H0 values only (the ones provided for each distinct model--because I did not get H0 and H1 values together in one output file). Thanks. MODEL FIT INFORMATION Number of Free Parameters 14 Loglikelihood H0 Value -1189.806 H0 Scaling Correction Factor for MLR 1.1902 To do difference testing you need to run two analyses. The first is the least restrictive model. It is referred to as H1 in the writeup. The nested model is referred to as H0 in the write up. In both cases, the H0 values are taken from the output to use in the computations. EFried posted on Tuesday, April 02, 2013 - 1:55 pm When comparing 2 models using the MLR estimator, each model provides 3 scaling correction factors and 2 loglikelihoods. I don't find it specified which one to use for model comparison (http://www.statmodel.com/chidiff.shtml). Looks like the only difference is that model 2 has a direct effect from m to y2. ri ri posted on Saturday, August 30, 2014 - 4:08 pm Yes, as far as I know one Needs to do a chi square difference test to compare the two models. In regular way, one just uses the chi square values. But since I have categorical data, I shall do it differently I suppose? I used the difftest command, but could not find the scale correction to calculate the difference with the formula provided at the Website. With only one parameter difference you can just look at the z-test for that parameter in the model that is less restrictive. In the general case you use DIFFTEST, first running the less restrictive model and then the more restrictive model. You don't need the scaling correction factors or the computations on the website. DIFFTEST does it for you. ri ri posted on Tuesday, September 02, 2014 - 12:16 am I tried the DIFFTEST to compare the contrained and uncontrained model, it worked wonderfully! Just I have another methodological question. In the user guide multiple Group Analysis, you wrote an example, Fixing the mean of the variables in Group 2 to Zero. If I compare constrained and unconstrained models, is it necessary to fix the mean to Zero? I also saw some People Center the means of the continous variables in order to minimize multicollinearity. If I compare two path models (such as the above mentioned model comparion), I wonder if mean centering is needed? To compare means across groups, use the model with means zero in all groups versus the model with means zero in one group and free in the others. Centering is not needed. Ari J Elliot posted on Wednesday, January 07, 2015 - 7:44 pm Hello Drs. Muthen, Regarding chi square difference testing with MLR, please confirm that the H0 scaling correction factor that should be used in the calculation is the one listed under Loglikelihood, NOT the one listed under the MLR chi-square test of model fit. Thus in the following output for the nested model, I would use 1.5862 as the scaling correction factor. H0 Value -15770.576 H0 Scaling Correction Factor 1.5862 for MLR H1 Value -15733.949 H1 Scaling Correction Factor 1.5549 for MLR ............................. .....Chi-Square Test of Model Fit Value 50.133* Degrees of Freedom 5 P-Value 0.0000 Scaling Correction Factor 1.4612 for MLR If you use chi-square for the difference testing, you should use the scaling correction factor under chi-square. If you use the loglikelihood for the difference testing, you should use the scaling correction factor under loglikelihood. Ari J Elliot posted on Thursday, January 08, 2015 - 11:37 am Ok thanks. To further clarify, the instructions on the webpage for difference testing using chi-square state: "Be sure to use the correction factor given in the output for the H0 model." Under the chi square I only see one scaling correction factor, whereas for loglikelihood there are correction factors provided for both H0 and H1. Given that correction factors for both the nested and comparison models are used in the calculation I'm not sure to what the reference to the H0 model in the instructions refers. You should use the H0 scaling correction factors. If only one is specified, it is the H0. Don't use the one for the baseline model. If you have further questions, send the output and your license number to firstname.lastname@example.org and we can tell you which number to use. Dear Muthen, I have a SEM model with these relationships (1) one latent and its three observed variables (2) Five observed variables that involved in path relationships with the latent in (1) (3) A correlation in two of the observed variables in (2). I use MLR estimator and would like to know the chi-square p-value. I follow the instruction given in your website on difference testing on chi-square, to compute the scaled difference in chi-square. My question is for the H0 (restricted model), which relationships in (1)to(3) I need to constrain? Should I just constrain the path relationships in (2) which is my main interest? Our goal in this test is to get a non significant p-value right? like the ML estimator's chi-square result for model fit test? is there any information how the Satorra-Bentler scaled chi-square difference test is influenced by high N´s (e.g. n > 30 000)? Is it influenced at all? I am investigating measurement invariance with large subpopulation-samples and complex survey data (students nested in teachers) and therefore I use the SB-chi-square to test nested models. Dear Linda, In ESEM invariance test for 2 groups, when I compare the metric (loading) invariance model with the configural (baseline) model, by using MLR estimator, can I still calculate the chi-square difference in Mplus using MLR? I have a problem to determine the constraint/more restrictive model for my configural and metric invariance ESEM models. The same thing for other invariance test (scalar, error variance). Can you give some advises. Thanks. Jiangang Xia posted on Thursday, November 02, 2017 - 9:41 am Dear Linda, I am confused by the instructions from the website and your previous responses to some above questions regarding the "Difference Testing Using Chi-square". For example, for the very first question in above, you said "Yes, you are doing this correctly." However, in the question, the "H0 Scaling Correction Factor for MLR" (5.722 and 5.750) under "Logliklihood" was used, not the "Scaling Correction Factor for MLR" (1.968 and 1.963)under "Chi-Square Test of Model Fit". In another question in above that was posted by Ari J Elliot posted on January 07, 2015, you responded that "If you use chi-square for the difference testing, you should use the scaling correction factor under chi-square." So I am not sure which "scaling correction factor" should we use for the "Difference Testing Using Chi-square". In order to avoid the confusion, here I want to use the output from the first question. If I want to compare the two models using chi-square, should I use H0 Scaling Correction Factor 5.722 for MLR ? or should I use Scaling Correction Factor 1.968 for MLR ? A related question: when should we use chi-square and when should we use Loglikelihood? The "Scaling Correction Factor for MLR" should be used with the "Difference Testing Using Chi-square". There should be no confusion about this since that correction factor is printed just below the chi-square value. In every case you can use chi-square or the log-likelihood. Both should give you exactly the same result (subject to round off error). Derek Boy posted on Saturday, January 20, 2018 - 9:44 am Dear Dr. Muthen In order to compare the two models as below, I have tried to do the deviance test using loglikelihood. However, I could not figure out the numbers of parameters (i.e., p0 and p1) which are required by the formula for computing the scaling correction (cd) = (p0*c0 - p1*c1)/(p0 - p1), where p0 is the number of parameters in the nested model and p1 is the number of parameters in the comparison model. Would you kindly help, please? Best regards. ------------------------------- MODEL A Number of Free Parameters 17 Loglikelihood H0 Value -215.987 H0 Scaling Correction Factor 0.4637 for MLR H1 Value -215.987 H1 Scaling Correction Factor 0.4637 for MLR … … … Chi-Square Test of Model Fit Value 0.000* Degrees of Freedom 0 P-Value 0.0000 Scaling Correction Factor 1.0000 for MLR ------------------------------- MODEL B Number of Free Parameters 19 Loglikelihood H0 Value -214.709 H0 Scaling Correction Factor 0.4195 for MLR H1 Value -214.715 H1 Scaling Correction Factor 0.4195 for MLR … … … Chi-Square Test of Model Fit Value 0.000* Degrees of Freedom 0 P-Value 1.0000 Scaling Correction Factor 1.0000 ------------------------------- You have zero degrees of freedom for both models so neither model is testable and no model comparison can be done. Derek Boy posted on Saturday, January 20, 2018 - 6:37 pm Dear Dr. Muthen So, my two models are saturated and having zero degrees of freedom. Many many thanks for your pointing it out to me. But, may I also ask whether fitting a saturated model is a bad thing to do? Should I do something to make it unsaturated? What is that something you would suggest me to do? Please kindly advice. With best regards. I intended to compare two nested model with the Satorra-Bentler scaled chi-square difference test (both models used MLR as the estimator) and based on the formula, the computed SB-chi-square is a negative number. I wonder if this is possible and how should I resolve this issue? I am comparing models in a TWOLEVEL MIXTURE (MLR) analysis to test if particular covariates should be included in the model. I am using the Loglikelihood to calculate the TRd value, as per the website (the Chi-square values do not appear in my output for some reason). I can do the calculation, however, how do I calculate the degrees of freedom in order to obtain the critical value of the TRd value? I comuted two CFAs, testing a three-fator model and a three-factor model with a general factor using Mplus editior 7.3. Now I'd like to do a DIFF test to see which model is better. However, my outputs do not give me a correction factor. How do I get the correction factor? A quick look says that the only difference between the models is that you put a second-order factor behind the 3 first-order factors. If that's the case, the models are the same because 3 indicators of 1 factor is a just-identified model - you are not restricting the covariance matrix for the first-order factors. You need 4 or more first-order factors for that. Q1: There is no way in Mplus or any software. The 2 models are the same when you have only 3 first-order factors. "The same" means that they produce the exact same covariances among the observed variables. The models have the same number of parameters and when you have estimated one, you can transform its parameter values to the parameter values of the other. Q2: There is no change that you can make - just accept the fact that this can't be tested - you can present the second-order factor model but you can't say that it fits better or worse. I am comparing two Path Analysis models using the chi-square difference testing (Satorra-Bentler scaled chi-square difference test) - I got the formula for it from this website: https://www.statmodel.com/chidiff.shtml - the first set of tests listed on the page. The nested model has 8 degrees of freedom, whereas the comparison model is saturated and has 0 degrees of freedom and a chi square value of 0. Is it okay to carry out the chi-square difference testing between the two models, given that one of the models is saturated? If it is indeed okay, I wonder if there is a citation you are aware of that I could cite to justify doing this (I am getting pushback on this from others). Hello, I am doing a Factor Analyisis and my goal is to test whether a hierachical model with a general factor or just a three-factor model fits my data best. When performing the hierachical model the input reading terminates normally, however, my factor loadings and the std errors leave me with questions. Why are the loadings and errors so high? Is there a problem in my code? Code: Data: FILE is Dr_E.inp; Variable: NAMES ARE EP1 EP2 EP3 EP4 EP5 EP6 EP7 EP8 EP9 EP10; USEVARIABLES ARE EP1 EP2 EP3 EP4 EP5 EP7 EP8 EP9 EP10; CATEGORICAL ARE EP1 EP2 EP3 EP4 EP5 EP7 EP8 EP9 EP10; MODEL: F1 BY EP1* EP2; F2 BY EP3* EP4 EP5; F3 BY EP7* EP8 EP9 EP10;[F1-F3@0];F1-F3@1; Y BY F1* F2 F3;[Y@0];Y@1; OUTPUT: SAMPSTAT tech1 MODINDICES; PLOT: TYPE=PLOT2; Output: Y BY F1 1.059 0.170 6.218 0.000 F2 5.140 7.062 0.728 0.467 F3 2.544 0.931 2.732 0.006 Means Y 0.000 0.000 999.000 999.000 Intercepts F1 0.000 0.000 999.000 999.000 F2 0.000 0.000 999.000 999.000 F3 0.000 0.000 999.000 999.000 Thank you for your insights! I need to compare nested models. I am using ML and I got the following loglikelihood statistics for the two models I want to compare. As I understood correctly for ML I don't need the scalling correction factors: Model 1: -6369.028 (67) Model 2: -6368.841 69 Loglikelihood ratio: .374 Difference in Degree of Freedom: 2 How do I now know if the difference is significant or not? Thank you very much for the help. Best regards Dinah With df=2, the 5% critical value for Chi-square is 5.991, so the difference is not significant at the 5% level. You know that 0.374 is small for df=2 because the expected value is 2 (the df). mdehne posted on Wednesday, May 20, 2020 - 3:06 am Hi Linda, Bengt, or Tihomir I have got two questions: 1) Am I correct that the computation of the chi square difference test for MLR (http://www.statmodel.com/chidiff.shtml) does not ensure the positiveness as proposed by Satorra and Bentler (2010) and is thus based on the original paper (Satorra, 2000; Satorra & Bentler, 2001)? If I am correct: 2) Is there any computational opportunity to use Satorra & Bentler's (2010) alternative?

IFUP-TH/2003-23, ITEP-TH-42/03, TIT-HEP/506 NONABELIAN SUPERCONDUCTORS: VORTICES AND CONFINEMENT Roberto AUZZI , Stefano BOLOGNESI , Jarah EVSLIN , Kenichi KONISHI , Alexei YUNG Scuola Normale Superiore - Pisa , Piazza dei Cavalieri 7, Pisa, Italy Dipartimento di Fisica “E. Fermi” – Università di Pisa , Istituto Nazionale di Fisica Nucleare – Sezione di Pisa , Via Buonarroti, 2, Ed. C, 56127 Pisa, Italy Dept. of Physics, Tokyo Inst. of Technology , 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551 Japan Petersburg Nuclear Physics Institute , Gatchina, 188300 St. Petersburg, Russia Institute of Theoretical and Experimental Physics , B. Cheryomushkinskaia 25, 117259 Moscow, Russia We study nonabelian vortices (flux tubes) in gauge theories, which are responsible for the confinement of (nonabelian) magnetic monopoles. In particular a detailed analysis is given of SQCD with gauge group deformed by a small adjoint chiral multiplet mass. Tuning the bare quark masses (which we take to be large) to a common value , we consider a particular vacuum of this theory in which an subgroup of the gauge group remains unbroken. We consider flavors so that the sub-sector remains non asymptotically free: the vortices carrying nonabelian fluxes may be reliably studied in a semi-classical regime. We show that the vortices indeed acquire exact zero modes which generate global rotations of the flux in an group. We study an effective world sheet theory of these orientational zero modes which reduces to an sigma model in (1+1) dimensions. Mirror symmetry then teaches us that the dual group is not dynamically broken. 1. Introduction and Discussion Some sort of nonabelian vortices are believed to be responsible for confinement in QCD. Although in string theory these objects appear naturally, they turn out to be somewhat elusive in four-dimensional field theories. The existing literature on the subject certainly provides an incomplete picture. There are several reasons for this unsatisfactory situation. One of the reasons is that boundstates of vortices are not generally stable. An example is the case of an gauge theory (e.g., gauge theory with all fields in the adjoint representation) broken completely by a Higgs mechanism, where possible vortices represent nontrivial elements of the fundamental group -charged objects cannot be BPS saturated [1, 2], and this fact, together with the unknown dependence of their properties on the form of the potential, number of the fields, etc., has obstructed investigations of such vortices. Secondly, often these theories become strongly coupled at low energies and therefore an analytical study of the vortex configurations is very difficult. For instance, confinement in QCD may be due to the vortices of electric fields appearing in a dual (magnetic) ( or ?) theory. Unfortunately, neither the true nature of the effective magnetic degrees of freedom nor their form of interactions is known at the moment. ’t Hooft’s suggestion that they be abelian monopoles of a gauge-fixed theory , must still be verified. On the other hand, there is no experimental indication that the gauge group is dynamically broken to . Finally, in the examples of classical solutions for “nonabelian vortices” discussed so far in the literature the vortex flux is actually always oriented in a fixed direction in the Cartan subalgebra, showing that they are basically abelian. Useful hints come from the detailed study of a wide class of softly broken supersymmetric gauge theories where the dynamics appears particularly transparent. It was shown that, in fact, different types of confining vacua are realized in these models [5, 6, 7, 8, 9]. It is possible that in some cases confinement is due to the condensation of monopoles associated with the maximally abelian subgroup (a dual Meissner effect), as in the vacua surviving the adjoint mass perturbation in the pure SYM [5, 7, 8]. These cases provided the first examples of four-dimensional gauge theory models in which the ’t Hooft-Mandelstam mechanism of confinement is realized and can be analysed quantitatively. A detailed study of these cases has shown however that dynamical abelianization takes place there, with a characteristically richer meson spectrum [8, 1, 10, 11]. Indeed the low-energy effective gauge group of the theory is and the meson spectrum is classified according to the number of possible abelian strings via (cfr. (1.1)). Thus vorties and therefore mesons come in infinite towers, a feature not expected in the real world QCD. However, such is not the typical situation in softly broken theories with fundamental matter fields (quarks) [12, 9]. Confining vacua in , and gauge theories with quark flavor, are typically described by effective nonabelian dual gauge theories. For instance, in the so-called -vacua of gauge theory with flavors and vanishing bare quark masses, the low-energy effective theory is a dual theory. Addition of the adjoint chiral multiplet mass term breaks supersymmetry to , and the dual quarks in the of condense. These “dual quarks” have been recently identified as the quantum Goddard-Nuyts-Olive-Weinberg monopoles [14, 15]. Their condensation is believed to give rise to nonabelian confinement via formation of nonabelian flux tubes. In fact, the problem of nonabelian vortices is very closely related to (in a sense, it is one and the same problem as) that of the nonabelian monopoles on which they end. A key feature found in is that the quantum behavior of the nonabelian monopoles, and in fact the vacuum properties themselves depend critically on the presence of massless flavors of matter. We shall find below that the existence of nonabelian vortices similarly requires the presence of massless flavors in the underlying theory. Inspired by these developments, and based on a work by Marshakov and one of the authors (A.Y.) , we present in this paper a study of nonabelian superconductors, concentrating our attention on the properties of the vortices appearing in these systems. In a companion paper , we shall explore more extensively the properties of nonabelian monopoles themselves. Our analyses are done in a context where the dynamics of the model is well understood and the transition from a theory with abelian vortices to one with nonabelian vortices can be studied in a weakly coupled semi-classical regime throughout. The model we consider is probably the simplest of such models, QCD with gauge group and hypermultiplets of fundamental matter (quarks). Upon deformation of this theory via a small mass term for the adjoint chiral multiplet, , the Coulomb branch of the theory shrinks to a number of isolated vacua. Generically the vacuum expectation value (VEV) of the adjoint field breaks the gauge symmetry down to . However, it was shown in [12, 9] (see also ) that some of the vacua of QCD preserve a nonabelian subgroup. These vacua are classified by an integer . In a semiclassical regime, which is valid at large bare quark masses the adjoint scalar VEVs in those vacua take the form, where quark masses out of possible masses are chosen to satisfy the vacuum equations. When the quark masses are tuned to a common value , the pattern of the spontaneous breaking changes to The sector is a pure Yang Mills theory 111Recall that the quark masses come from the superpotentials and becomes strongly interacting at low energies and gets dynamically broken to . The sector, on the other hand, having massless flavors, remains weakly coupled as long as . Furthermore, in the presence of the aforementioned adjoint mass perturbation, the light squark fields acquire VEVs of color-flavor diagonal form (“Color-Flavor Locking”), which breaks the gauge group completely at scales far below the bare quark masses: . The theory is now in the Higgs phase, and develops vortex configurations, representing nontrivial elements of The key fact is that the system has an exact global symmetry, respected both by the interactions and by the scalar VEVS (1.4) and (1.6). A given vortex configuration however breaks this symmetry: it turns out that the symmetry is broken as (see below.) As a result, exact orientation zero modes of are generated. To work things out concretely, we analyse the case of the vacua of the gauge theory ( above) in detail in the main body of this paper. The subgroup, classically restored in the limit of equal quark masses, stays unbroken in the full quantum theory, as the relevant sector of the theory is infrared free if , or is conformal invariant if . On the other hand, of course, the underlying gauge theory is asymptotically free for , so we shall take to be either or . This is one of the important points of our analysis: by working in the regime in which the interactions remain weak at all scales, the continuous transition from the theory with abelian vortices (unequal quark masses) to the theory with nonabelian vortices which are qualitatively different, can be studied explicitly and reliably. The unbroken gauge group is further broken at a much lower mass scale, yielding vortices representing the nontrivial elements of Indeed, as the bare quark masses are tuned to a common value, , starting from unequal and generic values, the low-energy gauge group gets enhanced from to . The set of abelian vortices appearing in the unequal mass cases acquires a certain degeneracy and at the same time some orientation (in the color space) zero modes appear which relate the vortices of the same tension by global rotations. These zero modes are associated with the diagonal global subgroup of color crossed with the flavor which is an exact symmetry of the system. More precisely, the vortex zero modes parametrize as each vortex solution breaks the exact symmetry to a subgroup. We then work out the effective world-sheet theory of the vortex zero modes, and show that it reduces to the sigma model in (1+1) dimensions. Classically the sigma model has spontaneous symmetry breaking and appears to yield massless Goldstone fields. In terms of strings in four dimensions this would mean that is spontaneously broken and the string flux is oriented in some particular direction inside the gauge subgroup. However the quantum physics of the sigma model in (1+1) is quite different. It is well understood using the mirror map , which relates it to a sine-Gordon theory. In particular it is known that the model has a mass gap and no spontaneous symmetry breaking. In terms of strings in 4D this means that the string is not oriented in any particular direction inside group. This ensures that our vortices are truly nonabelian. The sine-Gordon superpotential is generated dynamically in the effective (1+1)-dimensional worldsheet theory which produces exactly two vacua. Our considerations can be straightforwardly generalized to the vacua of the theory with , with unbroken group, although our analysis in these more general cases is less complete. In particular, in the case of an theory broken to the zero modes of the vortex are described by a 2-dimensional CP sigma model whose mirror is an affine Toda theory with the desired vacua. The vortices studied in this paper, though stable in the low-energy theory, are strictly speaking metastable as the underlying gauge group (e.g., ) is simply connected. Their decay rates are however small, being exponentially suppressed by ratios of heavy monopole masses squared to the string tensions [19, 20] . Our result provides, albeit indirectly, a couterexample to the no-go theorem on the existence of monopoles with nonabelian charges discussed earlier . These nonabelian monopoles do exist in our theory as stable solitons and act as the sources of the nonabelian vortices considered here, and are actually confined by them. We exhibit here explicitly the transformations among the vortices, which imply certain non-local transformations for their sources. We will see that the zero modes of the vortices are normalizable. To calculate the zero mode of a single monopole, which necessarily sources an infinite vortex, we must integrate that of the vortex along its infinite length. Thus we find, as was seen in the flavorless cases of Refs. , that the zero mode of a single monopole is nonnormalizable. In a color-neutral configuration of monopoles the total length of the vortices may be taken to be finite and so the integral is finite, yielding normalizable zero modes which again generalize those known to exist in the flavorless case. Throughtout this paper we limit ourselves to cases with large bare quark masses where the original electric subgroup remains weakly coupled. When the bare quark masses are tuned to small values or even to zero, the low-energy system is weakly coupled when described in terms of the magnetic variables instead of the electric ones. The excitations which are quarks in the electric description at large quark masses become monopoles in the magnetic description at small quark masses [6, 22]. The properties of the corresponding - vacua have been studied in detail in , and in the case of a SCFT vacua of theory, in . The properties of these corresponding vacua are closely related by holomorphy. The organization of the paper is as follows. In Sect. 2 we review QCD with equal quark masses, work out its low-energy description, vacuum structure and the low-energy spectrum. In Sect. 3 we derive nonabelian Bogomolny equations, construct vortices and study their zero modes. We discuss the generalization to the more general case of breaking in Sect. 4. In Sect. 5 we work out the effective world sheet theory for orientational zero modes and discuss its physics. While this work was in preparation Ref. appeared which considers vortices in the very similar three-dimensional theory with an FI term. While these vortices are not strings but particles, the worldvolume theories appear to be related by dimensional reduction, and the vacuum structures and spectra appear to be the same. Thus many of our results as well as an extensive analysis of the relevent moduli spaces may be found there. 2.1. The Model The field content of QCD with the gauge group SU(3) and flavors of chiral multiplets is as follows. The vector multiplet consists of the gauge field , two Weyl fermions , and the scalar field , all in the adjoint representation of the gauge group. Here is a spinor index while all adjoint fields are matrices in the Lie algebra . The chiral multiplets of the theory consist of complex scalar squarks and and Weyl fermion quarks and , all in the fundamental representation of the gauge group. Here is a color index while is a flavor index, . This theory has a Coulomb branch on which the adjoint scalar acquires the vacuum expectation value (VEV) generically breaking the gauge group down to . Here and are the Gell-Mann matrices of the Cartan subalgebra. In this paper we consider the special vacua for which For these vacua the low-energy gauge group is , at least classically. We perturb the above theory by adding a small mass term for the adjoint matter via the superpotential Generally speaking, the superpotential breaks down to . The Coulomb branch shrinks to a number of isolated vacua [12, 9]. In the limit these vacua correspond to special singular points on the Coulomb branch in which pairs of monopoles/dyons or quarks become massless. Three of these points are always at strong coupling. They correspond to vacua of the pure gauge theory. The massless quark points are at weak coupling if the quark masses are large, . The vacua in which quarks become massless will be referred to as the quark vacua. We shall be mainly interested in these quark vacua. It is important to note that supersymmetry is not broken to the leading order in the parameter in the effective theory [1, 11]. In the effective low-energy theory the superpotential (2.3) gives rise to a superpotential linear in plus higher order corrections. If only the linear term in ’s in the superpotential is kept and if we restrict our attention to the special vacua (2.2), then it reduces to a Fayet-Iliopoulos term which does not break the supersymmetry. 2.2. symmetric low-energy theory The gauge group is broken down to by the VEV of the adjoint scalar (2.1) at generic values of quark masses. However, in the equal quark mass limit () which we shall consider from now on, the VEV of the field vanishes (see Sec. (2.3.)), and the low-energy gauge group is . W-bosons which are charged with respect to both factors of the low-energy group acquire a large mass of order . The third color components of quarks also become heavy in this vacua with masses of order of . Let us consider now the scales of order , which are well below W-boson masses ( is taken small, ). There the low-energy theory contains the following light fields of the vector multiplet: four complex scalar light fields and where is the color index, one gauge field and one gauge field together with their fermionic superpartners. For example the gauge fields are defined as follows: where our notation corresponds to expanding gauge and adjoint fields in the orthogonal basis of the Gell-Mann matrices, being the first three Gell-Mann matrices normalized as Light quark multiplets contain complex scalar -doublets , together with their fermionic superpartners, . The bosonic part of the low-energy effective theory then acquires the form Here is the covariant derivative in the adjoint representation of gauge subgroup, while where we suppress the color indices and are Pauli matrices. The coupling constants and correspond to and sectors respectively. The potential in the Lagrangian (2.5) is given by the D and F terms where other D-terms involving the adjoint scalar fields and () (which vanish at and ) are left implicit. The term in the second line arises when we expand fields and in the superpotential (2.3) around their VEV’s and keep only terms linear in fluctuations of these fields. As we have already noted, this means that the theory in (2.5) is a bosonic part of a supersymmetric theory. In particular this ensures that our theory has BPS vortices [1, 11, 16] (see also the seventh ref. in ). Below the scale the gauge group is broken and we have two coupling constants and which run according to the and renormalization group flows respectively. Note that with a logarithmic accuracy we can neglect mixing of these two coupling constants. In the case with four flavors the coupling does not run ( theory with is conformal) and is given by its value at the scale Since at large the sector is weakly coupled, it remains so at low energies. The coupling undergoes an additional renormalization from the scale to the scale determined by the masses of light states in the low-energy theory (which are of the order of , see next subsection). Thus we have where we use the fact that the one loop coefficient of the -function for theory is and substitute and the electric charge , see (2.6). Clearly, this coupling is even smaller than the one in the sector. If the number of the quark flavors is taken to be five, the gauge coupling constant also runs to smaller values towards the infrared. In general therefore one has , both small, and we shall not need more details in the analyses below. 2.3. Vacuum structure and low-energy spectrum In this subsection we review the vacuum structure and low-energy mass spectrum of QCD [12, 9] generalizing the analysis made in to the case of the low-energy group. To find the vacua of the effective theory (2.5) we have to look for the zeros of the potential (2.7). At generic large values of quark masses solutions have the following structure [9, 16]. Besides the three strong coupling vacua which exist already in the pure gauge theory there are vacua and vacua, were is the number of quark flavors which develop non-zero VEV’s. Here we are mostly interested in vacua, which have an unbroken gauge group which becomes exact in the case of equal quark masses. Clearly the minimal number of flavors for which we can have a vacuum is . Let us consider this case first. The adjoint scalar matrix is given by where is the common mass of both flavors. In the above notation (2.10) reads For real values of and we can use gauge rotations to make squark VEV’s real. We write the squark field as a matrix where is a color index and is a flavor one. Then the squark VEV’s are given by where we have used color-flavor mixed matrix notation, and we have introduced which acts as the Fayet-Iliopoulos parameter of the . sets the scale of the low-energy theory (2.5). Only the two upper color components and the first two flavors are shown in Eq.(2.12): all other components have vanishing VEVS. Now consider the spectrum of light fields in this vacuum. The low-energy gauge group is broken completely by squark VEV’s and all gauge bosons acquire masses. The mass matrix for the gauge fields , can be read off of the kinetic terms for the quarks in (2.5). It turns out that it is diagonal in the basis , . In particular, the mass of is given by while the mass of the W-boson is The mass matrix for squarks is now of size including four real components of complex fields and for each color and flavor. It has four zero eigenvalues associated with the four states “eaten” by the Higgs mechanism for and gauge factors and two non-zero eigenvalues coinciding with gauge boson masses (2.14) and (2.15). The eigenvalue (2.14) corresponds to three squark eigenvectors while the one in (2.15) corresponds to nine squark eigenvalues. Altogether we have one long multiplet with mass (2.14), containing eight bosonic states (3 states of the massive field plus 2 states of plus 3 squark states) and eight fermionic states. In addition we have three long multiplets with mass (2.15) labeled by the color index also containing eight bosonic and eight fermionic states each 222 See for a discussion of the emergence of long multiplets in Seiberg-Witten theory upon adjoint mass term deformation.. Note that no Nambu-Goldstone multiplets appear in this vacuum: all phases associated with broken symmetries are ”eaten” by Higgs mechanism. Actually, in the theory with discussed above, the gauge interactions become strong below the scale , and the properties of the theory at low energies (at mass scales of order of ) cannot be determined from the Lagrangian (2.5) only. For this reason, we introduce more flavors into our theory and consider the theory with or . The low-energy then remains in the weak coupling regime. This theory has vacua of the type described above, for unequal quark masses. Each of these vacua corresponds to choosing two flavors out of which develop VEV’s. This gives choices for . In the limit of equal masses all six vacua coalesce and a Higgs branch develops from the common root. The dimension of this Higgs branch is [12, 16]. To see this note that we have real variables subject to four -term and eight -term conditions in the potential (2.7). Also 3+1 gauge phases are eaten by the Higgs mechanism. Thus we have remaining degrees of freedom. We consider below a special submanifold of the Higgs branch which admits BPS flux tubes (cf. [1, 25, 16, 26]). This base submanifold is compact and has the minimal value of the quark condensate . One point on this submanifold which corresponds to non-zero VEV of the first flavor and non-zero VEV of the second flavor while all other components are zero is given in (2.12). Other points on the base of the Higgs branch are given by a flavor rotation of (2.12). The dimension of the base submanifold of the Higgs branch is . To see this note that VEV’s of two flavors break symmetry down to . Thus the number of ”broken” generators is and also we have to subtract four phases “eaten” by the Higgs mechanism. Other points on the dimensional Higgs branch correspond to non-zero VEV’s of massless moduli fields, and these points do not admit BPS strings. In particular, the ANO strings on the Higgs branch were studied in [28, 26], they correspond to a limiting case of type I strings with the logarithmically thick tails associated with massless scalar fields. We shall not discuss here strings at generic points on the Higgs branch. Before ending this subsection, we need to comment on the soliton sector. In the monopole sector, all solitonic states associated with the symmetry breaking (2.10) are massive. In particular, one finds an exactly degenerate doublet of BPS monopoles of minimum mass . Apparently, a set of “monopole” states become massless as the bare quark masses are tuned to a common value, , at which point the low-energy gauge group gets enhanced from to . For instance, the BPS monopole carrying magnetic charge (1,-1) with respect to two factors above has mass proportional to , and appears to become massless in the limit of restoration. Classically this “state” becomes infinitely extended in space in such a limit, and at the same time the fields degenerate into trivial vacuum configuration 333This is analogous to the fate of the ’t Hooft - Polyakov monopole of the spontaneously broken theory, in the limit . . More importantly, as the topological structure of the theory changes in the restoration limit (from Eq.(3.6) to Eq.(3.13), see below) such a “massless monopole” is no longer topologically stable. 3. Non-abelian Vortices We will now construct (BPS) vortex solutions in the theory described above and show that they possess exact zero modes. 3.1. Non-abelian Bogomolny Equations As we have already anticipated, by restricting ourselves to a particular base submanifold of the Higgs branch of the theory with four flavors, we are able to deal with BPS strings throughout. By gauge and flavor rotations the squark VEVS can be taken to be of the form (2.12). Then classically only the two flavors which develop VEV’s will play a role in the vortex solution. Other flavors remain zero on the solution, and one can consider the squark fields to be matrices. Note however that the additional two flavors are crucial in the quantum theory, in keeping the interactions weakly coupled 444In fact the additional flavors are important even classically. In the presence of additional flavors strings can turn into semilocal strings, see for a review on semilocal strings. We shall not study this issue here.. Let us make an ansatz, and a convenient redefinition of the squark fields The low-energy action (2.5) then reduces ( and stand for the and coupling constants at the scale , respectively) to where , leading to the following first order equations for strings Here is the sign of the total flux specified below. The string solutions found in the case of unequal quark masses can be readily recognized as particular solutions of these equations. To construct them we further restrict the gauge field to the single color component (by setting ), and consider only squark fields of the color-flavor diagonal form: by setting all other components to zero. For unequal masses the relevant topological classification was and the allowed strings formed a lattice labeled by two integer winding numbers. In particular, assume that the first flavor winds times while the second flavor winds times and look for solutions of (3.4) using the following ansatz 555We use a notation slightly different from the one used in : instead of ; instead of . The cylindrical coordinates are here denoted as ), the vortex center extending along the axis. where are polar coordinates in the (1,2) plane while the profile functions , for scalar fields and , for gauge fields depend only on . The profile functions in these equations are determined by the following boundary conditions for the gauge fields, and the requirement that the squark fields be everywhere regular. The behavior of the latter at , and that at (e.g., if ), follow from these requirements. Here the sign of the string flux is The tension of a -string for the case of equal quark masses is determined by the flux of the gauge field alone and is given by Note that and -strings are exactly degenerate. Note also that does not enter the central charge of the algebra and so does not affect the string tension. The stability of the string in this case is due to the factor of the low-energy group only. The equations (3.4) represent a nonabelian generalization of the Bogomolny equations for the ANO string . For a generic -string equations (3.8) do not reduce to the standard Bogomolny equations. For instance, for the -string these equations reduce to two Bogomolny equations while for the and strings they do not. The charges of -strings can be plotted on the Cartan plane of the algebra. We shall use the convention of labeling the flux of a given string by the magnetic charge of the monopole which produces this flux and must be attached to each end. This is possible since both string fluxes and monopole charges are elements of the group . This convention is convenient because specifying the flux of a given string automatically fixes the charge of the monopole that it confines. Our strings are formed by the condensation of squarks which have electric charges equal to the weights of algebra. The Dirac quantization condition tells us that the lattice of -strings is formed by roots of the algebra . The lattice of -strings is shown in Fig. 3. Two strings and are the “elementary” or “minimal” BPS strings. If we plot two lines along charges of these “elementary” strings (see Fig. 3) they divide the lattice into four sectors. It turns out that the strings in the upper and lower sectors, which are labeled by black circles in Fig. 3, are BPS but they are marginally unstable at real quark mass ratios. Instead, strings in the right and left sectors, which are labeled in Fig. 3 by white circles, are bound states of the “elementary” ones but they are not BPS. 3.2. Minimal vortex of generic orientation: zero modes Actually, the relevant homotopy group here is instead of (3.6), as we are working with the case of equal quark masses where the low-energy gauge group is . The generator of the fundamental group is a loop which encircles the once , and thus to calculate the tension of a string, or to determine whether it is stable, it suffices to simply count the winding number around this circle. This means that the lattice of -strings reduces to a tower labeled by one integer . For instance, the -string becomes completely unstable as it winds forward once and then backward once, and so there is no net winding and so no topological charge. On the restored group manifold it is also trivial, as it goes half way around the equator and then goes back. The string goes all of the way around the equator, making a contractible loop, but is stable because it wraps the twice (it wraps the original once). On the other hand, the and strings cannot be shrunk because they correspond to a half circle along the equator. They have the same tension (see (3.12)) for equal quark masses and thus apparently belong to doublet of an . In general non-BPS strings on the -lattice (see Fig. 3) become unstable as they have tensions above their BPS bounds and we are left with BPS strings at each winding number . The reduction of the string lattice is illustrated Fig. 4. Most importantly, this suggests that there be a continuously infinite number of vortices of minimum winding and with the same tension, of which the and vortices discussed above are just two particular cases (Fig. 5). Below we show that this is indeed correct, by a continuous deformation of the -string solution transforming it into a -string. This deformation leaves the string tension unchanged and therefore corresponds to an orientational zero mode. First let us separate physical variables from the gauge phases eaten by the Higgs mechanism in the quark fields. To do so we use the following parametrization of the quark matrix Here and are matrices from the and gauge factors respectively while and are real. The parametrization (3.15) represents eight real variables in terms of 3+1=4 gauge phases eaten by the Higgs mechanism and four physical variables and . In particular, (2.12) corresponds to Now let us fix the unitary gauge (at least globally, which is enough for our purposes) by imposing the condition that squark VEV’s are given precisely by (3.16) and so all gauge phases are zero. Now transform the -string solution (3.7) into unitary gauge, which corresponds to the singular gauge in which the string flux comes from the singularity of the gauge potential at zero. In this gauge the solution (3.7) for the -string takes the form Note that a global diagonal subgroup in the product of gauge and flavor symmetries is not broken by the squark VEV. Namely, where is a global rotation in while the squark VEV matrix is given by (2.12). We call this unbroken group . Now let us apply this global rotation to the string solution (3.17). We find where we use a matrix notation for the gauge field, . Using the representation 666Explicitly, if , the rotation matrix is given by . where is a unit vector on , , we can rewrite the gauge potential of (3.19) in the form Since the symmetry is not broken by squark VEV’s it is physical and does not correspond to any of the gauge rotations eaten by the Higgs mechanism. To see this explicitly let us rewrite the quark field of our solution (3.19) using the parametrization (3.15). We get We see that all gauge phases are zero while physical variables acquire an -dependence. Clearly the solution (3.19) interpolates between and strings. In particular it gives a -string for and a -string for . The symmetry is exact and the tension of the string solution (3.19) is independent of : see (3.12). However, an explicit vortex solution breaks the exact as the two angles associated with vector - two orientational bosonic zero modes of the string - parametrize the quotient space . In the regular gauge, the minimal nonabelian vortex of generic orientation (3.19) takes the form where is given by Eq. (3.20) and the profile functions are solutions of Eq.(3.8) for . In this gauge it is particularly clear that this solution smoothly interpolates between the and solutions: if the first flavor squark winds at infinity while for the second flavor does. To further convince ourselves that the rotation considered above corresponds to physical zero modes we can construct a gauge invariant operator which has -dependence on our solution. One example is which is a matrix in flavor indices. Inserting the solution (3.25) this operator reads We see that is a gauge invariant operator which has -dependence localized near the string axis where is non-zero. As we have already mentioned the central charge of the algebra reduces to the component of the flux If we define a gauge invariant flux it reduces to the one in (3.28) so the

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Austin Grocers recently reported the following income statement (in millions of dollars): Sales $700 Operating costs 500 EBIT $200 Interest 40 EBT $160 Taxes (40%) 64 Net income $ 96 ------ Dividends $ 32 Addition to retained $ 64 earnings This ye I do not understand Pro-forma income statements & I have to create one. Here is the problem: Style Right Company makes hair dryers. During the past few days, its accountants have been preparing the master budget for the coming year, 2006. To date, they have gathered the following projected data: Sales revenue (at $20 per u Uncle Ralph's sells cookies in a large shopping mall. The following multi-step income statement was prepared for the year ending December 31, 2004. Sales $36,000 Cost of Good Sold 4,000 Gross Profit The net income reported on the income statement for the current year was $275,000. Depreciation recorded on fixed assets and amortization of patents for the year were $40,000 and $9,000, respectively. Balances of current asset and current liability accounts at the end and at the beginning of the year are as follows: Preparation of Income Statement (What are the Net Income and Earning per Share?) and Retained Earnings Statement (What is the Retained earnings on June 30, 2006?). Prepare an income statement and a statement of retained earnings for Big Sky Corporation for the year ended June 30, 2006, based on the following information---see at Please assist with the attached problem. The following data pertain to Brown Brother's Company operations for July. Total Product X Product Y Number of units sold... 10,000 12,000 Selling price per unit... $20,000 $25.00 Variable cost per unit: Production... $9 $10.00 Selling & administration... $3 The following selected information is taken from the records of Beckstrom Corporation. Accounts payable . . . . . . . . . . . . . . . . . . . . . . . . $ 35,000 Accounts receivable . . . . . . . . . . . . . . . . . . . . . . 65,000 Advertising expense . . . . . . . . . . . . . . . . . . . . . . 15,000 Cash . . . . . . . . . Find the most recent annual report of a company of your interest and answer the following questions regarding the company's Income Statement Please see the attached file. ? add a copy of the Income Statement you used In the library or on the Internet, find the most recent annual report of a company of your interest and answer the following questions regarding the company's Income Statement:  What accounting period(s) does the Income Statement (IS) cove Please help me with the following 3 problems. Fundamentals of Financial Management 3-1 Income Statement little Books Inc. recently reported $3 million of net income. Its EBIT was $6 million, and its tax rate was 40 percent. What was its interest expense? [Hint: Write out the headings for an income statement and then fill Following are three separate events affecting the managerial accounting systems for different companies. Match the management concept(s) that the company is likely to adopt for the event identified.There is some overlap in the meaning of customer orientation and total quality management and, therefore, some responses can include I need help with Exercise 18-5 (Management concepts) and questions 1 and 2 of Problem 18-8A (Manufacturing and income statements; inventory analysis.) Please see attached document for full description. Thank you!! Please assist with this problem to include reference sources. Mohican Inc. has the following items on its GAAP financial statements: Net income per the financial statements is $5,000,000. A $7,500,000 contingent liability and related expense for legal damages is included in the financial statements. The footnote ind Pro-Forma Income Statement Silver Company has asked you to prepare a pro-forma income statement for the coming year.The following information is available: Expected sales revenue . . . . . . . . . . . . . . . .. . . $1,240,000 Manufacturing costs: Variable cost of goods sold . . . . . . . . . . . . . . 625,000 Fixed ove The income statement and supplemental information for Xtra Company are provided below: (see attachment) a. Acquired equipment for $87,000 by paying cash of $75,750 and is issuing a note payable for the difference. b. Beginning cash balance, $15,020. Increase in cash, $86,490 c. Collections from customers were $5,250 more than The sustainable Chemical Corporation produced a specialty chemical called SC. At the beginning of each year, the company estimated what the cost of SC would be for the coming year as one factor in the development of its pricing and promotion strategies and as a benchmark against which to compare the actual costs of production. T Please help me prepare an income statement and answer the following questions for this situation: Diane Maynard was grateful for the balance sheets that her friend prepared. ( I attached the balance sheet I did) In going over the numbers, she remarked, "it's sort of surprising that cash increased by $31677, but net income was 2. An S corporation's profit and loss statement shows net profits of $90,000(book income). The corporation has three equal shareholders. From supplemental data, you obtain the following information about some items that are included in the $90,000. Selling expenses (21,200) Municipal bond interest income 2,000 Dividends re Listed below are the results of Rulon Candies' operations for 2005 and 2006. (Assume 4,000 shares of outstanding stock for both years.) 2006 2005 Sales $300,000 $350,000 Utilities expenses 15,000 8,500 Employee salaries 115,000 110,000 Advert The trial balance as at 31 March 2007 (1) Office equipment, at cost 2,750,000 (DR) Motor vehicles, at cost 937,500 (DR) Land, at valuation 2,487,500 (DR) Accumulated depreciation at 1 April 2006 Office equipment 2,016,800 (CR) Motor vehicles 126,572 (CR) Additional information relates to the above: Depreciation is t A contribution income statement for the Nantucket Inn is shown below. (Ignore income taxes). Revenue.............................................$500,000 Less: Variable expenses............................300,000 Contribution margin................................$200,000 Less: Fixed expenses.............................. 1. Build a Statement of Cash Flows using the Indirect Method 2. Determine Income Tax owed and amount paid to Vendors Please see attachment. My next question is on Sales. I need to list the Total Sales (as revenue) at the beginning of my Income Statement. Will that include the following? Sales Cash (from start up in November 2005 Balance sheet) $10,172 accounts receivable $870 minus ending cash (what shows up as cash on hand in March 20 I do not understand the role of the income statement on an accrual basis. If you are given a balance sheet from November 2005 (which is the company start up), and one from March 2006 (when the company is no longer open), how do you calculate the income statement. Do you use the information given from the November 2005 as the beg Problem 16-6 Contribution Margin and Functional Income Statements The following information is available for Dabney Company for 2006: Sales revenue (at $20 per unit) $151,200 Fixed manufacturing costs 24,000 Variable manufacturing costs (at $8 per unit) 60,480 Fixed selling expenses 70,000 Variable selling expenses (at $2 See attached file for full problem description. Burrel Manufacturing Company established the following standard price and cost data Sales price 7.50 per unit Variable Manufacturing cost 3.00 per unit fixed manufacturing cost 3,000 total fixed selling and administrative cost 1,2 Using the trial balance given at the top of the next page, prepare an income statement. Debit Prepare an income statement and a statement of retained earnings for Big Sky Corporation forthe year ended June 30, 2006, based on the following information: Capital stock (1,500 shares @ $100) . . . . . . . . . . . . . . . . . . . . . . . . . . $150,000 Retained earnings, July 1, 2005 . . . . . . . . . . . . . . . Income statement for Shirley Company for 2005 and 2006 follow: 2006 2005 Sales 240,000 200,000 Cost of Goods Sold 147,900 108,000 Selling Expenses 40,100 I am having trouble preparing this income statement can you please sow me how to do it and explain why it is done this way? Problem 2-5 Income Statement Preparation The following information is taken from the records of Hill, Dunn, & Associates for the year ended December 31, 2006. P.A.S.S. Power Accounting System Softwar

