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How to Solve Systems of Equations in Seven Steps Solving systems of linear equations is really hard for many students because it's simply not taught in a way that students can understand. The average curriculum teaches a few "methods" for how solve these systems, but these methods can't really be applied to more complicated systems. Moreover, these "methods" are really cookie-cutter in nature, and require perfectly-formulated examples to work well. Instead of following the "traditional" method of teaching this subject, we have a much easier way to solve this stuff. Step 1: Get Both Equations in Slope-Intercept Form This method requires that both of your equations be written in slope-intercept form, which looks something like y = mx + b. Many of your problems and exercises will give you equations in other formats, like the standard form that looks like Ax + Bx = C. Before you can solve a system of equations, however, you must check to make sure that the lines in question aren't parallel (which means there are no solutions) and that they aren't actually the same line (which means there are infinite solutions). To do this, we need the equations in the slope-intercept form. Step 2: Compare the Slopes and Y-Intercepts If the slopes of the two equations are different, then there is exactly one solution to the system of equations. If the slopes are the same, then there will not be exactly one solution. When you have slopes that are the same, you have to look at the y-intercepts of each equation to see if there are zero or infinite solutions. If the y-intercepts are the same, then the two equations are the same (which should be obvious), and there are infinite solutions. Otherwise, there will be no solutions because the lines are parallel. This much should have been a review for most students. Now that we've verified that there is one solution, it's time to find that solution with a simple method! How Many Solutions? Step 3: Set the Equations Equal to Each Other