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Affine Mirković-Vilonen polytopes Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie algebra has a crystal in the sense of Kashiwara, which describes its combinatorial properties. For a given , there is a limit crystal, usually denoted by , which contains all the other crystals. When is finite dimensional, a convex polytope, called the Mirković-Vilonen polytope, can be associated to each element in . This polytope sits in the dual space of a Cartan subalgebra of , and its edges are parallel to the roots of . In this paper, we generalize this construction to the case where is a symmetric affine Kac-Moody algebra. The datum of the polytope must however be complemented by partitions attached to the edges parallel to the imaginary root . We prove that these decorated polytopes are characterized by conditions on their normal fans and on their -faces. In addition, we discuss how our polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody algebras. Our main tool is an algebro-geometric model for constructed by Lusztig and by Kashiwara and Saito, based on representations of the completed preprojective algebra of the same type as . The underlying polytopes in our construction are described with the help of Buan, Iyama, Reiten and Scott’s tilting theory for the category . The partitions we need come from studying the category of semistable -modules of dimension-vector a multiple of . Let be a symmetrizable generalized Cartan matrix, with rows and columns indexed by a set . We denote by the Kac-Moody algebra defined by . It comes with a triangular decomposition , with a root system , and with a Weyl group . The simple roots are indexed by and the group is a Coxeter system, generated by the simple reflections . We denote the length function of by and the set of positive (respectively, negative) roots by (respectively, ). The root lattice is denoted by and we set . The canonical pairing between and its dual will be denoted by angle brackets. Lastly, we denote by the set of linear combinations of the simple roots with nonnegative coefficients and we set . The combinatorics of the representation theory of is captured by Kashiwara’s theory of crystals. Let us summarize quickly this theory; we refer the reader to the nice survey for detailed explanations. A -crystal is a set endowed with maps , , , and , for each , that satisfy certain axioms. This definition is of combinatorial nature and the axioms stipulate the local behavior of the structure maps around an element . This definition is however quite permissive, so one wants to restrict to crystals that actually come from representations. In this respect, an important object is the crystal , which contains the crystals of all the irreducible lowest weight integrable representations of (see Theorem 8.1 in ). This crystal contains a lowest weight element annihilated by all the lowering operators , and any element of can be obtained by applying a sequence of raising operators to . The crystal itself is defined as a basis of the quantum group in the limit . Working with this algebraic construction is cumbersome, and there exist other, more handy, algebro-geometric or combinatorial models for . One of these combinatorial models is Mirković-Vilonen (MV) polytopes. In this model, proposed by Anderson , one associates a convex polytope to each element . The construction of is based on the geometric Satake correspondence. More precisely, the affine Grassmannian of the Langlands dual of contains remarkable subvarieties, called MV cycles after Mirković and Vilonen . There is a natural bijection from onto the set of all MV cycles [11, 12, 23], and is simply the image of by the moment map. Using Berenstein and Zelevinsky’s work , the second author showed in that these MV polytopes can be described in a completely combinatorial fashion: these are the convex lattice polytopes whose normal fan is a coarsening of the Weyl fan in the dual of , and whose -faces have a shape constrained by the tropical Plücker relations. In addition, the length of the edges of is given by the Lusztig data of , which indicate how , viewed as a basis element of at the limit , compares with the PBW bases. 1.2 Generalization to the affine case This paper aims at generalizing this model of MV polytopes to the case where is an affine Kac-Moody algebra. Obstacles pop up when one tries to generalize the above constructions of to the affine case. Despite difficulties in defining the double-affine Grassmannian, the algebro-geometric model of using MV cycles still exists in the affine case, thanks to Braverman, Finkelberg and Gaitsgory’s work ; however, there is no obvious way to go from MV cycles to MV polytopes. On the algebraic side, several PBW bases for have been defined in the affine case by Beck , Beck and Nakajima , and Ito , but the relationship between the different Lusztig data they provide has not been studied111Recently, Muthiah and Tingley have considered this problem in the case . They have shown that the resulting combinatorics matches that produced in the present paper, in the sense that the MV polytopes coming from the Lusztig data provided by the PBW bases match those defined here. It should be easy to extend this result to the case of an arbitrary symmetric affine Kac-Moody algebra.. As recalled above, in finite type, the normal fan of an MV polytope is a coarsening of the Weyl fan, so the facets of an MV polytope are orthogonal to the rays in the Weyl fan. Therefore an MV polytope is determined just by the position of theses facets, which form a set of numerical values dubbed “Berenstein-Zelevinsky (BZ) data”. In the case , a combinatorial model for an analog of these BZ data was introduced by Naito, Sagaki, and Saito [51, 52]. Later, Muthiah related this combinatorial model to the geometry of the MV cycles. However, the complete relationship between this combinatorial model and our affine MV polytopes is not yet clear. 1.3 The preprojective model Due to the difficulties in the MV cycle and PBW bases models, we are led to use a third construction of , recently obtained by the first two authors for the case of a finite dimensional . This construction uses a geometric model for based on quiver varieties, which we now recall. This model exists for any Kac-Moody algebra (not necessarily of finite or affine type) but only when the generalized Cartan matrix is symmetric. Then is the incidence matrix of the Dynkin graph ; here our index set serves as the set of vertices and is the set of edges. Choosing an orientation of this graph yields a quiver , and one can then define the completed preprojective algebra of . A -module is an -graded vector space equipped with linear maps. If the dimension-vector is given, we can work with a fixed vector space; the datum of a -module then amounts to the family of linear maps, which can be regarded as a point of an algebraic variety. This variety is called Lusztig’s nilpotent variety; we denote it by , where is the dimension-vector. Abusing slightly the language, we often view a point as a -module. For , let be the set of irreducible components of . We set . In , Lusztig endows with a crystal structure, and in , Kashiwara and Saito show the existence of an isomorphism of crystals from onto . This isomorphism is unique since has no non-trivial automorphisms. Given a finite-dimensional -module , we can consider the dimension-vectors of the -submodules of ; they are finitely many, since they belong to a bounded subset of the lattice . The convex hull in of these dimension-vectors will be called the Harder-Narasimhan (HN) polytope of and will be denoted by . The main result of is equivalent to the following statement: if is finite dimensional, then for each , the set contains a dense open subset of . In other words, is the general value of the map on . This result obviously suggests a general definition for MV polytopes. We will however see that for of affine type, another piece of information is needed to have an complete model for ; namely, we need to equip each polytope with a family of partitions. Our task now is to explain what our polytopes look like, and where these partitions come from. 1.4 Faces of HN polytopes Choose a linear form and let denote the maximum value of on . Then is a face of . Moreover, the set of submodules whose dimension-vectors belong to has a smallest element and a largest element . The existence of and follows from general considerations: if we define the slope of a finite dimensional -module as , then is the semistable subquotient of slope zero in the Harder-Narasimhan filtration of . Introducing the abelian subcategory of semistable -modules of slope zero, it follows that, for each submodule , In other words, the face coincides with the HN polytope of , computed relative to the category , and shifted by . Our aim now is to describe the normal fan to , that is, to understand how , and depend on . For that, we need tools that are specific to preprojective algebras. 1.5 Tits cone and tilting theory One of these tools is Buan, Iyama, Reiten and Scott’s tilting ideals for . Let be the simple -module of dimension-vector and let be its annihilator, a one-codimensional two-sided ideal of . The products of these ideals are known to satisfy the braid relations, so to each in the Weyl group of , we can attach a two-sided ideal of by the rule , where is any reduced decomposition of . Given a finite-dimensional -module , we denote the image of the evaluation map by . Recall that the dominant Weyl chamber and the Tits cone are the convex cones in the dual of defined as We will show the equality for any finite dimensional -module , any and any linear form . This implies that is a vertex of and that the normal cone to at this vertex contains . This also implies that is contained in When runs over the Tits cone, it generically belongs to a chamber, and we have just seen that in this case, the face is a vertex. When lies on a facet, is an edge (possibly degenerate). More precisely, if lies on the facet that separates the chambers and , with say , then . Results in and moreover assert that is the direct sum of a finite number of copies of the -module . There is a similar description when is in ; here the submodules of that come into play are the kernels of the coevaluation maps , where again . 1.6 Imaginary edges and partitions (in affine type) From now on in this introduction, we focus on the case where is of symmetric affine type, which in particular implies is of untwisted affine type. The root system for decomposes into real and imaginary roots ; the real roots are the conjugate of the simple roots under the Weyl group action, whereas the imaginary roots are fixed under this action. The Tits cone is . We set . The projection maps onto the “spherical” root system , whose Dynkin diagram is obtained from that of by removing an extending vertex. The rank of is , which is also the multiplicity of the imaginary roots. The vector space identifies with the hyperplane of the dual of . The root system defines an hyperplane arrangement in , called the spherical Weyl fan. The open cones in this fan will be called the spherical Weyl chambers. Together, this fan and the hyperplane arrangement that the real roots define in make up a (non locally finite) fan in the dual of , which we call the affine Weyl fan and which we denote by . Each set of simple roots in is a basis of ; we can then look at the dual basis in , whose elements are the corresponding fundamental coweights. We denote by the set of all fundamental coweights, for all possible choices of simple roots. Elements in are called spherical chamber coweights; the rays they span are the rays of the spherical Weyl fan. Now take a -module . As we saw in the previous section, the normal cone to at the vertex (respectively, ) contains (respectively, ). Altogether, these cones form a dense subset of the dual of : this leaves no room for other vertices. This analysis also shows that the normal fan to is a coarsening of . Thus, the edges of point in directions orthogonal to one-codimensional faces of , that is, parallel to roots. In the previous section, we have described the edges that point in real root directions. We now need to understand the edges that are parallel to . We call these the imaginary edges. More generally, we are interested in describing the faces parallel to . Let us pick and let us look at the face . As we saw in section 1.4, this face is the HN polytope of , computed relative to the category . It turns out that and only depend on the face of the spherical Weyl fan to which belongs. We record this fact in the notation by writing for . We need one more definition: for , we say that a -module is a -core if it belongs to for all sufficiently close to . In other words, the category of -cores is the intersection of the categories , taken over all spherical Weyl chambers such that . For each , the set of indecomposable modules is a constructible subset of . It thus makes sense to ask if the general point of an irreducible subset of is indecomposable. Similarly, the set of modules that belong to is an open subset of , so we may ask if the general point of an irreducible subset of is in . In section 7.4, we will show the following theorems. For each integer and each , there is a unique irreducible component of whose general point is an indecomposable -core. We denote by this component. Let be a positive integer and let be a spherical Weyl chamber. There are exactly irreducible components of whose general point is an indecomposable module in . These components are the , for . In Theorem 1.2, the multiplicity of the root materializes as a number of irreducible components. Now let and pick in a spherical Weyl chamber . Let be a general point of and let , an object in . Write the Krull-Schmidt decomposition of as , with , …, indecomposable; then each is in , so for a certain integer . Moreover, it follows from Crawley-Boevey and Schröer’s theory of canonical decomposition that each is the general point of an irreducible component . Using Theorem 1.2, we then see that each is a component for a certain . Gathering the integers according to the coweights , we get a tuple of partitions . In this context, we will show that the partition depends only on and , and not on the Weyl chamber . We are now ready to give the definition of the MV polytope of : it is the datum of the HN polytope , for general in , together with the family of partitions defined above. 1.7 -faces of MV polytopes Let us now consider the -faces of our polytopes . Such a face is certainly of the form where belongs to a -codimensional face of . There are three possibilities, whether belongs to , or . Suppose first that . Then the root system is finite of rank , of type or type . More precisely, let be of minimal length such that and let ; then the element maps onto the root system . The full subgraph of defined by gives rise to a preprojective algebra . The obvious surjective morphism induces an inclusion , whose image is the category . Further, the tilting ideals provide an equivalence of categories whose action on the dimension-vectors is given by . Putting all this together, we see that is the image under of the HN polytope of the -module . In addition, genericity is preserved in this construction: if is a general point in an irreducible component of a nilpotent variety for , then is a general point in an irreducible component of a nilpotent variety for . When is of type , this implies that the -face obeys the tropical Plücker relation from . A similar analysis can be done in the case where is in . It then remains to handle the case where , that is, where belongs to a face of codimension one in the spherical Weyl fan. Here is an affine root system of type . The face separates two spherical Weyl chambers of , say and , and there are spherical chamber coweights and such that and . Choose and . Assume that is the general point of an irreducible component. As we saw in section 1.6, the modules and are then described by tuples of partitions and , respectively. Both these modules are subquotients of , so this latter contains the information about the partitions for all . Let be the polytope obtained by shortening each imaginary edge of the -face by . Then , equipped with the two partitions and , is an MV polytope of type . A partition can be thought of as an MV polytope of type , since the generating function for the number of partitions equals the graded dimension of the upper half of the Heisenberg algebra. Thus, the family of partitions can be thought of as an MV polytope of type . We can therefore regard the datum of the face and of the partitions as an MV polytope of type . Theorem 1.3 will be proved in section 7.5. Our method is to construct an embedding of into , where is the completed preprojective algebra of type ; this embedding depends on and its essential image is large enough to capture a dense open subset in the relevant irreducible component of Lusztig’s nilpotent variety. In this construction, we were inspired by the work of I. Frenkel et al. who produced analogous embeddings in the quiver setting. So the final picture is the following. Let be the set of all lattice convex polytopes in , equipped with a family of partitions , such that: The normal fan to is a coarsening of the Weyl fan . To each spherical Weyl chamber corresponds a imaginary edge of ; the difference between the two endpoints of this edge is equal to . A -face of is an MV polytope of type , or ; in the type case, this means that its shape obeys the tropical Plücker relation. At the end of section 1.6, we associated an element of to each . The map is bijective. 1.8 Lusztig data As explained at the end of section 1.1, for a finite dimensional , the MV polytope of an element geometrically encodes all the Lusztig data of . In more detail, let be the number of positive roots. Each reduced decomposition of the longest element of provides a PBW basis of the quantum group , which goes to the basis at the limit . To an element , one can therefore associate many PBW monomials, one for each PBW basis. In other words, one can associate to many elements of , one for each reduced decomposition of . These elements in are called the Lusztig data of . A reduced decomposition of specifies a path in the 1-skeleton of that connects the top vertex to the bottom one, and the corresponding Lusztig datum materializes as the lengths of the edges of this path. With this in mind, we now explain that when is of affine type, our MV polytopes provide a fair notion of Lusztig data. To this aim, we first note that a reasonable analog of the reduced decompositions of is certainly the notion of “total reflection order” (Dyer) or “convex order” (Ito), see [15, 28]. By definition, this is a total order on such that (Unfortunately, the convexity relation implies that for any positive integers and . We therefore have to accept that is only a preorder; this blemish is however limited to the imaginary roots.) A convex order splits the positive real roots in two parts: those that are greater than and those that are smaller. One easily shows that the projection maps onto a positive system of . Thus, there exists such that Given such a convex order , we will construct a functorial filtration on each finite dimensional -module , such that each is a vertex of . The family of dimension-vectors are the vertices along a path in the -skeleton of connecting the top vertex and bottom vertices. The lengths of the edges in this path form a family of natural numbers , defined by the relation . Further, if we choose satisfying (1.1), then and . Fix and take a general point in . Besides the family of natural numbers mentioned just above, we can construct a tuple of partitions by applying the analysis carried after Theorem 1.2 to the module , where is the spherical Weyl chamber containing . To , we can thus associate the pair consisting in the two families and . All this information can be read from . We call the Lusztig datum of in direction . Let us denote by the set of all partitions and by the set of finitely supported families of non-negative integers. The map is bijective. The MV polytope contains the information of all Lusztig data of , for all convex orders. This is in complete analogy with the situation in the case where is finite dimensional. The conditions on the -faces given in the definition of say how the Lusztig datum of varies when the convex order changes; they can be regarded as the analog in the affine type case of Lusztig’s piecewise linear bijections. The knowledge of a single Lusztig datum of , for just one convex order, allows one to reconstruct the irreducible component . This fact is indeed an ingredient of the proof of injectivity in Theorem 1.4. Through the bijective map , the set acquires the structure of a crystal, isomorphic to . This structure can be read from the Lusztig data. Specifically, if is the smallest element of the order , then is the -coordinate of , and the operators and act by incrementing or decrementing this coordinate. As mentioned at the beginning of section 1.3, Beck in , Beck and Nakajima in , and Ito in construct PBW bases of for of affine type. An element in one of these bases is a monomial in root vectors, the product being computed according to a convex order . To describe a monomial, one needs an integer for each real root and a -tuple of integers for each imaginary root , so in total, monomials in a PBW basis are indexed by . Moreover, such a PBW basis goes to at the limit . (This fact has been established in for Beck’s bases, and the result can probably be extended to Ito’s more general bases by using or .) In the end, we get a bijection between and . We expect that this bijection is our map . 1.9 Plan of the paper Section 2 recalls combinatorial notions and facts related to root systems. We emphasize the notion of biconvex subsets, which is crucial to the study of convex orders and to the definition of the functorial filtration mentioned in section 1.8. Section 3 is devoted to generalities about HN polytopes in abelian categories. In section 4, we recall known facts about preprojective algebras and Lusztig’s nilpotent varieties. We also prove that cutting a -module according to a torsion pair is an operation that preserves genericity. In section 5, we exploit the tilting theory on to define and study the submodules and mentioned in section 1.5. An important difference with the works of Iyama, Reiten et al. and of Geiß, Leclerc and Schröer is the fact that we are interested not only in the small slices that form the categories (notation of Iyama, Reiten et al.) or (notation of Geiß, Leclerc and Schröer), but also at controlling the remainder. Moreover, we track the tilting theory at the level of the irreducible components of Lusztig’s nilpotent varieties and interpret the result in term of crystal operations. In section 6, we construct embeddings of into , where is the completed preprojective algebra of type . The data needed to define such an embedding is a pair of rigid orthogonal bricks in satisfying . The key ingredient in the construction is the -Calabi-Yau property of . The final section 7 deals with the specifics of the affine type case. All the results concerning the imaginary edges, the cores, or the partitions are stated and proved there. We thank Claire Amiot for suggesting to us that the reflection functors in are related to those in , which allowed us to take the current literature [14, 27, 59] into account in section 5. We also thank Thomas Dunlap for sharing his ideas about affine MV polytopes and for providing us with his PhD thesis . We thank Alexander Braverman, Bernhard Keller, Bernard Leclerc, and Dinakar Muthiah for very helpful discussions. Finally, we thank two anonymous referees for thorough and insightful reports which led to significant improvements in the presentation. P. B. acknowledges support from the ANR, project ANR-09-JCJC-0102-01. J. K. acknowledges support from NSERC. P. T. acknowledges support from the NSF, grants DMS-0902649, DMS-1162385 and DMS-1265555. 1.11 Summary of the main notations the set of partitions. the Grothendieck group of , an essentially small abelian category. the set of isomorphism classes of simple objects in . the HN polytope of an object . the face of a HN polytope , where and is the maximal value of on . the base field for representation of quivers and preprojective algebras. a finite graph, wihout loops (encoding a symmetric generalized Cartan matrix). the corresponding symmetric Kac-Moody algebra. the upper nilpotent subalgebra of . an orientation of (thus is a quiver). the set of edges of the double quiver . the source and target maps. the completed preprojective algebra of . the category of finite dimensional left -modules. the root system of . the standard basis of . the positive and negative roots with respect to this basis. the Weyl group. the length function. the root lattice. , the -vector space with basis . the dual vector space. the -th coordinate on ; thus is the dual basis of . the -invariant symmetric bilinear form (real roots have square length ). the dominant Weyl chamber. the Tits cone. , for . and , the root subsystem and the parabolic subgroup defined by . the longest element in , when the latter is finite. the linear form such that for each . , for ; thus for any reduced decomposition . the preprojective algebra of type . In the case of an affine root system: the positive primitive imaginary root. the projection modulo . the spherical (finite) root system. the “minimal” lift, a right inverse of . the rank of . the dual of . the set of all spherical chamber coweights. the Weyl fan on , completed on by the spherical Weyl fan. the coroot lattice, spanned over by the elements . the translation, for ; thus for each . the image of in . And after having chosen an extending vertex in the extended the vertices of the (finite type) Dynkin diagram. a preferred system of simple roots for . the spherical fundamental coweights, a basis of . the dominant spherical Weyl chamber. The set of irreducible components of a topological space is denoted by . If is an irreducible topological space, then we say that a propriety depending on a point holds for general in if the set of points of at which holds true contains a dense open subset of . We sometimes extend this vocabulary by simply saying “let be a general point in ”; in this case, it is understood that we plan to impose finitely many such conditions . 2 Combinatorics of root systems and of MV polytopes In this section, we introduce our notations and recall general results about root systems and biconvex subsets. Starting from section 2.3 onwards, we focus on the case of an affine root system. 2.1 General setup Let be a finite graph, without loops: here is the set of vertices and is the set of edges. We denote by the free abelian group on and we denote its canonical basis by . We endow it with the symmetric bilinear form , given by for any , and for , is the negative of the number of edges between the vertices and in the graph . The Weyl group is the subgroup of generated by the simple reflections ; this is in fact a Coxeter system, whose length function is denoted by . Lastly, we denote by the set of all linear combinations of the with coefficients in and we denote by the linear form that maps each to . The matrix with entries is a symmetric generalized Cartan matrix, hence it gives rise to a Kac-Moody algebra and a root system . The latter is a -stable subset of , which can be split into positive and negative roots and into real and imaginary roots . Given a subset , we can look at the root system . Its Weyl group is the parabolic subgroup of . If is finite, then it has a longest element, which we denote by . An element is called -reduced on the right if for each . If is -reduced on the right, then for all . Each right coset of in contains a unique element that is -reduced on the right. The Weyl group acts on and on its dual . The dominant chamber and the Tits cone are the convex cones in defined as The closure is the disjoint union of faces for . The stabilizer of any point in is precisely the parabolic subgroup . Thus The disjoint union on the right endows with the structure of a (non locally finite) fan, which we call the Tits fan. To an element , we associate the subset . If is a reduced decomposition, then The following result is well-known (see for instance Remark in ). For , the following three properties are equivalent: Let and let . If is -reduced on the right, then . Let and let be such that . Then there exists such that . Since is a nonnegative linear combination of the roots for , it follows that there exists such that . For this , we have , whence by Lemma 2.1. Therefore is not -reduced on the right. 2.2 Biconvex sets (general type) A subset is said to be clos if the conditions , , imply (see , chapitre 6, §1, nº 7, Définition 4). A subset is said to be biconvex if both and are clos. We denote by the set of all biconvex subsets of and endow it with the inclusion order. An increasing union or a decreasing intersection of biconvex subsets is itself biconvex. Each finite biconvex subset of consists of real roots and is a , with (see , Proposition 3.2). For convenience, we will say that a biconvex set is cofinite if its complement is finite. Given , we set and

By Kenji Ueno Sleek algebraic geometry is equipped upon basic notions: schemes and sheaves. the idea of schemes was once defined in Algebraic Geometry 1: From Algebraic types to Schemes, (see quantity 185 within the comparable sequence, Translations of Mathematical Monographs). within the current booklet, Ueno turns to the speculation of sheaves and their cohomology. Loosely conversing, a sheaf is a fashion of keeping an eye on neighborhood info outlined on a topological area, resembling the neighborhood holomorphic services on a posh manifold or the neighborhood sections of a vector package deal. to check schemes, it truly is beneficial to review the sheaves outlined on them, specially the coherent and quasicoherent sheaves. the first instrument in knowing sheaves is cohomology. for instance, in learning ampleness, it's often priceless to translate a estate of sheaves right into a assertion approximately its cohomology. The textual content covers the real issues of sheaf idea, together with different types of sheaves and the basic operations on them, similar to ... coherent and quasicoherent sheaves. right and projective morphisms. direct and inverse photographs. Cech cohomology. For the mathematician unexpected with the language of schemes and sheaves, algebraic geometry can look far away. besides the fact that, Ueno makes the subject look common via his concise variety and his insightful causes. He explains why issues are performed this fashion and supplementations his factors with illuminating examples. therefore, he's capable of make algebraic geometry very available to a large viewers of non-specialists. Read Online or Download Algebraic geometry 2. Sheaves and cohomology PDF Best algebraic geometry books Written for graduate scholars and researchers alike, this set of lectures presents a established creation to the idea that of equidistribution in quantity conception. this idea is of becoming significance in lots of parts, together with cryptography, zeros of L-functions, Heegner issues, best quantity concept, the speculation of quadratic types, and the mathematics facets of quantum chaos. The speculation of singularities is a vital a part of numerous branches of arithmetic: algebraic geometry, differential topology, geometric optics, and so forth. right here the point of interest is at the singularities of soft maps and purposes to dynamical structures - particularly, bifurcations. This contains the examine of bifurcations of intersections of strong and risky cycles. The concept that of a interval of an elliptic crucial is going again to the 18th century. Later Abel, Gauss, Jacobi, Legendre, Weierstrass and others made a scientific learn of those integrals. Rephrased in sleek terminology, those provide how to encode how the advanced constitution of a two-torus varies, thereby exhibiting that definite households include all elliptic curves. - Introduction to the Classical Theory of Abelian Functions (Translations of Mathematical Monographs) - Intersection Theory, 2nd Edition - Topological Dynamics and Applications: A Volume in Honor of Robert Ellis : Proceedings of a Conference in Honor of the Retirement of Robert Ellis, ... of Minnesota (Contemporary Mathematics) - Algebraic curves Additional info for Algebraic geometry 2. Sheaves and cohomology A general series of equivalence is then defined as an aggregate of virtual sets obtained by addition and subtraction of a finite number of elementary series. Series (of points) and systems (of curves) of equivalence on a threefold are defined in an analogous manner. The theory of equivalence on a threefold V which, in all that follows, is assumed to be non-singular, requires for its development a knowledge of the corresponding results for curves and surfaces; in particular, the establishment of the invariant series and systems of V rests on the theory of invariants of curves and surfaces. Chapter III. Systems of Surfaces. 1. The RIEMANN-RoCH theorem. We consider m this section the problem of determining the freedom of the complete linear system characterised by a given non-singular surface on a non-singular threefold V. If C is such a surface, with virtual characters n, n, p, we define the virtual freedom d of the system ICI by the formula d = n - n + p - Pa + 2, where Pa denotes the arithmetic genus of V. In the case where C is non-special, with effective freedom r = d, we say that ICI is regular. For example, B. SEGRE [IJ uses the present methods to obtain equivalences for the invariant series of any surface of the form c1 Sl + C2 S2 ' where Cv C2 are integers, in terms of the invariant series of Sl and S2' In the same work SEGRE finds the covariant systems of one or two nets of surfaces on V, of two or more pencils, and also of linear systems of freedom three or four, thereby establishing many interesting relations between the entities in question. One of the most striking of these is the following: given two pencils IAI, IBI of general character, generically situated, the number of pairs A, B which have stationary contact with one another is 48(P,.

Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2021View this Special Issue A Conservative Crank-Nicolson Fourier Spectral Method for the Space Fractional Schrödinger Equation with Wave Operators In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved Crank-Nicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and second-order accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results. The Schrödinger equation is one of the most basic equations in quantum mechanics, which was proposed by Austrian physicist Schrödinger in 1926. The equation can correctly describe the quantum behaviors of wave function, which has made great contributions to the study of quantum mechanics. Since then, the Schrödinger system has attracted a large number of mathematicians and physicists to explore the characteristics of its solution and physical applications. The study of conservative methods for the Schrödinger equation is one of the most popular research fields. Over the past decades, most of the researches on the conservative method of the Schrödinger equation focus on the integer-order Schrödinger equation (e.g., see Refs. [1–7]). As models of science and engineering are needed to be more realistic, the fractional-order Schrödinger equation becomes one of the most important models in the fields of Bose-Einstein condensation, plasma, nonlinear optics, fluid dynamics [8, 9], etc. However, few studies have been investigated on conservative methods for the fractional Schrödinger equation. Besides that, most of the existing fractional-order conservative methods are finite element and finite difference methods [10, 11]. From the viewpoint of mathematics, the solution of the Schrödinger system has important geometric structures such as energy conservation and multisymplectic structure. Therefore, these properties should be maintained as much as possible in the construction of numerical methods. In this paper, we consider the following nonlinear fractional Schrödinger equation: subject to the boundary condition and the initial conditions where and are positive real constants, , and . and are given real functions. The fractional Laplacian operator can be defined as a pseudo-differential operator with the symbol : where is the Fourier transform and is the Fourier transform of . The spectral method is a generalization of a standard separation variable method, for which Chebyshev polynomials and Legendre polynomials are generally used as the basic functions of approximate expansions. And the Fourier series is convenient to deal with the periodic boundary conditions. Bridges and Reich first put forward the Hamiltonian system using the Fourier spectrum discrete method in 2001. Based on their theoretical ideas, Chen and Qin in the same year proposed the Fourier pseudo-spectral method for the Hamiltonian partial differential equation and used it to integrate the nonlinear Schrödinger equation with periodic boundary conditions. For more comprehensive work on the different conservative Fourier pseudo-spectral methods, refer to [2, 14–16] and their references. Since the equation is calculated on a finite interval , it is converted into periodic boundary conditions in this paper and studied on and below. Let The outline of the remainder of this paper is as follows. In Section 2, a conserved Crank-Nicolson Fourier Galerkin method and a conserved Crank-Nicolson Fourier collocation method are constructed to discrete time variables and spatial variables. Energy-preserving and mass-preserving properties of the new method are investigated, and the error estimate is derived in Section 3. In Section 4, numerical experiments are presented to illustrate the theoretical results. Finally, the conclusions are given in Section 5. 2. Crank-Nicolson Fourier Spectral Method and Conservation Laws Let be the set of all complex-valued and -periodic -functions on . Denote as the inner product on the space with the norm (abbreviated as ): For as a nonnegative real number, let be the closure of . Note that . For any function , the following equations can be developed easily: where the Fourier coefficients are arranged as For the Fourier transform of fractional Laplacian , we have In order to discretize the equation in the temporal direction, the time step is defined by . Denote difference operator where is a positive integer (). Therefore, the Crank-Nicolson method was used to discretize equation (6) in the time axis. 2.1. Crank-Nicolson Fourier Galerkin Method For positive even number , the basis function space can be constructed as where the norm and seminorm of are characterized by The orthogonal operators are defined as follows: Lemma 1 [18, 19]. Suppose that for all ; it holds that Denote The time variables of equation (6) are discretized by the Crank-Nicolson method. And the discrete Fourier Galerkin approximation for equation (6) has a modified scheme as follows: where , . 2.2. Crank-Nicolson Fourier Collocation Method For positive even number , consider the points , , as collocation nodes. The discrete Fourier coefficients of a function on with respect to the collocation points are the following form: Using the inversion formula, we have Define the interpolation operator at the collocation points: According to (27), Corollary 3. For any , , there exists a constant independent of and , such that Using the Fourier collocation method to discrete the spatial variables of the equation, we get the fully discrete scheme for equations (6)–(8) as the following forms: Applying the Fourier transformation to (24), we get the following form: where . 2.3. Theory Analysis of Conservation Proof. We derive the full discrete Fourier Galerkin method: Let in equation (37); it holds that Taking the imaginary part of equation (38), due to The above equality indicates that the method (24) maintains the conservation of the discrete mass. The following items consider the conservation of the discrete energy. Let ; according to equation (37), we also get Taking the real part of (42), due to therefore, using (43)–(46), we obtain thus, Based on the above analysis, the method (24) also maintains the conservation of the discrete energy.☐ 3. Theory Analysis of Convergence In order to simplify the notation, we always assume that is a positive constant in this article, which might be different in every formula. Lemma 5 . For any discrete function , it holds that Lemma 6. For , there exists a positive constant , such that Proof. Using Theorem 4, it yields thus, Because of , it satisfies Sum the inequalities of Lemma 5 from 0 to yields Adding (53) and (54), we can obtain the following items: For is sufficiently small (), this implies According to the discrete Gronwall’s inequality, there is Therefore, ☐ Proof. Let , , , and ; then, . From triangle inequality and Lemma 1, it yields According to the orthogonality of the projection operator , we get The authors derive the full discrete Fourier Galerkin method: Subtracting equation (62) from equation (61), due to thus, According to the orthogonality of operator , i.e., . Therefore, Let in (64), and taking the real part, due to therefore, using (66)-(68), this implies where Thus, according to Lemma 6, we can get Note Lemma 1; it gives that Then, Thus, (69) becomes Because of and from Lemma 5, it gives that Then, combining (74) and (76) leads to Summing above inequalities (77) from 1 to yields Hence, using the discrete Gronwall’s inequality gives thus, Substituting (80) into (60) can yield which immediately gives conclusion.☐ Similar to the proof of Theorem 7, we can obtain the following theorem. 4. Numerical Example Numerical examples will be proposed in this section to verify the correctness of the theoretical analysis, that is, the convergence of the numerical method and its ability to maintain discrete mass and discrete energy. Example 1. Consider the nonlinear fractional Schrödinger equation with the wave operator: Let , , and . Figures 1 and 2 present the numerical solutions for and . We can find that the order of will affect the shape of the solution. There is no exact solution of (83) known for . Therefore, numerical solution calculated by the method (24) with and is taken as the reference solution. Let be the numerical solution, and calculate the error at in the sense of the discrete norm:

A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2017; you can also visit the original URL. The file type is Cai and VS. Pless, Orphan structure of the first-order Reed-Muller codes, Discrete Mathematics 102 (1991) 239-247. We investigate a method of combining two codes which we call the outer product. ... First-order Reed-Muller codes are outer products of a number of copies of the full binary space of length 2, and we apply our results to obtain cosets of the Reed-Muller codes which have no ancestors, ... The first-order Reed-Muller code R(1, m) is the outer product of m first order Reed-Muller codes R(1, 1). ...doi:10.1016/0012-365x(92)90118-y fatcat:ldwuv7uotbfltgmqq2f73mmmg4 A deep result about the Reed Muller codes, proved by Mykkeltveit in 1980, is that the covering radius of the Reed Muller code R(1, 7) equals 56. ... Among the results about \(1, m), that \(1, 7)=56, proved by Mykkeltveit , is one of the most difficult. ... Introduction Let R(r, m) be the rth order Reed Muller code of length 2 m , and let \(r, m) be its covering radius. ...doi:10.1006/jcta.1996.0055 fatcat:aphetd6usbay7dkzi52mfsrk3i We characterize cosets that are orphans, and then prove the existence of a family of orphans of first order Reed-Muller codes R(1,m). ... Richard A. (1-WI); Pless, Vera S. (1-ILCC) Orphans of the first order Reed-Muller codes. ... Cosets of the Reed-Muller code R(m-3,tn) are classified under the actions of GL(m,2) and GA(m, 2), the latter being the automorphism group of R(m -3, m) for m > 4. ... The number of cosets in each class is calculated. Orphans of R(m-3, m) are identified, and the normality of R(m-3, m) is established. ... G-equivalent cosets have the same coding theoretic properties. In this paper, we concentrate on the cosets of the (m -3)rd order Reed-Muller code R( m -3, m) of length 2". ...doi:10.1016/0012-365x(94)90113-9 fatcat:5rkol7flgfgibg7yfabbzxao7a We study a family of particular cosets of the first-order Reed-Muller code R(1; m): those generated by special codewords, the idempotents. ... Thus we obtain new maximal weight distributions of cosets of R(1; 7) and 84 distinct almost maximal weight distributions of cosets of R(1; 9), that is, with minimum weight 240. ... Charpin and C. Carlet for their motivating discussions and N. Sendrier for valuable improvements concerning the implementation. We wish to thank H. F. Mattson and E. F. ...doi:10.1109/18.761276 fatcat:rfor5ftkqjhb5i5iteq3ukmhl4 cosets of the Reed-Muller code R(m — 3, m). ... The covering radius is defined as the weight of the coset of greatest weight; it is the maximum distance of any vector in the space from the code. In a paper on RM, G. Seroussi and A. ... II (224-233); Philippe Langevin, On the orphans and covering radius of the Reed-Muller codes (234-240); Yuan Xing Li and Xin Mei Wang, A joint authentication and encryption scheme based on algebraic coding ... Janwa, On the parameters of algebraic geometric codes (19-28); Erich Kaltofen and B. David Saunders, On Wiedemann’s method of solving sparse linear systems (29-38); Simon N. Litsyn [S. N. ... (English summary) 94b:94033 Langevin, Philippe On the orphans and covering radius of the Reed-Muller codes. (English summary) (see 94b:68002) Litsyn, S. ... ., 94m:94030 Hou, Xiang Dong Some results on the covering radii of Reed-Muller codes. (English summary) 94f:94020 Katsman, G. L. Bounds on covering radius of dual product codes. ... Orphans of the first order Reed-Muller codes. 91b:94044 Bruck, Jehoshua (with Naor, Moni) The hardness of decoding linear codes with preprocess- ing. (Not in MR) Bussemaker, F.C. ... A new table of constant weight codes. 91h:94028 Brualdi, Richard A. (with Pless, Vera S.) On the length of codes with a given covering radius. 91¢:94025 — (with Pless, Vera S.) ... On the covering radius of Reed-Muller codes. (English summary) 93¢e:94016 Conway, J. H. (with Pless, Vera S.; Sloane, N. J. A.) ... Orphan structure of the first-order Reed-Muller codes. 93b:94021 Cai, Ning see Brualdi, Richard A.; et al., 93b:94021 Calderbank, A. R. Covering machines. ... Proceedings of 1995 IEEE International Symposium on Information Theory The proofs of Propositions 2.5 and 2.6, by induction on t, are based on the fact that codes with length n, covering radius t, and containing K(n, t) words (i.e., optimal covering codes) cannot be too unbalanced ... Table A gives bounds for K(The so-called sphere-covering bound states that if C is a code of length n and covering radius t, then the volume of a sphere of radius t, multiplied by the cardinality of C, ... Specific Classes of Codes Covering Radius of Reed-Muller Codes Reed-Muller (RM) codes are among the most interesting families in the study of covering radius. ...doi:10.1109/isit.1995.535749 fatcat:vqartrp3offzdj6ols6lnqeht4 (From the text) 91a:94027 94B05 — (with Pless, Vera S.) Orphans of the first order Reed-Muller codes. JEEE Trans. Inform. Theory % (1990), no. 2, 399-401. ... On the length of codes with a given covering radius. Coding theory and design theory, Part I, 9-15, IMA Vol. Math. Appl., 20, Springer, New York, 1990. ... On the length of codes with a given covering radius. Coding theory and design theory, Part I, 9-15, IMA Vol. Math. Appl., 20, Springer, New York, 1990. ... (From the text) 91a:94027 94B05 — (with Brualdi, Richard A.) Orphans of the first order Reed-Muller codes. JEEE Trans. Inform. Theory % (1990), no. 2, 399-401. ... Boolean Models and Methods in Mathematics, Computer Science, and Engineering Acknowledgement We thank Caroline Fontaine for her careful reading of a previous draft of this chapter. ... of the covering radius of the Reed-Muller code of order 1 if n is even; indeed, in the case of the Reed-Muller code of order 1, the covering radius coincides with the maximum nonlinearity of Boolean functions ... (the nonlinearity of f ) equals 2 n−1 − 2 n/2−1 (the covering radius of the Reed-Muller code of order 1). ...doi:10.1017/cbo9780511780448.011 fatcat:dtgopxbkmjgahepxw46xgo6rd4 Langevin, Philippe On the orphans and covering radius of the Reed-Muller codes. ... (see 94i:00015) 65M06 (76M25) Langsetmo, Lisa The K-theory localization of loops on an odd sphere and applications. Topology 32 (1993), no. 3, 577-585. ... « Previous Showing results 1 — 15 out of 35 results

In a stretched hydrogen molecule the two electrons that are paired at equilibrium forming a bond become un-paired and localized on the individual H atoms. In singlet diradicals or doublet triradicals such a weak paring exists even at equilibrium. At a single-determinant SCF level of the theory the valence electrons of a singlet system like H remain perfectly paired, and one needs to include non-dynamical correlation to decouple the bond electron pair, giving rise to a population of effectively-unpaired (“odd”, radicalized) electrons.946, 84, 906 When the static correlation is strong, these electrons remain mostly unpaired and can be described as being localized on individual atoms. These phenomena can be properly described within wave-function formalism. Within DFT, these effects can be described by broken-symmetry approach or by using SF-TDDFT (see Section 7.3.1). Below we describe how to derive this sort of information from pure DFT description of such low-spin open-shell systems without relying on spin-contaminated solutions. The first-order reduced density matrix (1-RDM) corresponding to a single-determinant wave function (e.g., SCF or Kohn-Sham DFT) is idempotent: where is the electron density of spin at position , and is the spin-resolved 1-RDM of a single Slater determinant. The cross product reflects the Hartree-Fock exchange (or Kohn-Sham exact-exchange) governed by the HF exchange hole: When 1-RDM includes electron correlation, it becomes non-idempotent: The function measures the deviation from idempotency of the correlated 1-RDM and yields the density of effectively-unpaired (odd) electrons of spin at point .946, 791 The formation of effectively-unpaired electrons in singlet systems is therefore exclusively a correlation based phenomenon. Summing over the spin components gives the total density of odd electrons, and integrating the latter over space gives the mean total number of odd electrons : The appearance of a factor of 2 in Eq. (10.132) above is required for reasons discussed in Ref. 791. In Kohn-Sham DFT, the SCF 1-RDM is always idempotent which impedes the analysis of odd electron formation at that level of the theory. Ref. 788 has proposed a remedy to this situation. It was noted that the correlated 1-RDM cross product entering Eq. (10.131) reflects an effective exchange, also known as cumulant exchange.84 The KS exact-exchange hole is itself artificially too delocalized. However, the total exchange-correlation interaction in a finite system with strong left-right (i.e., static) correlation is normally fairly localized, largely confined within a region of roughly atomic size.58 The effective exchange described with the correlated 1-RDM cross product should be fairly localized as well. With this in mind, the following form of the correlated 1-RDM cross product was proposed:788 The function is a model DFT exchange hole of Becke-Roussel (BR) form used in Becke’s B05 method.47 The latter describes left-right static correlation effects in terms of certain effective exchange-correlation hole.47 The extra delocalization of the HF exchange hole alone is compensated by certain physically motivated real-space corrections to it:47 The BR exchange hole is used in B05 as an auxiliary function, such that the potential from the relaxed BR hole equals that of the exact-exchange hole. This results in relaxed normalization of the auxiliary BR hole less than or equal to unity: The expression of the relaxed normalization is quite complicated, but it is possible to represent it in closed analytic form.790, 789 The smaller the relaxed normalization , the more delocalized the corresponding exact-exchange hole.47 The exchange hole is further deepened by a fraction of the exchange hole, , which gives rise to left-right static correlation. The local correlation factor in Eq.(10.134) governs this deepening and hence the strength of the static correlation at each point:47 The final formulas for the spin-summed odd electron density and the total mean number of odd electrons read: It is informative to decompose the total mean number of odd electrons into atomic contributions. Partitioning in real space the mean total number of odd electrons as a sum of atomic contributions, we obtain the atomic population of odd electrons () as: Here is a subregion assigned to atom in the system. To define these atomic regions in a simple way, we use the partitioning of the grid space into atomic subgroups within Becke’s grid-integration scheme.52 Since the present method does not require symmetry breaking, singlet states are calculated in restricted Kohn-Sham (RKS) manner even at strongly stretched bonds. This way one avoids the destructive effects that the spin contamination has on and on the Kohn-Sham orbitals. The calculation of can be done fully self-consistently only with the RI-B05 and RI-mB05 functionals. In these cases no special keywords are needed, just the corresponding EXCHANGE rem line for these functionals. Atomic population of odd electron can be estimated also with any other functional in two steps: first obtaining a converged SCF calculation with the chosen functional, then performing one single post-SCF iteration with RI-B05 or RI-mB05 functionals reading the guess from a preceding calculation, as shown on the input example below: $comment Stretched H2: example of B3LYP calculation of the atomic population of odd electrons with post-SCF RI-BM05 extra iteration. $end $molecule 0 1 H 0. 0. 0.0 H 0. 0. 1.5000 $end $rem SCF_GUESS CORE METHOD B3LYP BASIS G3LARGE PURECART 222 THRESH 14 MAX_SCF_CYCLES 80 PRINT_INPUT TRUE SCF_FINAL_PRINT 1 INCDFT FALSE XC_GRID 000128000302 SYM_IGNORE TRUE SYMMETRY FALSE SCF_CONVERGENCE 9 $end @@@ $comment Now one RI-B05 extra-iteration after B3LYP to generate the odd-electron atomic population and the correlated bond order. $end $molecule read $end $rem SCF_GUESS READ EXCHANGE BM05 PURECART 22222 BASIS G3LARGE AUX_BASIS riB05-cc-pvtz THRESH 14 PRINT_INPUT TRUE INCDFT FALSE XC_GRID 000128000302 SYM_IGNORE TRUE SYMMETRY FALSE MAX_SCF_CYCLES 0 SCF_CONVERGENCE 9 DFT_CUTOFFS 0 $end @@@ $comment Finally, a fully SCF run RI-B05 using the previous output as a guess. The following input lines are obligatory here: PURECART 22222 AUX_BASIS riB05-cc-pvtz DFT_CUTOFFS 0 $end $molecule read $end $rem SCF_GUESS READ EXCHANGE BM05 PURECART 22222 BASIS G3LARGE AUX_BASIS riB05-cc-pvtz THRESH 14 PRINT_INPUT TRUE INCDFT FALSE IPRINT 3 XC_GRID 000128000302 SYM_IGNORE TRUE SCF_FINAL_PRINT 1 SYMMETRY FALSE MAX_SCF_CYCLES 80 SCF_CONVERGENCE 8 DFT_CUTOFFS 0 $end Once the atomic population of odd electrons is obtained, a calculation of the corresponding correlated bond order of Mayer’s type follows in the code, using certain exact relationships between , , and the correlated bond order of Mayer type . Both new properties are printed at the end of the output, right after the multipoles section. It is useful to compare the correlated bond order with Mayer’s SCF bond order. To print the latter, use SCF_FINAL_PRINT = 1.

Add your company website/link to this blog page for only $40 Purchase now!Continue FutureStarrCalculate a Percentage From a Fraction If you need to know the percentage of a number, you can use a calculator to find the answer. You can also try to find the percentage from the fraction. In this article, we will look at how to calculate a percentage from a fraction. This is an important step to solve problems, so you should know how to calculate this. To calculate a percentage of four of twenty, you must first figure out how many people are in your group. Then multiply each person's share by the number of people in the group. For example, if six people are on the payroll, and one is absent, then you can multiply the shares of the other five to come up with 63%. By using the percentage formula, you can easily calculate the percentage of four of twenty as well as the fraction of a group of people. When you're working with a fraction, you'll want to convert it to a decimal form. Adding a percent symbol after the decimal will make it much easier to calculate percentages. This method is especially helpful if you're working with large numbers. To make percentage calculations easier, you can use a calculator. Most calculators have a section for decimal fractions. This way, you won't have to figure out how to divide a whole number by a decimal. Also, a percentage calculator makes it easy to compare numbers. Another way to figure out percentages is to use a general guideline. Decimals have a special symbol for percentages, so when you see 1%, you can easily convert it to 60% or 75%. Likewise, 60% is 60/100, which means 0.6 x $700. Percentage change calculators are often used to compare measurements from two different time periods. For instance, you might want to compare how many people lived in a village before and after industrialization. Likewise, you can compare the differences in the number of people in two different countries using census data. When working with fractions, students should learn how to convert them to percentages. The process is simple and helps students understand fractions. To convert a fraction to a percentage, multiply the answer by the fraction's numerator, which is the number above the fraction line. There are three basic ways to find the percentage of a number. One is to multiply it by its lowest value, and the other is to divide it by a base of 100. Once you have determined the proportion, you can use the calculator to convert it to a decimal, if necessary. You can also use the percentage formula to solve equations involving percentages. A common formula to use is X/Y = P x 100. This formula is the most basic and can be used to solve many percentage problems. For more complicated percentage problems, you can use the following formulas: Another way to find the percentage is to use a percentage calculator. By entering a fraction, the calculator will calculate the result. For example, entering 5% of 20 would yield a result of 1 in the result box. This tool can be used for discount applications or to calculate different values, like body fat. Converting fractions to percentages is a simple and straightforward process that works for any fraction. Once you've mastered the basic method, it will be easier to apply to any fraction. Make sure to round your final percentage to two decimal places. You will become more confident with percentages over time as you practice and get more practice. You can also use a percentage worksheet generator to generate randomized differentiated percentafe problems. The first method involves dividing the value by the total value, or the denominator. Then, you multiply this value by 100. This is known as the unitary method. The second method is by changing the denominator of the equation to one hundred. The denominator must be equal to the value of the numerator. If you're working with data, you may be wondering how to convert a fraction to a percentage. To convert a fraction to a percentage, first convert it to decimal form. To do so, divide the denominator by the numerator. Then, multiply the result by 100 to get the percentage. Percentages are useful because they make calculations much easier. They also help you work with parts of a whole, such as one hundred percent. Many fractions do not have an exact decimal equivalent, so it's easier to work with percentages. In addition, many countries use a decimal system for currency and measurement. The process to convert a fraction to a percentage is simple. Once you have your numeric value, you need to divide it by 100 to find the percentage. Once you have the percentage, place the percent symbol (%) after the final product. The higher the percentage is, the more it is of that quantity. Once you've converted the fraction to a percentage, you can use your calculator to convert the number to decimal form. To do this, you need to make sure that all the digits of the number are higher than ten. In this way, 89/100 is 89%. In addition, it's important to remember that the percent symbol means "per one hundred" and should be represented in decimal form. You can also use this symbol to convert fractions into percentages. In some cases, rounding will be necessary to ensure that the result is as precise as possible. A percentage can be made up of different parts, or parts of a whole. For example, if an apple is cut into four equal pieces, every piece will be a half of one hundred. Then, if an apple is sliced into eight quarters, you could write a fraction such as "3/4" on each half. This would be six thirds of one apple. In the same way, you can divide the rest of the apple into eighteen equal parts. Getting a percentage is easier if you know the formula for multiplying two values by 100. If you want to convert a fraction to a percentage, you need to take the first half as a decimal, then divide the remainder by 100 to get the percentage. Once you've done that, you can use your calculator to multiply the numerator by the original value. Using the decimal place, we can write the fraction as 28.57142857142857 percent or 28.6%. Round to the nearest decimal place to get the correct result. Similarly, we can also write it as 1/9 of 72 or 1640. Visual estimations are a great way to represent proportions and to communicate them to others. They are especially useful when you want to compare fractions. The human eye is trained to compare things, so visual models are perfect for this purpose. Also, they are a great choice for a casual setting. Visual fractions make it easier to estimate, because the numbers are smaller. They also keep the proportions intact. For example, if we need to calculate 5% of 14, we'll divide 5% by 2. This will give us the answer of 5/14. This will be easier than attempting to multiply or divide by whole numbers. Suppose you need to find the equivalent fraction of 1/9 of 96 in a given visual fraction. How would you find the answer to this question? You could use the visual fraction model. This method can help you solve problems related to fractions and whole numbers. For example, if you need to find how much one pound of roast beef will serve four people, you can look for the equivalent fraction. There are two ways to find 1/9 of 72. The first way is to divide 72 by a number with the same number of terms. Then, use this number to determine the equivalent fraction. For example, 42/70 equals 6/10, and 72 times ten equals 140. The second way is to multiply the fractions with the same denominator. For example, 8% of 60 is 4.8. Similarly, 1/5 of something means to multiply it by five. Therefore, 4.8 becomes 1.6. This way, you can calculate the number of minutes in eight minutes. You can find 1/9 of 1640 in Visual Fr fractions by multiplying 8 by 3. You will find that this number is equal to 24. Then, you can add these fractions together and get 33/56. Now, you can use these same steps to find 1/9 of 1640 in decimal form. To find 3/14 of 96, multiply 163/24 by 6. This gives you a fraction of 12. This fraction is the same as 3/14, but is less than the number. Likewise, divide 21/30 by 20/30 to get a fraction of 7/10. The fraction 4/7 of 14 is the same as 2 multiplied by 4. For example, imagine that a table has two substances in an 8:6 ratio. The first substance is 60mg and the second substance is 80mg. If a tablet has these two substances, then it has a total of 140mg. The other substance is 40mg, which is half the tablet's total dosage. If you want to find out the percentage of a number, you can use this calculator. It will simplify a complex number into an easy-to-understand format, and it will automatically update. Enter the number, 4, and the calculator will automatically divide it by 14 and multiply it by 100. There are a number of methods for simplifying percentages. The most common one involves converting a fraction to a decimal number. The easiest way to do this is to move the decimal point two places to the left and multiply the result by 100. The second method involves finding the greatest common divisor of the fraction and dividing by that number. If you have a mixed number, the proper fraction should be entered. For example, 5% of twenty will be written as x/100. Once you have the answer, you need to subtract one from the whole number to get the proper percentage. Next, you can use the percentage calculator to calculate different values such as a discount application. Percentages can be calculated by using the formula: Percentage = (number to find percentage of) x 100. You can simplify 4 out of 14 percents by moving the decimal point two places to the left and right. This will simplify the calculation of percentages. The second method involves simplifying 4 out of 14 percentages into its smallest terms. This way, you can get detailed information about the lowest term, in this case, 2/7. It turns out that 2 is the greatest common factor between 4 and 14. In this way, you can simplify 4 out of 14 into a much simpler form. Another way to simplify percentages is to use decimals instead of the fractions. When you're dealing with the percentages of a number, the decimal will be easier to understand and convert. Once you've mastered this, you can use percentages to simplify your calculations. You can also simplify the percentage to a fraction if you want to use it in a calculation. First, you need to convert the percent to the fractional form. For example, 50% is equal to 50/100. To simplify it further, you can make it 1/2. Finally, you can also simplify it to a fraction by dividing it by 100. Learning how to simplify fractions to percentages is an important math skill for students, which will help them understand fractions better. If you're a struggling student, practice these skills and it will get easier. The best way to practice converting fractions to percentages is to use a percentage worksheet generator. This program allows you to generate randomized differentiated percentafe problems. The first step in converting 4 out of 14 to a percentage is to determine the decimal form of the fraction. To do this, divide the fraction by fourteen and multiply the result by 100. The result will be the percentage value. For example, if 4/14 is 57%, the decimal form of the fraction would be 5/14. Once you have a decimal and a fraction, you'll need to figure out how to convert each. The easiest way to do this is to multiply the numerator by 100. If the fraction is smaller than 100, the answer will be lower. Then, divide the decimal value by 100 and multiply the result by 100 again. Similarly, you can convert mixed numbers to percentages by adding the decimal to the fraction portion of the number. You can also multiply the result with 100 to get the percentage value. However, if you're not sure which method works, you can use a decimal to percentage calculator. Percentage is a useful concept to know for daily arithmetic. It's also useful when making investments and purchasing expensive items. You can save time and money by knowing what percentages are and how to use them. The more you understand how percentages work, the more you can make your calculations more accurate. Percentage is a value that shows how much one number is equal to another number. It is also useful for calculating the results of tests. For example, if you are given a score of sixteen out of twenty, you can find the percentage of the score by dividing the score by twenty and multiplying the result by 100. There are many situations in which you might need to calculate the percentage of something. For instance, you may need to know how much a certain number is worth in relation to a certain discount. Knowing how to convert a percentage from a decimal to a fraction will simplify calculations. Decimal numbers are easier to convert to percent than fractions, but it's not impossible to convert a fraction to percent. Simply divide the fraction by 100 to get the percentage you're looking for. This way, you can find the percentage of 4 out of 14 in relation to another number in less than one second. When you need to calculate the percentage of a number, you first need to determine which numbers are being compared. A number can be a fraction, decimal, or both. You can also find out the percent of a number by multiplying the number by the percent symbol. In academics, a percentage of a number represents its value. For instance, a student can calculate a grade by dividing the total number of marks from a test by the percentage of that grade. Once you have this information, you can use it to calculate how much a student scored on the final exam. In everyday life, people use the percentages in everyday language. For instance, 150% means that someone gives more than what is required. In the context of a percentage, it's often easier to use a decimal. A calculator is helpful in most situations. In the first example, it's simple to understand the three parts of a percent problem. First, you must identify the base number, the unknown number, and the amount. Then, you can solve for the percentage by writing an equation. In addition, you can also write the percent as a fraction. Similarly, you can use the percentage of four out of 14 in relation to another number. For example, if you're comparing women's shoe sizes, a US seven to US eight difference of 0.25 inches is the same as the percentage difference between US 8 and US seven. In addition to the decimal system, ancient Romans used fractions that were 1/100. The Romans used them when assessing the value of goods and taxes. The practice became more widespread in the Middle Ages. By the 17th century, it became the standard for interest rates. Then, mathematicians began abbreviating the word "p" to represent its percentage. First, consider the number 2/3, which is 24. Then divide it by 5. This will give you 2.4. Next, multiply 2.4 times four to get 9.6. Finally, 4/1 can be written as a fraction, so 1/7 x 4 will give you 4/7. If we want to write 1/7 of 4 out of 14 as a fraction, we must first know the number. We can do this by dividing 160 by 6. This will give us a number of 26.7. We can also write it as 1/9 of 96 which is 10.7. One of the best ways to learn how to write fractions is to make sure that the numbers you are using are proper. Proper fractions have a numerator that is smaller than the denominator, so the top number will be smaller than the bottom. We must also remember to simplify fractions, so that the bottom number will be smaller than the top one. To multiply 1/8 by 4, you will first divide the fraction by its numerator. Then divide the result by the denominator to get the answer in decimal form. This process can be repeated many times as long as you are careful. After you have answered the question correctly, you will be able to divide the result into smaller fractions. For example, the fraction 2/3 of a number is equal to 24. Therefore, you can divide 24 by three. Likewise, dividing 12 by five will yield 2.4. This value will then be multiplied by four to get 4.9. This is the same as multiplying 5/9 by four to get 5. In addition, you can write 1/8 x 4 as a fraction. If you are looking for a solution to the question: "How to convert 4/14 to percent", you have come to the right place. We have provided you with a step-by-step explanation and calculator that will help you convert 4/14 to percent. Just type the number in the box below, and multiply it by 100. Afterwards, type the symbol % to complete the conversion. If you want to convert 3/4 of 4/14 to a percent, you'll need to use a fraction to percent calculator. The calculator will accept input values in b/c format, meaning four tenths should be typed as 4/10, and one and three-half should be typed as 1 3/2. It is important to leave a space between the integer and the fraction. Once you have entered your input values, you can multiply them by 100 to get the decimal equivalent. Finally, you'll need to add the percent sign to get the final result. In addition, the decimal equivalents for the most common fractions can be found using element14. For example, 3/10 is equal to 37 percent, while 7/8 is equal to 87.5 percent. A fraction's numerator is the number of parts it contains, while the denominator indicates the number of parts that make up the whole. Another common way to convert fractions to percentages is to multiply the decimal by 100. You can also use a decimal to percent calculator to convert decimals to percentages. Once you know the right formula for converting a fraction to a percentage, you can use it to solve other common problems. Percents are easier to remember than fractions. For example, if there are 38 students in a class, then the number of female students is 23. This is a percentage, and you'll know that 23 female students are the majority of the class. To convert a fraction to a percent, use the following formula: If you need to know how to convert fractions to percents, you will need a calculator that can help you do so. The formula for a percent is based on the numerator of a number and the denominator. A fraction is equal to the number divided by the number of digits in the denominator. Decimal numbers have digits to the right of the decimal point. For example, 3/8 is 37.5 percent. Similarly, 7/8 is 87.5 percent. In addition, 5/8 is 62.5 percent, and 3/10 is 30 percent. These examples demonstrate how to convert fractions to a percent. When you need to convert a fraction to a percent, you must first know how to calculate the smallest and largest fractions. Then, you can use the table below to find the right proportion. It's a two-step process that is often used when comparing different size quantities.

In the following sections we will discuss models for the excitation of various kinds of wind instrument, and for that we will need to know a little about fluid flow. This section gives a qualitative overview of some important fluid phenomena and terminology, introduced through pictures. A side link will give some technical background about how the subject can be approached mathematically. A. Laminar and turbulent flow So far our only contact with the world of fluid dynamics has been through the very special case of linear acoustics. However, we will need to go beyond this to understand such things as how a clarinet or flute mouthpiece works. If you think about it for a moment, you already know that everyday fluid dynamics must involve nonlinear effects. Look at the small waterfall in Fig. 1. You see smooth water coming over the lip of the fall, but it turns into complicated turbulent flow at the bottom. The smooth initial flow is known as “laminar flow”. Figure 2 shows another example. This is a schlieren image of the plume of hot air rising from a candle flame. At the bottom the flow is laminar, but there is a rather abrupt transition to turbulent flow in the middle of the image, without anything obvious happening there to trigger the transition (nothing analogous to the waterfall edge, for example). This is a spontaneous transition to chaotic behaviour, in the sense discussed in section 8.4. Chaotic behaviour like this can only occur in nonlinear systems, although if you recall the example of the double pendulum from section 8.4 you will appreciate that it may only take a rather simple-looking nonlinear system to exhibit such behaviour. The origin of the main nonlinearity in the governing equation for fluid flow (known as the “Navier-Stokes equation”) is nothing more than a quadratic term, as shown in the next link, but this is enough to allow the full complexities of turbulence. The possibility of turbulence is surely enough to tell us that we should not expect easy mathematical solutions to the governing equations for fluid flow. It then makes sense to explore approximations of different kinds: these can simplify the mathematics under certain circumstances and allow us to make some progress with understanding the physics behind fluid behaviour. B. Incompressible flow We have already met one approximation: when we derived the linear wave equation back in section 4.1.1, we assumed that all relevant quantities like pressure and flow velocity were small in some appropriate sense, and that allowed us to ignore a lot of complicating factors. However, in order to treat sound waves we certainly had to allow our fluid (usually air) to be compressible. But in the waterfall seen in Fig. 1, the water behaves as if it were incompressible. This turns out to make a major simplification in the governing equations, and for many problems it is a useful approximation to make. The previous link gives some details. You might think that the incompressible approximation would have nothing to do with the acoustics of wind instruments, since we are surely always talking about sound waves? But this would be misleading. Compressibility is indeed always important for the acoustic resonances of instrument tubes, and for the internal pressure waveform when an instrument is played. But when we are thinking about how a mouthpiece works, we are mainly concerned with a different aspect of fluid flow, associated with the air blown into the instrument by the player. We can usually get a rather good approximation to the behaviour by ignoring the compressibility of the air moving through the mouthpiece. The underlying reason for this is that the air-flow is very slow compared to the speed of sound (around 340 m/s), or in other words the Mach number is very small. This is the mathematical condition for compressibility effects to be unimportant. We will make use of an important result which takes its simplest form in the case of incompressible flow. This is called Bernoulli’s principle. The mathematical details are given in the previous link, but in words Bernoulli’s principle says that if you follow a laminar air-flow along its streamlines, a region where the flow speeds up is automatically associated with the pressure going down, and vice versa. So if a jet of air is squeezed through a narrow gap, for example between the reed and the lay of a clarinet mouthpiece, the pressure will be lower there. This low pressure tends to make the reed close towards the lay. We will see in section 11.3 that this is an important ingredient of how a reed mouthpiece works. There is another physical property of fluids we need to think about, called viscosity. Figure 3 shows a familiar sight, honey running slowly off a spoon. If you imagine doing the same thing with a spoonful of water, the behaviour would be quite different: almost all the water would fall off the spoon as soon as you tip it up. This contrasting behaviour happens because the viscosity of honey is far greater than the viscosity of water. But you may have noticed that I said “almost all water would fall off the spoon”. Actually, a few drops of water continue to fall, after most of it has gone. All normal fluids, including water, have some viscosity. When you tip the spoon up, a thin layer of the water is reluctant to run off. It clings to the surface, and only runs down gradually — very much like the honey, except that with water it is only a very thin layer that shows the behaviour. The same thing happens when you wash up crockery: if you leave it to drain for a while, that gives the last thin layer of water time to drip off. So what exactly is viscosity? It is the property of a fluid that resists shear deformation. Figure 4 shows a sketch of a layer of fluid between two metal plates. (In essence, this is a sketch of an apparatus you could use to measure viscosity.) The upper plate is forced to move to the right. This plate does not slide over the fluid, it carries it along with it: fluid in contact with a solid surface is “stuck” to it by molecular forces. So the fluid at the top of the layer moves rightwards with the plate, but the fluid at the bottom is anchored to the bottom plate. Provided the motion is slow enough, the fluid in between will show a linear velocity profile as sketched: this can be described as uniform shearing flow. Now the viscosity relates to the force necessary to move the plate, at a given speed and with a layer of given thickness and area. So now come back to our spoonful of water. What happens when you tip the spoon? The force of gravity acts in the same way on every particle of the water, so it all “wants” to move downhill at the same speed. But the water in contact with the spoon surface is stuck to it, and cannot flow away. The result is sketched in Fig. 5, which shows a magnified view of a small region near the spoon surface. Most of the water flows down at the same velocity, but a thin layer near the surface experiences shearing motion, which is resisted by the viscosity of the water. The result is a velocity profile of the kind sketched, featuring a boundary layer: a layer very close to the solid surface in which significant shear flow occurs. The thickness of this layer is determined by a balance between the force of gravity and the viscous force resisting the shear. The lower the viscosity, the thinner the layer. It is the lower part of this boundary layer that is left behind after the washing up, to drain away slowly in your dish-drainer. A boundary layer related to what has just been described will form on the wall of a wind instrument when it is played. The acoustic field inside the tube involves cyclical longitudinal motion of the air. Very close to the tube wall, though, the “no-slip boundary condition” applies: the air is anchored to the wall. A viscous boundary layer is the inevitable result. Since shear motion resisted by viscosity always dissipates energy, this is one of the two main mechanisms for damping of the acoustic resonances of the pipe, as mentioned in section 11.1. The boundary layer will be at its thinnest if the internal wall of the tube is very smooth, but if that surface is rough then the boundary layer is likely to be thicker, and the damping increased. D. Reynolds Number (and its relatives) Two effects we have described, nonlinearity and viscosity, are each described by one term in the governing Navier-Stokes equation (see the previous link). Fluid dynamicists use a quantity called the Reynolds Number to quantify the relative importance of these two terms (we described it briefly back in section 10.1.1). This number captures the ratio of strengths of the nonlinear term to the viscosity term. The formula for Reynolds Number turns out to involve the typical flow speed, multiplied by the typical length-scale, and divided by the viscosity. In a flow with very low Reynolds number, nonlinear effects can be neglected. The flow will be dominated by viscosity effects, and it will be laminar because viscosity has a stabilising effect on the instabilities leading to turbulence. Examples would be our honey running off the spoon (because the viscosity is large), or the swimming of a micro-organism in water (because the length-scale is very small). A flow with very high Reynolds Number will have the opposite behaviour: viscosity can be neglected, nonlinear effects will be strong, and the flow is likely to be turbulent. Examples would be the air flow through a jet engine (because the flow speed is very high), or the swimming of a human being in water (because this time the length-scale is much bigger than for the micro-organism, and the viscosity of water is quite small). The effective Reynolds Number will vary with depth through a structure like a boundary layer. There will always be a viscous-dominated layer very close to the wall, but it is perfectly possible (and indeed quite common) for the outer part of a boundary layer to become turbulent. See this video for a striking demonstration of a turbulent boundary layer. Fluid dynamicists really like to use non-dimensional numbers like the Reynolds Number. Another example, which is easy to visualise, is the Mach Number. This is the ratio of the flow speed to the speed of sound: so slow flows have low Mach Number, and a supersonic flow has a Mach Number bigger than 1. E. Flow separation and vortices Another phenomenon that can occur is that the shape of the solid object may create the conditions for the boundary layer to separate from the wall at some position, and then give rise to a turbulent wake. Figure 6 shows an example, in a wind-tunnel image of flow past a wing cross-section. The flow below the wing remains in contact with the surface, but above the wing it separates and produces a complicated wake structure. Under some circumstances this separation process varies cyclically with time, giving rise to a strikingly beautiful “Kármán vortex street”, like the example in Fig. 7. The image is taken from this Wikipedia page, where you can find more detail. In the process of shedding vortices on alternate sides, an alternating force is exerted on the solid body, in the direction perpendicular to the flow. If the object is something like a stretched string or a power cable, this force can set it into vibration. This is the underlying mechanism of the Aeolian harp, for example. Another phenomenon involving vortices is more directly relevant to musical wind instruments. If a jet of air is blown out of a slot, as in a recorder mouthpiece for example, each side of this air jet will be a shear layer: a sudden jump in air flow speed. Well, there is a famous mathematical result in fluid dynamics proving that an ideal shear layer is unstable. If the interface between the two flow speeds is initially straight, but is then perturbed a little, the perturbation will grow. With a narrow jet, the instability usually carries the whole jet up and down — we will say more about this in section 11.8 when we discuss air-jet instruments. For now, we can see one image for a rather wide air jet, in Fig. 8. In this schlieren image, we can see that the two shear layers on either side of the jet have generated elegant vortex shapes, in an alternating pattern. The image is taken from the doctoral thesis of Sylvie Dequand . Sylvie Dequand, “Duct aeroacoustics: from technological applications to the flute”, Doctoral dissertation, Eindhoven University of Technology (2001), https://pure.tue.nl/ws/portalfiles/portal/3429613/445111.pdf

Financial Management Case Essay After reading this chapter, students should be able to: Explain what is meant by a firm’s weighted average cost of capital. Define and calculate the component costs of debt and preferred stock. Explain why the cost of debt is tax adjusted and the cost of preferred is not. Explain why retained earnings are not free and use three approaches to estimate the component cost of retained earnings. Briefly explain the two alternative approaches that can be used to account for flotation costs. Briefly explain why the cost of new common equity is higher than the cost of retained earnings, calculate the cost of new common equity, and calculate the retained earnings breakpoint—which is the point where new common equity would have to be issued. Calculate the firm’s composite, or weighted average, cost of capital. Identify some of the factors that affect the WACC—dividing them into factors the firm cannot control and those they can. Briefly explain how firms should evaluate projects with different risks, and the problems encountered when divisions within the same firm all use the firm’s composite WACC when considering capital budgeting projects. List some problems with cost of capital estimates. Chapter 10 uses the rate of return concepts covered in previous chapters, along with the concept of the weighted average cost of capital (WACC), to develop a corporate cost of capital for use in capital budgeting. We begin by describing the logic of the WACC, and why it should be used in capital budgeting. We next explain how to estimate the cost of each component of capital, and how to put the components together to determine the WACC. We go on to discuss factors that affect the WACC and how to adjust the cost of capital for risk. We conclude the chapter with a discussion on some problems with cost of capital estimates. What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 10, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the “Lecture Suggestions” in Chapter 2, where we describe how we conduct our classes. DAYS ON CHAPTER: 3 OF 58 DAYS (50-minute periods) Answers to End-Of-Chapter Questions 10-1Probable Effect on rd(1 – T)rsWACC a.The corporate tax rate is lowered.+0+ b.The Federal Reserve tightens credit.+++ c.The firm uses more debt; that is, it increases its debt/assets ratio.++0 d.The dividend payout ratio is increased.000 e.The firm doubles the amount of capital it raises during the year.0 or +0 or +0 or + f.The firm expands into a risky new area.+++ g.The firm merges with another firm whose earnings are counter-cyclical both to those of the first firm and to the stock market.––– h.The stock market falls drastically, and the firm’s stock falls along with the rest.0++ i.Investors become more risk averse.+++ j.The firm is an electric utility with a large investment in nuclear plants. Several states propose a ban on nuclear power generation.+++ 10-2An increase in the risk-free rate will increase the cost of debt. Remember from Chapter 6, r = rRF + DRP + LP + MRP. Thus, if rRF increases so does r (the cost of debt). Similarly, if the risk-free rate increases so does the cost of equity. From the CAPM equation, rs = rRF + (rM – rRF)b. Consequently, if rRF increases rs will increase too. 10-3Each firm has an optimal capital structure, defined as that mix of debt, preferred, and common equity that causes its stock price to be maximized. A value-maximizing firm will determine its optimal capital structure, use it as a target, and then raise new capital in a manner designed to keep the actual capital structure on target over time. The target proportions of debt, preferred stock, and common equity, along with the costs of those components, are used to calculate the firm’s weighted average cost of capital, WACC. The weights could be based either on the accounting values shown on the firm’s balance sheet (book values) or on the market values of the different securities. Theoretically, the weights should be based on market values, but if a firm’s book value weights are reasonably close to its market value weights, book value weights can be used as a proxy for market value weights. Consequently, target market value weights should be used in the WACC equation. 10-4In general, failing to adjust for differences in risk would lead the firm to accept too many risky projects and reject too many safe ones. Over time, the firm would become more risky, its WACC would increase, and its shareholder value would suffer. The cost of capital for average-risk projects would be the firm’s cost of capital, 10%. A somewhat higher cost would be used for more risky projects, and a lower cost would be used for less risky ones. For example, we might use 12% for more risky projects and 9% for less risky projects. These choices are arbitrary. 10-5The cost of retained earnings is lower than the cost of new common equity; therefore, if new common stock had to be issued then the firm’s WACC would increase. The calculated WACC does depend on the size of the capital budget. A firm calculates its retained earnings breakpoint (and any other capital breakpoints for additional debt and preferred). This R/E breakpoint represents the amount of capital raised beyond which new common stock must be issued. Thus, a capital budget smaller than this breakpoint would use the lower cost retained earnings and thus a lower WACC. A capital budget greater than this breakpoint would use the higher cost of new equity and thus a higher WACC. Dividend policy has a significant impact on the WACC. The R/E breakpoint is calculated as the addition to retained earnings divided by the equity fraction. The higher the firm’s dividend payout, the smaller the addition to retained earnings and the lower the R/E breakpoint. (That is, the firm’s WACC will increase at a smaller capital budget.) Solutions to End-Of-Chapter Problems 10-1rd(1 – T) = 0.12(0.65) = 7.80%. 10-2Pp = $47.50; Dp = $3.80; rp = ? rp= = = 8%. 10-340% Debt; 60% Common equity; rd = 9%; T = 40%; WACC = 9.96%; rs = ? WACC= (wd)(rd)(1 – T) + (wc)(rs) 0.0996= (0.4)(0.09)(1 – 0.4) + (0.6)rs 0.0996= 0.0216 + 0.6rs 10-4P0 = $30; D1 = $3.00; g = 5%; rs = ? a.rs = + g = + 0.05 = 15%. b.F = 10%; re = ? re = + g= + 0.05 = + 0.05 = 16.11%. 10-5Projects A, B, C, D, and E would be accepted since each project’s return is greater than the firm’s WACC. 10-6a.rs = + g = + 7% = 9.3% + 7% = 16.3%. b.rs= rRF + (rM – rRF)b = 9% + (13% – 9%)1.6 = 9% + (4%)1.6 = 9% + 6.4% = 15.4%. c.rs = Bond rate + Risk premium = 12% + 4% = 16%. d.Since you have equal confidence in the inputs used for the three approaches, an average of the three methodologies probably would be warranted. rs = = 15.9%. 10-7a.rs= + g = + 0.06 b.F = ($36.00 – $32.40)/$36.00 = $3.60/$36.00 = 10%. c.re = D1/[P0(1 – F)] + g = $3.18/$32.40 + 6% = 9.81% + 6% = 15.81%. 10-8Debt = 40%, Common equity = 60%. P0 = $22.50, D0 = $2.00, D1 = $2.00(1.07) = $2.14, g = 7%. rs = + g = + 7% = 16.51%. WACC= (0.4)(0.12)(1 – 0.4) + (0.6)(0.1651) = 0.0288 + 0.0991 = 12.79%. 10-9Capital SourcesAmountCapital Structure Weight Common Equity 1,728 60.0 WACC= wdrd(1 – T) + wcrs = 0.4(0.13)(0.6) + 0.6(0.16) = 0.0312 + 0.0960 = 12.72%. 10-10If the investment requires $5.9 million, that means that it requires $3.54 million (60%) of common equity and $2.36 million (40%) of debt. In this scenario, the firm would exhaust its $2 million of retained earnings and be forced to raise new stock at a cost of 15%. Needing $2.36 million in debt, the firm could get by raising debt at only 10%. Therefore, its weighted average cost of capital is: WACC = 0.4(10%)(1 – 0.4) + 0.6(15%) = 11.4%. 10-11rs= D1/P0 + g = $2(1.07)/$24.75 + 7% = 8.65% + 7% = 15.65%. WACC = wd(rd)(1 – T) + wc(rs); wc = 1 – wd. 13.95%= wd(11%)(1 – 0.35) + (1 – wd)(15.65%) 0.1395= 0.0715wd + 0.1565 – 0.1565wd wd= 0.20 = 20%. 10-12a.rd = 10%, rd(1 – T) = 10%(0.6) = 6%. D/A = 45%; D0 = $2; g = 4%; P0 = $20; T = 40%. Project A: Rate of return = 13%. Project B: Rate of return = 10%. rs = $2(1.04)/$20 + 4% = 14.40%. b.WACC = 0.45(6%) + 0.55(14.40%) = 10.62%. c.Since the firm’s WACC is 10.62% and each of the projects is equally risky and as risky as the firm’s other assets, MEC should accept Project A. Its rate of return is greater than the firm’s WACC. Project B should not be accepted, since its rate of return is less than MEC’s WACC. 10-13If the firm’s dividend yield is 5% and its stock price is $46.75, the next expected annual dividend can be calculated. Dividend yield= D1/P0 Next, the firm’s cost of new common stock can be determined from the DCF approach for the cost of equity. re= D1/[P0(1 – F)] + g = $2.3375/[$46.75(1 – 0.05)] + 0.12 10-14rp = = = 11.94%. 10-15a.Examining the DCF approach to the cost of retained earnings, the expected growth rate can be determined from the cost of common equity, price, and expected dividend. However, first, this problem requires that the formula for WACC be used to determine the cost of common equity. WACC= wd(rd)(1 – T) + wc(rs) 13.0%= 0.4(10%)(1 – 0.4) + 0.6(rs) rs= 0.17667 or 17.67%. From the cost of common equity, the expected growth rate can now be determined. rs= D1/P0 + g 0.17667= $3/$35 + g g= 0.090952 or 9.10%. b.From the formula for the long-run growth rate: g= (1 – Div. payout ratio) ROE = (1 – Div. payout ratio) (NI/Equity) 0.090952= (1 – Div. payout ratio) ($1,100 million/$6,000 million) 0.090952= (1 – Div. payout ratio) 0.1833333 0.496104= (1 – Div. payout ratio) Div. payout ratio= 0.503896 or 50.39%. 10-16a.With a financial calculator, input N = 5, PV = -4.42, PMT = 0, FV = 6.50, and then solve for I/YR = g = 8.02% 8%. b.D1 = D0(1 + g) = $2.60(1.08) = $2.81. c.rs = D1/P0 + g = $2.81/$36.00 + 8% = 15.81%. 10-17a.rs= + g 0.09= + g 0.09= 0.06 + g Less: Dividends per share 3.600 Retained earnings per share$1.800 Rate of return 0.090 Increase in EPS$0.162 Plus: Current EPS 5.400 Next year’s EPS$5.562 Alternatively, EPS1 = EPS0(1 + g) = $5.40(1.03) = $5.562. 10-18a.rd(1 – T) = 0.10(1 – 0.3) = 7%. rp = $5/$49 = 10.2%. rs = $3.50/$36 + 6% = 15.72%. Debt [0.10(1 – T)]0.157.00%1.05% c.Projects 1 and 2 will be accepted since their rates of return exceed the WACC. 10-19a.If all project decisions are independent, the firm should accept all projects whose returns exceed their risk-adjusted costs of capital. The appropriate costs of capital are summarized below: RequiredRate ofCost of Therefore, Ziege should accept projects A, C, E, F, and H. b.With only $13 million to invest in its capital budget, Ziege must choose the best combination of Projects A, C, E, F, and H. Collectively, the projects would account for an investment of $21 million, so naturally not all these projects may be accepted. Looking at the excess return created by the projects (rate of return minus the cost of capital), we see that the excess returns for Projects A, C, E, F, and H are 2%, 1.5%, 0.5%, 2.5%, and 3.5%. The firm should accept the projects which provide the greatest excess returns. By that rationale, the first project to be eliminated from consideration is Project E. This brings the total investment required down to $15 million, therefore one more project must be eliminated. The next lowest excess return is Project C. Therefore, Ziege’s optimal capital budget consists of Projects A, F, and H, and it amounts to $12 million. c.Since Projects A, F, and H are already accepted projects, we must adjust the costs of capital for the other two value producing projects (C and E). RequiredRate ofCost of If new capital must be issued, Project E ceases to be an acceptable project. On the other hand, Project C’s expected rate of return still exceeds the risk-adjusted cost of capital even after raising additional capital. Hence, Ziege’s new capital budget should consist of Projects A, C, F, and H and requires $15 million of capital, so $3 million of additional capital must be raised. 10-20a.After-tax cost of new debt: rd(1 – T) = 0.09(1 – 0.4) = 5.4%. Cost of common equity: Calculate g as follows: With a financial calculator, input N = 9, PV = -3.90, PMT = 0, FV = 7.80, and then solve for I/YR = g = 8.01% 8%. rs = + g = + 0.08 = + 0.08 = 0.146 = 14.6%. Debt [0.09(1 – T)]0.405.4%2.16% Common equity (RE)0.6014.6 8.76 Note to Instructors: The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD. Skye’s WACC will be 11.22% so long as it finances with debt, preferred stock, and common equity raised as retained earnings. If it expands so rapidly that it uses up all of its retained earnings and must issue new common stock with a cost of 13.83% (average of DCF and CAPM estimates), then its WACC will increase to 11.51%. Coleman Technologies Inc. Cost of Capital Coleman Technologies is considering a major expansion program that has been proposed by the company’s information technology group. Before proceeding with the expansion, the company must estimate its cost of capital. Assume that you are an assistant to Jerry Lehman, the financial vice president. Your first task is to estimate Coleman’s cost of capital. Lehman has provided you with the following data, which he believes may be relevant to your task: 1.The firm’s tax rate is 40%. 2.The current price of Coleman’s 12% coupon, semiannual payment, noncallable bonds with 15 years remaining to maturity is $1,153.72. Coleman does not use short-term interest-bearing debt on a permanent basis. New bonds would be privately placed with no flotation cost. 3.The current price of the firm’s 10%, $100 par value, quarterly dividend, perpetual preferred stock is $111.10. 4.Coleman’s common stock is currently selling for $50 per share. Its last dividend (D0) was $4.19, and dividends are expected to grow at a constant rate of 5% in the foreseeable future. Coleman’s beta is 1.2, the yield on T‑bonds is 7%, and the market risk premium is estimated to be 6%. For the bond-yield-plus-risk-premium approach, the firm uses a risk premium of 4%. 5.Coleman’s target capital structure is 30% debt, 10% preferred stock, and 60% common equity. To structure the task somewhat, Lehman has asked you to answer the following questions. A.(1)What sources of capital should be included when you estimate Coleman’s WACC? Answer:[Show S10-1 through S10-3 here.] The WACC is used primarily for making long-term capital investment decisions, i.e., for capital budgeting. Thus, the WACC should include the types of capital used to pay for long-term assets, and this is typically long-term debt, preferred stock (if used), and common stock. Short-term sources of capital consist of (1) spontaneous, noninterest-bearing liabilities such as accounts payable and accrued liabilities and (2) short-term interest-bearing debt, such as notes payable. If the firm uses short-term interest-bearing debt to acquire fixed assets rather than just to finance working capital needs, then the WACC should include a short-term debt component. Noninterest-bearing debt is generally not included in the cost of capital estimate because these funds are netted out when determining investment needs, that is, net operating rather than gross operating working capital is included in capital expenditures. A.(2)Should the component costs be figured on a before-tax or an after-tax basis? Answer:[Show S10-4 here.] Stockholders are concerned primarily with those corporate cash flows that are available for their use, namely, those cash flows available to pay dividends or for reinvestment. Since dividends are paid from and reinvestment is made with after-tax dollars, all cash flow and rate of return calculations should be done on an after-tax basis. A.(3)Should the costs be historical (embedded) costs or new (marginal) costs? Answer:[Show S10-5 and S10-6 here.] In financial management, the cost of capital is used primarily to make decisions that involve raising new capital. Thus, the relevant component costs are today’s marginal costs rather than historical costs. B.What is the market interest rate on Coleman’s debt and its component cost of debt? Answer:[Show S10-7 through S10-9 here.] Coleman’s 12% bond with 15 years to maturity is currently selling for $1,153.72. Thus, its yield to maturity is 10%: Enter N = 30, PV = -1153.72, PMT = 60, and FV = 1000, and then press the I/YR button to find rd/2 = I/YR = 5.0%. Since this is a semiannual rate, multiply by 2 to find the annual rate, rd = 10%, the pre-tax cost of debt. Since interest is tax deductible, Uncle Sam, in effect, pays part of the cost, and Coleman’s relevant component cost of debt is the after-tax cost: rd(1 – T) = 10.0%(1 – 0.40) = 10.0%(0.60) = 6.0%. Should you use the nominal cost of debt or the effective annual cost? Answer:Our 10% pre-tax estimate is the nominal cost of debt. Since the firm’s debt has semiannual coupons, its effective annual rate is 10.25%: (1.05)2 – 1.0 = 1.1025 – 1.0 = 0.1025 = 10.25%. However, nominal rates are generally used. The reason is that the cost of capital is used in capital budgeting, and capital budgeting cash flows are generally assumed to occur at year-end. Therefore, using nominal rates makes the treatment of the capital budgeting discount rate and cash flows consistent. C.(1)What is the firm’s cost of preferred stock? Answer:[Show S10-10 and S10-11 here.] Since the preferred issue is perpetual, its cost is estimated as follows: rp = = = = 0.090 = 9.0%. Note (1) that since preferred dividends are not tax deductible to the issuer, there is no need for a tax adjustment, and (2) that we could have estimated the effective annual cost of the preferred, but as in the case of debt, the nominal cost is generally used. C.(2)Coleman’s preferred stock is riskier to investors than its debt, yet the preferred’s yield to investors is lower than the yield to maturity on the debt. Does this suggest that you have made a mistake? (Hint: Think about taxes.) Answer:[Show S10-12 and S10-13 here.] Corporate investors own most preferred stock, because 70% of preferred dividends received by corporations are nontaxable. Therefore, preferred often has a lower before-tax yield than the before-tax yield on debt issued by the same company. Note, though, that the after-tax yield to a corporate investor and the after-tax cost to the issuer are higher on preferred stock than on debt. D.(1)Why is there a cost associated with retained earnings? Answer:[Show S10-14 through S10-16 here.] Coleman’s earnings can either be retained and reinvested in the business or paid out as dividends. If earnings are retained, Coleman’s shareholders forgo the opportunity to receive cash and to reinvest it in stocks, bonds, real estate, and the like. Thus, Coleman should earn on its retained earnings at least as much as its stockholders themselves could earn on alternative investments of equivalent risk. Further, the company’s stockholders could invest in Coleman’s own common stock, where they could expect to earn rs. We conclude that retained earnings have an opportunity cost that is equal to rs, the rate of return investors expect on the firm’s common stock. D.(2)What is Coleman’s estimated cost of common equity using the CAPM approach? Answer:[Show S10-17 here.] The CAPM estimate for Coleman’s cost of common equity is 14.2%: rs= rRF + (rM – rRF)b = 7.0% + (6.0%)1.2 = 7.0% + 7.2% = 14.2%. E.What is the estimated cost of common equity using the DCF approach? Answer:[Show S10-18 through S10-20 here.] Since Coleman is a constant growth stock, the constant growth model can be used: = = = = + 0.05 = 0.088 + 0.05 = 8.8% + 5.0% = 13.8%. F.What is the bond-yield-plus-risk-premium estimate for Coleman’s cost of common equity? Answer:[Show S10-21 here.] The bond-yield-plus-risk-premium estimate is 14%: rs = Bond yield + Risk premium = 10.0% + 4.0% = 14.0%. Note that the risk premium required in this method is difficult to estimate, so this approach only provides a ballpark estimate of rs. It is useful, though, as a check on the DCF and CAPM estimates, which can, under certain circumstances, produce unreasonable estimates. G.What is your final estimate for rs? Answer:[Show S10-22 here.] The following table summarizes the rs estimates: MethodEstimate rd + rp 14.0 At this point, considerable judgment is required. If a method is deemed to be inferior due to the “quality” of its inputs, then it might be given little weight or even disregarded. In our example, though, the three methods produced relatively close results, so we decided to use the average, 14%, as our estimate for Coleman’s cost of common equity. H.Explain in words why new common stock has a higher cost than retained earnings. Answer:[Show S10-23 here.] The company is raising money in order to make an investment. The money has a cost, and this cost is based primarily on the investors’ required rate of return, considering risk and alternative investment opportunities. So, the new investment must provide a return at least equal to the investors’ opportunity cost. If the company raises capital by selling stock, the company doesn’t receive all of the money that investors contribute. For example, if investors put up $100,000, and if they expect a 15% return on that $100,000, then $15,000 of profits must be generated. But if flotation costs are 20% ($20,000), then the company will receive only $80,000 of the $100,000 investors contribute. That $80,000 must then produce a $15,000 profit, or a $15/$80 = 18.75% rate of return versus a 15% return on equity raised as retained earnings. I.(1)What are two approaches that can be used to adjust for flotation costs? Answer:The first approach is to include the flotation costs as part of the project’s up-front cost. This reduces the project’s estimated return. The second approach is to adjust the cost of capital to include flotation costs. This is most commonly done by incorporating flotation costs in the DCF model. I.(2)Coleman estimates that if it issues new common stock, the flotation cost will be 15%. Coleman incorporates the flotation costs into the DCF approach. What is the estimated cost of newly issued common stock, considering the flotation cost? Answer:[Show S10-24 and S10-25 here.] re= + g = + 5.0% = + 5.0% = 15.35%. J.What is Coleman’s overall, or weighted average, cost of capital (WACC)? Ignore flotation costs. Answer:[Show S10-26 here.] Coleman’s WACC is 11.1%. WACC= wdrd(1 – T) + wprp + wcrs = 0.3(10%)(0.6) + 0.1(9%) + 0.6(14%) = 1.8% + 0.9% + 8.4% = 11.1%. K.What factors influence Coleman’s composite WACC? Answer:[Show S10-27 here.] There are factors that the firm cannot control and those that they can control that influence WACC. Factors the firm cannot control: Factors the firm can control: Capital structure policy L.Should the company use the composite WACC as the hurdle rate for each of its projects? Explain. Answer:[Show S10-28 here.] No. The composite WACC reflects the risk of an average project undertaken by the firm. Therefore, the WACC only represents the “hurdle rate” for a typical project with average risk. Different projects have different risks. The project’s WACC should be adjusted to reflect the project’s risk.

What is an equation What is the difference between an expression and an equation? Difference Between Expression and Equation Equations and expressions are common in understanding mathematical concepts. Compared to language, expressions are like sentences and similes like complete sentences. The expressions are not related and the equations show relationships. You must solve equations when expressions cannot be calculated. What is the difference between formula and equation? As the name suggests, the difference between a formula and an equation. is that the formula (mathematics) is a mathematical rule expressed symbolically while the equation (senseid) (mathematics) is a statement that two expressions are equal, expressed by writing two expressions separated by an equal sign, from which you get a certain Measure. What is an example of a math equation? Equations in Mathematics. Definition. An equation is a statement that expresses the equality of two mathematical expressions. An equation has an equal sign, an expression on the right, and an expression on the left. Examples of equations. 3x + 3 = 2x + 4: The left side of the equation is 3x + 3 and the right side is 2x + 4. What is equation in simple terms? A math equation is an expression that contains at least one variable (= unknown value) and an equal sign (=) with a math expression on each side. An equal sign indicates that both sides have the same meaning. The equation can be as simple as x = 0 or as complex as 4 (3y^99) + 76 = 42 ÷ 3x or more. How does an equation differ from an expression? Answer: An expression is a mathematical statement that combines numbers, variables and operators, and an equation is a mathematical statement in which two expressions are equal. What is an expression vs an equation? The following points summarize important differences between an expression and an equation: A mathematical expression that groups numbers, variables, and operators to indicate the value of something is called an expression. An expression is part of a sentence that represents a single numerical value. The expression is simplified by a calculation in which the values are replaced by variables. What is different about equation and expression? The main difference between an expression and an equation is that the expression does not contain the equals sign and the equation does not contain the equals sign. What are variables, expression and equation? An equation defines two equal expressions. A variable is a character that represents a number. In the expression, 8x + 17x is a variable. In the equation = 25 and is a variable. For example, variables do not have to be letters, the expression 6 + 2 contains a variable. An expression or equation can contain more than one variable. What is the difference between an expression and an equation third grade math An equation is described as a mathematical operator with two identical expressions. An expression is part of a sentence that represents a single numerical value. A comparison, on the other hand, is a sentence that shows similarity between two expressions. What do you learn in expression vs equation? In the Expression vs. Equation section, students explore the keywords used to describe expressions and equations, such as coefficient, coefficient, term, variable, and constant. Then they will practice defining and naming parts of the expression with their new terms! Sign up to start collecting! When is the equality of two expressions True? If the equation is conditional, the equality of the two expressions applies to the specific value of the variables involved. However, if the equation is identical, then equality holds for all values of the variable. There are four types of equations, which are described below:. Which is an example of an equation in math? Examples: 9x + 2, x - 9, 3p + 5, 4m + 10 In mathematics, the term equation means the statement of equality. This is a sentence where two expressions match. To satisfy the equation, it is important to determine the value of the variable in question, this is called the solution or square root of the equation. When is an equation called a quadratic equation? Quadratic Equation: When the highest power in an equation is 2, it is called a quadratic equation. For example: The following points summarize important differences between an expression and an equation: A mathematical set that combines numbers, variables, and operators to indicate the value of something is called an expression. What is an expression in math Microsoft Word makes it easy to enter math expressions. Click in the text to place the cursor where you want to write the algebraic expression. On the ribbon, click the Insert tab. On the ribbon, click the Equation button, represented by the pi symbol. How do you solve math expression? A linear algebraic equation is nice and simple, it contains only first degree constants and variables (no exponentials or oddities). To solve this problem, simply use multiplication, division, addition, and subtraction to isolate the variable and solve for x. What does 'evaluating expressions' mean in math? Evaluating any expression is a very simple mathematical concept. When you substitute a specific value for each variable, you perform an operation called evaluating an expression. A mathematical expression is algebraic, it contains a finite set of variables and numbers, and then algebraic operations. Evaluation makes printing easy. What does express mean in a math problem? In mathematics, if you express a quantity or a mathematical problem in a certain way, write it down with certain symbols, numbers or equations. This equation can be expressed as follows. It is indicated as a percentage. An order or an urgent order is an order that is clearly and deliberately expressed. Is an inequality an expression or an equation? Equations and inequalities are mathematical statements formed by relating two expressions to each other. In an equation, two expressions are considered equal, which is denoted by =. Although, as with inequality, the two terms do not necessarily coincide, as indicated by the symbols: >, a greater sign that is a symbol indicating a serious inequality between two particular values, than a value greater than the one on the left . What are the steps to solving inequalities? Steps to solve inequalities: • Select the variable to the LEFT side of the inequality. • Addition or subtraction to isolate the variable term. • Multiply or divide to isolate the variable. • Remember to flip the symbol when multiplying or dividing by a negative number! How do I write a mathematical equation? To write a mathematical equation, you must first press the \ key. A menu opens where you can choose what to write, draw, or paste. What does solving equation Mean in math? In mathematics, solving an equation means finding its solutions, which are values (numbers, functions, quantities, etc.) that satisfy the condition given by the equation, which generally consists of two expressions that are connected by an equal sign. What does X mean in a math equation? The letter x is often used in algebra to denote an unknown value. This is called a variable or sometimes unknown. At x + 2 = 7, x is a variable, but you can determine its value if you try! The variable does not have to be x, it can be y, w or any letter, name or symbol.

3 edition of Applied Mathematics for Technical Programs found in the catalog. Applied Mathematics for Technical Programs Robert G. Moon by Prentice Hall Written in English |The Physical Object| The source of all great mathematics is the special case, the con-crete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.1 We begin by describing a rather general framework for the derivation of PDEs. An introductory course in applied mathematics for Industrial and Technical Programs that covers arithmetic operations with signed numbers, fractions and decimals; measuring methodology and unit conversion; and basic algebraic concepts including solving linear equations and systems of equations. Engineering Mathematics – I Dr. V. Lokesha 10 MAT11 8 Leibnitz’s Theorem: It provides a useful formula for computing the nth derivative of a product of two functions. Statement: If u and v are any two functions of x with u n and v n as their nth derivative. Then the nth derivative of uv is. For courses in technical and pre-engineering technical programs or other programs for which coverage of basic mathematics is required. recognizing the importance of these topics for success in solving applied problems. Also available with MyLab Math MyLab(TM) Math is an online homework, tutorial, and assessment program designed to work with. Applied mathematics and statistics is an integral part of emerging fields such as computational medicine/biology, language processing, information security, and computer science. In today’s data-intensive world, it is used to answer questions and solve problems in areas as diverse as finance, government and law, medicine, and national defense. The undergraduate curriculum in Applied Mathematics is designed to give training in the applications of mathematics in engineering and science. The use of computational methods and implementation of algorithms on computers is central. Required technical electives should be selected after consultation with an Applied Mathematics advisor. They. English maritime writing Speech by M. Francois Mitterrand, President of the French Republic, Before the Ubersee-Club in Hamburg The Slipperdown chant Everyday Spelling Teachers Edition Grade 2 Library automation and information networks 1988 G.B. stamps + varieties [One on you] Northwest Style *Wrong * Practical microstrip design and applications The secret keeper. Survey and perspectives of Chiles economic development, 1940-1965 In addition to MATHEMATICS FOR MACHINE TECHNOLOGY, 8E, Dr. Peterson has authored or co-authored four other Cengage books: INTRODUCTORY TECHNICAL MATHEMATICS, TECHNICAL MATHEMATICS, TECHNICAL MATHEMATICS WITH CALCULUS, and MATH FOR THE AUTOMOTIVE TRADE/5(9). Get this from a library. Applied mathematics for technical programs: algebra. [Robert G Moon]. This book stands out among applied mathematics books. I like it very much, although it is certainly not easy to read. The equations and mathematics in the book are not particularly intimidating; the book is challenging because it gets you to think about difficult concepts at a sort of "meta" level. Book Description: The Mathematics of Technical Analysis by Clifford J. Sherry and Jason W. Sherry promises to revolutionize how we think about the markets. In this ground-breaking work, the authors challenge the random walk hypothesisthe idea that there is /5(4). The idea was to produce a book that gives a flavour of what applied mathematics is about, what it’s used for, and what the future might be. In no sense can it be comprehensive, even in about pages, but we tried to have a selection of topics that we thought were the most interesting and to get the very best authors we could from around. Download Applied Mathematics - III By G.V. Kumbhojkar - The book has been rebinded and is useful for mechanical, automobile, production and civil engineering. "Applied Mathematics - III By G.V. Kumbhojkar PDF File" "Free Download Applied Mathematics. Best Colleges for Applied Mathematics. Approximately 8, applied mathematics degrees were awarded to students last year in the United States. With so many choices it can be a daunting task finding the right fit. This year's Best Colleges for Applied Mathematics ranking analyzed 55 colleges and universities that offer a bachelor's degree in applied mathematics. Developmental Math; Finite Math & Applied Calculus; Liberal Arts Math / Teacher's Math; Math for Careers; Mathematics; Dual Enrollment Programs. Preview; Purchase; Get Started; Training and Support; Technical Mathematics. Find resources for working and learning online during COVID PreK–12 Education. The best-seller in technical mathematics gets an “Oh, wow!” update. The 11th Edition of Basic Technical Mathematics with Calculus is a bold revision of this classic bestseller. The text now sports an engaging full-color design, and new co-author Rich Evans has introduced a wealth of relevant applications and improvements, many based on user s: 6. Applied Technical Mathematics Doug Gardner Rogue Community College Rogue Community College Redwood Highway Grants Pass, OR () An Innovative Math in CTE Curriculum: Funded by an ATE grant from the National Science Foundation: About this Book: This book is designed to be read. Harvard University offers 3 Applied Mathematics Degree programs. It's a large private university in a mid sized city. Instudents graduated in the study area of Applied Mathematics with students earning 91 Bachelor's degrees, 12 Master's degrees, and 4 Doctoral degrees. Find Technical Mathematics Textbooks at up to 90% off. Plus get free shipping on qualifying orders $25+. Choose from used and new textbooks or get instant access with eTextbooks and digital materials. Advising Note: This suggested schedule represents one possible scenario for taking the minimum number of Applied Math courses required for the the 24 credit option requirement is satisfied, further technical electives can be selected from additional APPM. Media related to Applied mathematics at Wikimedia Commons; The Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to promoting the interaction between mathematics and other scientific and technical communities. Aside from organizing and sponsoring numerous conferences, SIAM is a major publisher of research journals and books in applied mathematics. Modeling and Approved Electives: Applied Mathemat 91r, ; Economics ; or an approved advanced technical elective from outside of the student’s application area Application: Five courses from an area of application in which mathematics has been substantively applied, selected to provide a coherent and cumulative introduction to. SIAM publishes high-quality textbooks and monographs for applied mathematicians, computational scientists, and engineers working in academia, government labs, and industry. Check out our books, which highlight the many different aspects of applied mathematics. Contact us if you are interested in becoming a SIAM book author. Courses in applied mathematics vary in the STEM fields they concern, but any of them can prepare students for advanced technical careers. Ordinary and partial differential equations are critical knowledge for engineers and physicists, and information and coding theory form the backbone of most computer scientists’ expertise. The language of mathematics consists of many dialects, or subdisciplines. These include arithmetic, algebra, geometry, trigonometry, and statistics, to name a few. This book concentrates on the rudimentary skills needed to study mathematics and solve practical problems encountered in technical. Lecturer / Mathematics Rajagopal Polytechnic College Gudiyatham This book has been prepared by the Directorate of Technical Education This book has been printed on 60 G.S.M Paper Through the Tamil Nadu Text book and Educational Services Corporation Convener Thiru Lecturer (S.G) / Mathematics Rajagopal Polytechnic College Gudiyatham. Mathematics plays a major role in the bottom line of industrial organizations, and helps companies perform better in today’s data-driven marketplace. A career in applied mathematics is more than just crunching numbers. It's being able to use mathematics to. For courses in technical and pre-engineering technical programs or other programs for which coverage of basic mathematics is required. The best-seller in technical mathematics gets an "Oh, wow!" update The 11th Edition of Basic Technical Mathematics is a bold .Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics (Princeton Puzzlers) by Robert B. Banks and a great selection of related books, art and collectibles available now at. In addition to MATHEMATICS FOR MACHINE TECHNOLOGY, 8E, Dr. Peterson has authored or co-authored four other Cengage books: INTRODUCTORY TECHNICAL MATHEMATICS, TECHNICAL MATHEMATICS, TECHNICAL MATHEMATICS WITH CALCULUS, and MATH FOR THE AUTOMOTIVE s:

About This Chapter Quadratic & Polynomial Equations - Chapter Summary If you're seeking to learn or gain additional information about quadratic and polynomial equations, this chapter can help! Top instructors have developed entertaining lessons you can use to discover or refresh your knowledge of the mathematical operations associated with quadratic equations, polynomial expressions and much more. By taking time to review this chapter, you will have the ability to: - Understand the meaning of polynomials, binomials, quadratics and basic polynomial graphs - Factor polynomial expressions - Write polynomial equations by using rational and complex zeros - Solve a quadratic equation by factoring - Define and understand the equation for the quadratic formula - Complete the square, then write the standard form of an equation - Choose the correct method for solving quadratic equations - Solve quadratic equations in a real-world context Feel confident that you have a good grasp of quadratic and polynomial equations by tailoring this chapter's lessons to your schedule and study habits. Access the lessons anytime from any computer, smartphone or tablet. Review them in any sequence and revisit them as often as needed. When ready, assess your knowledge by taking short lesson quizzes and a chapter exam. If you need additional details about lesson topics, be sure to submit your questions to our experts. 1. What are Polynomials, Binomials, and Quadratics? What do polynomials, binomials, and quadratics have in common? What are their differences? How can you identify each of them? Watch this video lesson to find the answers. 2. Factoring Polynomial Expressions When it comes to factoring polynomials, there are several methods to choose from depending on what kind of polynomial you are looking at. Watch this video lesson to learn how to identify and use the appropriate method. 3. Using Rational & Complex Zeros to Write Polynomial Equations In this lesson, you will learn how to write a polynomial function from its given zeros. You will learn how to follow a process that converts zeros into factors and then factors into polynomial functions. 4. Understanding Basic Polynomial Graphs This lesson will cover understanding basic polynomial graphs. The lesson focuses on how exponents and leading coefficients alter the behavior of the graphs. 5. How to Solve a Quadratic Equation by Factoring If your favorite video game, 'Furious Fowls,' gave you the quadratic equation for each shot you made, would you be able to solve the equation to make sure every one hit its target? If not, you will after watching this video! 6. What is the Quadratic Formula? - Definition, Equation & Proof The quadratic formula is a method that is used to find the roots of a quadratic equation. In this lesson, you will learn about the history of the quadratic formula, how to use it, and prove it. A quiz at the end will solidify what you've learned. 7. How to Complete the Square Completing the square can help you learn where the maximum or minimum of a parabola is. If you're running a business and trying to make some money, it might be a good idea to know how to do this! Find out what I'm talking about here. 8. Write the Standard Form of an Equation by Completing the Square In math, we have a process called completing the square where you take your quadratic equation and rewrite it to make it easier to solve. Watch this video lesson to learn how you can do this process easily yourself. 9. Deciding on a Method to Solve Quadratic Equations There are several ways to solve quadratic equations, so how do you pick a good approach? In this lesson, we'll discuss a systematic method that helps you pick the best way to solve any quadratic equation that might cross your path. 10. Real World Interpretations of Quadratic Equations You have probably heard the saying, 'What goes up must come down?' But did you ever wonder why? The physical force that causes this behavior is gravity. And the mathematical relationship that describes it is a quadratic equation. Earning College Credit Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level. To learn more, visit our Earning Credit Page Transferring credit to the school of your choice Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you. Other chapters within the NY Regents Exam - Algebra I: Test Prep & Practice course - Algebraic Units & Modeling - Structure of Mathematical Expressions - Polynomial Operations Basics - Creating Equations, Inequalities & Expressions - Solving & Graphing Equations & Inequalities - Understanding Data Using Statistics - Interpreting Linear Models - Understanding Functions - Interpreting, Analyzing & Graphing Functions - Building Linear & Exponential Functions - Basics of Rational & Irrational Numbers - About the NY Regents Examinations - NY Regents Exam - Algebra I Flashcards

Solved by verified expert:Upon graduating from UCI, you start a job in finance earning $72,000 per year and decides to buy a home. You can afford to pay no more than $3,000 per month for the monthly payment on 30-year fixed rate home loan(mortgage) and the current conventional 30-year fixed mortgage interest rate you qualify to borrow at is 5%What is the maximum size home loan you can afford?Suppose you borrow the maximum loan amount found in part(a). How much total interest do you pay in the first 2 month?Suppose black rock stock is trading at $400 and pays a discrete dividend of $3.00 to its sharebolders once per quarter, the first dividend of $3 to be paid in 0.25 years from today, the second dividend of $3 to be paid in 0.5 years from today, the third dividend of $3 to be paid in 0.75 years from today, and the fourth dividend of $3 to be paid in 1 year from today. The continuously compound risk-free interest rate that you can lend or borrow at is 8%. In this problem, you are to use continuous compounding of interest in all computations.A)What is the fair value price for a one-year prepaid forward contract on BLK?B)Suppose the market prices for a normal(not prepaid) 1-year forward contract on Blackrock are as follows:a)Bid price for BLK forward contract:$520.95b)Ask price for BLK contract: $530.95Does an arbitrager opportunity exist? If so, specify the exact components of the portfolio you need to capture the arbitrage, and the exact arbitrage profit.3. Your are a bond portfolio manager at Blackrock and the investment committee has asked you to buy a bond with price B1, and sell short a certain quantity N of a second bond with price B2 Bond with price B1 is a 1-year zero coupon bond with a yield-to-maturity’s of 1% Bond with price B2 is a 2-year zero coupon bond with a yield-to-maturity of 2%The resulting portfolio value is II = B1-N*B2A)How would you choose N to optimally hedge the interest rate exposure of the portfolio II and thus minimize its sensitivity to interest rate changes? Find a numerical value for N.B)Using the value of N that you found in part (a), what is your portfolio’s profit or loss if both of the yield-to-maturities of bond B1 and B2 suddenly decrease by 1%? Round your numerical answer to the nearest 4th-decimal place.4. Consider this table of risk-free zero coupon bond (ZCB) prices:Quarter12345678ZCB 0.98040.96120.94230.92380.90570.88800.87060.8535A)Given the zero coupon bond prices in the table above, what is the 8th quarter interest rate swap price?B)Suppose there is an abrupt stock market crash and resulting flight to quality which investors sell Rosalyn assets such as stocks and aggressively buy risk-free bond. Consequently all of the zero coupon bond prices in the table suddenly increase by 0.01. What os the new 8th quarter interest rate swap price?5. An investor enters into a 2-year swap agreement to purchase crude oil at $51.25 per barrel. Soon after the swap is created, forward prices rise and the new 2-year swap price is $61.50. If interest rate are 1% and 2% on 1st and 2nd year zero coupon government bonds, respectively, what is the gain or loss to be made from unwrapping the original swap agreement?6. An iron butterfly, also known as the iron fly, is the name of an advanced, neutral-outlook options trading strategy that involves buying and holding four different option at three different strike prices. Suppose you construct an iron butterfly position on Blackrock stock as follows, with Blackrock currently trading at $400. All options have an expiration date of one year from today and the constant a>0. 1. Long one out-of-the-money put option with premium $5 and strike price of 400-a 2. Short one at-the-money put with premium $35 3.Short one at-the-money call with premium $45 4.Long one out-of-the-money call with with premium $15 and strike price of 400+aA) Draw the profit diagrams at expiration for the Blackrock iron butterfly, with the constant a as a free parameter, assuming the risk-free interest rate is zero.B) For what range of values for a does an arbitrage opportunity exist. Suppose the risk-free rate is zero and the market price for Blackrock stock option that expire in one year are as followStrike Price$350$450Call Premium$25$75Put Premium$15$60A)Specify the component and present cost of a long synthetic one-year forward agreement on Blackrock stock using only the option with strike price $350 in the table. The position should have a payoff at expiration that os identical to a one-year forward agreement with a forward agreement with a forward price of $350B)Using any of the options in the table, does an arbitrage opportunity exist? If so, specify the exact components of the portfolio you need to capture the arbitrage, and the exact arbitrage profit.8. Consider an investment that is long ten S&P 500 index futures contracts at a price $2700.00. The initial margin requirement is $36.000 per contract and the maintenance margin is $30.000 per contract. The risk-free rate is zero.A) What is the notional value of your long exposure to the S&P 500 index?B) Upon opening the position you deposit, the minimum initial margin required in your future trading account. At what S&P trading account. At what S&P 500 index future price will a margin call occur?9. For this problem, please circle True or false for each statement concerning derivative market.True or false. Consider 1-year European put and call option on Blackrock stock with the same strike price. If the strike price is equal to the fair value price for the 1-year forward price on Blackrock stock, the put premium and call premium must be equal, otherwise an arbitrage opportunity exists.True or False. The maximum loss of a short put at expiration is unbounded.True or False. The slope coefficient computed by regressing a particular stock’s historical returns on the S&P 500 index returns is call beta, the stock’s beta with respect to the S&P index, beta is important in finding variance minimizing hedged portfolios.Bond price move in the opposite direction as bond yields.The maximum loos of a short stock position is equal to the price at which you enter the short position.The premium of a call option increase as its strike price increase.A swap provides a means for replacing a stream of uncertain and charitable payment with a fixed, non-variable payment stream that is certain.The zero cost collar options trading strategy involves buying a call option and selling a put option on the same stock with the same expiration date, same premium, and higher stroke price than the call option. Suppose your go long 10 Eurodollar futures contracts at a price of 97. When the future contract settle at expiration, 3-month LIBOR is 1%. Ignoring commissions and margin interest, your position results in a $50,000 loss. The current U.S. dollar / Chinese yuan currency spot rate is $0/130 per yuan. The fair value price for the U.S. dollar / Chinese yuan exchange rate for a 2-year forward contract is $0.1408. If the U.S. dollar denominated annual interest rate is 6.0%, the Chinese yuan-denominated annual interest rate must be 3%.Fei opened a 100-share long position in Blackrock stock using the Robinhood iPhone app when Blackrock had a bid price of $390 and an offer price of $400. A few minutes later he added another 100-share long position in Blackrock stock when the bid price was $440 and the offer price was $450. He sold all 200 shares at the end of the day when Blackrock stock had a bid price of $475 and offer price of $485. The Robinhood iPhone app features zero trading commissions. Mingkang was also trading Blackrock stock on the same day, using the interactive breakers a platform that was used in the Math 133A trading competition, his first 100 shares were bought when Blackrock had a bid price of $394.99 and an offer price of $395.00, and his second 100-share were bought when Blackrock stock had a bid price of $444.99 and an offer price of $445.00. He sold all 200 shares at the end of the day when Blackrock stock had a bod price of $480.00 and offer price of $480.01. Interactive Brokers charge a $0.005 per share commission. Who I earned a higher net trading profit, Fei or Mingkang? Make sure you carefully account fort trading profit and cost. Don't use plagiarized sources. Get Your Custom Essay on Expert answer:MATH133A California Irvine Derivative Market Probl

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[SR][SR] ¸¸²¹¸¸´¸¸¸²¸ homeWorks EXPO 2016 ¸¡¸²¸¹¸¸¸¸¸¹¸¹¹¸¸¸±¸¹¸¸£¸¸¹¸²¸¹¸«¹¸¡¸±¸¸¹¸²¸¸¢¸¹¹¸¸¶¹¸¸¸±¸¸¸µ¸¸§¹¸². 8. Natural Indicators. ¹¸¸¸¸¸¸¸¸£¸¹¸ ¸²¸¢¹¸¸¹¸²¸ 4. Essay About Of Vietnam. ¹¸¸ Living world ¹¸¸£¸¸¹ kassa. 5. Natural. ¹¸¸¸¥¸¹¸¢¸¸° ¸ª¸²¸§¹ ¸¸¥¸²¸¹¸¡¹¹¸¹ Buy Essay Online For Cheap - Natural Indicators - University of Manitoba doc Nov 12, 2017 Natural indicators, high quality custom essay writing service - narrative essay with So, your professor just gave you a new assignment, and it looks like an interesting topic. The problem is you dont know how to write a narrative essay. Relax (but dont procrastinate)! Narrative essays are actually pretty fun to natural indicators, write. Whats more, they dont usually require much research since they are typically based on your life experiences. All that said, there are some important rules to follow. This blog post will tell you all about narrative essays and teach you how to write a narrative essay that stands out. Narration is writing that tells a story. Walking Across Movie. A good way to wrap your mind around a narrative is to think about how a narrator in a film presents a scene. He tells the story from a particular perspective, giving a detailed account of natural indicators what happened. Consider the narration in this clip from How the Grinch Stole Christmas : So, how is the narrators recounting of the Grinchs failure to steal Christmas related to learning how to Essay, write a narrative essay? As the narrator in your essay, you set the scene and tell the natural story from your viewpoint, giving a detailed report of events. Chances are, you narrate stories every day. I mean, didnt you just tell your friend all about that funny thing that happened in walking egypt movie class earlier? You know how to narrate. Natural Indicators. So, writing a narrative essay should be easy, right? Well, hold on, its not that simple. One of the walking across movie challenges with writing narrative essays is that you often have to distill a complex story into a limited (and to-the-point) number of natural indicators words. At the same time, you have to garner enough interest to walking, keep the reader engaged in your story. Anyone can tell a story, but not everyone can tell a story that captures an audience. Its important to indicators, keep some rules in egypt mind as you learn how to write a narrative essay. The best way to learn how to indicators, write a narrative essay is to see an example. Im going to pretend that Im the character Rudy (from the 1993 film Rudy ), and Im going to write a narrative essay about something that happened in Spectrum Techniques Phone Essay my (Rudys) life. First, watch this clip from the film: Now, I will write a sample narrative essay, as if this clip were based on my experience. Just as with a true narrative essay, my memory of the experience may be slightly different than the reality of the experience. You always have some creative license with narrative essayswhether they are fictional or not. Read this sample essay first, and then Ill break it down into natural indicators its elements: A janitor changed my life. I was at a low point, ready to quit everythingeven when I had it all. Role In The Heian Period. I didnt realize how lucky I was. At 5 foot nothing, 100 and nothing pounds, I was hardly your typical football player. But, that didnt stop me from believing that I could play for Notre Dame. It turns out, the most important part of achieving my dreams is believing in natural indicators myself. After two years of Role Period Essay trying hard to prove that I was worthy of indicators playing, I found out that I hadnt made the dress list for our kickoff game.After fighting to be on the team and sweating through every practice, I was going to competitive, sit on the bench again. So, I decided to call it quits. Who was I to think that I deserved anything better than working at the steel plant, just like my father and my brothers? If that life was good enough for natural them, why wasnt it good enough for southwest competitive me? As I stood there in section five, staring out at natural, the empty stadium, I thought of Poetry's In The Heian Period how proud my dad would have been to natural, see me out there on Spectrum, the field playing for the team we both loved so much. I felt so stupid. I wasnt a football player. I was a bench warmer nothing more. Thats when the team janitor found me standing there. Hey, he said. Dont you have to be at practice? Not anymore, I said, annoyed. I quit. Whyd you quit? You dont seem like the natural quitting type. I dont know, I said. Heian Period Essay. I just dont see the point anymore. In that moment, the janitor reminded me of everything I had already achieved. Against all odds, I had stuck with the natural indicators team for two years, and southwest competitive advantage, I was going to graduate with a degree from natural indicators, Notre Dame. What he said next drove his point home. Southwest Competitive. He said, In this lifetime, you dont have to natural indicators, prove nothin to nobody except yourself. He had a point. I had already proven myself to everyone except for me. If I didnt believe in myself who would ever believe in me? Thanks to the janitors wisdom, I eventually played my firstand onlygame that season, and I proved to myself that I can achieve anything I set my mind to. Okay, now lets pick this thing apart. Spectrum Techniques Essay. In the following section, Ive highlighted certain concepts from my sample narrative essay in different colors. Their explanations follow. A janitor changed my life. I was at a low point, ready to quit everythingeven when I had it all. I didnt realize how lucky I was. At 5 foot nothing, 100 and nothing pounds, I was hardly your typical football player. But, that didnt stop me from believing that I could play for Notre Dame. It turns out, the most important part of achieving my dreams is believing in myself. Lets break it down. Start with a strong hook . Natural. Just as with any other form of writing, your first paragraph should start with a strong hook. The sentence, a janitor changed my life , sets up the fahrenheit story with a bold statement meant to natural, capture the attention of my readers. The goal is to make readers ask, How did a janitor change your life? What happened?For more information on hook sentences, read my blog post, How to Poetry's Role Period, Write Good HookSentences. Set the scene . In this section of my first paragraph, I set the scene. I give the reader some context for my story (I was at a low point. I was a struggling football player for Notre Dame etc.). Define the natural indicators purpose . Have you ever heard anyone talk on In The Essay, and on about something without making a point? This is natural indicators a common trap for writers attempting a narrative essay. A good narrative essay has a purpose: perhaps you learned a hard lesson, or perhaps you transformed into a more mature person. Whatever the case, that purpose should be stated in the first paragraph. In the example narrative, my purpose is to make the point that the most important part of achieving my dreams is believing in myself . As you can see, the first paragraph is advantage critical to setting up a good story. Now, lets talk about natural, what goes on in your body paragraphs. After two years of trying hard to prove that I was worthy of playing, I found out that I hadnt made the dress list for our kickoff game. After fighting to be on the team and sweating through every practice , I was going to Hybrid Techniques for Cell Essay, sit on the bench again. So, I decided to call it quits. Who was I to indicators, think that I deserved anything better than working at fahrenheit 451 clarisse death, the steel plant, just like my father and my brothers? If that life was good enough for them, why wasnt it good enough for me? As I stood there in section five, staring out at natural, the empty stadium , I thought of how proud my dad would have been to see me out there on movie, the field playing for the team we both loved so much. Natural. I felt so stupid. I wasnt a football player. Egypt Movie. I was a bench warmer nothing more. Thats when the indicators team janitor found me standing there. Hey, he said. Dont you have to fahrenheit 451 clarisse death, be at indicators, practice? Not anymore, I said, annoyed . I quit. Whyd you quit? You dont seem like the southwest quitting type. I dont know, I said. Natural. I just dont see the point anymore . In that moment, the The Landscapes of Vietnam janitor reminded me of everything I had already achieved. Against all odds, I had stuck with the natural team for two years, and I was going to graduate with a degree from Notre Dame. What he said next drove his point home. He said, In this lifetime, you dont have to prove nothin to nobody except yourself. Lets break it down. Use vivid and southwest, appropriate detail . The goal here is to indicators, recreate the 451 clarisse story for natural indicators your reader just like it happened. Make the story vivid and fahrenheit 451 clarisse, full of detail. Note, however, that this is not a descriptive essay, so only include the details that matter most to natural indicators, your story . Use dialogue . Sometimes, a great story cant be told without dialogue. Its definitely okay to incorporate dialogue, as necessary, especially if its a natural part of fahrenheit death your story.In my sample essay, the conversation with the janitor is critical to the story, so including the dialogue from this interaction is indicators appropriate. Southwest Advantage. Write chronologically . Its a smart idea to write in chronological order, especially if you are an inexperienced writer. What happened first, next, and last?This will help you to stay true to your story and not wander. In this sample, I focus on the sequence of natural indicators events that led me to my moment of truth, how the janitor talked me into staying on the team, and how this changed my perspective on life. Maintain consistency in narration . Poetry's Role Period Essay. In this example narrative essay, I chose to write in the first-person narrative voice and in indicators the past tense.I chose first person because I was telling a story that happened to me (remember, Im pretending to walking across, be Rudy in this sample). I chose past tense because Im telling a story that happened in the past .Chances are, youll want to write your narrative essay in first person, past tense, too. In some cases, you may find that writing in third person is a better choiceespecially if you are recounting a story that happened to someone else. Natural. But, whatever you choose, keep it consistent throughout. Okay! Lets move on to the last paragraph. He had a point. I had already proven myself to everyone except for me. If I didnt believe in myself who would ever believe in competitive advantage me? Thanks to the janitors wisdom, I eventually played my firstand onlygame that season, and I proved to natural, myself that I can achieve anything I set my mind to . Lets break it down. Restate your purpose . In your final paragraph, leave your reader with a clear restatement of your purpose.Remember, I began this sample narrative essay by In The Heian Essay stating my purpose: The most important part of achieving my dreams is believing in myself. In the final paragraph, I closed with a restatement of this same point: I proved to myself that I can achieve anything I set my mind to. Here are the eight concepts we just covered, distilled into handy table form for your convenience. Final Thoughts on natural indicators, How to Write a Narrative Essay. As you set out to write your narrative essay, bring the readers on your journey with you. Give them a reason to across egypt movie, listen to indicators, your story. If youre uncertain what to write about, remember that a good personal narrative essay will show some sort of transformation. Across. For example, you started out as a shy person, but had an natural indicators, interesting experience that made you more outgoing. Find a story of transformation, and then write about competitive, what happened. If you need more ideas, check out these example narrative essays. Finally, always be sure to indicators, edit your personal narrative essay before you submit it! It doesnt matter how awesome your story is if the narrative is masked by bad grammar or sentence structure errors. Psst. 98% of Kibin users report better grades! Get inspiration from over 500,000 example essays. About the Author. Naomi Tepper is a former Kibin editor, the former content manager for the Kibin blog, and fahrenheit death, forever a word nerd. Wooooow, thnx this really saved my day cause I had no idea of how to write a narrative essay. Happy to help and thanks for reading! Youre welcome! Thanks for reading. Hey anyone wants to indicators, learn the best way to write an essay without tutoring . here it it. this stuff is as good as perfect to learn an essay. +Naomi Tepper thanks a ton.. this will help me improve my narrative writing skills.. Thanks again #128578; Aww, shucks thank you! More than happy to Hybrid Spread Spectrum Techniques Essay, help. Thank you I really appreciate your help. Natural. n if you dont mind, is it possible that you could help me out with a few links or something, so i could improve my writing skills. (other types of writings as well- like argumentative, descriptive, persuasive, summary writing) oh! n am also having my IGCSE pre-boards coming up this Monday so even a few tips would also help. Thanks:) Truthfully, I dont know much about the southwest competitive IGCSE boards, but I imagine there might be some timed writing tests involved, in indicators which case this post might come in handy: https://www.kibin.com/essay-writing-blog/how-to-write-a-timed-essay/ Thanks alot all of them are really helpful #128578; Can you please guide me to walking egypt, another link with summary writing.. Indicators. i think the The Scale of Cyberspace above link doesnt seem to be working.. Thanks in indicators advance. #128578; Hi Naomi. Found this really helpful, thank you. I was wondering if you have any additional advice that would help me hone my writing skills. #128578; Hi again, Sowmya! Woot! Looks like your finding your way around our blog. Nice to see you reading this post as well. Yes, I have lots of advice on writing better essays. Check out the comment I left for competitive Joshua (below), it links to some of natural my most useful blog posts. Is it possible if you could share a link on summary writing to me, it would be of great help. Thanks in advance. thank for the awesome help. You are so welcome! Thanks for the comment. #128578; THANK YOU SO MUCH . YOU HELPED ME A LOT ! Awesome! Happy to southwest competitive, help. #128578; I want to help my son write a narrative. He has to choose to be something from indicators, our timeline of study with our coop. He has chosen the Black Death. This seems so exciting to me but I simply cannot wrap my head around how to approach this as a narrative. Movie. He actually wants to natural indicators, BE the black death. Ideas? Wow! Your son sounds very creative. I love this concept. Your son should choose the characteristics of the Black Death that he wants to competitive advantage, personify. Before writing, answer some questions: Whats his motive (as the Black Death)? How does he feel about the natural indicators work he is across doing? Does he have a visible form or is he invisible? If visible, how do humans see him? Try to keep the indicators personification consistent throughout. Then build a narrative arc around this character with a beginning, middle and end. Across Movie. Perhaps tell the story of visiting on a particular family. What does he learn from the indicators experience? What is the point (thesis)? Most importantly have fun! I love this! Do you have an fahrenheit, example with a guide just like the one you did for narrative essay but for an illustration essay? Thanks so much Erin! Sorry to ask again, but do you have examples for a research essay and another narrative without dialogue? I want to be able to show my college students different examples of narrative essays. Pay for Exclusive Essay - Natural Indicators - University of Manitoba doc Nov 12, 2017 Natural indicators, custom essay order - Andrew Carnegie and the Rise of Big Business Lesson Plans for Teachers. The Andrew Carnegie and the Rise of natural indicators, Big Business lesson plan contains a variety of teaching materials that cater to all learning styles. Inside you'll find 30 Daily Lessons, 20 Fun Activities, 180 Multiple Choice Questions, 60 Short Essay Questions, 20 Essay Questions, Quizzes/Homework Assignments, Tests, and more. The lessons and activities will help students gain an intimate understanding of the text, while the tests and quizzes will help you evaluate how well the The Scale Essay students have grasped the material. View a free sample. Target Grade: 7th-12th (Middle School and High School) Length of Lesson Plan: Approximately 144 pages. Page count is natural, estimated at 300 words per page. Length will vary depending on format viewed. Browse The Andrew Carnegie and the Rise of Big Business Lesson Plan: The Andrew Carnegie and of Vietnam the Rise of indicators, Big Business lesson plan is downloadable in PDF and Word. The Word file is viewable with any PC or Mac and can be further adjusted if you want to mix questions around and/or add your own headers for things like Name, Period, and Date. The Word file offers unlimited customizing options so that you can teach in the most efficient manner possible. Once you download the file, it is yours to keep and print for across egypt, your classroom. View a FREE sample. The Lesson Plan Calendars provide daily suggestions about what to natural indicators teach. They include detailed descriptions of when to assign reading, homework, in-class work, fun activities, quizzes, tests and more. Use the entire Andrew Carnegie and the Rise of Big Business calendar, or supplement it with your own curriculum ideas. Calendars cover one, two, four, and eight week units. Determine how long your Andrew Carnegie and the Rise of Big Business unit will be, then use one of the fahrenheit 451 clarisse calendars provided to plan out your entire lesson. Chapter abstracts are short descriptions of events that occur in each chapter of Andrew Carnegie and the Rise of Big Business . They highlight major plot events and detail the natural indicators important relationships and death characteristics of important characters. The Chapter Abstracts can be used to review what the students have read, or to prepare the natural students for what they will read. Hand the abstracts out in class as a study guide, or use them as a key for Spread Techniques for Cell Phone, a class discussion. They are relatively brief, but can serve to indicators be an The Landscapes excellent refresher of Andrew Carnegie and indicators the Rise of The Scale of Cyberspace, Big Business for either a student or teacher. Indicators? Character and Object Descriptions. Character and across egypt Object Descriptions provide descriptions of the significant characters as well as objects and places in natural Andrew Carnegie and the Rise of Big Business . Of Cyberspace Essay? These can be printed out and used as an indicators individual study guide for students, a key for leading a class discussion, a summary review prior to exams, or a refresher for an educator. The character and object descriptions are also used in of Cyberspace Essay some of the quizzes and tests in this lesson plan. The longest descriptions run about natural indicators, 200 words. They become shorter as the importance of the character or object declines. This section of the lesson plan contains 30 Daily Lessons. Daily Lessons each have a specific objective and southwest offer at least three (often more) ways to teach that objective. Lessons include classroom discussions, group and partner activities, in-class handouts, individual writing assignments, at natural least one homework assignment, class participation exercises and other ways to teach students about Andrew Carnegie and the Rise of across movie, Big Business in a classroom setting. You can combine daily lessons or use the natural indicators ideas within them to create your own unique curriculum. They vary greatly from across movie day to day and offer an array of creative ideas that provide many options for an educator. Fun Classroom Activities differ from Daily Lessons because they make fun a priority. The 20 enjoyable, interactive classroom activities that are included will help students understand Andrew Carnegie and the Rise of Big Business in indicators fun and entertaining ways. Fun Classroom Activities include group projects, games, critical thinking activities, brainstorming sessions, writing poems, drawing or sketching, and countless other creative exercises. Many of the activities encourage students to of Cyberspace interact with each other, be creative and think outside of the natural box, and ultimately grasp key concepts from the text by doing rather than simply studying. Fun activities are a great way to keep students interested and engaged while still providing a deeper understanding of Spread Techniques for Cell, Andrew Carnegie and the Rise of Big Business and its themes. Essay Questions/Writing Assignments. These 20 Essay Questions/Writing Assignments can be used as essay questions on a test, or as stand-alone essay topics for a take-home or in-class writing assignment on Andrew Carnegie and the Rise of Big Business . Students should have a full understanding of the natural indicators unit material in Poetry's In The Heian Essay order to answer these questions. They often include multiple parts of the work and ask for a thorough analysis of the overall text. They nearly always require a substantial response. Essay responses are typically expected to be one (or more) page(s) and consist of indicators, multiple paragraphs, although it is possible to write answers more briefly. These essays are designed to challenge a student's understanding of the broad points in competitive advantage a work, interactions among the characters, and main points and themes of the text. But, they also cover many of the other issues specific to natural the work and to the world today. The 60 Short Essay Questions listed in this section require a one to southwest competitive advantage two sentence answer. They ask students to demonstrate a deeper understanding of Andrew Carnegie and the Rise of Big Business by describing what they've read, rather than just recalling it. The short essay questions evaluate not only indicators, whether students have read the material, but also how well they understand and can apply it. They require more thought than multiple choice questions, but are shorter than the essay questions. The 180 Multiple Choice Questions in this lesson plan will test a student's recall and Spread Spectrum for Cell Essay understanding of Andrew Carnegie and the Rise of Big Business . Use these questions for quizzes, homework assignments or tests. The questions are broken out into sections, so they focus on specific chapters within Andrew Carnegie and the Rise of Big Business . This allows you to test and review the book as you proceed through the unit. Typically, there are 5-15 questions per chapter, act or section. Use the Oral Reading Evaluation Form when students are reading aloud in natural class. Pass the Poetry's In The Period Essay forms out before you assign reading, so students will know what to expect. You can use the forms to provide general feedback on audibility, pronunciation, articulation, expression and rate of natural, speech. You can use this form to Essay about The Landscapes grade students, or simply comment on their progress. Use the Writing Evaluation Form when you're grading student essays. This will help you establish uniform criteria for grading essays even though students may be writing about different aspects of the material. Natural? By following this form you will be able to 451 clarisse evaluate the thesis, organization, supporting arguments, paragraph transitions, grammar, spelling, punctuation, etc. of each student's essay. The Quizzes/Homework Assignments are worksheets that can be used in a variety of ways. They pull questions from the multiple choice and short essay sections, the character and object descriptions, and the chapter abstracts to create worksheets that can be used for pop quizzes, in-class assignments and homework. Periodic homework assignments and natural quizzes are a great way to encourage students to stay on top of their assigned reading. They can also help you determine which concepts and Role In The Period Essay ideas your class grasps and which they need more guidance on. By pulling from the natural different sections of the lesson plan, quizzes and homework assignments offer a comprehensive review of fahrenheit 451 clarisse death, Andrew Carnegie and the Rise of Big Business in manageable increments that are less substantial than a full blown test. Use the Test Summary page to determine which pre-made test is most relevant to your students' learning styles. This lesson plan provides both full unit tests and mid-unit tests. You can choose from several tests that include differing combinations of multiple choice questions, short answer questions, short essay questions, full essay questions, character and object matching, etc. Some of the tests are designed to be more difficult than others. Some have essay questions, while others are limited to short-response questions, like multiple choice, matching and short answer questions. If you don't find the combination of questions that best suits your class, you can also create your own test on Andrew Carnegie and the Rise of Big Business . You have the option to Create Your Own Quiz or Test. If you want to integrate questions you've developed for your curriculum with the questions in this lesson plan, or you simply want to create a unique test or quiz from the questions this lesson plan offers, it's easy to do. Cut and paste the information from the natural indicators Create Your Own Quiz or Test page into fahrenheit 451 clarisse a Word document to get started. Scroll through the sections of the lesson plan that most interest you and indicators cut and paste the fahrenheit 451 clarisse death exact questions you want to indicators use into your new, personalized Andrew Carnegie and the Rise of Big Business lesson plan. Buy Essay Papers Here - Natural Indicators | Howtosmile Nov 12, 2017 Natural indicators, how to buy an essay online - Foreign aid vs. International trade Essay. Foreign aid vs. international trade is indicators, a long lasting debate as to which strategy leads to the greatest level of economic development. Foreign Aid is defined as any assistance that is Poetry's Role Heian Period, given to a country not provided through normal market forces. Natural Indicators? There are numerous forms of The Scale Essay, aid, from humanitarian emergency assistance, to natural indicators food aid, military assistance, etc. Development aid has long been recognized as crucial to of Vietnam help poor developing nations grow out of poverty. International trade is the exchange of goods or services across international borders. Economic development as defined by AmartyaSen, 1998 Nobel prize laureate, requires the removal of major sources of unfreedom: poverty as well as tyranny, poor economic opportunities as well as systematic social deprivation. (1) In 1970, the worlds affluent countries agreed to give 0.7% of their GNI (Gross National Income) as official international development aid, annually. Since then, these rich nations have rarely met their actual promised targets. The US is often the largest donor in dollar terms, but ranks amongst the lowest in natural terms of meeting the of Vietnam, stated 0.7% target. The two charts below, reproduced from the OECD publications (2012) shows aid granted in constant dollars and as a percentage of natural, GNI. Both support the conclusion regarding the failure to meet agreed upon aid commitments and the level of decreasing donations. (2)Billions have been donated, but it appears that Africa which has received the most aid remains a continent impoverished. As Dambisa Moyo in Poetry's the Wall Street Journal writes money from rich countries has trapped many African nations in a cycle of corruption, slower economic growth and poverty. Cutting off the flow would be far more beneficial .the insidious aid culture has left African countries more debt-laden, more inflation-prone, more vulnerable to the vagaries of the currency markets and more unattractive to higher-quality investment. Its increased the natural, risk of civil conflict and fahrenheit unrest .. Aid is an unmitigated political, economic and natural humanitarian disaster. (3) Wall Street Journal, March 21st, 2009. Over the past 60 years at least $1 trillion of development aid has been granted to Africa. And unfortunately real per-capita income in 2014 is less than it was in the 1970s. More than 50% of the population over 350 million people live on less than a dollar a day, a figure that has nearly doubled in about two decades (4) Wall Street Journal, March 21st, 2009. In 2005, the indicators, International Monetary Funds report Aid Will Not Lift Growth in Africa. concluded that governments, donors and campaigners should be more modest in their claims that increased aid will solve Africas problems. (5) Historically Asia was underdeveloped too. Southwest Competitive? Yet various policies by governments to enhance international trade have resulted in many Asian countries i.e. Korea, Taiwan, Malaysia, and Singapore achieving spectacular economic growth and along with it, higher standard of living for its citizens. Dato Kim Tan, the co-founder and natural trustee for the Transformational Business Network (TBN) wrote growing up din Asia, I saw the Asian tiger economies that 30-40 years ago had a lower GDP than Uganda or Kenya, transform themselves through enterprise, not through aid and philanthropy. (3)Paul Kagame, President of the Republic of Rwanda and Poetry's Role In The Essay that countrys first democratically elected president wrote: There is bad aid and there is good aid. The bad aid is that one which creates dependencies, as weve known for a long time now. But good aid is natural indicators, that which is targeted to create capacities in people so that they are able to live on their own activities. In the The Scale of Cyberspace, long-term they have to depend on themselves rather than depend on aid.(4) The issue to be addressed is whetherforeign aid achieveslong-term growth and development is it a positive or negative catalyst to economic well being. And the corollary to this statement is indicators, whether international trade is a better alternative to Essay economic achievement. The historical basis for concluding that there is gain from trade originates from David Ricardos work: Principals in natural indicators Political Economy and Taxation. The historical context of the time was the protectionist English corn laws, restricting wheat imports. Ricardos conclusion, arrived at by his theory of comparative advantage, showed that countries could benefit if they specialized and promoted free trade. Figure 1: Comparative Advantage. Country A has an absolute production advantage of both goods A and southwest competitive B because it can produce more of these goods. Since the natural, PPC is not parallel we can presume that the opportunity cost are both different. As a consequence specialization and mutually beneficial trade can occur. Essay About Of Vietnam? Country A has a comparative advantage in the production of good B and indicators Country B has a comparative advantage in the production of fahrenheit 451 clarisse death, good A. Country B PPF curve has a lower slope therefore its comparative advantage of production is the good on the horizontal axis. To establish a policy of natural indicators, international trade, governments need to develop programmes that require important changes in the society. For example, states must develop competitive market economies based on environmental sustainability, infrastructure, establish a stable currency, a transparent political framework and legal system, security, and educated populace as prerequisites. Good A= Capital Goods. Good B= Consumer Goods. Figure 2: PPF curve with improved production via exports. Country A allocates its resources to The Scale of Cyberspace Essay mostly consumption. Country C allocates its resources towards investment/capital goods. Countries that allocate their resources mainly in natural indicators investment/capital goods are greater to Role Essay have long-term growth than those countries who are more focused on consumption who are using up more current resources. With respect to Aid, money that is indicators, allocated to current consumption will not improve future conditions where as aid money allocated for such production of infrastructure, health, etc. Essay? will in the long run exhibit economic growth. Interestingly,South Koreas economic growth from the natural indicators, early 1960s to the late 1990s was one of the worlds fastest, and South Korea remains one of the fastest growing developed countries in the 2000s.This experience is known as the Miracle on the Han River. Like many underdeveloped societys, S. Korea possessesno natural resources and suffers from overpopulation.In addition, the Korean War destroyed much of its infrastructure. The solution was South Korea adapting an export-oriented economic strategy.Remarkably in 1980, the advantage, South Korean GDP per capita was $2,300, about one-third of nearby developed Asian economies such as Singapore and Japan. Indicators? In 2010 South Korean GDP per capita advanced to $30,000 almost thirteen times since 1980. (See two charts below on Korean Economic Growth and South Korean Economic Recovery after 1997.) It is acknowledged that the Korean GDP per capita in 1960 was lower than some sub-Saharan African countries. The growth of the Koreanindustrial sector was the principal stimulus to across movie economic development. It is true that initially some foreign aid was granted by the U.S. given the contentious communist threat, nevertheless, it was from strong domestic government support and natural indicators a move toward competitive markets and international trade that resulted in the investment of modern technology and newly built facilities at a rapid pace. Fahrenheit 451 Clarisse? The export to foreign markets and the plowing of the foreign currency back into further industrial expansion was the strategy for growth. In addition, this economic strategy suited South Korea given its poor natural resource endowment, low savings rate, and small domestic market. Natural Indicators? Labour-intensive manufactured exports, permitted South Korea to develop a competitive advantage. No doubt government initiatives were essential to this process. About? The inflow of foreign capital was greatly encouraged to supplement the shortage of domestic savings. Indicators? These efforts enabled South Korea to achieve rapid growth in exports and subsequent increases in income.(5) Finally, the later development of a unique multinational firm known as the chaebol family-controlled firms owning numerous international enterprises,enhanced Korean economic growth. There is strong cooperation with government receiving financial support and guidance i.e. innovation and research. Today of these multinationals are Samsung, Hyundai, and LG. Essay Of Vietnam? Thus, it is evident that South Korean expansion and natural indicators wealth for its citizenry has derived from the adoption of a competitive market manufacturing economy, and the development of a high-tech commercethat emphasizes international trade. Essay? Foreign loans, supportive government policies, and not aid have been the critical catalyst to such a Han miracle. Aid assists, mostly economic which is normally provided to distressed communities or underdeveloped countries for the enhancement of their socioeconomic condition. Aid is linked to need, is often not linked to the ability to enhance trade. Natural Indicators? Trade can result in inefficient distribution of income while if aid is mishandled the distribution of aid will be unfair, with people being deprived of the benefits of aid and effect economic growth of the country. Does Aid lead to The Scale of Cyberspace the establishment of good infrastructure? Development aid is given by governments, i.e. the US being the largest aid donor in the world (As of natural indicators, 2010) and Poetry's In The Period other organizations such as the World Bank. Aid to indicators all countries in the world declined (2007, World Bank) An overview of a number of African countries that have been recipients of aid will consider the use and allocation of the foreign aid, and assess the outcome of these donations upon the populace of the continent. One would note initially that there is a need for humanitarian aid to assist in Spread for Cell Essay emergencies and alleviate suffering but long term growth and development requires a different approach. Besides charity aid is a minimal portion of the over-all aid from governments. In the Democratic Republic of Congo, according to a report by indicators, Transparency International, the Zairian president Mobutu Sese Seko (1965-1977) is The Scale, listed as having embezzled at least $5 billion form the nation. (6) Transparency International In 2009, the natural indicators, former president of 451 clarisse, Malawi Bakili Muluzi was prosecuted for indicators, stealing $12 million allocated for aid. And Zambias former president, Frederick Chiluba is under investigation for taking millions from movie money that was set aside for health, infrastructure, and education. Thus, these examples of corruption pervert the opportunity for aid to assist the average African improve their conditions. Economic theory will emphasize that inefficiency will result from government intervention and in particular free funds. Natural? Bad government worsens the Essay, situation as they are often bureaucratic in addition to corrupt. The civil service is inefficient and vulnerable to cronyism which has consequence for responsible governance. Natural Indicators? Economic incentives are nullified and governments need not respond to citizen needs. Investors require transparency in government and business but such a prerequisite is not required when donations are the source of funds. Policies such as taxation can be deferred since donations flow in to the country. The priority is to remain in power. Africa continues to The Scale Essay be the most unstable region. There are numerous examples of natural indicators, civil strife as exemplified by the current war in Essay Southern Sudan and The Congo. According to the Stockholm International Peace Research Institute, Africa had a greater number of wars than elsewhere in the world. Obviously, such strife has immense impact upon indicators daily life and thwarts any possibility of private investment. Aid thus fills the vacuum. As stated in his book, Dead Aid: Why Aid Is Not Working and How There is a Better Way for fahrenheit, Africa, by D. Moyo, aid-financed efforts to force-feed democracy to economies facing poverty and indicators difficult economic prospects remain, at Role Heian Period best, precariously vulnerable. (7) Further evidence of indicators, endemic problems is observed in Nigeria. This government is one of the most mismanaged in Africa. For example, during the World Economic Forum of 2005 in Davos Switzerland, four state governors were being investigated in the U.K. for money laundering. Large amounts of aid are looted. Speaking at the New Partnership for walking across egypt movie, African Development (NEPAD) meeting in Abuja, Nigeria, in December 2003, the former British secretary of state for natural indicators, international development, Lynda Chalker, noted that 40 percent of the wealth created in Africa is invested outside the continent. (8) Advantages and Disadvantages to Trade: Trade provides development countries with an walking across egypt important base for their own improvement. Countries can develop their own strategies and outcomes to achieve their development objectives i.e. focusing on indicators, agriculture of across egypt, manufacturing services. Countries are the determinants of their own economic destiny. One difficulty of natural indicators, trade is the international economic system imposes impediments i.e. trade barriers. The global market is not an competitive advantage international free trade market i.e. tariffs, taxes and natural indicators subsidies, regulations and such restrictions operate to the disadvantage of the Techniques Phone Essay, developing countries. The donor country puts restrictions on indicators, the recipient country i.e. the requirement for government transparency, human rights, political support at the UN or Ideological support. Japan gains support from small countries on whale killing/harvesting using trade. Developing countries have weak economic power to challenge economic injustices i.e. EU and USA have large program of subsidies and economical protectionism.One argument states that trade is a foundation for walking across egypt movie, international cooperationi.e. one country is the resources base and indicators the other importing country is the manufacturing base. Trade would benefits those countries who are engaged in trade (based on the market). 451 Clarisse? Trade relationships would result in natural indicators a more equitable partnership and thus might be a catalyst for investment and growth. Developing country could receive direct foreign investments and the benefits that would sue from that, technology, employment etc. The result of this would lead to lower prices, greater choice, more efficient allocation of resources, foreign exchange and idea flowing between countries. Once getting passed trade impediments it can seen as long-term development strategy for a developing country. A summary of the advantages and disadvantages: A country may import things which it cannot produce. Maximum utilization of resources. Benefit to egypt consumer. Reduces trade fluctuations. Utilization of surplus produce. Fosters international trade. Import of harmful goods. It may exhaust resources. Danger of starvation. One country may gain at the expensive of another. It may lead to war. Advantages and natural indicators Disadvantages to Hybrid Spectrum for Cell Phone Aid: Aid has been said to decrease the development for countries with their own improvement. Aid has many advantages but as it can be seen using examples such as African countries it can be seen that aid has decreased the development for countries. Countries can be giving aid for ethical reasons, if there was a natural disaster and were in need of assistance. Aid is based on need, aid permits to countries to retain their dignity because aid if often perceived as condescending. There are different types of aid, tied and untied. Tied aid is foreign aid that must be spent in the country providing the aid or in a group of selected countries. Untied aid is assistance given to developing countries, which can be used to purchase goods, and services in virtually all countries. Jeffery Sachs (UN advisor) argued that aid is important, it is essential to break the poverty cycle and indicators a determinant of long term growth. Spread Spectrum Techniques? His argument stated that poor countries are not wealthy because of diseases such as malaria, they need to remove problems such as disease because the free market will not due this. Giving of aid is perceived as an inducement to behave or support certain policies. Natural Indicators? Aid can be used as blackmailing or getting the upper hand on another country in order to gain their vote in the UN. William Easterly (Professor of economics, NY university), aid does more harm than good, its often wasted, it inhibits peoples inventiveness, devising their old solutions. Aid is corrupt because its given to corrupt officials and often its fragmented that there is many donors and Role Essay misallocations of the natural indicators, funds. Aid often comes with a lot of preconditions. Furthermore, aid has often come with a price of its own for the developing nations. Aid is often wasted on conditions that the recipient must use overpriced goods and services from donor countries. Most aid does not actually go to the poorest who would need it the most. Aid amounts are dwarfed by rich country protectionism that denies market access for poor country products, while rich nations use aid as a lever to open poor country markets to their products. Large projects or massive grand strategies often fail to help the Hybrid Spectrum Phone, vulnerable as money can often be embezzled away. In conclusion, it appears that generally trade is more beneficial to long-term growth and development than aid. African examples discussed are proof that aid is counterproductive and the road to economic failure. Natural? The opposite appears true. Those countries able to divorce themselves from walking across aid dependency are more likely to succeed as evidenced by China, India, and even South Africa. Natural? A strategy of transparent government, efficient civil service to meet social needs i.e. education, incentives, entrepreneurship, the fahrenheit 451 clarisse death, rule of law, patent protection, institutions to natural attract foreign direct investment by creating attractive tax structures and reducing the red tape and complex regulations for businesses. Private capital investment into sustainable companies, with aid directly to community based non-profitable organizations is recommended. African nations should also focus on increasing trade, but there have been cases where rapid growth in international trade has led to bottlenecks and impediments to The Scale of Cyberspace growth and development. There appears to be no panacea but one does conclude that in order to natural indicators achieve economic, social, and political progress there are essential non-economic and economic prerequisites cultural (i.e. individualism, attitudes of personal achievement, hard work, etc.) political (i.e. transparent governments, stability, the rule of law,) and economic (i.e. infrastructure, communications network, merit goods i.e. education and health etc.). Each nation is unique with regard to trade and aid. The answer is not either or aid or trade .but accountable and transparent aid in southwest conjunction with sustainable trade and honest and efficient government. Bibliography: (Complete bibliography after check by indicators, Ms. Kerr) 1. Amartya Sen. 1999. Development as Freedom. Fahrenheit 451 Clarisse? Oxford university press 2. http://www.globalissues.org/article/35/foreign-aid-development-assistance 3. wall street journal. 4. wall street journal. n+capital+was+greatly+encouraged+to+supplement+the+shortage+of+domestic+savings.+These+efforts+enabled+South+Korea+to+achieve+rapid+growth+in+exports+and+subsequent+increases+in+income.%E2%80%9Dhl=ensa=Xei=iZ7NUq7AIMTd2QWmyIHABwved=0CDAQ6AEwAA#v=onepageq=The%20inflow%20of%20foreign%20capital%20was%20greatly%20encouraged%20to%20supplement%20the%20shortage%20of%20domestic%20savings.%20These%20efforts%20enabled%20South%20Korea%20to%20achieve%20rapid%20growth%20in%20exports%20and%20subsequent%20increases%20in%20income.%E2%80%9Df=false 8. OECD Development Statistics Online, last accessed April 7, 2012 9. http://www.bbc.co.uk/news/uk-20405140. 10. http://www.guesspapers.net/1593/advantages-and-disadvantages-of-international-trade/ 11. http://www.moeasmea.gov.tw/ct.asp?xItem=72CtNode=263mp=2 12. http://www.globalissues.org/article/35/foreign-aid-development-assistance 13. http://www.ryanallis.com/sustainable-capitalism-and-the-role-of-aid-vs-trade-in-prosperity-creation/ 14. http://www.cato.org/publications/economic-development-bulletin/african-perspectives-aid-foreign-assistance-will-not-pull-africa-out-poverty. University/College: University of California. Type of paper: Thesis/Dissertation Chapter. Date: 6 May 2016. Let us write you a custom essay sample on Foreign aid vs. International trade. for only $16.38 $13.9/page. Haven't found the natural, Essay You Want? Get your custom essay sample. For Only $13.90/page. 3422 Old Capitol Trail, Suite 267, Wilminton, DE 19808, USA. Essay Writing Service - Natural Indicators explained with an experiment - Nov 12, 2017 Natural indicators, online cheap custom essay - A Surprise Visit by Gubernatorial Candidate, Charlie Crist, St Lucie River/Indian River Lagoon. Charlie Crist visits the natural, St Lucie River/Indian River Lagoon October 3, 2014. Of Cyberspace Essay? (Photo by Jacqui Thurlow-Lippisch.) As you probably saw in the paper, Charlie Crist, Democratic gubernatorial candidate for Florida, visited Stuart on Wednesday, October 3, 2014. Very exciting! I have been waiting for the governors race to crank up and for the St Lucie River/Indian River Lagoon to be in the spotlight as it should be. In my dream of dreams Governor Scott and natural indicators Charlie Crist would come to Stuart debate. For now, I will be satisfied that this year both have visited and spoken and are aware of fahrenheit 451 clarisse health issues facing the indicators, St Lucie River/Indian River Lagoon. I had seen Charlie Crist speak before at the Florida League of Cites, in Hollywood, but it was fun to see the man, the myth, our former governor, up close, face to face, along with his beautiful wife Carol. I shall tell you of my short experience As I did not learn of the visit until the day before, I already had commitments and about The Landscapes of Vietnam thus was one hour and a half late to Mr Crists planned arrival to tour the lagoon at the invitation of Marty Baum, Indian Riverkeeper. This was inconsequential as the Crist team was late itself, so when I arrived at Sandsprit Park, Mr Crist and his team had just gone out in the boat with Marty Baum, Mark Perry of natural Florida Oceanographic, and fahrenheit 451 clarisse death others. Coming in off the boat. Marty Baum and friends. As Mr Crist came off the boat I moved to natural greet him and grabbed his hand to help him onto southwest competitive the dock. Natural? I thanked him for his visit. His wife, Carol was at his side holding a sign that read, A FAIR CHANCE FOR FLORIDA. I knew that the couple had married in 2008 and southwest advantage that she was a very successful business woman who ran the family business, Franco American Novelty Co . and was most well-known for inventing the slogan: Where Fashion Meets Halloween, and creating sexy Halloween costumes that were big sellers. She is a talented marketer. I wondered what she thought so far of natural indicators Stuart. She smiled, stepping off the boat in high heels. Charlie Christs wife, Carol, was at his side on the boat ride. Poetry's Period? (Public photo) Once off the indicators, boat they got their sea legs as apparently a big wave had hit the small crowd loved him and gathered around. Mr Crist composed himself, a calm and Techniques for Cell Phone confident speaker. This was obviously a man used to being in the public eye. The questions from the natural indicators, reporters about the lagoon were many, and Mr Crist did a good job answering. Nonetheless there was an awkward moment that must be noted. Tyler Treadway from the Stuart News asked Mr Crist about about of Vietnam, how he would stop the natural, discharges from Lake Okeechobee and The Landscapes Mr Crist answered that he would put the right people on the South Florida Water Management Board and the discharges would stop; like they did before, when he was governor There was silence. Natural Indicators? Tylers lips quivered; he said something like Sir I dont know how you can say that? That you stopped the discharges . Charlie Crist repeated again that he had. I watched this crowd who adored and appreciated Mr Crist but they were too educated and too far into this not to know that stopping the Heian Essay, discharges is a lot more complicated that that .and that they have NEVER stopped for natural, long. The was the pregnant pause .Silence .Tyler looked at his writing pad .the crowd stared into space. As usual I could not stop myself and I finally blurted out: Temporarily .. Charlie Crist looked straight at Poetry's Role In The Heian Period, me; I respectfully held my gaze, head slightly down. We all looked at each other .and stared. It was time to go. We all thanked him. It was a great visit. I believe Mr Crists intentions are good, I just dont think, he like all politicians, is used to speaking to such an indicators, educated public. We know the truth, we need a third outlet from the 451 clarisse, lake and other water holding areas, there are no saviors, no boards that will fix our terrible St Lucie River/Indian River Lagoon/Lake Okeechobee problem. Why didnt he speak about his US Sugar buyout plan from 2008? Said someone as we were all walking away? He didnt know who he was taking to, I answered. The most educated public in the state. Politicians are not used to that We waved goodbye. And Charlie flashed us that winning smile. Natural? We hope hell come again. Pam Joy hold a sign SAVE OUR RIVER. St Lucie River Kidz were there to meet Charlie Crist. Walking Across Egypt Movie? Their mission is to speak out, get involved and raise awareness . Sugar Hill Sector Plan/Airglades Airports Location and How it Affects the Indian River Lagoon. Sugar Hill Sector Plan and Airglades Airports location in reference to indicators Lake Okeechobee, (Maps, iPhone, 2014.) Since last week, you may have seen press on Sugar Hill and Hybrid Spectrum for Cell Airglades Airport, a land use change proposal located in indicators, Hendry County southwest of Lake Okeechobee. This is a highly controversial, approximately 67 square miles, of present farmland that could change to residential, (up to 18,000 homes), and commercial lands, built around an airport that is already in place with the potential to Essay expand. Just for comparison, the Airglades Airport runway is 5900 feet long while Witham Airports in natural indicators, Stuart is 5800 according to my husband Ed. (http://en.wikipedia.org/wiki/Airglades_Airport) Anyway, this morning I do not have time to debate this issue in detail, but I will say of course that it is a true game changer. I wanted to SHOW where these lands are located in reference to lands that are still available for southwest, purchase by indicators the state of Florida due to an option you may have heard of Hybrid Spread Spectrum Techniques as well. If purchased, these option lands would be key in Everglades restoration from Lake Okeechobee to indicators the Everglades and restoring some pathetic remnant of its historic flow. Presently, the Techniques Phone, state does not want to buy these lands because politicians claim there is no money to maintain them and if they were bought the natural, lands will just end up sitting there, at great expense until a possible time they could be utilized in the future, like 2060. This argument may sound reasonable but in order to save the Everglades and the St Lucie River/Indian River Lagoon, purchasing these lands is critical and across should be done now because if these lands are not purchased now, as the Sugar Hill Sector Plan shows, their land use could be changed and then the lands will be too expensive for the state to ever purchase. Market value for agricultural lands is less than residential. Sometimes life demands you spend money now to natural indicators save in of Cyberspace Essay, the future. So, just so we know where we are talking about, where are these Sugar Hill and Airglades lands located? See map below. Airglades Airport and natural the Sugar Hill Sector plan are located south west of Clewiston by fahrenheit 451 clarisse death about five miles on the west side of Lake O along Hwy. 27. They are located on Highway 27 west of Clewiston. Indicators? Highway 27 runs through the fahrenheit 451 clarisse death, Everglades Agricultural Area south of Lake Okeechobee connecting both coasts. The Airglades Airport and Sugar Hill Sector Proposal are located right before the highway shoots north. See red dots above and below. Indicators? The 67 square miles of Sugar Hill Proposed Sector lands are around the airport. I do not know exactly where, but I would think mostly south. At closer view, one can see the Airglades Airport amongst the sugar fields. Now if we look at a partial map of the option lands we can see that the Airglades Airport and Hybrid Spectrum for Cell Sugar Hill Sector Proposal are located in lands that were designated for purchase to one day benefit Everglades restoration for posterity. (Dark green is of most importance and yellow is of importance, both are option lands ) Map showing option lands close to Clewiston. Overall EAA option lands: Option Lands marked for natural, purchase for Everglades restoration under former Gov. In The Period? Charlie Crist. Indicators? The deal fell apart due to The Scale politics and the financial crisis of 2008a much smaller land purchase ensued. OK so how would this fit into the Plan 6, River of Grass restoration? Please keep in mind Plan 6 and all plans are fluid as they have not happened yet The amount of water we are talking about it almost beyond comprehension and indicators requires great areas of land beyond lines on a sheet of paper. So even though this Plan 6 chart concentrates flow between the Miami and New River Canals the lands west of death this area where Sugar Hill would be located are part of the overall restoration plan for this area as we can tell from the option lands maps above. Plan 6 flow, River of Grass. In conclusion, and to repeat myself: all the indicators, lands marked as option lands are important for the overall Everglades/Northern Estuaries restoration project. The Sugar Hill Sector Plan, if successful, is Poetry's In The Period Essay setting a precedent for changes in agricultural land use in Florida. There may be no turning back on natural this at this point as the Scott Administration gutted the Department of Community Affairs that used to keep such land use changes in check. Across Egypt? As usual the state of Florida has put development before restoration of natural lands and waters protection. When Floridas future waters are just one big toxic algae bloom, and people do not want to live here, I wonder if some of our politicians will wish they had voted another way? Oh but they will be dead like me, so I guess it doesnt matter . It does matter. It matters almost more than anything in the world. Please make your voice known and lets leave something to natural the children of the future other than cookie cutter homes. Not Only Have We Become a Political Force for the Indian River Lagoon; We are in the Drivers Seat, So Drive! Gubitorial candidates wrote directly to Hybrid for Cell Phone Essay the residents of the indicators, St Lucie River/Indian River Lagoon area on May 8th, 2014 in the Stuart News. I didnt pick up Sundays Stuart New s until Monday, as I had been out of town. Sipping my coffee and holding the old fashioned paper, I love so much, my lips curled in a broad smile. The opinion page juxtaposed articles by two candidates running for Florida governor: Nan Rich and Rick Scott. These letters were not just broad sweeping letters; they were thoughtful and personal, they mentioned the The Scale of Cyberspace Essay, River Warriors and natural direct stories of inspiration from local residents. Dont get me wrong, I know it is an election year, but nonetheless, it is simply amazing. In one year, since the discharges from Lake Okeechobee and our local canals turned our world upside down, and Poetry's Role In The Heian Essay news of natural indicators such went viral, the Role In The Heian Essay, people have accomplished the most impressive of our forefathers American expectations. Expectations that years of natural indicators social conformity and acceptance of over-development and pollution had overridden. Death? The people of the natural indicators, Indian River Lagoon have stood up to their government. The rebellion of the southern lagoon corresponded to the northern lagoons massive deaths of manatees, dolphins, pelicans and loss of 451 clarisse death almost 100% of its seagrasses. These die-offs and toxic algae blooms in the north, actually began happening in 2011 but did not come out natural indicators, publicly until the Hybrid Spread Spectrum Techniques Phone, uprising in the southern lagoon blended the two tragedies. I know for many of indicators my friends the politics of the River Movement is The Scale hypocritical, frustrating and painful. I feel the natural indicators, same way. In fact lately I have been a bit depressed over the whole thing. But I am getting out of fahrenheit 451 clarisse it. Boy is it a pleasure to natural see that paper, to be in the drivers seat, to have them writing letters, visiting, and actually thinking that there is no longer a golden ticket to Hybrid pollute. I have been in Martin County many years, and on this level, this has never happened before. Finally, even Senator Rubio is getting heat in indicators, the press; long standing Senator Nelson is happy hes been around, but also nervous he is part of the The Scale of Cyberspace, lagoon establishment; Charlie Crist is taking out his old notes about US Sugar; Senator Negron is promising more for natural, the lagoon in 2015; Congressman Murphy is regrouping and studying the Farm Bill after the ACOE refused CEPP on his watch; and the future speakers of the Florida house and senate are making their cases for the future of water. Many times the lagoon has been defiled by our government, in fact 2013 was not the worst its ever been. But I am telling you, this time it is different because of us. This time we have exposed them. This time we are asking truly for government to do what it is supposed to: protect the health; safety and welfare of Poetry's Heian Essay its people. This time we are united in indicators, a brotherhood and sisterhood of diverse backgrounds and interests. This time we have reached a tipping point, as has the Hybrid Spectrum Techniques for Cell Phone, lagoon. And most important for change, this time, they are watching and listening to us. Please take advantage of this opportunity. Dont turn your back because the politics are so repulsive to watch. Look to the skylook to the river write a letter or make a call and say : thank you; we are happy you are taking an interest in the lagoon; I will be weighing who to vote for indicators, based on who really has the desire, passion and an honest heart. These politicians may never be able to reach perfection, their world is pretty insane, but be grateful they are paying attention, and know you are a force for change in a way never before. Drive your points home!

A Sudoku with clues as ratios. A Sudoku with a twist. A Sudoku with clues as ratios or fractions. Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells. Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this A Sudoku based on clues that give the differences between adjacent cells. A Sudoku that uses transformations as supporting clues. This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule. This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it? 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. Two sudokus in one. Challenge yourself to make the necessary This sudoku requires you to have "double vision" - two Sudoku's for the price of one A Sudoku with clues given as sums of entries. 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? Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card? Four small numbers give the clue to the contents of the four 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? This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning. Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem. You need to find the values of the stars before you can apply normal Sudoku rules. Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal. This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid. A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both? Use the clues about the shaded areas to help solve this sudoku The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . . This Sudoku combines all four arithmetic operations. Given the products of diagonally opposite cells - can you complete this Sudoku? In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9. Solve the equations to identify the clue numbers in this Sudoku problem. in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same? Each clue number in this sudoku is the product of the two numbers in adjacent cells. This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set. 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? A pair of Sudoku puzzles that together lead to a complete solution. A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article. 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. Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4. This Sudoku, based on differences. Using the one clue number can you find the solution? The clues for this Sudoku are the product of the numbers in adjacent squares. Can you use your powers of logic and deduction to work out the missing information in these sporty situations? 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? Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for? The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid. You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku. 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? Use the differences to find the solution to this Sudoku. Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

The BUG strategy is one of our more conservative strategies. The strategy does not attempt to predict prices or the future state of the economy. It holds a broad diversified number of assets that complement each other, each performing well in a different economic environment such as inflation, deflation, growth and stagnation. It is meant for long term, steady growth and low risk. It inherits part of its logic from Harry Browne's tried-and-true Permanent Portfolio and the publicized workings of the All-Weather portfolio. 'The total return on a portfolio of investments takes into account not only the capital appreciation on the portfolio, but also the income received on the portfolio. The income typically consists of interest, dividends, and securities lending fees. This contrasts with the price return, which takes into account only the capital gain on an investment.'Applying this definition to our asset in some examples: 'Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression ratio that provides a constant rate of return over the time period. CAGR is not an accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry.'Using this definition on our asset we see for example: 'Volatility is a rate at which the price of a security increases or decreases for a given set of returns. Volatility is measured by calculating the standard deviation of the annualized returns over a given period of time. It shows the range to which the price of a security may increase or decrease. Volatility measures the risk of a security. It is used in option pricing formula to gauge the fluctuations in the returns of the underlying assets. Volatility indicates the pricing behavior of the security and helps estimate the fluctuations that may happen in a short period of time.'Using this definition on our asset we see for example: 'Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference. Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in our definition is the semi-deviation, that is the standard deviation of all negative returns.'Applying this definition to our asset in some examples: 'The Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) is a way to examine the performance of an investment by adjusting for its risk. The ratio measures the excess return (or risk premium) per unit of deviation in an investment asset or a trading strategy, typically referred to as risk, named after William F. Sharpe.'Applying this definition to our asset in some examples: 'The Sortino ratio improves upon the Sharpe ratio by isolating downside volatility from total volatility by dividing excess return by the downside deviation. The Sortino ratio is a variation of the Sharpe ratio that differentiates harmful volatility from total overall volatility by using the asset's standard deviation of negative asset returns, called downside deviation. The Sortino ratio takes the asset's return and subtracts the risk-free rate, and then divides that amount by the asset's downside deviation. The ratio was named after Frank A. Sortino.'Using this definition on our asset we see for example: 'Ulcer Index is a method for measuring investment risk that addresses the real concerns of investors, unlike the widely used standard deviation of return. UI is a measure of the depth and duration of drawdowns in prices from earlier highs. Using Ulcer Index instead of standard deviation can lead to very different conclusions about investment risk and risk-adjusted return, especially when evaluating strategies that seek to avoid major declines in portfolio value (market timing, dynamic asset allocation, hedge funds, etc.). The Ulcer Index was originally developed in 1987. Since then, it has been widely recognized and adopted by the investment community. According to Nelson Freeburg, editor of Formula Research, Ulcer Index is “perhaps the most fully realized statistical portrait of risk there is.'Applying this definition to our asset in some examples: 'A maximum drawdown is the maximum loss from a peak to a trough of a portfolio, before a new peak is attained. Maximum Drawdown is an indicator of downside risk over a specified time period. It can be used both as a stand-alone measure or as an input into other metrics such as 'Return over Maximum Drawdown' and the Calmar Ratio. Maximum Drawdown is expressed in percentage terms.'Using this definition on our asset we see for example: 'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has seen between peaks (equity highs). Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover from the largest loss), but it doesn’t have to be'Which means for our asset as example: 'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks (equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of all.'Using this definition on our asset we see for example:

Here we are concerned with the motion of a vehicle, i.e., its acceleration, deceleration, and the amount of weight it can carry. Also we are concerned with the selection and sizing of propulsion systems for a vehicle. To look at propulsion and resistance, use Newton's laws of motion and Concerned with net force on a vehicle or the difference between what is available from the propulsion system and the resistance encountered. There is a circular problem. Since force and resistance are functions of velocity, velocity is a function of acceleration, acceleration is a function of force. So you need to know V to compute F, F to compute a, a to compute V. Resistance to Motion - Elements of resistance to motion 1) Friction, not a function of velocity--fixed in quantity, between wheel and surface, bearings, etc., friction depends upon (weight, form, and type of surface) 2) Losses varying with speed, i.e., sway, flange friction, bumps, etc. 3) Air resistance depends upon cross section area, shape, length, etc. of vehicle,varies with the square of speed. Note curve can drop when vehicle reaches a more drag-free regime, i.e., break sound barrier, hydrofoil. 4) Grade resistance, from going up or down grade can be positive or negative; doesn't vary with speed. Grade resistance is equal to the component of vehicle weight which is parallel to the grade line. Grade resistance = W = weight of vehicle G = percent grade 5) Curve Resistance: From friction of rail flanges against rails as a train goes around a cure, etc.. Railroad cure resistance is 0.8 lb. per ton of car per degree of curvature. I.e., a 100 ton car on a 2 degree curve has a resistance of 160 lb.. or R = K1 + K2 * V + K3*V2 + W * G/100 Resistance equations for various types of vehicles. Rflat ground = (1.3wn+29n) + bwnV + CAV2 Rt = (1.3wn+29n) + bwnV + CAV2 + 20wn*G Rt = total resistance in lb. including grade resistance V = speed - mph. w = weight per axle (tons) n = number of axles; Note, wv = w*n = gross vehicle weight b = coefficient of moving friction: .03 locomotives; .045 freight cars C = drag coefficient of air .0017 streamlined locomotives, .0025 other locomotives, .0005 for trailing freight cars and .00034 for trailing passenger cars. A = cross sectional area of vehicle, 120 sq. ft. for locomotives, 90 sq. ft. for freight cars and 120 sq. ft. for passenger cars. G = % grade (upgrade +, downgrade -) . . for Davis formula K1 = 1.3wn + 29n K2 = bwn K3 = CA Modified Davis Equation (FE booklet) Road vehicle resistance for Trucks on a gradient R = 7.6T + .09TV + .002AV2 + 10*T*G T = weight of vehicle in 1000 lb. V = velocity = mph. A = cross sectional area - ft.2 R = resistance in lb. G = percent grade for Autos on a gradient R = 10T + .1TV + .0026CAV2 + 10*T*G C = air resistance parameter Auto C = .4 to .5, new autos C = .35 Convertible C = .6 to .65 Bus C = .6 to .7 Comparison of equations These equations are all similar in form, but show different effects from the various components of motion. |Truck||Auto||Bus||Four Axle Locomotive| (T is weight in kips in all of the above.) Need propulsive force to overcome resistance, accelerate vehicle. We are interested in the force needed to accelerate a given mass of vehicle at a given rate. Again, simple physics used: F = ma The rate at which work is done is power. P = dw from work equation dw = F(x)dx/dt P = F(x)dx/dt; units in the English system are ft. - lb/sec. when F is less than R, the vehicle decelerates. Force applied over a distance, x, is work: Fuel consumption is directly related to the amount of work done. Fuel = consumption rate * work done The fuel consumption rate could be given in gallons of fuel per ft.-lb. For example, 1 gallon of gasoline contains 120,000 BTU and 1 BTU is equivalent to 778 ft.-lb. of work. Therefore 1 gallon of gasoline has the energy equivalent of 9.33 * 107 ft.-lb. of work. 1) A 200 ton, 4 axle locomotive has the following characteristics: b = .03, C = .0025, A = 120 ft.2 What is the resistance at 12 mph? 50 mph? 70 mph? 100 mph? Weight/axle = 50 tons at 12 mph: R = 1.3 * 200 + 29 * 4 + .03 * V * 200 + .0025 * 120 * V2 R = 376 + 6V + .3V2 R = 376 + 72 + 43.2 = 491.2 lb. at 50 mph: R = 260 + 116 + .03 * 200 * 50 + .0025 * 120 * 50 * 50 R = 376 + 300 + 750 = 1426 lb. at 70 mph: R = 376 + 6 * 70 + .3 * 702 R = 376 + 420 + 1470 = 2266 lb. at 100 mph: R = 376 + 600 + 3000 = 3976 lb. 2) What is the maximum traction force of a 2500-HP locomotive with an efficiency of .83 at 12 mph? 50 mph? 70 mph? 100 mph? at 12 mph: 1 HP = 550 ft.-lb./sec = 550 * 3600/5280 = 375 lb.-mi./hr. F = 375 * P(HP) * E/V(mph) = 375 * 2,500 * .83/12 F = 64,800 lb. at 50 mph: F = 15,600 lb. at 70 mph: F = 11,100 lb. at 100 mph: F = 7,800 lb. 3) At what speed does the resistance of the locomotive equal its maximum propulsive force? (i.e., what is the maximum speed assuming proper gearing and suitable track?) F = R 778,100/V = 376 + 6V + .3V2 Solving the above (by trial and error), Vmax = 128 mph. 4) At what gradient will the locomotive coast downhill at a constant 50 mph? R50 mph = Rgrade 1426 = 20 * wn * g g = 1426/(20 * 200) = 0.356 g = -0.36% 5) The locomotive moving alone travels 10 miles at a constant 50 mph, on level track. If the energy conversion rate is 30%, what is the fuel consumption rate? At a constant speed a force equal to the resistance must be applied: (zero acceleration) F = R = 1426 lb. (from Davis equation) Work = 1426 * 5280 * 10 ft. lb. Fuel used = 1426 * 5280 * 10/(.3 * 9.33 * 107) = 2.69 gallons Rate = .269 gal./mile = 3.72 mpg 6) Very small automobile Engine power = 12 hp. Weight = 1200 lb. Cross sectional area = 20 ft.2 Standard suspension, aerodynamic characteristics Wind resistance parameter: C = 0.4 Engine efficiency = 0.3 Flat road, no curves Question: How fast is the maximum steady state speed and what is the fuel economy at that speed? R = Ra + Rr R = 0.01T + 0.0001TV + 0.0026CAV2 = 0.01(1200) + 0.0001(1200)V + 0.0026 * 20 * .4 * V2 R = 12.0 + 0.12V + 0.0208V2 At steady state speed F = R Propulsive force = R so substituting F for M in the horsepower relationship R*V = 375HP = 4500 Substituting for R 12V + 0.12V2 + 0.0208V3 = 4500 Which can be solved for V (trial and error) V ~ 55.1 mph R = 81.7 lb. Is this car fuel efficient? Let's use these relationships. 1 btu = 778 ft.-lb. 1 gallon of gas = 120,000 btu Fuel conversion efficiency = 0.3 Drive the car for one mile: 7) What is the maximum acceleration from a speed of 30 mph of a 250 hp truck operating up a 5% grade? The truck weighs 20,000 lbs., is 85% efficient and has a cross-sectional area of 100 sq. ft.. (15) Torque: twisting moment applied to a wheel Torque = F * b, units are ft.-lb. Horsepower = 0.00019*t*N t = torque of engine N = rpm of engine For a fixed gearing, torque is directly proportional to tractive effort. Tractive effort = tGE/r t = torque at output shaft of engine G = total gear ratio between output shaft and axle E = drive line efficiency r = radius of wheel under loaded conditions concern with propulsion system which has two elements: prime mover and transmission. For prime mover we can look at horsepower and torque curves The above curve is typical of a diesel engine; for a gasoline engine, maximum torque is reached at a higher RPM. This difference is why diesel engines are good for weight hauling, since they can apply high torque at low speeds, as used in locomotives. Gasoline engines are good for rapid acceleration, as used in automobiles. Electric motors, Horsepower v. Engine Speed Curves can have a number of different shapes depending upon requirements; i.e., need high starting torque--you can get practically any shape you want depending on the design of the engine and transmission. Advantages of straight electric locomotive: you can temporarily overload the locomotive when starting out and start up a heavy train. Same idea used in diesel-electric locomotive (and also heavy off-road trucks). Steam locomotive did not have nearly the same starting torque as a diesel. Diesel can pull more than a steam locomotive with the same horsepower. This and lower maintenance and operating costs led to the replacement of steam locomotives by diesel-electrics. Transmissions. Want to get torque and power of engine transmitted to the wheels. Purpose of transmissions transfer power from engine to drive wheels while producing a reduction in engine speed and providing flexible control. Vehicle movement is concerned is the variation of net propulsive force with speed (unfortunately, this is not a constant). Also, net force = propulsive force - resistance. Maximum propulsive force limited by: characteristics of the propulsion system, i.e., for above at low speeds motors and generators heat up. To maintain high torque and prevent stalling. Also adhesion of wheels to ground limits maximum pull. F = µ N µ = coefficient of friction between wheel and rail, roughly .30 N = weight on one wheel Minimum tractive effort is determined by maximum safe operating speed of motor, i.e., red line, above that engine tears apart from centrifugal force. Look at propulsive force of vehicle as it varies with speed and also resistance. Note when F = R the vehicle is at maximum speed (i.e. net force = 0) and there is no force available to accelerate the vehicle. Use curves such as this to describe motion of vehicle. Note: It is analytically impossible to predict speed, distance, acceleration of most vehicles because: You can, however, solve for maximum speed, maximum grade, maximum weight, etc. This has to be analyzed by simulation. Use small increments of time, Dt. Using t, beginning--find v, then F, then solve for new a, find v, . . ., etc. Go to speed limit. Two types of limits on speed. - Artificial (i.e. speed limit) - Curvature (i.e. maximum safe speed around a curve) V curve = 429.53 * (F+e)/degree curvature Deceleration of vehicles usually occurs at constant rate. Vehicle motion then simulated for a short period of time t. K1, K2, K3 Side friction factor

Compactly Supported Curvelet-Type Systems We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. Second-generation curvelets were introduced by Candès and Donoho, who also proved that curvelets give an essentially optimal sparse representation of images (functions) that are except for discontinuities along piecewise -curves . It follows that efficient compression of such images can be archived by thresholding their curvelet expansions. Curvelets form a multiscale system with effective support that follows a parabolic scaling relation . Moreover, they also provide an essentially optimal sparse representation of Fourier integral operators and an optimal sparse and well organized solution operator for a wide class of linear hyperbolic differential equations . However, curvelets are band-limited, and contrary to wavelets it is an open question whether compactly supported curvelet-type systems exist. In this paper we study a flexible method for generating curvelet-type systems with the same sparse representation properties as curvelets (when sparseness is measured in curvelet-type sequence spaces). The method uses a perturbation principle which was first introduced in , further generalized in , and refined for frames in . We give a constructive description of how to construct curvelet-type system consisting of finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. This gives the flexibility to construct curvelet-type systems with a prescribed nature (see Section 6) such as compact support in direct space. For the sake of convenience the construction will only be done in , but it can easily be extended to . The main results can be found in Sections 4 and 5. The curvelet-type sequence spaces we use are associated with curvelet-type spaces which were introduced in . Here was constructed by applying a curvelet-type splitting of the frequency space to a general construction of decomposition spaces, thereby obtaining a natural family of smoothness spaces for which curvelets constitute frames (see Section 2). Originally, this construction of decomposition spaces based on structured coverings of the frequency space was introduced by Feichtinger and Gröbner and Feichtinger . For example, the classical Triebel-Lizorkin and Besov spaces correspond to dyadic coverings of the frequency space (see ). The outline of the paper is as follows. In Section 2 we define second-generation curvelets and curvelet-type spaces. Furthermore, we introduce curvelet molecules which will be the building blocks for our compactly supported curvelet-type frames. Next, in Section 3 we use the properties of curvelet molecules to show that the “change of frame coefficient” matrix is almost diagonal if the curvelet molecules have sufficient regularity. With the machinery of almost diagonal matrices, we can then in Section 4 show that curvelet molecules which are close enough to curvelets constitute frames for the curvelet-type spaces. Finally, in Section 5 we give a constructive description of how to construct these curvelet molecules from finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. We conclude the paper with a short discussion in Section 6 of the possible functions which can be used to construct the curvelet molecules. 2. Second-Generation Curvelets We begin this section with a brief definition of curvelets and curvelet molecules which will later be used to construct curvelet-type frames. Furthermore, we define the curvelet-type spaces for which curvelets constitute frames. For a much more detailed discussion of the curvelet construction, we refer the reader to [1, 3], and for decomposition spaces, of which the curvelet-type spaces are a subclass, we refer to [7, 8]. Let be an even window that is supported on such that its -periodic extension obeys Define for and . Next, with the angular window in place, let obey with supported in a neighborhood of the origin. We then define Notice that the support of is contained in a rectangle given by where is determined uniquely for a minimal and ( depends weakly on , see [1, Section 2.2]). With and the system is an orthonormal basis for . We let , and by duality extend it uniquely from to . Finally, we define where is rotation by the angle , and as coarse-scale elements we define , where and is sufficiently small. The system is called curvelets, . It can be shown that curvelets constitute a tight frame for (see [1, Section 2.2]). To later construct curvelet-type frames, we need a system of functions which share the essential properties of curvelets. As we will see, curvelet molecules, which were introduced in and used there to study hyperbolic differential equations, have all the properties we need. For , we define , and for suitably differentiable functions we define . Definition 2.1. A family of functions is said to be a family of curvelet molecules with regularity , , if for they may be expressed as where and all functions satisfy the following.(i)For there exist constants independent of such that (ii)There exist constants independent of such that The coarse-scale molecules, , must take the form , where satisfies (2.8). It can be shown that curvelets constitute a family of curvelet molecules with regularity for any (see [3, page 1489]). To define the curvelet-type spaces which together with the associated sequence spaces will characterize the sparse representation properties of curvelets, we need a suitable partition of unity. Definition 2.2. Let for , where was defined in (2.3). A bounded admissible partition of unity (BAPU) is a family of functions satisfying:(i); (ii);(iii). Definition 2.3. Let be a BAPU and . For and , we define as the set of distributions satisfying We also need the sequence spaces associated with the curvelet-type spaces. For the sake of convenience, we write instead of when the index set is clear from the context. Definition 2.4. For and , we define the sequence space as the set of sequences satisfying where the -norm is replaced with the -norm if . Notice that the sequence spaces are special cases of as we have . Next, we introduce frames for and use the notation when there exist two constants , depending only on “allowable” parameters, such that . Definition 2.5. We say that a family of functions in the dual of is a frame for if for all we have The following is called the frame expansion of when it exists: in the sense of , where is the frame operator , . Proposition 2.6. Assume that , and . For any finite sequence , one has Furthermore, is a frame for with frame operator , Notice that frame expansions for two frames and have the same sparseness when measured in the associated sequence space if and also constitute frames for , Hence, to get a curvelet-type system with the same sparse representation properties as curvelets , it suffices to prove that constitutes a frame for . 3. Almost Diagonal Matrices To generate curvelet-type frames in the following sections we introduce the machinery of almost diagonal matrices in this section. Almost diagonal matrices where used in on Besov spaces, and here we find an associated notion of almost diagonal matrices on . The goal is to find a definition so that the composition of two almost diagonal matrices gives a new almost diagonal matrix and almost diagonal matrices are bounded on . To help us define almost diagonal matrices we use a slight variation of the pseudodistance introduced in which was constructed in . For this we need the center of in direct space, , and the “direction” of . Definition 3.1. Given a pair of indices and , we define the dyadic-parabolic pseudodistance as where The dyadic-parabolic distance was studied in detail in , and from there we can deduce the following properties.(i)For there exists such that (ii)For there exists such that (iii)For and there exists such that (iv)Let and be two families of curvelet molecules with regularity , . Then there exists such that These properties lead us to the following definition of almost diagonal matrices on . Definition 3.2. Assume that , and . Let and . A matrix is called almost diagonal on if there exists such that Remark 3.3. Note that by using (3.5), we get that the composition of two almost diagonal matrices on gives a new almost diagonal matrix on . We are now ready to show the most important property of almost diagonal matrices; they act boundedly on the curvelet-type spaces. Proposition 3.4. If is almost diagonal on , then is bounded on . Proof. We only prove the result for as the result for follows in a similar way with replaced by . Let , and assume for now that . We begin with looking at the -norm of . By using Minkowski’s inequality, Hölder’s inequality and (3.3) we get We then have For we use Hölder’s inequality and (3.4) to get For the result follows by a direct estimate. The case remains, and here we first observe that is almost diagonal on . Furthermore, if we let we have Before we can put these two observations into use, we need that We then have 4. Curvelet-Type Frames In this section we study a family of curvelet molecules which is a small perturbation of curvelets . The goal is first to show that if is close enough to , then is a frame for . Next to get a frame expansion, we show that is also a frame. The results are inspired by where perturbations of frames were studied in Triebel-Lizorkin and Besov spaces. Let be a system that is close to in the sense that there exists such that for where , and all functions satisfy the following.(i)For we need (ii)Furthermore we need Then is a family of curvelet molecules with regularity and motivated by being a tight frame for , we formally define as It follows from (3.6) and Proposition 3.4 that is a bounded linear functional on ; in fact we have Furthermore, is a norming family for as it satisfies . This can be used to show that is a bounded operator on , and for small enough this will be the key to showing that is a frame for . Theorem 4.1. There exists such that if satisfies (4.1) for some , then one has Proof. That is a norming family gives the upper bound; thus we only need to establish the lower bound. For this we use that is also a norming family so we have It then follows that By choosing we get the lower bound. As one might guess from Theorem 4.1, the boundedness of the matrix on is the key to showing that is also a frame for . Proposition 4.2. There exists such that if satisfies (4.1) for some and furthermore is a frame for , then is bounded on . Proof. The fact that is a frame for ensures that is a bounded operator on . We first show that is bounded on . This will follow from showing that choosing small enough and using the Neumann series. Assume for a moment that satisfies By using that is self-adjoint, we then have So to show (4.9) it suffices to prove (4.10). Note that By setting we have the decomposition Since is a family of curvelet molecules with regularity , we have from (3.6) that are almost diagonal on . Next, we use Remark 3.3, and by Proposition 3.4, Consequently, (4.9) holds, and for sufficiently small the operator is bounded on . Finally, let and . By using (2.9) we have that , and as is a frame for , we have that is self-adjoint which gives If we combine this with being a norming family (4.5), we get which proves that is bounded on . That is a frame for now follows as a consequence of being bounded on . We state the following results without proofs as they follow directly in the same way as in the Besov space case. The proofs can be found in . First, we have the frame expansion. Lemma 4.3. Assume that is a family of curvelet molecules with regularity and a frame for . If is bounded on , then for one has in the sense of . Next, we have that is a frame for . Theorem 4.4. Assume that is a family of curvelet molecules with regularity and a frame for . Then is a frame for if and only if is bounded on . It follows from Proposition 4.2, Lemma 4.3, and Theorem 4.4 that if is a family of curvelet molecules which is close enough to curvelets, then the representation , has the same sparse representation properties as curvelets when measured in . As a final result we also have a frame expansion with . Lemma 4.5. Assume that is a family of curvelet molecules with regularity and a frame for . If the transpose of is bounded on , then for one has in the sense of . All that remains now is to construct a flexible family of curvelet molecules which is close enough to curvelets in the sense of (4.1). 5. Construction of Curvelet-Type Systems In this section we construct a flexible curvelet-type system. We do this by showing that finite linear combinations of shifts and dilates of a function with sufficient smoothness and decay can be used to construct a system that satisfies (4.1). From the previous section, we then have that the representation , , has the same sparse representation properties as curvelets when measured in . First we take , , which for fixed satisfies Next, for we define , where . It then follows that We recall that curvelets (2.6) are a family of curvelet molecules for any regularity . From the definition of a family of curvelet molecules (Definition 2.1), we have that for curvelet molecules can be expressed as where must satisfy (2.8) and (2.9). So to construct a family of curvelet molecules which satisfy (4.1), we need to construct a family of functions such that satisfy (4.2) and (4.3). We define as for and to construct we also need the following set of finite linear combinations: We have omitted the construction of for as it follow in a similar way. Proposition 5.1. Let and . If , fulfills (5.1) and fulfills then for any there exist ( independent of ) and such that Proof. Let and , be given. We construct the approximation of in direct space in three steps. First by a convolution operator , then by which is the integral in taken over a dyadic cube , and finally by a discretization over smaller dyadic cubes . From (5.2) we have Define , where . For , we use the mean value theorem to get Inserting this in (5.8) we have For , we split the integral over and . If , then , and we have Integrating over with gives So by choosing sufficiently large in (5.10)–(5.12), we get For the next step we fix and choose . Let denote the dyadic cube with sidelength , sides parallel with the axes and centered at the origin. We then approximate with defined as In which case we have and it follows that We first estimate the integral for which gives and . Hence we obtain For , we split the integral over and . If , then , and we get Similar for we have By choosing sufficiently large in (5.17)–(5.19), we obtain For the final step we fix , choose , and approximate by a discretization where is the center of the dyadic cube and is the set of dyadic cubes with sidelength which together give . Note that , . We introduce which gives By using the mean value theorem, we then get where is the line segment between and . If and , then we have For and , we have , and hence for , it follows that By choosing sufficiently large, we obtain by combining (5.23)–(5.25) that Finally by combining (5.13), (5.20) and (5.26), we get To approximate in frequency space we use three steps similar to the approximation in direct space. Note that still fulfills (5.27) if we choose even larger. First we use to approximate in which case we have By choosing such that and such that for , we get Next, we fix , choose and limit the Fourier integral of to from the approximation in direct space, This gives In the last step, we fix and approximate by . We introduce which gives By combining (5.29)–(5.32) for sufficiently large , we get It follows that by choosing large enough fulfills both (5.7) and (5.4). Furthermore, we have , . In this paper we studied a flexible method for generation curvelet-type systems with the same sparse representation properties as curvelets when measured in . With Proposition 4.2, Lemma 4.3, and Theorem 4.4 we proved that a system of curvelet molecules which is close enough to curvelets has these sparse representation properties. Furthermore, with Proposition 5.1 we gave a constructive description of how such a system of curvelet molecules can be constructed from finite linear combinations of shifts and dilates for a single function with sufficient smoothness and decay. Examples of functions with sufficient smoothness and decay are the exponential function and the rational functions with sufficiently large. An example with compact support can be constructed by using a spline with compact support. Furthermore as the system is constructed using finite linear combinations of splines, we get a system consisting of modulated compactly supported splines. H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, switzerland, 1983.

We're told. An article on the room in microbial system reports that a fraction of the soluble material passing from the Roman without being fermented during the first hour after it's ingestion could be calculated by the following. Integral and where K measures the rate that the material is fermented but we want to do is first determine what above in nickel is and evaluated or the following values of K that were used in the article. So one twelve, one, twenty four and one forty eight of an hour Looking at center girl, we have a polynomial one minus team more supplied by an exponential. And whenever we have this situation, we want our polynomial to be set to you. And then we want our exponential to br Steve, when we're going to use integration by parts to solve this. So let's go ahead and find what are the new and are being should be. So we have you is equal to one my misty. Take the derivative of this with respect. T wind up, Do you over the teen will be negative. Not just negative one and then solving or you will do is negative. Dt, I got a little bit now to solve her. See, we have Devi is equal to okay, he to the negative k clear e to the negative. Okay, Team guillotine and integrating each side of this will give the music to Kay He to the negative, Katie. And then we divide by the power in the experiment. So Kay and this here will simplify. So the case cancel out. I will have negative e to the negative kay tea for And now that we have these, we can go ahead and used the integration by parts. Remember, integration by parts for this will be a UV. And since it's a definite integral, we need evaluate this from zero one and then we subtract off the integral zero one of So we had you here, so I won't be We had do you or DVD. Now I want you so plugging everything and we have you was one minus. Team V is Magda groups Rachael negative e to the negative, Katie and this year's evaluate. Your mom's here with one, and then we subtract off until the whole crumbs, if you want of being, which is negative. Negative. Katie haven't multiplied by D Yu and Yu is negative. Katie So no, this these negatives in inter broke cancel out and also notice that when we plug in one in tow, one minus table, get zero So the zero mine it's So we go ahead zero into here. So one mind zero and the negative e to the negative k time zero So again, that negative that negative, they're slow and then e to the negative. K zero is one. So this is one. And so this first part will simplify down to just what now for the integral Here we have e to the negative, Katie. So integrating this will be e to the negative a team divided by its power Or that's point in the power. So and we evaluate this from zero to one so we can go ahead and get rid of these negatives here. And then we'LL just have to plug in one and zero So end up with wass So we plug won in and I will be e to the negative. So one time's kay divided by K and then we subtract awful the club zero in and eat the negative K time zero. Once again it was one just like what we have here. So that would be one minus. And just for a little bit of convenience factor on one over cake here. Oh, I have one plus one over. Okay, I need to the leg. Okay. Minus one. So this is R definite inter role for any value. Okay, As long this cave, it's not zero, because otherwise we'll have one over zero, and that's an undefined expression. Now, we wanted to find this for certain bodies. Okay? And this is just what we had on the last page re Burton. So we first want to do Carrie is equal to one over twelve. So plugging this and well, I really just want to do is plug everything into the right hand side here. So we'll have one loss. So one over K in this case will be won over one twelve or just twelve. They have e to the negative one of twelve minus one. And when we plug this into a calculator, we end up with approximately zero point mr zero zero point zero four zero five. So this is the value we get when K is one twelve now. Kay is equal to one over twenty four. So one plus So we're gonna plug in one over Kay. So that will turn that to twenty four e to the negative came. So want over twenty four minus one. And then this value here is approximately zero point zero two o five. And lastly, we want to look at when k is one over. So what? One plus so one over k. So again, this is just forty eight, then e to the negative one over forty eight minus one. And this here is approximately equal to zero point zero one zero three four. So, beans or the three values we get or each of those cave values and home for the second part of the question, we're now given that the fraction of intermediate material left in the room and at one hour that this, for some reason, he's getting cut. All there we go. So we're told that the fraction of intermediate material left in government at one hour that escapes digestion, buy passage between one and six hours is given by that interval there, and we essentially want to do the same thing that we did in part so we personally to find the integral and then put in all the values for Kay. So again, we have a pollen Nobile six minus t and also group in this one fifth here. So you can call this are you? And we have this exponential, which is the same exponentially had before so that this will be our d. So the DVD is the same as before, so we can go ahead and just say V is negative e to the negative. Okay, Teeth. But are you this time or D? You'Ll be slightly different. So what have you is equal to six minus t over. By taking the derivative of this with respect, team will get you a tea is equal to negative one fifth of them solving for you or solving differential to get to you. Why do you is negative one fifth time's duty. So once again, we could go ahead and rewrite alteration. Buy parts for this so be moving evaluated from one to six, minus that definite integral from one to six. Oh, they you Now let's plug everything So you is going to be sit Cover six minus t over by ravine is negative into negative equity. We evaluate this from six of one, then subtract off after integral from one to six of so bean once again will be negative. And do you is negative one there two and life before these negatives will cancel integral. And now there's still hadn't start plugging cancel plug. So if I plunk six in just like before, that first term is going to give us zero. So if zero minus when you plug it ones of six minus one, divided by divided by eye multiplied by I'm not going to be the negative K Times one and then we subtract off and let's go ahead and integrate this here. So the one fifth let's go ahead and pull the one out on DH, even like this house pulled the one without and then definitely grow from one sixth of need to like Katie. You too. Now, this negative there and that their contempt slow and then six minus one is five divided by five. Give one so we can rely so more as e to the negative. Okay, and integrating that there will do negative one. You need to the negative kay team and we need to divide this by a constant in the moon later. So negative. Kay. And we evaluate this from one to six. So again, these negatives Well, cat's out and we could go ahead and write this here. That e to the negative. Okay. Plus so plugging in six will give he to the negative six charity over five k and then we subtract Oh e to the negative one times k over five K and knows that we have this need to the negative kay here. So just help make this a little bit better, although ahead and factor out and e to the negative. Okay. And doing that will give one minus one over five k and then we'LL have Plus, I need to the negative six k over five. So this here is general anti derivatives and this is as long as Cadiz and like we get before. So this his what he had on last page just re Britain and we want to solve or each of the values of K and the first part. So we have K is it what, one twelve again and doing this will give So we only care about this right hand side of the equation here. So we'LL have okay is one twelve Someone overcame again will be, well, so well divided by five he to the negatives of six times one twelve we'll be one how And then we add that to one minus. So again one of the K will be well, we divide this five bye and then we have to eat the negative one twelve And so this here is approximately equal to zero point one six seven six war. You're heading into this for K using one of twenty four. Well, so now everywhere that there was a twelve there'LL be twenty four twenty four five and to the negative six times one over twenty four will now be negative for so negative one over four. Then we're gonna add this toe one minus me or one for over five and into the negative one over twenty four. And this here is approximately zero point zero nine green too. And then for last value. So Okay, people too One over forty eight. We'LL have a forty eight over five and then he to the length six times one forty eight. Okay, negative one over eight and we had that one money nous or bye, he to the negative one over four feet. And this here is approximately zero point or no, no, no. And got toe. Add this zero right here. So point zero four nine to it. And so these three values are what we get for you to Katie's or the second part of the question..

Law a body of rules of conduct of binding legal force and effect, prescribed, recognized, and enforced by controlling authority in us law, the word law refers to any rule that if broken subjects a party to criminal punishment or civil liability. Calculators for ohms law calculate voltage, current, resistance, and power convert watts, volts, and amps. Our ohm's law calculator is a neat little app which helps you use the relationships between voltage, current and resistance across a given conductor. What is ohms law – formula, equation & triangle ohm's law is one of the most fundamental of laws for electrical theory the ohms law formula links voltage and current to the properties of the conductor in a circuit. Ohm's law georg simon ohm (1787-1854) formulated the relationships among voltage, current, and resistance as follows: the current in a circuit is directly proportional to the applied voltage and inversely proportional to the resistance of the circuit. Ohm's law: ohm’s law, description of the relationship between current, voltage, and resistance the amount of steady current through a large number of materials is directly proportional to the potential difference, or voltage, across the materials. Ohm's law is the mathematical relationship among electric current, resistance, and voltage. Vaping is all about ohm’s law if you're looking to get the best hit out of your apv or e-cig mod, this is a useful guide to read. A video explaining my philosophy and principles for teaching ohm's law people sometimes dismiss the law, or teach it poorly using outdated methods this is . Electronics tutorial about ohms law and power in a dc circuit including its relationship between voltage, current and resistance. read reviews, compare customer ratings, see screenshots, and learn more about ohm's law download ohm's law and enjoy it on your iphone, ipad, and ipod touch. Contents[show] history no, ohm's law has nothing to do with calm humming or yoga whatsoever rather, ohm's law is a relationship vital to the understanding of circuitry. Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:. Kids learn about ohm's law in the science of electricity and physics including current, resistance, voltage, example problems, and v=ir. The term ohm’s law refers to one of the fundamental relationships found in electronic circuits: that, for a given resistance, current is directly proportional to voltage in other words, if you increase the voltage through a circuit whose resistance is fixed, the current goes up if you decrease . Electricity basics when beginning to explore the world of electricity and electronics, it is vital to start by understanding the basics of voltage, current, and resistance. Ohm's law is fundamental for resistors resistance is measured in ohm use an easy calculator to determine current, voltage and resistance. Learning objectives explain the origin of ohm’s law calculate voltages, currents, or resistances with ohm’s law explain what an ohmic material is. A voltage divider circuit is a very common circuit that takes a higher voltage and converts it to a lower one by using a pair of resistors the formula for calculating the output voltage is based on ohms law and is shown below. For those of you unfamiliar with ohms law, this calculator allows you to determine whether the coil, tank and mod you're using will work safely together. Ohm's law calculator with real-time results as you change volts, current (amperage), resistance or power (wattage) the fields can be individually locked. In this calculator/table, you may enter two of the four factors in ohm's law they are power (p) or (w), measured in watts, voltage (v) or (e), measured in volts, current or amperage (i), measured in amps (amperes), and resistance (r) measured in ohms. Download this app from microsoft store for windows 10, windows 81, windows 10 mobile, windows phone 81 see screenshots, read the latest customer reviews, and compare ratings for ohm's law. : a law in electricity: the strength of a direct current is directly proportional to the potential difference and inversely proportional to the resistance of the circuit. This is called ohm’s law let’s say, for example, that we have a circuit with the potential of 1 volt, a current of 1 amp, and resistance of 1 ohm using ohm’s law we can say: let’s say this represents our tank with a wide hose the amount of water in the tank is defined as 1 volt and the . Simple to use ohm's law calculator calculate power, current, voltage or resistance just enter 2 known values and the calculator will solve for the others. Ohm’s law is a formula used to calculate the relationship between voltage, current and resistance in an electrical circuit. The ohm is defined as an electrical resistance between two points of a conductor when a constant potential difference of one volt, applied to these points, produces in the conductor a current of one ampere, the conductor not being the seat of any electromotive force. The ohm's law wheel is a simple tool that gives you a formula base on the variable you want and the variables you have. 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Improve your math knowledge with free questions in slope-intercept form: write an equation and thousands of other math skills. Example 1: write the equation of the line in slope-intercept form with a slope of -5 and a y-intercept of 3 the needed information to write the equation of the line in the form y = mx + b are clearly given in the problem since. Slope intercept calculator finds the equation in slope intercept form enter 2 points or 1 point and the slope, and we'll do the rest. To write the equation of a line given the slope and a point on the line, you can use the slope-intercept form of the line and plug in the given values plug in the value of the slope for m, and the coordinates of the given point into the x and y variable. This algebra lesson gives an overview of equations of lines and shows how to graph a line using the slope-intercept (y-intercept) form of a line linear equations. The slope-intercept form is written as y = mx+b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis) it's usually easy to graph a line using y=mx+b other forms of linear equations are the standard form and the point-slope form. This gives the basic outline of the dreaded y=mx+b formula, for linear equations how to solve for b (y-intercept), and how to solve for any other part my goodness, this is getting intense. ©l b2e0 e1r2n gkxuvt4a a lszotfmtvwxavrpe j elglsc vo x pa el hl y urving2hft xs9 8r neosweruv aezd hv 7 am ra cd ief sw 6i9tnh7 ailn 7fqi0n1ihtbe h 4p uryeu-kafl wgwegbfr pakw worksheet by kuta software llc. Learn how to solve problems involving writing an equation in slope-intercept form 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 kastaticorg and kasandboxorg are unblocked. This is called the slope-intercept form because m is the slope and b gives the y-intercept (for a review of how this equation is used for graphing, look at slope and graphing ) i like slope-intercept form the best. Linear equations can take several forms, such as the point-slope formula, the slope-intercept formula, and the standard form of a linear equationthese forms allow mathematicians to describe the exact same line in different ways. In the worked examples in the next section, i'll use the point-slope formula, because that's the way i was taught and that's what most books wantbut my experience has been that many students prefer to plug the slope and a point into the slope-intercept form of the line, and solve for b. Section 41 writing equations in slope-intercept form 177 using points to write equations write an equation of each line that passes through the given points a (−3, 5), (0, −1) b. 1 write the equation in slope-intercept form identify the slope and y-intercept show all work 2x - 3y = 9 2 write the equation in slope-intercept form. These linear equations worksheets will produce problems for practicing graphing lines in slope-intercept form you may select the type of solutions that the students must perform these linear equations worksheets are a good resource for students in the 5th grade through the 8th grade. The slope intercept form equation is expressed as y = mx + c, where 'm' represents the slope of the line and 'c' represents the y-intercept of a line you can find the equation of a straight line based on the slope and y-intercept using this slope intercept form calculator. In this form only (when your equation is written as y = ) the coefficient of x is the slope and the constant is the y intercept when an equation is written in this form, you can look at the equation and have enough information to graph the equation. Name date period 4-1 skills practice graphing equations in slope-intercept form write an equation of a line in slope-intercept form with the given slope. - how to write a slope intercept equation for a line on a graph for more practice and to create math worksheets, visit davitily. To write an equation in slope-intercept form, given a graph of that equation, pick two points on the line and use them to find the slope this is the value of m in the equation next, find the coordinates of the y -intercept--this should be of the form (0, b ). The slope intercept form calculator tells you how to find the equation of a line for any given two points that this line passes through it will help you find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas. This can be done by calculating the slope between two known points of the line using the slope formula find the y-intercept this can be done by substituting the slope and the coordinates of a point (x, y) on the line in the slope-intercept formula and then solve for b. Writing linear equations date_____ period____ write the slope-intercept form of the equation of each line 1) 3 x write the slope-intercept form of the equation. Equations that are written in slope intercept form are the easiest to graph and easiest to write given the proper information all you need to know is the slope (rate) and the y-intercept. Identify the slope m and y-intercept from the assortment of linear equations in slope-intercept form the equations are written in the form y = mx + c where m is the slope, and c is the y-intercept download the set. Slope intercept form showing top 8 worksheets in the category - slope intercept form some of the worksheets displayed are graphing lines in slope intercept, writing linear equations, , practice for slope y intertcept and writing equations, slope intercept form word problems, infinite algebra 1, model practice challenge problems vi, lines lines lines slope intercept form lesson plan. Identify the initial value (y-intercept) from a table, graph, equation, or verbal description use the slope and y-intercepts to write a linear function in the form from any representation (table, graph, or verbal description. Free slope intercept form calculator - find the slope intercept form of a line given two points, a function or the intercept step-by-step line equations functions. 8 in order to join a yoga club there is a $100 annual fee and a $5 fee for each class you attend write an equation in slope-intercept form that.

Since gluons are located within nucleons and immediately outside of them, how do experiments determine parameters like their speed? Is it possible we could be assuming they travel at the speed of light since they are massless, but in reality they travel faster/slower than light? $\begingroup$ I'm simply wondering, is there any way to know for certain? I have read that we still understand a very limited amount about what actually goes on in the nucleus, in terms of the positions of nucleons being undefined, etc. $\endgroup$– snowgNov 14, 2018 at 22:44 $\begingroup$ Yes. I'd like to learn more about gluons and the nuclear force. I've watched many youtube videos, obviously not a credible academic source. I know of the color charge and I know that the strong force binds quarks into nucleons, while also overcoming the electromagnetic force to create the nucleus itself. $\endgroup$– snowgNov 14, 2018 at 22:47 Gluons, like quarks, are bound inside nucleons. However, it's not quite correct to think of either quarks or (especially) gluons as being little particles inside a nucleon. Before addressing questions about their speed, it's important to appreciate the limitations of the idea that they are particles — or classical waves, or anything else to which we might normally apply a concept like "speed". Quantum chromodynamics (QCD) is expressed in terms of quark and gluon fields, not particles. Particles are phenomena that the model predicts when the conditions are right. The only particles that QCD predicts under ordinary conditions are mesons (like pions) and baryons (like protons and neutrons), both of which are "color neutral". In that context, I don't know of any natural sense in which qualities like "speed" can be attributed to individual gluons, because I don't know of any natural sense in which a meson or baryon is made of individual gluons! Mesons and baryons are a different kind of structure composed of quark and gluon fields. Even the usual cartoon of a proton being made of two up-quarks and a down-quark isn't quite accurate. It's good enough for some purposes (like the Bohr model of the atom), but a proton is more accurately described as a quantum superposition of many different combinations of quarks (and gluons), like three up-quarks and one anti-up-quark and a down-quark. One of the most direct manifestations of individual quarks and gluons is in a phenomenon called jets. This phenomenon occurs, for example, when an electron and positron (anti-electron) are smashed together with a center-of-mass energy between about 5 GeV and 45 GeV (the GeV is a convenient unit of energy is particle physics; it stands for "giga electron volts"). In this energy range, the result is often two back-to-back "jets" of hadrons (mesons and baryons). QCD predicts that this will occur as the result of the electron and anti-electron annihilating each other and producing a quark and anti-quark moving away from each other in opposite directions. But since quarks (and anti-quarks) are confined under ordinary conditions, they don't get very far before they become "clothed" with other quarks and gluons, resulting in two back-to-back jets (bunches) of many color-neutral particles instead. At the higher end of this energy range, above about 20 GeV, we occasionally see three-jet events, with three jets of color-neutral particles propagating away from the point where the electron and anti-electron annihilated each other. This is also predicted by QCD, which describes it as the creation of a quark, and anti-quark, and one gluon, all initially flying away from each other. This can't last for long, though; they quickly "clothe" themselves with other quarks and gluons, resulting in three jets of color-neutral particles instead. (By the way, to check these things while I was typing them, I referred to page 24 in chapter 1 of Renton's book Electroweak Interactions. But I still accept responsibility if I've said anything inaccurate.) So, how fast does a gluon move? Well, considering the high energies involved in the collisions that produce these jets, the final particles tend to be moving away from the collision point at very nearly the speed of light, even though most of them have mass. I'm not sure the "bare" gluon lasts long enough to really constitute a well-defined particle (maybe an expert can chime in and quantify this — or correct me if I'm wrong), in which case the concept of "speed" is once again not quite appropriate. So, do gluons move at the speed of light? Before we can answer that in a meaningful way, we have to find a condition under which gluons exist as individual particles long enough for the concept of "speed" to make sense... and that's not as easy as it sounds. $\begingroup$ I'm so glad someone is mentioning his source! Everyone here should do that. Well written too. $\endgroup$ Jun 10, 2020 at 19:36 $\begingroup$ Why do you describe a proton as a quantum superposition of quarks and gluons? Isn't that somehow the same thing as the cartoon-picture of a proton? $\endgroup$ Jun 10, 2020 at 19:59 1$\begingroup$ @VeronicaNoordzee Thank you for the compliment! There are different cartoons, some more complicated and/or accurate than others. What I meant by "the usual cartoon" is the picture of a proton as being made of smaller particles, but then I wasn't consistent in the way I used the word "quark." I started out talking about quark and gluon fields, but then I implicitly reverted to talking about quarks as particles when I said that a proton is a quantum superposition of different numbers of quarks (3, 5, 7, and so on). So I guess my description was still cartoon-ish. Is that what you mean? $\endgroup$ Jun 11, 2020 at 23:40 1$\begingroup$ @VeronicaNoordzee Our current understanding of everything except gravity is based on quantum fields. Particles are one of the phenomena that quantum fields can exhibit. When learning QM, we start by learning models in which particles are prescribed as inputs to the theory. That's often a good approximation (quantum chemistry uses it), and the math is more manageable. In quantum field theory, particles are derived instead of prescribed, which is why they can be created/destroyed. Do we experience quantum fields directly? That's a tricky question... Depends on what you mean by "directly"! $\endgroup$ Jun 14, 2020 at 1:18 1$\begingroup$ I just love your replies, C.A. So much is said in so few words. Well, well done! In my mind, I distinguish between the visible world (using light(microscopes)) and the non-visible world. Therefore, we can experience the non-visible world only indirectly (in cloud-chambers, LHC-collisions, etc.). Yes, I'm a latecomer in Physics, but I find it fascinating to see how mankind succeeded in explaining its environment. This stuff really makes me happy! :-) P.S. Fortunately, nature gave a brain to understand things to some level. $\endgroup$ Jun 15, 2020 at 14:08 This is complementary to the answer by Chiral Anomaly, extending it: At extreme energies in particle interactions, QCD predicts quark gluon plasma: A quark–gluon plasma (QGP) or quark soup is a state of matter in quantum chromodynamics (QCD) which exists at extremely high temperature and/or density. This state is thought to consist of asymptotically free strong-interacting quarks and gluons, which are ordinarily confined by color confinement inside atomic nuclei or other hadrons. This is in analogy with the conventional plasma where nuclei and electrons, confined inside atoms by electrostatic forces at ambient conditions, can move freely. This at the moment is studied in ion collisions at the LHC: For a few millionths of a second, shortly after the Big Bang, the universe was filled with an astonishingly hot, dense soup made of all kinds of particles moving at near light speed. This mixture was dominated by quarks – fundamental bits of matter – and by gluons, carriers of the strong force that normally “glue” quarks together into familiar protons and neutrons and other species. In those first evanescent moments of extreme temperature, however, quarks and gluons were bound only weakly, free to move on their own in what’s called a quark-gluon plasma. Jets are “hard probes”, by nature strongly interacting but moving so fast and with so much energy that they are often not completely absorbed by the surrounding quarks and gluons in the quark-gluon plasma. The degree of jet quenching – a figure that emerges in data from millions of collision events – plus the jets' orientation, directionality, composition, and how they transfer energy and momentum to the medium, reveal what’s inside the fireball and thus the properties of the quark-gluon plasma. One has to realize that particle physics is a mixture of theory and experimental validation. The validation of QCD at low energy, which has the analogue of the photon, called a gluon, with a zero mass and thus limited to the fixed velocity c of special relativity , hypothetically, because it cannot be measured in the lab, only the consequences are measured : mesons, baryons, jets ... which validate the QCD model quite well . A QCD model with a massive gluon would not fit the data in any way. But as the theory is asymptotically free, more data come from high energies where in a plasma quarks and gluons can be modeled as particles, and as of the present, the zero mass of the gluon is validated by data at the LHC because QCD predictions fit the data, and by observations of the quark gluon plasma state of the early universe, which also can be modeled by asymptotically free quarks and gluons. The basic point is that one trusts the mathematics in the model , if one cannot measure directly the hypothesis, and checks the hypothesis by fitting data, which validate the model . In this sense a measurement at second (third fourth ... as there are integrals involved) hand. $\begingroup$ Really nice addition! Thank you, Anna. $\endgroup$ Jun 10, 2020 at 19:44

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(Click on logo to download). |BOWIE RING TONES FOR NOKIA AND SIEMENS MOBILE PHONES| All The Young Dudes: 8#f2 8d2 8a1 8#f2 16e2 16#f2 16e2 16#f2 8e2 8#c2 8d2 8b1 8#f1 8d2 16#c2 16d2 16#c2 16d2 8#c2 8b1 8a1 8#c2 8#c2 8#c2 8#c2 8c2 8b1 4a1 8#c2 8#c2 8#c2 8#c2 8b1 8a1 4#f2 Miracle Goodnight: 4a1 4b1 4c2 4d2 4g1 4- 8g1 8g1 8g1 2- 4- 8- 8g1 8g1 8g1 4- 8- 8g1 8g1 8g1 8- 4a1 4b1 4c2 4d2 4g1 4- 8g1 8g1 8g1 2- 4- 8- 8g1 8g1 8g1 4- 8- 8c2 8c2 8c2 8- 8b2 8g2 8c3 8a2 8b2 8c3 4d3 4g1 Quicksand: 4e1 8e2 2#c2 8b1 4#c2 4.a1 4- 4a2 8#g2 2#f2 4e2 8#d2 4.e2 4- 8#g2 4#f2 4.e2 4#f2 4d2 8#c2 4.d2 4- 2a2 2#f2 2#d2 4e2 4- 2a2 2e2 2#f2 1#c2 1b1 Rebel Rebel: 4d1 4e2 4d2 8#c2 4b1 4b1 8b1 8#c2 8b1 8#g1 8e1 4d1 4e2 4d2 8#c2 4b1 4b1 8b1 8#c2 8b1 8#g1 8e1 4d1 4e2 4d2 8#c2 4b1 4b1 8b1 8#c2 8b1 8#g1 8b1 4#c1 Seven: 4e2 8g2 4g2 4e2 8g2 4a2 8g2 4g2 2- 8d2 8d2 8d2 8#d2 8e2 4c2 4a1 2- 4e2 8g2 4g2 8a2 4b2 4a2 8g2 4g2 8e2 4g2 4a2 8c2 4c2 4d2 4c2 2- 8c2 8#d2 4.#d2 8d2 4c2 4e2 4c2 4- 8d2 8e2 4.e2 8g2 4c3 4c2 4- 8c2 4d2 Starman: 8g1 8a1 2f1 2f2 8e2 8d2 8c2 8d2 4c2 8- 8d2 8e2 8d2 8c2 8d2 8c2 8d2 8c2 8d2 8e2 8d2 8c2 8d2 4c2 8g1 8a1 2f1 2f2 8e2 8d2 8c2 8d2 4c2 8- 8d2 8e2 8d2 8c2 8d2 8c2 8d2 8c2 8d2 8e2 8d2 8e2 8f2 4g2 The Man Who Sold The World: 8a1 8a1 8a1 4g1 16a1 16#a1 8a1 8g1 8a1 8a1 8a1 4g1 16a1 16#a1 8a1 8g1 8a1 8a1 8a1 4g1 16a1 16#a1 8a1 8g1 4a1 4b1 4c2 4a2 The Man Who Sold The World: (Motorola) 4 A4 A4 A4 G4 R2 A2 A#2 A4 G4 A4 A4 A4 G4 R2 A2 A#2 A4 G4 A4 |ANIMATED DAVID BOWIE GIFS| Ziggy Stardust 1972 (13Kb) - Aladdin Sane (11Kb) - Aladdin Sane Side (12Kb) - Pin Ups 1973 Saxophone (138Kb). DO NOT LINK DIRECTLY TO ANY OF THE GRAPHICS! |Created: July 1997 © Paul Kinder||Last Updated: 26/5/11|

QUANTITATIVE GENETICS BPSC 148 Popular in Course Popular in Botany This 11 page Class Notes was uploaded by Keshawn Howell on Thursday October 29, 2015. The Class Notes belongs to BPSC 148 at University of California Riverside taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/231737/bpsc-148-university-of-california-riverside in Botany at University of California Riverside. Reviews for QUANTITATIVE GENETICS Report this Material What is Karma? Karma is the currency of StudySoup. You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more! Date Created: 10/29/15 Chapter 14 Basic Concepts of Selection Evolutionary forces Evolutionary forces are factors that can change the genetic properties of a population There are four evolutionary forces selection mutation migration and genetic drift Selection is considered to be the most important one Selection is de ned as a phenomenon that parents who contribute to the next generation are not a random sample but selected based on some criteria Genetic properties of a population include gene frequencies and genotype frequencies However these are not observable what we can observe are the population mean and variance that re ect gene and genotype frequencies of the population Change in gene and genotype frequencies Here we use deleterious recessive gene as an example Assuming that Al is completely dominant overAz allele so that A1141 and11A are indistinguishable in terms of selection Genotype 141A1 141A2 A2 A2 Total Initial freq p2 2pq q2 1 Coef cient of 0 0 3 Selection Fitness 1 1 ls Garnetic p2 qu 13q2 13q2 contribution The tness denoted by w is de ned as w 1 s where s is the selection coef cient The garnetic contribution of each genotype is the initial frequency multiplied by the tness After selection the genotype frequencies will change ie the genotype frequencies in the selected parents will be different from the frequencies of the initial unselected population Genotype 141A1 141A2 A2 A2 Total New freq in p2 2 pg 1 3M2 1 selected parents l sq2 1 59 l sq2 De ne the initial frequency of A1 and A2 by p0 and q0 respectively then the new allelic frequencies forll and A2 are 2 2 p1PlH1 p0 p0q0 p0 1 qu 2 1 qu l sqg lW2 1 Mar 617 sq2 q1Q1H1 1 20 0t 20 20 sq0 2 1 qu 1 qu The change of gene frequency in one generation of selection is 2393090 1 M q1 qo M 0 S 0 So ql S qo meaning that frequency of A2 has been decreased due to selection against this allele If the selected parents are randomly mated the genotype frequencies in the offspring will be Genotype Frequency A1141 1 11 2 A1142 2q11q1 Aw q Therefore after one generation of selection both the gene and genotype frequencies have changed As a result the population mean and genetic variance have also changed 162 Changes of a metric character Truncation selection An individual is selected if its phenotypic value is greater than a threshold T otherwise it is culled Insert Fig 171 here The tness of an individual is de ned as 1 if y 2 T 0 if y lt T Selection dz erentials De ne 7 and J75 as the means of the parental population before and after selection respectively The selection differential is de ned by S J73 y Response to selection Response to selection is de ned as the difference between the mean of offspring and the mean of the parental generations Denote the response to selection by S the relationship between R and S is given by R hZS This equation also called the selection equation is important in plant and animal breeding as well as in evolution Selection intensity Selection differentials cannot be compared for different traits A large value of selection differential does not mean a strong selection The strength of a selection is measured by selection intensity Selection intensity denoted by i is also called standardized selection differential It is defined as the selection differential divided by the phenotypic standard deviation ie where UP is the standard deviation of phenotypic value ie 6P 1 6 46 JV We now express the selection response as a function of the selection intensity RhZSh2ioP hioA Proportion selected p The proportion selected p is the ratio of the number of selected individuals to the total number of individuals measured Theoretically if y is normally distributed ie y N NL6p is defined by T TLUP t p 1Lofydy 1 Lo fy H6pdy H6P1liwfxdx 1 0 where x y u OP is the standardized normal variable and t T u OP is the standardized truncation point in the N 01 scale Relationship between i and p If the selected character is normally distributed there is a simple relationship between i and p 1 1 2 Z tmexp7t t 1 2 pl q3tl Lomexp x dx Z in See the appendix of this chapter for the numerical values of the relationship among i tand p 164 When males and females have different selection differentials their average should be used S S selection d1fferent1al for males lead1ng to im 039 m P S S f selection differential for females leading to if f 039 P S Sm Sf leading to 139 im z39f Therefore R hZS h2Sm h2Sf hzimcp h2ifcp An example Given 6p 5 u 25 and p 020 find the truncation point selection intensity selection differential and response to selection 1 From the appendix table we get I 084 and z39 140 whenp 020 Note that tis the truncation point in the standardized scale What is the truncation point in the real scale T Because I THwe have Tt6Pu0842X525292l 6 F 2 Selection differential S 1390 l40gtlt5070 3 Assume that both the males and the females have the same selection intensity and h2 050 then the predicted selection response is R h2 S 05gtlt70 35 The population mean ofthe offspring will be u R 25 35 285 165 Truncation selection an AD YNuio 9 n1 5 gm gum Em a DUI p y u T Phenotypic value in the original scale y m m g xN g ml 01 Emma E 1 t gm p lt1gt Em a 76 4 72 u 2 a a XY 39 MVG Phenotypic value in the standardized normal scale x Chapter 3 Population Mean Phenotypic value the value observed when a character is measured on an individual For example the body weight of a particular pig is 300 lb The value of 300 1b is the phenotypic value of this pig for body weight Genotypic value average of the phenotypic values of all individuals who have the same genotype For example there are 1000 individuals in a population who have the same genotype A1142 For a particular trait the phenotypic values of the 1000 individuals are 125 089 210 Because all the 1000 individuals have the same genotype 1000 the average values of their phenotypic values 025 089 210 is the genotypic value of genotype A1142 Environmental deviation the difference between the phenotypic value and the genotypic value The rst genetic model P G E phenotypicwlue genotypicvalue environmentaldeviation Because environmental deviations can be positive and negative the average of E s for all individuals will be zero and thus the average phenotypic values is the genotypic value For the purpose of deduction we must assign arbitrary values to the genotypes under discussion This is done in the following way Considering a single locus with two alleles A1 and A2 we call the genotypic value of one homozygote a that of the other homozygote 7a and that of the heterozygote d We shall adopt the convention that11 is the allele that increases the value We thus have a scale of genotypic values as in Fig 31 Genotype A2142 141142 A1f1 Genotypic a 0 d a value Fig 31 Arbitrarily assigned genotypic values The genotypic value of the heterozygote al indicates the dominance effect If there is no dominance effect the genotypic value of A 1A2 should be half way between A1141 and 142142 ie al 0 With dominance d at 0 and the degree of dominance is usually expressed by the ratio d a Note the value 0 re ects the midpoint in the arbitrary scale The actual midpoint value can be different from 0 as shown in the following example The example also shows how to calculate the arbitrarily scaled genotypic values from the actual values Example dwar ng gene in the mouse Genotype A1A1 pgA1Az pg pgAzAz Actual weight g 14 12 6 14 6 10 The From this table we can calculate the actual midpoint value which is arbitrarily scaled genotypic values are then obtained by subtracting the actual midpoint value from the actual values of the genotypes After the subtractions we have al4 104 a6 10 4 d12 102 The degree of dominance is 6161 24 05 Note that midpoint is just a convenient way to express the position of the population There are several other ways to express the population position Remember to adjust the data by the midpoint before conducting any data analysis Population mean Let p and q be the frequencies of allele A1 and A2 respectively The population mean expectation is calculated using the following table Genotype Frequency y Value y y X f y 141A1 p2 a ap2 A1141 2m d dam AM q2 a aq2 The population mean in the arbitrary scale is the expectation of Y M EY 0P2 dam aq2 6100 q 2106161 The population mean I has been expressed as deviation from the actual midpoint Thus the actual population mean should be M midpoint Example dwar ng gene in the mouse Assume that the frequency of allele is p 09 and that ofpg allele is q 01 We have Map q2pqd4X09 0l2X09X0lx2356 The actual population mean isM midpoint 356 10 1356 Average effect of gene Genotypes are not transmitted from generation to generation it is the gene that is inherited from parent to offspring Average effect of a gene is the mean deviation from the population mean of individuals which received that gene from one parent the gene received from the other parent having come at random from the population First let us look at the average effect of allele A1 Consider one allele beingll and the other allele being randomly sampled from the population The probability that the other allele is A1 equals p If this happens the genotype will be A1141 with a genotypic value a On the other hand the probability that the other allele is A2 equals a Ifthe other allele is sampled then the genotype will be A1142 with a genotypic value 51 So in the population containing individuals with one A1 allele being fixed a proportion of p will be A1141 and a proportion ofq will be A1142 Define the mean ofthis population by M Al we have M A1 ap dq We then express this mean by the deviation from the population mean M resulting in the defined average e ect of genell all MA1Mqadqp The average ef ect of A2 is similarly defined by first obtaining M A pd qa and then 0 2 MA M padqp Average effect of gene substitution In a biallelic system the average effect of gene substitution is more informative than the average effect of gene The average effect of gene substitution is defined by a a1 a2 adq p The average gene effects and average effect of gene substitution can be derived using the conditional distribution and conditional mean approach Let us define a 2X2 table as Amp AM Amp A1A1apz A1A2d pq a A2Q 14214103 7 14214200 92 0 2 0L1 0L2 M A1 2 ap d apdq P P 001MAfM apdqap q2pqd qadqp Appendix E Linear Combination Quadratic Forms and Covariance Matrix Linear combination Let a be an ngtltl vector of constants Xan ngtltl vector of variables The scalar l aTX is called a linear combination ofX For example x1 13x1 4x2 5x3 3 4 5 x2 aTX x3 3 x1 where a 4 andX x2 5 x3 Quadratic forms LetA be an 71er constant matrix usually symmetrical X be an ngtltl vector of variables then Q X TAX is a quadratic form of X For example let x1 an a12 a13 x x2 A 6121 6122 6123 a 31 a 32 a 3 33 x1 all all 6113 X x2 and A all 6122 6123 x3 6131 6132 6133 the quadratic form is an all an 1 3 3 QXTAXC1 x2 x3 6121 6122 6123 2 ZZaUx1xj 11 1 a3 l X X X 3 Variance covariance matrix Let X x1 xn T be an ngtltl vector of multiple variables Denote the variance 2 and covariance by 61 Varxland GU C0vxlxy respectively for 1 l n The variancecovariance matrix of X is de ned as El 2 612 62 39 39 39 6m x VarX Var xn 6m 6H2 39 39 39 n VarX is a compact way of writing an array of variances and covariances Let Xx1xnT a vector 71 variables and Y y1ymT be a vector ofm variables De ne COVx1y1 C0Vx1y2 COVx1 ym Covx Covx Covx m CMXJT zyl 2y2 2y COVOCWJl OVOCWJz COVOCWJm W as the covariance matrix betweenX and Y We can de ne the following partitioned matrix X VarX E CovX Y T Var J E Y CovYXT VarY where VarX is an n X n variance matrix VarY is an m X m variance matrix and CovYXT CovXYT T is an m X n covariance matrix Expectation of linear combination Let Xx1xnT and Ex1 H1 fori ln De ne u1 HAT then x1 190 H1 EXE x E99 2 p 96 Exn H If a a1 any is a vector of constants then EaTX aTEX Let Y y1 ynT be another vector of variables we have EXYEXEY An example E2 Are 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[su_list icon=”icon: angle-double-right” icon_color=”#3498db”] - Part 1 – Tyre as a vertical spring - Part 2 – Road Interaction & Contact Patch Grip - Part 3 – Longitudinal Forces - Part 4 – Lateral Forces - Part 5 – Load Sensitivity We have now analyzed a bit of the main aspects of tyres behavior, including how a tyre produces longitudinal and lateral forces and load sensitivity. However, a question still remains: how does a tyre behave, when it had to exchange with the road lateral and longitudinal forces at the same time? Any time a tyre interacts with the road with a planar force having both a lateral and a longitudinal component, we talk about combined grip. A typical situation where combined grip is used by the driver is when he/she starts to steer, while still pushing on the brake pedal (trail braking), or when he/she accelerates out of a corner while the car has not yet completely been realigned. Both of these situations take place regularly on a track if a driver explores the full potential of car and tyres. In general, it is common for the car to negotiate a section of a track and experience both a lateral and longitudinal acceleration, exchanging with the road forces with both a lateral and a longitudinal component. From a pure mathematical perspective, a force can be represented by a vector, which is an entity defined by a magnitude (force’s intensity) and a direction (if we talk about planar forces, a combination of a lateral and longitudinal portion). A vector (and, hence, a force) is normally represented graphically with an arrow, this making much easier also for who is not too versed with mathematic to grasp its meaning. The picture above describes a pure braking situation on the left side and a pure cornering situation, on the right side. From a more rigorous perspective, we can imagine that the two vectors/forces refers to a known and fixed vertical load. In both of these cases, the tyre can exploit its full potential in one direction only, hence maximizing either car’s lateral or longitudinal acceleration. Since we now know that tyre forces can be represented as vectors, we can also conclude that, in cases where a combination of a lateral and a longitudinal force are exchanged with the road, the magnitude of the combined force can be obtained combining (with a vector sum) lateral and longitudinal components. In a first approximation, useful here to explain the concept of combined grip, we can imagine the maximum possible magnitude of lateral and longitudinal force that a tyre can exchange with the road (for a fixed vertical load) to be the same. This will also mean that if we ask the tyre to provide at the same time a lateral and a longitudinal force, the resultant force vector (which is, as we saw, the vector sum of lateral and longitudinal component) cannot have a bigger magnitude than the one of the pure lateral / longitudinal force maximum magnitude. If all of this is true, we can then define the maximum combined force for a given vertical load by using a circle, with its radius given by the maximum pure longitudinal /pure lateral force: This means, on one side, that the magnitude of our maximum combined force is the same as the maximum magnitude of the pure lateral or longitudinal force. But this also means that, in combined grip situations, the maximum allowed lateral and longitudinal forces will be smaller than in pure cornering or in pure braking: The circle defining the maximum combined force the tyre can exchange with the road is often called the friction circle. The reality of things is actually a bit more complex that what we explained, as very often, for a given vertical load, the maximum magnitude of the longitudinal force a tyre can achieve is not necessarily the same as the maximum magnitude of the lateral one. This is why engineers very often talks about friction ellipse, instead of friction circle. Another important point to mention is that, depending on tyre construction and the goals that want to be achieved with a certain design, it is possible that the shape of the envelope defining the maximums, is not a regular shape and cannot be defined either with a circle or an ellipse. However, this goes beyond the scope of this short overview. It is also interesting to analyze how a slip curve changes its shape, in a combined grip situation. As we have seen, anytime we have a non-zero slip ratio and slip angle at the same time, the amount of longitudinal and lateral force that the tyre can produce drops. The shape of a plot of longitudinal force with respect to slip ratio, when the tyre also experiences a slip angle, depends very much on tyre construction. In general, anyway, the bigger the slip angle, the smaller the available longitudinal force for a given slip ratio. This effect of having a bigger slip angle is stronger at smaller slip ratios and tends to lose intensity as we move toward bigger and bigger slip ratios, up to 1 (or -1). We can identify a similar trend if we look at the plot of lateral force with respect to slip angle, when the tyre also experiences a slip ratio. For a given slip angle, the tyre will produce smaller and smaller cornering forces, as the slip ratio gets bigger. It is also interesting to cross these plots, looking for example at how the curve of lateral force with respect to slip ratio looks like, for growing slip angles: We see the same tendencies we already identified, with the lateral force (at a given slip angle) dropping as the slip ratio increase. This plot gives maybe a better idea of how much cornering potential the tyre loses if the slip ratio increases, with the lateral forces coming very close to zero if the slip ratio gets to the value of one. Similar tendencies can be seen if we look at the plot of longitudinal force with respect to slip angle, with varying slip ratio. It is interesting to notice that the longitudinal force drops much slower with respect to the slip angle, compared to what the lateral force does with respect to the slip ratio. This tells us that, while it is difficult to steer when a front tyre experiences high slip ratios (or, in an extreme case, is locked), it is still possible brake the car (up to a point) even when the front tyres are steered. The reason for this is that, normally, pretty small slip ratios are required to produce high braking forces. The final part next week will look at cambers, temperatures and pressures.

This is an ambitious and substantial book on a classical subject area that has been extensively covered in the past. The authors are well-known mathematicians; Rubinstein pere is the father figure of moving boundary problems and his son has a substantial reputation. Such a book invites comparisons with its predecessors, such as Courant and Hilbert's standard work. What do the present authors have to add? The authors set out their programme in the preface. They intend to study partial differential equations as arising naturally out of continuum models of physical processes. Moreover, they will give a rigorous treatment of "classical" methods, which they imply have been somewhat unjustly neglected by "modern mathematical education". And since that education often contains little or no physical background, they will begin with a description of commonly occurring models in the natural sciences, and their formulation in terms of partial differential equations. Finally, they intend their book to be used by graduate students in mathematics, engineering and natural sciences. They begin, then, with a chapter on the ideas of continuum modelling. Several consistent features of the book are immediately clear. The first is, quite simply, that there is an enormous amount of information. After a discussion of the basics of models of continuous media, we find outlines of the usual models for linear elasticity, viscous and inviscid fluids, electrostatics and electrodynamics, chemical kinetics, equilibrium thermodynamics, thermodynamics of irreversible processes, together with a preliminary discussion of conservation laws. All this, in 28 pages. Second, there is no lack of rigour. The tone is set by the definition of a continuum, on page one; the comments on thermodynamics indicate that the authors have an inclination to a "rational mechanics" approach. It is perhaps no coincidence that the notation is rather dense - altogether, there is more than a whiff of the formidable and comprehensive Soviet school of textbook writing. Finally, there is a degree of idiosyncrasy in the subject matter and the way it is treated. This has positive and negative aspects. On the downside, one could cite their definition of a rotational fluid flow as one in which the velocity is the curl of a vector potential (thereby making stationary fluids rotational, and forcing all rotational flows to have zero divergence), their use of the notation (Greek) nu(x) for the Heaviside function, or their frankly eccentric preference for a version of the Laplace transform that is p times the usual definition. On the other hand, subjects such as chemical kinetics, electrodiffusion, the Buckley- Leverett equation and the Stefan problem (which, of course, reflect the particular interests of the authors) make a welcome appearance. There follows a chapter in which the relationship between continuum models and partial differential equations is further developed -- and then we come to the heart of the matter, a detailed analysis, in the classical style, of the mathematical properties of some of these equations. The ordering is quite conventional, beginning with quasilinear and nonlinear first-order equations (there is a nice treatment of the latter but, oddly, Charpit's equations are not named as such, and neither are obvious applications such as geometric optics or the ship wave equation cited). The classification of second-order quasilinear equations is followed by about 50 pages on hyperbolic equations, 100 on second-order elliptic equations (but not the biharmonic equation), and nearly 200 on the parabolic case. We are led through extremely (perhaps overly) detailed treatments of the main classical theorems of the linear classical theory, and there is a more sporadic treatment of selected nonlinear problems. The final section turns to some more practical matters, with chapters on Fourier series, integral transforms and asymptotic analysis. The material of the first two of these has largely been better presented elsewhere, while in the third the authors more or less admit that they cannot cover the subject adequately. Lastly, a clutch of appendices about routine methods includes some interesting specialised applications to heat and mass transfer. There is no doubt that this is a work of considerable and thorough erudition. But the rather distinctive slant and style of the book has consequent drawbacks. One is that not many students will find it easy to develop their intuition for the central ideas. This is partly because informal - intuitive - treatments are largely absent, but also the theory of distributions has been, perhaps rather mistrustfully, relegated to a minor role. (The delta function is not even mentioned in the treatment of Green's functions for elliptic equations.) A second is that the enormous literature on functional-analytic approaches to applied mathematics might almost not exist, if we read only this book. A more balanced treatment would spell out the comparison; the reader could then judge whether the emphasis on classical methods was justified. On the positive side, in addition to, and hidden away in, the vast body of information are some fascinating topics that are impossible to find elsewhere. What we need is a user's guide to help us to find them. Sam Howison is a lecturer in applied mathematics, University of Oxford. Partial Differential Equations in Classical Mathematical Physics Author - Isaak Rubinstein and Lev Rubinstein ISBN - 0 521 41058 4 Publisher - Cambridge University Press Price - £60.00 Pages - 676

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions. I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to? Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece? Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game. Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too? Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf. Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total. 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? 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. The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece? What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself. Find the next number in this pattern: 3, 7, 19, 55 ... These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers? Where can you draw a line on a clock face so that the numbers on both sides have the same total? Can you make square numbers by adding two prime numbers together? What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros? 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 are three buckets each of which holds a maximum of 5 litres. Use the clues to work out how much liquid there is in each bucket. 48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers? Fill in the missing numbers so that adding each pair of corner numbers gives you the number between them (in the box). Fill in the numbers to make the sum of each row, column and diagonal equal to 34. For an extra challenge try the huge American Flag magic square. On a calculator, make 15 by using only the 2 key and any of the four operations keys. How many ways can you find to do it? How would you count the number of fingers in these pictures? Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens? In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first? If the answer's 2010, what could the question be? Mr. Sunshine tells the children they will have 2 hours of homework. After several calculations, Harry says he hasn't got time to do this homework. Can you see where his reasoning is wrong? On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are? Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was. Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing? Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes? If the numbers 5, 7 and 4 go into this function machine, what numbers will come out? 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? Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? Are these domino games fair? Can you explain why or why not? 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? In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time? Find at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100. The value of the circle changes in each of the following problems. Can you discover its value in each problem? Can you score 100 by throwing rings on this board? Is there more than way to do it? A group of children are using measuring cylinders but they lose the labels. Can you help relabel them? Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon? 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. Using the statements, can you work out how many of each type of rabbit there are in these pens? Can you substitute numbers for the letters in these sums? On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there? Use your logical reasoning to work out how many cows and how many sheep there are in each field. A game for 2 or more players with a pack of cards. Practise your skills of addition, subtraction, multiplication and division to hit the target score. If you have only four weights, where could you place them in order to balance this equaliser? 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?

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? This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo! 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 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 To avoid losing think of another very well known game where the patterns of play are similar. A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels? A game for 2 people. Take turns joining two dots, until your opponent is unable to move. Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . . 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 This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning? There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . . A game for 2 players Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star. A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly? On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible? Show that all pentagonal numbers are one third of a triangular number. Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet? Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow? A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . . Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling? This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . . This task depends on groups working collaboratively, discussing and reasoning to agree a final product. A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than Can you find a rule which relates triangular numbers to square numbers? A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be? Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal. What can you see? What do you notice? What questions can you ask? The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN? A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ? ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR? Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung. A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn. Can you find a rule which connects consecutive triangular numbers? Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them? Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end. Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum. In a three-dimensional version of noughts and crosses, how many winning lines can you make? Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians? Two intersecting circles have a common chord AB. The point C moves on the circumference of the circle C1. The straight lines CA and CB meet the circle C2 at E and F respectively. As the point C. . . . A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square? A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel? Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100? Simple additions can lead to intriguing results... A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Ward Identity and Homes’ Law in a Holographic Superconductor with Momentum Relaxation We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes’ law and Uemura’s law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes’ law does not hold but the Uemura’s law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected. a]Keun-Young Kim, a,b]Kyung Kiu Kim, c]and Miok Park Ward Identity and Homes’ Law in a Holographic Superconductor with Momentum Relaxation School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea Department of Physics, College of Science, Yonsei University, Seoul 120-749, Korea School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea Keywords: Gauge/Gravity duality, Holographic superconductor, Homes’ law - 1 Introduction - 2 AC conductivities: holographic model and method - 3 Conductivities with a neutral scalar hair instability - 4 Ward identities: constraints between conductivities - 5 Homes’ law and Uemura’s law - 6 Conclusion and discussions - A Two-point functions related to the real scalar operator Holographic methods have provided novel and effective tools to study strongly correlated systems [1, 2, 3, 4] and they have been applied to many condensed matter problems. In particular, holographic understanding of high superconductor is one of the important issues. After the first holographic superconductor model proposed by Hartnoll, Herzog, and Horowitz (HHH)111The HHH model is a class of Einstein-Maxwell-complex scalar action with negative cosmological constant. [5, 6], there have been extensive development and extension of the model. For reviews and references, we refer to [2, 3, 7, 8]. The HHH model is a translationally invariant system with finite charge density. Therefore, it cannot relax momentum and exhibits an infinite electric DC conductivity even in normal phase not only in superconducting phase. To construct more realistic superconductor models, a few methods incorporating momentum relaxation were proposed. One way of including momentum relaxation is to break translational invariance explicitly by imposing inhomogeneous (spatially modulated) boundary conditions on a bulk field [9, 10, 11, 12, 13]. Massive gravity models [14, 15, 16, 17, 18, 19, 20] give some gravitons mass terms, which breaks bulk diffeomorphism and translation invariance in the boundary field theory. Holographic Q-lattice models [21, 22, 23, 24, 25] take advantage of a global symmetry of the bulk theory. For example, a global phase of a complex scalar plays a role of breaking translational invariance. Models with massless scalar fields linear in spatial coordinate [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] utilize the shift symmetry. Some models with a Bianchi VII symmetry are dual to helical lattices [37, 38, 39]. Based on these models, holographic superconductor incorporating momentum relaxation have been developed [40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. In this paper, we study the HHH holographic superconductor model with massless scalar fields linear in spatial coordinate , where the strength of momentum relaxation is identified with the proportionality constant to spatial coordinate. The property of the normal phase of this model such as thermodynamics and transport coefficients were studied in [30, 26, 50, 28, 29, 51, 52]. The superconducting phase was analysed in [43, 44]. In particular, optical electric, thermoelectric and thermal conductivities of the model have been extensively studied in [29, 44, 51, 52]. Building on them, we further investigate interesting properties related to conductivities and momentum relaxation. There are three issues that we want to address in this paper: (1) conductivities with a neutral scalar hair instability, (2) Ward identities: constraints between conductivities, (3) Homes’ law and Uemura’s law. We explain each issue in the following. (1) In a holographic superconductor model of a Einstein-Maxwell-scalar action [6, 7], a superconducting state is characterized by the formation of a complex scalar hair below some critical temperature. In essence, the complex scalar is turned on by coupling between the maxwell field and complex scalar through the covariant derivative. Interestingly, it was also observed [6, 7] that a different mechanism for the instability forming neutral scalar hair222A neutral scalar may arise from a top-down setting [53, 54]. is possible. This instability was not associated with superconductivity because it does not break a symmetry, but at most breaks a symmetry . Therefore, in this system with a neutral scalar hair, it is natural to expect that DC electric conductivity will be finite contrary to the case with a complex scalar hair (superconductor). However, to our knowledge, it has not been checked yet. In the early models without momentum relaxation, this question is not well posed since electric DC conductivity is always infinite due to translation invariance and finite density. In this paper, in a model with momentum relaxation, we show that DC electric conductivity is indeed finite with a neutral scalar hair. (2) It was shown [3, 55], in normal phase without momentum relaxation, there are two constraints relating three transport coefficients: electric conductivity(), thermoelectric conductivity() and thermal conductivity(). The constraints can be derived by the Ward identity regarding diffeomorphism. Thanks to these two constraints, and can be obtained algebraically once is computed numerically. This is why only is presented in the literature . In our model, there is an extra field, a massless scalar for momentum relaxation, and it turns out there are three Ward identities of six two-point functions: , and and three more two-point functions related to the operator dual to a scalar field. Therefore, the information of alone cannot determine and . If we know three two-point functions then the Ward identities enable us to compute the other three two-point functions. In this paper, following the method in , we first derive the Ward identities for two-point functions analytically from field theory perspective. Next, we confirm them numerically from holographic perspective. This confirmation of the Ward identities also demonstrates the faithfulness of our numerical method. (3) Homes’ law and Uemura’s law are empirical and material independent universal laws for high-temperature and some conventional superconductors [56, 57]. The law states that, for various superconductors, there is a universal material independent relation between the superfluid density () at near zero temperature and the transition temperature () multiplied by the electric DC conductivity () in the normal state right above the transition temperature . where , and are scaled to be dimensionless, and is a dimensionless universal constant: or . They are computed in from the experimental data in [56, 57]. For in-plane high superconductors and clean BCS superconductors . For c-axis high superconductors and BCS superconductors in the dirty limit . Notice that momentum relaxation is essential here because without momentum relaxation is infinite. There is another similar universal relation, Uemura’s law, which holds only for underdoped cuprates [56, 57]: where is another universal constant. In the context of holography Homes’ law was studied in [58, 45]. It was motivated by holographic bound of the ratio of shear viscosity to entropy density () in strongly correlated plasma and its understanding in terms of quantum criticality or Planckian dissipation ,where the time scale of dissipation is shortest possible. Since Homes’ law also may arise in systems of Planckian dissipation there is a good chance to find universal physics in condensed matter system as well as in quark-gluon plasma. In Homes’ law was observed in a holographic superconductor model in a helical lattice for some restricted parameter regime of momentum relaxation, while Uemura’s law did not hold in that model. However, physic behind Homes’ law in this model has not been clearly understood yet. For further understanding on Homes’ law, in this paper, we have checked Homes’ law and Uemuras’ law in our holographic superconductor model. We find that Homes’ law does not hold but Uemura’s law holds at small momentum relaxation region, related to coherent metal regime. This paper is organised as follows. In section 2, we introduce our holographic superconductor model incorporating momentum relaxation by massless real scalar fields. The equilibrium state solutions and the method to compute AC conductivities are briefly reviewed. In section 3, the conductivities with a neutral scalar instability are computed and compared with the ones with a complex hair instability. In section 4, we first derived Ward identities giving constraints between conductivities analytically from field theory perspective. These identities are confirmed numerically by holographic method. In section 5, after analysing conductivities at small frequency, we discuss the Home’s law and Uemura’s law in our model. In section 6 we conclude. 2 AC conductivities: holographic model and method 2.1 Equilibrium state In this section we briefly review the holographic superconductor model we study, referring to [26, 29, 51, 52, 61] for more complete and detailed analysis. We consider the action333The complete action includes also the Gibbons Hawking term and some boundary terms for holographic renormalization, which are explained in [26, 29, 51, 52, 61] in more detail. where and is the holographic direction. is the Ricci scalar and is the cosmological constant with the AdS radius . We have included the field strength for a gauge field , the complex scalar field with mass , two massless scalar fields, . The covariant derivative is defined by with the charge of the complex scalar field. The action (2.1) yields equations of motion for which we make the following ansatz: In the gauge field, encodes a finite chemical potential or charge density and plays a role of an external magnetic field. is dual to a superconducting phase order parameter, condensate. Near boundary , with two undetermined coefficients and , which are identified with the source and condensate respectively. The dimension of the condensate is related to the bulk mass of the complex scalar by . In this paper, we take and to perform numerical analysis. is introduced to give momentum relaxation effect where is the parameter for the strength of momentum relaxation. For , the model becomes the original holographic superconductor proposed by Hartnoll, Herzog, and Horowitz (HHH) [5, 6]. First, if (no condensate), the solution corresponds to a normal state and its analytic formula is given by where is the location of the black brane horizon defined by , , and is interpreted as charge density. It is the dyonic black brane modified by due to . The thermodynamics and transport coefficients(electric, thermoelectric, and thermal conductivity) of this system was analysed in detail in . In the case without magnetic field, see . Next, if , the solution corresponds to a superconducting state with finite condensate and its analytic formula is not available444A nonzero induces a nonzero , which changes the definition of ‘time’ at the boundary so field theory quantities should be defined accordingly.. For , the solutions are numerically obtained in for and in for . For example we display numerical solutions for some cases in Figure 1, where we set and plot dimensionless quantities scaled by : , , and . 2.2 AC conductivities The purpose of this subsection is to briefly describe the essential points of a method to compute the AC thermo-electric conductivities. For more details and clarification regarding our model at , see [52, 51] for normal phase and for superconducting phase. At see for normal phase. In order to study transport phenomena holographically, we introduce small bulk fluctuations around the background obtained in the previous subsection. For example, to compute electric, thermoelectric, and thermal conductivities it is enough to consider where for and is enough for thanks to a rotational symmetry in space. For the sake of illustration of our method, we consider the case for and refer to for . In momentum space, the linearized equations of motion around the background are555For case, the bulk fluctuations to direction should be turned on so the number of equations of motion are doubled too. Near boundary () the asymptotic solutions are The on-shell quadratic action in momentum space reads Here is the coefficient of when is expanded near boundary and is charge density. The index in and are suppressed. The remaining task for reading off the retarded Green’s function is to express in terms of . It can be done by the following procedure. First let us denote small fluctuations in momentum space by collectively. i.e. Near black brane horizon (), solutions may be expanded as which corresponds to incoming boundary conditions for the retarded Green’s function and is some integer depending on specific fields, . The leading terms are only free parameters and the higher order coefficients such as are determined by the equations of motion. A general choice of can be written as a linear combination of independent basis , (), i.e. . For example, can be chosen as Every yields a solution , which is expanded near boundary as where and the leading terms are the sources of -th solutions and are the corresponding operator expectation values. and can be regarded as regular matrices of order , where is for row index and is for column index. A general solution may be constructed from a basis solution set : with arbitrary constants ’s. For a given , we always can find 666There is one subtlety in our procedure. The matrix of solutions with incoming boundary condition are not invertible and we need to add some constant solutions, which is related to a residual gauge fixing . so the corresponding response may be expressed in terms of the sources 3 Conductivities with a neutral scalar hair instability By the numerical method reviewed in the previous subsection, the electric, thermoelectric and thermal conductivities of the model (2.1) have been computed in various cases [29, 52, 44]. As an example, in Figure 2, we show the results for , which is reproduced here for easy comparison with new results in this paper. Figure 2 shows AC electric conductivity (), thermoelectric conductivity (), and thermal conductivity () for and at different temperatures. The colors of curves represent the temperature ratio, , where is the critical temperature of metal/superconductor phase transition. for dotted, red, orange,green, and blue curves respectively. In particular, the dotted curve is the case above and the red curve corresponds to the critical temperature. The first row is the real part and the second row is the imaginary part of conductivities. One feature we want to focus on in Figure 2 is pole in Im below the critical temperature. There is no pole above the critical temperature. By the Kramers-Kronig relation, the pole in Im implies the existence of the delta function at in Re. It means that in superconducting phase the DC conductivity is infinite while in normal phase the DC conductivity is finite due to momentum relaxation. Unlike the studies in , here we set . Between finite and zero , there is a qualitative difference in the instability of a Reissner-Nordstrom AdS black hole . The origin of the superconductor (or superfluidity) instability responsible for the complex scalar hair may be understood as the coupling of the charged scalar to the charge of the black hole through the covariant derivative . In other words, the effective mass of defined by can be compared with the Breitenlohner-Freedman (BF) bound. The BF bound for AdS is . The effective mass may be sufficiently negative near the horizon to destabilize the scalar field since becomes bigger at low temperature777As the temperature of a charged black hole is decreased, develops a double zero at the horizon.. Based on this argument one may expect that when the instability would turn off. However, it turns out that a Reissner-Nordstrom AdS black hole may still be unstable to forming neutral scalar hair, if is a little bit bigger than the BF bound for AdS. It can be understood by the near horizon geometry of an extremal Reissner-Nordstrom AdS black hole. It is AdS R so scalars above the BF bound for AdS may be below the bound for AdS. These two instability conditions can be summrized by one ineqaulity which reproduces the result for in Here, we see can be below the BF bound when . However, it was discussed in [6, 7] that the instability to forming neutral scalar hair for is not associated with superconductivity because it does not break a symmetry, but at most breaks a symmetry . Therefore, it would be interesting to see if the DC conductivity is infinite or not in the background with a neutral scalar hair.888We thank Sang-Jin Sin for suggesting this. Without momentum relaxation () this question is not well posed since the DC conductivity is always infinite with or without a neutral scalar hair due to translation invariance and finite density. Now we have a model with momentum relaxation (), we can address this issue properly. To have an instability at we choose the same parameters as Figure 2: and . For , , which is below the BF bound (3.1). Figure 3 shows our numerical results of conductivites, where all temperatures are below : for red, orange,green, and blue curves respectively. A main difference of Figure 3 from Figure 2 is the disappearance of pole in Im below . It confirms that the neutral scalar hair has nothing to do with superconductivity as expected. In Figure 3 it is not easy to see the conductivities in small regime, so we zoom in there in Figure 4. Contrary to the conductivity of normal component in superconducting phase, the DC electric conductivity is not so sensitive to temperature and increases as temperature decreases, which is the property of metal. The thermoelectric and thermal conductivities decrease as temperature increases except a small increase of thermoelectric conductivity near the critical temperature. As a cross check, we have also computed these DC conductivities analytically by using the black hole horizon data according to the method developed in . Since there is no singular behavior in the conductivities as we may regard the real scalar field here as the dilaton in and the conductivities read where , and are the entropy density, charge density and temperature in the dual field theory. They are given by , and . The analytic values are designated by the red dots in Figure 4 and they agree to the numerical values very well. For a special case with , in Figure 5, we see that , different from superconducting case ( shown in ), but , same as superconducting case. 3.1 Superfluid density with a complex scalar hair We have found that for there is no pole in Im, of which strength corresponds to superfluid density. To understand it better, we derive an expression for superfluid density for . Let us start with the Maxwell equation, Once we assume that all fields depend on and and the fluctuations are allowed only for the -direction, the -component of the Maxwell equation reads The integration of (3.5) from horizon to boundary gives the boundary current By hydrodynamic expansion for small , it turns out that the first term and the second term goes to zero as while the last term goes to constant. Here we used the expansions of the fileds near horizon where is a constant residual gauge parameter fixing , and , and can be expanded as With the following source-vanishing-boundary conditions except , the current (3.6) can be interpreted as This shows how the hairy configuration contributes to . If , vanishes, which confirms our numerical analysis. 4 Ward identities: constraints between conductivities In this section, we first analytically derive the Ward identities regarding diffeomorphism from field theory perspective. It gives constraints between conductivities() and two-point functions related to the operator dual to the real scalar field. Next, these identities are confirmed by computing all two-point functions numerically from holographic perspective. 4.1 Analytic derivation: field theory To derive the Ward identities, we closely follow the procedure in 999See for a holographic derivation. and extend the results therein to the case with real and complex scalar fields, which are and in (4.1). Our final results are (4.44)-(4.45) and (4.56)-(4.58). Let us start with a generating functional for Euclidean time ordered correlation functions: where , , , , and are the non-dynamical external sources of the stress-energy tensor , current , real scalar operators , and complex operators respectively. We define the one-point functions by functional derivatives of : where these expectation values are not tensors but tensor densities under diffeomorphism. One more functional derivatives acting on one-pint functions give us Euclidean time ordered two-point functions: We consider the generating functional invariant under diffeomorphism, , and the variation of the fields can be expressed in terms of a Lie derivative with respect to the vector field For diffeomorphism invariance, the variation of should vanish: which, after integration by parts, yields the Ward identity for one-point functions regarding diffeomorphism. By taking a derivative of (4.18) with respect to either , , or , we obtain the Ward identities for the two-point functions:

Archimedes: Biography, Scientific Accomplishments, Inventions, & Principle Born in Sicily, Magna Graecia, Archimedes was a critically acclaimed Greek scientist who attained so many feats in a host of scientific disciplines like astronomy, physics, mathematics, mechanics and engineering. He is famed for coming out with the Archimedes principle upon which fluid dynamics as a discipline partly rests on. Another spectacular accomplishment of him came in his accurate approximation of the mathematical constant pi (π), which he calculated to fall between 3 10/77 and 3 1/7. The fact that many of his inventions – like the Archimedes screw – are still in use to this day makes him an even more remarkable scientist of the classical age. Here is a complete biography, scientific contributions and major achievements of Archimedes of Syracuse (c. 287 bc – c. 212 bc), one of the most influential scientists in world history and a colossal Greek mathematician of the highest order. Birth and early life Archimedes’ place of birth, Syracuse, at the time was a buzzing city filled with many renowned scholars and artists. The city also benefited from its extensive trade ties with merchants from Greece, Phoenicia and Egypt. According to many historians and scholars in antiquity, Syracuse was at the time one of the most famous cities in the known world. Although not much is known about the early life of this great mathematician, many historians that came after him proposed that he most likely spent his early years in Egypt, particularly in the city of Alexandria. It was during this time that he invented a device known as the Archimedes’ screw (more on that later). The only thing we know about his family is that he was the son of Phidias, a renowned astronomer. One of his colleagues known as Heracleides is said to have written a biography about him. However, none of those biographies remain to this day. Archimedes and King Hieron II, the king of Syracuse What seems to be apparently clear is that he spent a huge chunk of his life in the Greek city-state of Syracuse on the island of Sicily. While in Sicily, he became a close friend of Hieron II (308-215 BC), the tyrant and later king of Syracuse. A number of accounts have stated that he helped Hieron II in his court. In one particular case he was able to calculate the proportion of silver and gold in a jewelry that had been gifted to King Hieron. This third century BC Sicilian-born scientist probably received much of his education in Alexandria in Egypt, where he studied geometry and astronomy from/with scholars and successors of Euclid. Alexandria at the time was perhaps the hub of the intellectual world as it housed a wide variety of scholars from different parts of the world. Archimedes’ Greatest Contributions Archimedes, a Greek mathematician, inventor, and scholar of critical acclaim, is praised as one of the greatest mathematicians of all time. He is praised due to his remarkable contributions to a host of disciplines such as mechanics, astronomy, geometry, arithmetic, and physics. Fictional stories about Archimedes Being an important figure from the classical age, his story and achievements often suffers a lot as the line between myth and factual detail gets blurred. “Heurēka!” (“Eureka! Eureka!”) Did Archimedes actually jump out of his bath and run out naked into the streets shouting “Heurēka!” (“I have found it!”)? Many claim that the story was anything but an exaggeration. That probably never happened when Archimedes conceived the principle of buoyant force. Like many scientists that make a breakthrough, Archimedes most likely would have been over the moon over his discovery of how to determine the proportion of silver and gold; however, it is unlikely that he stormed out of the bathroom stark naked into the streets. Similarly, it is highly doubtful that the Syracuse-based mathematician uttered the sentence: “Give me a lever and a place to stand and I will move the Earth”. To this day, there is no historical evidence to support such claim. Such a statement is demonstrative of the immense power of levers. Another made-up story is the one which says that he used a set of mirrors to reflect the sun’s rays to burn a number of Roman ships during the siege of Syracuse in 212 BC. Archimedes did indeed contribute immensely in producing war machines to defend Syracuse during the siege; however, he did not deploy some sort of sun ray-killing machines. Finally, the story that Archimedes met his end at the hands of a Roman soldier after he refused to abandon a mathematical diagram that he had been working on seems a bit too far-fetched. Although history has come to judge Archimedes as one of the greatest mathematician and inventors of all time, Archimedes never thought highly of some of his inventions. According to Plutarch, it was for the above reason why the mathematicians chose not leave written records of those works. However, he took an enormous amount of pride in his work in On the sphere and Cylinder, which shows the mathematical relationship between the volume of a sphere and the cylinder in which it is inscribed. He purposely instructed that his tomb carry image of a sphere inscribed in a cylinder. Archimedes thus was more pleased by his works in the development of mathematical theorems and proofs than his mechanical inventions. During his time, however, his fame predominantly came as a result of his mechanical inventions. READ MORE: 12 Greatest Ancient Greek Inventions Major achievements of Archimedes For someone to be described by ancient historians and modern historians alike as the greatest mathematician of all time means that Archimedes was truly indeed a gifted-mathematician. Below is a quick presentation of the major achievements of Archimedes: Archimedes’ Law of the Lever To demonstrate just how powerful levers can be, there have been some historians that claimed that the statement “Give me a lever and a place to stand, and I will move the world” was uttered by Archimedes. Regardless of whether he uttered it or not, the fact is that Archimedes made tremendous contributions to lever technology. He discovered that the same or even more work could be done when trade-offs are made between force and distance. Area, surface area and volume of sphere He used a method of integration to calculate the areas, surface areas and volumes of spheres and other shapes. His works are credited with laying the pillar for calculus, which would later be improved upon by modern mathematicians like Leibniz, Newton and Kepler. Nine of Archimedes’ treatise survived, including On the Sphere and Cylinder which showed that the surface area of any sphere of radius (r) is four times the value of the greatest circle. This theorem, which is contained in the treatise On the Sphere and Cylinder, can be expressed mathematically as S = 4πr2 Also in the work, the mathematician showed that the volume of the circumscribing cylinder is 2πr3 His mathematical theorem showed that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed. This can be expressed as V = 4/3πr3 Approximation of the value of pi (π) Another significant contribution of Archimedes to mathematics was the value of pi (π). The mathematician calculated that pi fell between 3 10/71 and 3 1/7. He is said to have used a very innovative technique to get the value. That technique was used until the 15th century CE. In addition to calculating pi, he also accurately approximated square roots. The mathematician also developed his system of showing large numbers mathematically. The Archimedes’ principle In addition to developing a host of theorems in mechanics, including in works related to the centre of gravity of solids and plane figures (in the treatise On Plane Equilibriums), he developed a law of buoyancy which is commonly known as Archimedes’ principle. The principle covers the weight of a body that is immersed in a liquid. Archimedes’ Principle, also known as the Law of Buoyancy, states that an object immersed in fluid will have an upward force equal to the weight of the displaced fluid. Archimedes’ Principle allows for the calculation of the volume or average density of an object immersed in a fluid. The principle is famed for helping measure the volume of irregular objects, such as jewelry, cutlery, and many others. In addition to that, it allows scantiest to understand how objects behave when immersed in any fluid. With Archimedes principle, one can explain how hot air balloons stay in the air, or how ships float. Applications of the Archimedes principle is vast and wide, including in disciplines like entomology, engineering, geology, medicine, dentistry, and engineering, among others. For example, in the medical field, the principle comes in useful when determining the densities of teeth and bones. The Archimedes screw The screw that Archimedes invented came to be known as the Archimedes’ screw. The screw, which is enclosed in a pipe, is used to lift water from one level to another. The Archimedes’ screw came in very handy when seamen removed water from their ship’s hull. He touched a bit on integral calculus His contributions to geometry were unparalleled. He even went as far as anticipating the method of exhaustion which is rudimentary form of modern-day calculus. Archimedes worked on finding the volumes of the segments of solids that emerged from the revolution of shapes such as circles, ellipse, hyperbola or parabola. This work of his, which is contained in the treatise On Conoids and Spheroids, fall under calculus in our modern time. His use of method of exhaustion (modern calculus) allowed him to come out with more mathematical theorems. The compound pulley This invention of Archimedes ended up bringing the scientist immense praise as it revolutionized the way large objects were lifted. According to Plutarch, the mathematician demonstrated this invention of his to King Hieron by effortless pulling a ship in a straight line. Archimedes’ passion and associations with other mathematicians of his era Archimedes was said to be completely devoted to the pursuit of scientific knowledge, particularly mathematics and mechanics. He often did this at a huge expense to his health as he neglected food and drink and basic personal hygiene so he could focus solely on his experiments and studies. Many of the critically acclaimed works of Archimedes came via his correspondence with mathematicians and colleagues of his in Alexandria, including Eratosthenes of Cyrene and Conon of Samos. A defense contractor for the city of Syracuse His knowledge in engineering and mechanics enabled him to come out with a number of inventions that he used to defend the city of Syracuse against the Romans in 213 BC. According to Plutarch, his close friend King Hieron was the one who encouraged him to build those war machines. His engines of war gave him more acclaim that his mathematical theorems. That comes as no surprise considering the fact that it was a BC era, an era when warfare was an extremely important. His war machines received enormous praise for being able to keep the Roman forces at bay for almost two years. Many of Archimedes works were theoretical in nature. Much of his work in mathematics was perhaps fanned on by his passion for mechanics. It’s been stated that his works in theoretical mechanics and hydrostatics enabled him to come out a number of mathematical theorems. As it is seen in his treatise Method Concerning Mechanical Theoremis, Archimedes used much of what he knew in mechanics to advance his knowledge in mathematics. Famous works of Archimedes that survived Many of his theorems in mechanics, including those on the center of gravity of plane figures, are contained in the treatise On Plane Equilibriums. In the treatise Quadrature of the parabola, Archimedes calculates the area of a segment of a parabola that had been cut off by any chord. Archimedes is credited with producing many works. Although many of them were lost, nine of his treaties survived. They are as follows: - On Floating Bodies - On Spirals - On Conoids and Spheroids - Measurement of a circle - The Sandreckoner - Quadrature of the Parabola - On Plane Equilibriums - On the Sphere and cylinder In the treatise The Sandrekoner, the mathematician shows how a number system could accommodate astonishing numbers of up to 8 x 1063. He goes on to say that with that number system, he could count every grain of sand which the universe could hold. He was one of the few scientists of his era that actually thrived to put his mathematical theorems into practice. This is one of the reasons why his works had/and continue to have a huge impact in the world of science. Archimedes favored the viewing his scientific experiments, including engineering problems, using the lens of mathematical theorems. His passion for mathematics is what led him to deploy mechanical experiments to gain greater understanding of mathematical theorems. The mathematician and inventor Archimedes of Syracuse died in 212 or 211 BC in Syracuse on the island of Sicily. He was killed during Roman siege (214-212 BC) on the city of Syracuse (during the Second Punic War). He was most likely in his mid-70s. Out of the strong respect and admiration General Marcus Claudius Marcellus had for Archimedes, he ordered that the scientist be giving a burial with honor. Marcellus had hoped to capture Archimedes alive so that he could perhaps benefit from the genius of the scientist. According to the Greek historian Plutarch, Archimedes was busy going about with some very important works in mathematics when a Roman soldier struck him down. The historian goes on to say that Archimedes, in spite of the order from the soldier, refused to halt his work. Another version of how Archimedes died (also from Plutarch) states that Roman soldiers wrongly thought that the mathematician carried on him a bag that contained gold. Unbeknownst to those soldiers, the bag actually contained mathematical instruments, spheres and angles that the mathematician was sending to Marcellus. Read More: 10 Most Famous Ancient Greeks and their Achievements More Archimedes Facts The story of Archimedes running naked into the crowded streets of Syracuse shouting “Eureka! Eureka!” was first written down by a Roman scholar and architect named Vitruvius. According to the accounts, the scientist had discovered the principle of buoyancy in order to help him assess whether King Hieron’s golden crown was pure gold or not. According to Greek historian Plutarch, Archimedes was probably related to Heiron II, the king of Syracuse. This point could be supported by the fact that he dedicated his treatise The Sandreckoner to Gelon, the son of Hieron. While in Alexandria, Egypt, he studied with a number of followers of the famous mathematician Euclid. He often corresponded with fellow mathematicians Conon of Samos and Eratosthenes of Cyrene, both lived in Alexandria. After realizing that some of his friends began to take credit for his mathematical proofs, he desisted from including proofs to theorems in his correspondence with mathematicians in Alexandria. Compared to other scientists and mathematicians of his, Archimedes has quite a lot more anecdotal details about his life. Much of what we know about the life and works of Archimedes comes from the accounts of Plutarch (c. 46-119 CE), the Greek historian and biography. Other sources came from the likes of Livy and other Greek historians. The roman general by the name of Marcus Claudius Marcellus – was saddened by his death as he intended to bring Archimedes into his service. The general was impressed by the machines that Archimedes built to defend the city from the Romans Following Rome’s siege of Syracuse in 212 BC, Roman General Marcellus is believed to have taken two spheres made by Archimedes back to Rome. The spheres were basically about the various planetary objects as well as their motions. Like many scientists and geniuses that came after Archimedes, the likes of Galileo and Newton were big admirers of the Sicilian scientist and mechanical engineer. Following his death, his mathematical works and treatise did not gain large acclaim as compared to the ones of the mathematician and geometer Euclid. Regardless, there were still a good number of mathematicians in Alexandria that were devout followers of his works. Those scholars included Theon, Pappus, and Heron. When the Roman statesman and scholar Cicero travelled to Sicily in 75 BC he searched for the tomb of Archimedes. Cicero found the tomb to be covered on all side by weeds.

Sunday, October 18, 2009 What Language Does Our Universe "Speak"? Many profound physicist have come to think of physics at the very tiniest length scale (the Planck length) as a computer or information processor. To name a few: John A. Wheeler, Richard Feynman, Roger Penrose, Gerard 't Hooft. Wheeler expressed this view as "it from bit". One of the main reasons for this view is the realization that physics at that scale will have to be discrete. If not, it becomes very hard to reconcile relativity and quantum mechanics into one theory. In the continuous domain calculation simply blow up: they cannot be re-normalized. In addition to that, the uncertainty principle of quantum mechanics demands that we can not even pinpoint things down to such precision without creating a black hole which would immediately render any measurement at a scale smaller than its horizon impossible....So these physicist think that physics at that scale is some sort of cellular automaton. Around the end of every century some people seem the need to make the rather absurd claim that science is coming to an end (I believe we have barely started, but anyway). This century this view is expressed in the book: "The End Of Science: Facing The Limits Of Knowledge In The Twilight Of The Scientific Age" by John Horgan. He argues that there are four recent theories that have shown the fundamental limitations of science: 1. Relativity: anything inside the horizon of a black hole will never get out. So we cannot study the inside of a black hole. 2. Quantum Mechanics: the world is irreducibly random. 3. Chaos: The dynamics of many real physical phenomenon displays extreme sensitivity to initial conditions. 4. Complexity Theory: Godel's theorem of incompleteness of formal systems. Let's see how these theories would fare in the face of a fundamental theory of "Physics as Computation" (PAC). I think the black hole issue is already close to being resolved. A quantum mechanical treatment of BHs will involve BH-radiation (or Hawking radiation). As such, in-falling matter will cause disturbances on the surface of the BH-horizon that encodes the information of the in-falling matter and which will eventually be radiated out again. No information is lost in the process. (Every BH will eventually die in an explosion that is more violent than the most energetic supernova, but it takes a while..) For the observer that stays outside the BH, the BH horizon is the edge of the universe in a very real sense. It will see his colleague that falls into the BH freeze onto the horizon, get disintegrated and eventually be radiated out again in bits and pieces. For the in-falling observer the edge of the universe is not the BH horizon, but a singularity at the center of the BH. In this case we have to deal with a singularity but it seems evident to me that the final PAC theory will describe that singularity not as an infinitely dense point but rather a sensible finite object. How the irreducibility of quantum mechanics may be resolved in terms of a cellular automaton was described in my previous blog on "Quantum Mechanics is not the Final Theory". The phenomenon of chaos in nonlinear dynamical systems makes claims on unpredictability of a more every day nature: for instance the weather patterns are unpredictable because a small error in the initial conditions may result in large differences a few days later (except in California where we don't need weather forecasting). The canonical example is this: x[t+1]=2*x[t] mod 1. This means that at every iteration we move all digits one decimal place to the left and set the number to the left of the dot to 0: 0.12345... Finally Godel's theorem. It says that within any sufficiently complex formal system there will be true theorems that cannot be proved. I am still thinking about these issues, but I seem to have an issue with the notion of "a true theorem". True can only acquire meaning as an interpretation of the formal system (say mapping sequences to mathematical or physical "truths"). But mathematics is itself a formal system. Truth does not exist outside any axiomatic system and the interpretation that Godel's theorem shows that truth is bigger than formal reasoning just doesn't sit well with me. Anyway, some future blogs will unquestionably be devoted to these deep issues. It will be very interesting to be able to answer the question: "what is the complexity class of the sequences generated by the cellular automaton that governs our universe". Or phrased more informally: "What language does our universe speak". Here is my prediction: Dutch ;-) (or maybe a language of the same complexity). It seems that Dutch is more complex than context-free languages due to cross-referencing but still decidable in polynomial time. It represents a possible level of complexity where things are not too regular but also not too unwieldy. Anyway, my prediction here should be taken with a huge grain of salt of course. Soooo, the universe is a huge computer that is computing "something". It is our task as scientists to figure what and how it is computing. Actually, we already know the answer: 42 ;-). But what was the original question? Let's leave that to religion.

Calculate The Bootstrap Standard Error Of The 75th Percentile Generated Wed, 05 Oct 2016 18:12:46 GMT by s_hv997 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection For our purposes here, these will be the 2.5th and 97.5th percentile, though generically these are the a/2 and 1-a/2 percentiles. We will be using the hsb2 dataset for all of the examples on this page. If we were calculating 95% confidence limits on the mean, SPSS could tell us that those limits were 61.01 and 68.19. check my blog Even if the population is not normal, the Central Limit Theorem tells us that the sampling distribution will be at least approximately normal, so we don't worry too much. Notice that it has a range of about 60 milliseconds, with a mean of about 65 milliseconds (the median was 62). Your cache administrator is webmaster. The median is not as well behaved as the mean relative to the central limit theorem, which does not apply to medians. Bootstrap Percentile Confidence Interval install.packages("boot") library(boot) hsb2<-read.table("http://www.ats.ucla.edu/stat/data/hsb2.csv", sep=",", header=T) Using the boot commandThe boot command executes the resampling of your dataset and calculation of your statistic(s) of interest on these samples. Preview this book » What people are saying-Write a reviewWe haven't found any reviews in the usual places.Selected pagesPage 13Title PageTable of ContentsIndexReferencesContentsChapter 1 Graphical Methods 1 Chapter 2 Regression 23 Chapter 3 The system returned: (22) Invalid argument The remote host or network may be down. dch: David C. However, there are two important features of this approach. How to cite this page Report an error on this page or leave a comment The content of this web site should not be construed as an endorsement of any particular The percentile method would take these to be the upper and lower cutoffs for the 95% confidence interval. The Bootstrap Method Of Constructing Confidence Intervals Can Be Used To Estimate To get the standard error of the median, we have to have the empirical standard deviation of a bunch of medians. Howell University of Vermont [email protected]Welcome to the Institute for Digital Research and Education Institute for Digital Research and Education Home Help the Stat Consulting Group by giving Better intervals I could say the same things here that I said for confidence limits on the mean, with respect for corrections for bias and acceleration. Then do your resampling. The system returned: (22) Invalid argument The remote host or network may be down. But we need one more thing--we need the standard error of the median that corresponds to the standard error of the mean in the traditional formula. Bootstrap Confidence Interval Calculator Diagram of the bootstrapped t method: Original Sample: 2 2 3 4 5 5 5 6 7 9 --> Med Sample 1: 2 2 2 5 6 6 6 7 7 The second argument can be an index vector of the observations in your dataset to use or a frequency or weight vector that informs the sampling probabilities. This book is meant for graduate students in statistics, economics, policy analysis, and social sciences, especially, but certainly not exclusively, those interested in the challenges of economic development in the Third Percentile Method Confidence Intervals However, SPSS cannot give us limits on the median If we use our program to calculate confidence limits on the median, we obtain the following results. We will demonstrate a few of these techniques in this page and you can read more details at its CRAN package page. Bootstrap Percentile Confidence Interval Generated Wed, 05 Oct 2016 18:12:46 GMT by s_hv997 (squid/3.5.20) Bootstrap Confidence Interval Example I will use the data from the condition in which 5 comparison digits were first presented, and the test stimulus actually was one of those digits. From these samples, you can generate estimates of bias, bootstrap confidence intervals, or plots of your bootstrap replicates. http://iembra.org/confidence-interval/calculate-confidence-intervals-from-mean-and-standard-error.php Your cache administrator is webmaster. Generated Wed, 05 Oct 2016 18:12:46 GMT by s_hv997 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection His Handbook on Poverty and Inequality (with Shahidur Khandker) was published by the World Bank in 2009, his articles have appeared in over 30 scholarly journals, and he has written numerous Bootstrap Percentile Confidence Interval In R error t1* 0.6174493 -0.004455323 0.04169738While the printed output for bootcorr is brief, R saves additional information that can be listed:summary(bootcorr) Length Class Mode t0 1 -none- numeric t 500 -none- The system returned: (22) Invalid argument The remote host or network may be down. When the sampling distribution is perfectly symmetric, the percentile method is quick, easy to comprehend, and accurate. news There is nothing sacred about these values, but they should give you the general idea. This method computes . What Is The Mean Difference In Credit Card Debt Of The Two Groups In The Original Data? This is analogous to what we did with the mean. Med1a, Med1b, Med1c, etc--inner set of bootstrapped medians, which will be used to calculate t*1. Let B represent the number of bootstrap samples we calculate in the outer loop, and let b represent the number of bootstrap samples we draw based on each outer bootstrap samples. Generated Wed, 05 Oct 2016 18:12:46 GMT by s_hv997 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.5/ Connection The procedures for bootstrapping almost any statistic follow a very predictable pattern, and I am not going to repeat much of that here. This is based on a study by Sternberg (1966), in which he asked subjects to view a set of digits for a brief time (measured in milliseconds) and then see a Bootstrap Confidence Interval R Her major areas of interest are applied statistics, statistics and marketing, the analysis of living standards surveys, data mining, and model selection. Your cache administrator is webmaster. Your cache administrator is webmaster. Then our confidence limits become. Notice that these limits are somewhat narrower (57.5 and 65.0) and that they are slightly asymmetric around the sample median. Each new sample contains n elements. We would expect a positive skew because of the nature of the task. This is a book that can serve as a reference work, to be taken down from the shelf and perused from time to time. Additionally, the book will be useful to academics and practitioners who work closely with survey data. IDRE Research Technology Group High Performance Computing Statistical Computing GIS and Visualization High Performance Computing GIS Statistical Computing Hoffman2 Cluster Mapshare Classes Hoffman2 Account Application Visualization Conferences Hoffman2 Usage Statistics 3D It is not the standard error of the median. Please try the request again. Before using commands in the boot package, you must first download the package and load it in your workspace. The system returned: (22) Invalid argument The remote host or network may be down. Your cache administrator is webmaster. A fifth type, the studentized intervals, requires variances from each bootstrap sample. Dominique is a Fellow of the American Statistical Association.Jonathan Haughton (Ph.D. The system returned: (22) Invalid argument The remote host or network may be down. Having drawn B bootstrap samples, we sort them as before from low to high. A specialist in the areas of economic development, international trade, and taxation, and a prize-winning teacher, he has lectured, taught, or conducted research in over a score of countries on five We can illustrate the result of this method using an example that I have used elsewhere. We can obtain an estimate of that by taking the medians of our B samples, and simply calculating the standard deviation of that distribution. Harvard 1983) is Professor of Economics at Suffolk University, and Senior Economist at the Beacon Hill Institute for Public Policy, both in Boston. The only method that I have programmed as of the time of this original writing is "Lunneborg's" method.

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AP Calculus AB Exam Overview The AP calc AB exam is split into 4 sections. The table below breaks down how the exam is split up: |Section 1, Part A |Section 1, Part B |Section 2, Part A |Section 2, Part B How to Use an AP Calculus Practice Exam We suggest the following when using our AP calculus AB exam: - Find a Quiet Room & Set Aside Some Time: Make sure you are in a quiet environment and have set aside ample time to take the practice exam. You will need to focus when taking the exam. - Take the Exam Like You Would on Test Day: Do not use a calculator on sections you are not allowed. Do not cheat and check your notes. Take the exam like you would on the test day to see how you would score. - Utilize Answer Explanations: Use our detailed answer explanations to learn how to solve the problems and learn important concepts. - Answer Our Practice Sets: The practice sets were designed to help you study in a stress free environment. 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« AnteriorContinuar » globe to the latitude thereof, 25 deg. where fix the quadrant of altitude, and place it so as to make an angle with the meridian of 43 deg. in the horizon, and observe where the edge of the quadrant interseets the parallel of 30 deg. south latitude, for that is the place of the port B. Then count the number of degrees on the edge of the quadrant intersected between the two ports, and there will be found 73 deg. which, multiplied by 60, gives 4380 miles for the distance sailed. As the two ports are now known, let each be brought to the meridian, and observe the difference of longitude in the equator respectively, which will be found 50 deg. N. B. Had this problem been solved by loxodromics, or sailing on a rhumb, the difference of longitude would then have been 52 deg. 30 min. between the two ports, PROBLEM XLV. Given the difference of latitude and distance run, to find the difference of longitude and angle of the course. Example. Admit a ship sails from a port A, in latitude 50 deg. to another port B, in latitude 17 deg. 30 min. and her distance run be 2220 miles. Rectify the globe to the latitude of the place A, then the distance run, reduced to degrees, will make 37 deg. which are to be reckoned from the end of the quadrant lying over the port A, under the meridian ; then is the quadrant to be moved, till the 37 degrees coincide with the parallel of 17 deg. 30 min. north latitude ; then will the angle of the course appear in the arch of the horizon, intercepted between the quadrant and the meridian, which will be 32 deg. 40 min.; and by making a mark on the globe for the port B, and bringing the same to the meridian, you will observe what number of degrees pass under the meridian, which will be twenty, the difference of longitude required. PROBLEM XLVI. Given the difference of longitude and course, to find the difference of latitude and distance sailed. Example. Suppose a ship sails from A, in the latitude 51 deg. on a course making an angle with the meridian of 40 deg. till the difference of longitude be found just 20 degrees ; then rectifying the globe to the latitude of the port A, place the quadrant of altitude so as to make an angle of 40 deg. with the meridian; then observe at what point it intersects the meridian passing through the given longitude of the port B, and there make a mark to represent the said port; then the number of degrees intercepted between that and the port A, will be 28, which will give 1680 miles for the distance run: and the said mark for the port B, being brought to the meridian, will have its latitude there shewn to be 27 deg. 40 min. Given the course and distance sailed, to find the difference of longitude, and difference of latitude. Example. Suppose a ship sails 1800 miles from a port A, 51 deg. 15 min. south-west, on an angle of 45 deg. to another port B. Having rectified the globe to the port A, fix the quadrant of altitude over it in the zenith, and place it to the south-west point in the horizon ; then upon the edge of the quadrant under 30 deg. (equal to 1800 miles from the port A) is the port B; which bring to the meridian, and you will there see the latitude; and, at the same time, its longitude on the equator, in the point cut by the meridian. In all these cases, the ship is supposed to be kept upon the arch of a great circle, which is not difficult to be done, very nearly, by means of the globe, by frequently observing the latitude, measuring the distance sailed, and (when you can) finding the difference of longitude ; for one of these being given, the place and course of the ship is known at the same time; and therefore the preceding course may be altered, and rectified without any trouble, through the whole voyage, as often as such observations can be obtained, or it is found necessary. Now if any of these data are but of the quantity of four or five degrees, it will suffice for correcting the ship’s course by the globe, and carrying her directly to the intended fort, according to the following problem. PROBLEM XLVIII. To steer a ship upon the arch of a great circle by the given difference of latitude, or difference of longitude, or distance sailed in a given time. Admit a ship sails from a port A, to a very distant port Z, whose latitude and longitude are given, as well as its geographical bearing from A; then, First, having rectified the globe to the port A, lay the quadrant of altitude over the port Z, and draw thereby the arch of the great circle through A and Z; this will design the intended path or track of the ship. Secondly, having kept the ship upon the first given course for some time, suppose by an observation you find the latitude of the present place of the ship, this added to, or subducted from the latitude of the port A, will give the present latitude in the meridian ; to which bring the path of the ship, and the part therein, which lies under the new latitude, is the true place B of the ship in the great arch. To the latitude of B rectify the globe, and lay the quadrant over Z, and it will shew in the horizon the new course to be steered. Thirdly, suppose the ship to be steered upon this course, till her distance run be found 300 miles, or 5 degrees ; then, the globe being rectified to the place B in the zenith, laying the quadrant from thence over the great arch, make a mark at the 5th degree. from B, and that will be the present place of the ship, which call C; which being brought to the meridian, its latitude and longitude will be known. Then rec. tify the globe to the place C, and laying the quadrant from thence to Z, the new course to be steered will appear in the horizon. Fourthly, having steered some time upon course, suppose, by some means or other, you come to know the difference of longitude of the present place of the ship, and of any of the preceding places, C, B, A ; as B, for instance; then bring B to the meridian, and turn the globe about, till so many degrees of the equator pass under the meridian as are equal to the discovered difference of longitude; then the point of the great arch cut by the meridian is the present place D of the ship, to which the new course is to be found as before, And thus, by repeating these observations at proper will find future places, E,F,G, &c. in the great arch; and by rectifying the course at each, your ship will be conducted on the great circle, or the nearest way from the port A to Z, by the use of the globe only

Presentation on theme: "Introduction to Electrical Machines"— Presentation transcript: 1 Introduction to Electrical Machines Mechanical and Electrical Systems SKAA 2032Introduction to Electrical MachinesDr. Asrul Izam AzmiFaculty of Electrical EngineeringUniversiti Teknologi Malaysia 2 IntroductionOne of energy can be obtained from the other form with the help of converters.Converters that are used to continuously translate electrical input to mechanical output or vice versa are called electric machines.The process of translation is known as electromechanical energy conversion. 3 Conversion from mechanical to electrical: generator Electrical systemMechanical systemElectric Machinee, iT, nMotorEnergy flowGeneratorAn electrical machine is link between an electrical system and a mechanical system.Conversion from mechanical to electrical: generatorConversion from electrical to mechanical: motor 4 DC machines (generators or motors) if the electrical system is DC. Induction machineElectrical MachinesSynchronous machineAC machineMachines are called AC machines (generators or motors) if the electrical system is AC.DC machines (generators or motors) if the electrical system is DC. 5 Coupling magnetic fields Electrical systemMechanical systemCoupling magnetic fieldse, iT, nTwo electromagnetic phenomena in the electric machines:When a conductor moves in a magnetic field, voltage is induced in the conductor.When a current-carrying conductor is placed in a magnetic field, the conductor experiences a mechanical force. 6 AC Rotating Machines DC machine Induction machine Electrical Machines Synchronous machineAC machine 8 Basic Idea A motor uses magnets to create motion. The fundamental law of all magnets:Opposites attractLikes repel.Inside an electric motor, these attracting and repelling forces create rotational motion 9 Basic Idea Magnetic field of a straight conductor The magnetic field lines around a long wire which carries an electric current form concentric circles around the wire.Right hand rule-1 10 Basic Idea Magnetic field of a circular conductor Right hand rule-1 gives the direction of the magnetic field inside and outside a current-carrying loop. 11 Basic Idea Magnetic field of a coil of wire A solenoid is a long coil of wireThe field inside a solenoid can be very uniform and very strong.The field is similar to that of a bar magnet. 12 Basic Idea The use of soft metal increases the magnetic field strength Use right hand rule-2 / eye rule to determine direction of magnetic field in a coil 13 Basic Idea Fleming’s left hand rule for motors Don’t be confused with Fleming’s right hand rule for generator 14 Working Principle Elementary AC motor Consider a rotor → formed by permanent magnet.Consider a stator → formed by coil of conductor to create AC electromagnetic field 15 Working PrincipleAn AC Current flowing through conductors energize the magnets and develop N and S poles.The strength of electromagnets depends on current.First half cycle current flows in one direction.Second half cycle it flows in opposite direction. 16 Working PrincipleConsider the AC voltage at 0 degrees, then, no current will flow, and there is no magnetism.Supplied voltageInitial position of the rotor 17 Working PrincipleAs voltage increases, current starts to flow and electromagnets gain strength and North and South poles appear.The rotor magnet is pushed CW, and the rotor and motor starts to rotate. 18 Working PrincipleWhen voltage decreases, the current decreases also, the electromagnet loses the strength, and when V=0 there is no magnetism. 19 Working PrincipleNow, AC voltage builds up as part of the negative cycle.Then, current flows in opposite direction, and the magnets reverse polarity.Therefore, the CW rotation continues. 25 Practical AC MotorWe can see that the poles rotate around the circumference of the motor.The rotor, no matter how it is positioned at rest, will be locked-in with the magnetic field and will turn in one direction only.(Same rotation as the poles). 26 Induction Motor Most AC motors are induction motors Induction motors are favored due to their ruggedness (no brush), simplicity and cheap.90% of industrial motors are induction motor.Application(1-phase): washing machines, refrigerators, blenders, juice mixers, stereo turntables, etc.(2-phase) induction motors are used primarily as servomotors in a control system.(3-phase): pumps, compressors, paper mills, textile mills, etc. 27 Induction MotorThe single-phase induction motor is the most frequently used motor in the worldMost appliances, such as washing machines and refrigerators, use a single-phase induction machineHighly reliable and economical 28 Induction MotorFor industrial applications, the three-phase induction motor is used to drive machinesLarge three-phase induction motor. (Courtesy Siemens). 29 Construction of Induction Motor An induction motor is composed of a rotor, (armature)A stator containing windings connected to a poly-phase energy sourceThe pair of coils correspond to the phases of electrical energy available.Each pair connected in series creating opposite poles:1 pole for North and 1 pole for South. 30 Induction Motor Stator with (a) 2-phase and (b) 3-phase windings. Stator frame showing slots for windings. 31 Induction Motor It has a stator and a rotor like other type of motors. 2 different type of rotors:Squirrel-cage winding,Wound-rotorBoth three-phase and single-phase motors are widely used.Majority of the motors used by industry are squirrel-cage induction motorsA typical motor consists of two parts:An outside stationary stator having coils supplied with AC current to produce a rotating magnetic field,An inside rotor attached to the output shaft that is given a torque by the rotating field. 32 Squirrel-cage Rotor Rotor is from laminated iron core with slots. Metal (Aluminum) bars are molded in the slots instead of a winding.Two rings short circuits the bars.–Most of single phase induction motors have Squirrel-Cage rotor.One or 2 fans are attached to the shaft in the sides of rotor to cool the circuit. 33 Wound Rotor It is usually for large 3 phase induction motors. Rotor has a winding the same as stator and the end of each phase is connected to a slip ring.Three brushes contact the three slip-rings to three connected resistances (3-phase Y) for reduction of starting current and speed control.Compared to squirrel cage rotors, wound rotor motors are expensive and require maintenance of the slip rings and brushes, so it is not so common in industry applicationsWound rotor induction motor was the standard form for variable speed control before the advent of motor 34 SlipsIt is virtually impossible for the rotor of an AC induction motor to turn at the same speed as that of the rotating magnetic field.If the speed of the rotor were the same as that of the stator, no relative motion between them would exist, and there would be no induced EMF in the rotor.Without this induced EMF, there would be no interaction of fields to produce motion. The rotor must, therefore, rotate at some speed less than that of the stator if relative motion is to exist between the two.The percentage difference between the speed of the rotor and the speed of the rotating magnetic field is called slip.The smaller the percentage, the closer the rotor speed is to the rotating magnetic field speed. 35 Slips The synchronous speed (NS) of a motor is given by: where NS : synchronous speed or the rotating magnetic field (rpm)NR : rotor speed (rpm)The synchronous speed (NS) of a motor is given by:whereF : frequency of the rotor current (Hz)NP : number of poles 36 Example ProblemA two pole, 60 Hz AC induction motor has a full load speed of 3554 rpm. What is the percent slip at full load?NP 37 Torque Torque is a rotational force. The torque of an AC induction motor is dependent upon the strength of the interacting rotor and stator fields and the phase relationship between them.whereT : torqueK: constantΦ: stator magnetic flux (Wb)IR : rotor current (A)cos θR : power factor of rotor 38 Voltage and frequency induced in the rotor The voltage and frequency induced in the rotor both depend on the slip. They are given by the following equation.f2 = s fE2 = s Eoc (approx.)f2 = frequency of the voltage and current in the rotor [Hz]f = frequency of the source connected to the stator [Hz]s = slipE2 = voltage induced in the rotor at the slip sEoc = open-circuit voltage induced in the rotor when at rest [V] 39 Active Power in a Induction Motor Efficiency () =PoutputPinput 40 Example 1Calculate the synchronous speed of a 3-phase induction motor having 20 poles when it is connected to a 50 Hz source.120 fpns =120 x 5020=300 r/minSource frequency = 50 Hz, number of poles = 20Synchronous speed ns = 41 Example 2A 0.5 hp, 6-pole induction motor is excited by a 3-phase, 60 Hz source. If the full-load is 1140 r/min, calculate the slip.Source frequency = 60 Hz, number of poles = 6Full load/rotor speed = 1140 r/min120 fpns =120 x 606=1200 r/minSynchronous speed ns = 42 Induction Motor Slip speed: ns – nR = 1200 – 1140 = 60 r/min Slip: s = (ns - nR) / ns= 60/1200= 0.05 or 5% 43 Example 3A single phase, 4 poles induction motor gives the following data:Output 373 W ; 230 VFrequency : 50 Hz., Input current 2.9 APower factor: 0.71 ; Speed: 1410 r.p.m.a) Calculate the efficiency of the motorb) Determine the slip of the motor when deliveringthe rated output

- Open Access Finite time synchronization problems of delayed complex networks with stochastic perturbations © Cui et al.; licensee Springer. 2014 - Received: 10 November 2013 - Accepted: 20 March 2014 - Published: 19 May 2014 The paper is concerned with the finite-time synchronization problem of delayed complex networks with stochastic perturbations. Based on the finite-time stability theorem, some sufficient conditions are obtained to ensure finite-time synchronization for the Markovian jump complex networks with time delays and partially unknown transition rates. Finally, the effectiveness of the proposed method is demonstrated by illustrative examples. - stochastic perturbations - complex networks - time delays Over the past decades, the dynamics analysis of complex networks has witnessed rapidly growing research interests since the pioneering work of Watts and Strogatz . On the one hand, complex networks exist in our daily life with examples including the Internet, the World Wide Web, food webs, electric power grids, cellular and metabolic networks, etc. . And on the other hand, the dynamical behaviors of complex networks have found numerous applications in various fields such as physics, technology, and the life sciences . In fact, synchronization is a basic motion in nature that has been studied for a long time [4–6]. Recently, synchronization of complex networks has received increasing research attention [7–10]. It is important to note that most of the above research results on network synchronization are based on the asymptotic process of an infinite time. That is, network synchronization only occurs when the time tends to infinity. Thus in theory, it is impossible for a network to achieve synchronization in a limited time. However, in actual physical or engineering systems, complex networks usually achieve synchronization state in a limited time, which is finite-time synchronization. On the one hand, in the existing literature on finite time synchronization is not treated often. And on the other hand, finite-time synchronization is a very important bridge for a complex network to succeed in the actual application. In addition, more and more researchers begin to realize the important role of finite-time synchronization, and there are some related research results [11–14]. Time delays often occur in complex networks because of the limited speed of signals traveling through the links and the frequently delayed couplings in biological neural networks, gene regulatory networks, communication networks, and electrical power grids [9, 10]. It has been well known that time delays can cause complex dynamics such as periodic or quasi-periodic motions, Hopf bifurcation, and higher-dimensional chaos. It should be noted that [11, 12] and did not consider the time-delay problem. Although there are some works that have been reported on the finite synchronization on delayed networks systems, they are mainly concerned with the finite-time boundedness . It is checked that the finite-time boundedness is conservative rather than the finite time convergence. In addition, stochastic perturbation becomes one of the main sources for causing instability and poor performance of networks . In reality, it has been revealed that a neural network sometimes has finite modes, so that switching from one to another at different times may occur [10, 16]. And such a switching (or jumping) can be governed by a Markovian chain [17, 18]. This is partly because a Markovian jump is a suitable mathematical pattern to represent a class of complex networks subject to random abrupt variations in the structures . Moreover, Markovian jump complex networks can be regarded as a special class of stochastic network systems. So a great number of significant results on synchronization of Markovian switching networked systems have been available in the literature [8, 10, 16]. Unfortunately, almost all of the above mentioned works on the synchronization problem of complex networks are built upon the assumption that switching probabilities are known precisely. However, in most cases the transition probabilities of Markovian jump systems or networks are not exactly known [19–21]. Moreover, the estimated values of the mode transition rates may also lead to instability or at least degraded system performance as the partially unknown mode transition rates in system matrices do . Some extended results concerning the uncertain transition probabilities have been reported in [22, 23]. However, such uncertainties require the knowledge of a bound or structure of uncertainties, which is conservative to some extent. Although the finite-time stability or stabilization problems of the control systems has received much attention [15, 24, 25], finite-time synchronization of the delayed complex networks has attracted comparatively less attention primarily due to the lack of an appropriate control method, and secondly due to the difficulty residing in the mathematical derivation. Besides, how to tackle the coexistence of finite-time synchronization and the other two typical networked-induced constraints, stochastic disturbance and time delays in Markovian jump complex networks with partially unknown transition rates, still remains open. In this paper, finite-time synchronization problems are studied for the delayed complex networks with stochastic perturbations and incomplete description of their transition rates. The main features of this paper are twofold: (1) Based on the finite-time stability theorem, some sufficient conditions are obtained to ensure finite-time synchronization for the Markovian jump complex networks with time delays and partially unknown transition rates. (2) For finite-time synchronization research, the model in this paper is more practical, because the network model involves time delays and stochastic perturbations. Notation Throughout this paper, and denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. The superscript ‘T’ denotes the transpose and the notation (respectively, ) where X and Y are symmetric matrices, means that is positive semi-definite (respectively, positive definite); I is the identity matrix with compatible dimension. refers to the Euclidean vector norm; the notation stands for the Kronecker product of matrices A and B. If A is a matrix, denotes the minimum eigenvalue. stands for a block-diagonal matrix E. means the expectation of the random variable x. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. with , and . Here, is the transition rate from to if , while . where ‘?’ represents the unknown transition rate. For notational clarity, , we denote with , . where , . The diffusive couplings mean that the coupled networks (1) are decoupled when the systems are synchronized. So, the coupling terms satisfy , , where denotes zero matrix of n dimension. where is the particular solution of the system (3). We need the following assumption to study the finite-time synchronization of the complex network (1). , , . where , , , . Assumption 3 where denotes the set of continuous functions mapping the interval into . Before ending this section, let us recall the following results, which will be used in the next section. Lemma 1 (Finite-time stability theorem ) Suppose that function is differentiable (the derivative of at 0 is in fact its right derivative) and , where and . Then will reach zero at finite time and for all . Lemma 2 (Jesen inequality ) If are positive number and , then . Lemma 3 If and , then . In this section, we deal with the synchronization problem in finite time for the Markovian jump complex networks with time delays by designing pinning controller. By constructing new stochastic Lyapunov-Krasovskii functionals and using the finite-time stability theorem, sufficient conditions for the finite-time synchronization control problems are derived. where , , . where and , , otherwise, , , . and is the sign function, , the real number β satisfies , . where and , , otherwise, , , . , . where , , , , . , is the initial condition satisfying Assumption 2. According to , for (), it is easy to get . According to (12), one obtains . For any , there is . Therefore, . By Lemma 1, converges to zero in a finite time, and the finite time is estimated by . Hence, the error vector () will stochastically converge to zero within . According to the Definition 1, the coupled complex network (1) is finite-time synchronization in the finite time . The proof is hence completed. □ Investigated the finite-time boundedness synchronization problems for complex networks with time delays. Different from this literature, this paper studied the network synchronization to achieve in a finite time. Therefore, the result of this paper shows more advantages. where , , , otherwise, , . , . We can obtain the following corollary. where is the diagonal matrix, , , then, under the set of controllers (20), the complex network (1) () is synchronization at finite time , where , , and , . , is the initial condition satisfying Assumption 2 of , . In this section, the example is given to demonstrate the effectiveness of the proposed approach. This paper has introduced a general delayed complex networks model with stochastic perturbations and the finite-time synchronization problem of Markovian switching complex networks with stochastic disturbance. Based on the finite-time stability theorem and inequality technique, easily testable conditions have been established to ensure finite-time synchronization for the addressed complex networks. Moreover, conditions that guarantee the finite-time synchronization of the delayed complex networks without switching have also been established. With variable time delays or random delays or mixed delays, finite-time synchronization research of Markovian switching complex network remains open. And it is hard for us to solve such problems, which is our future research. 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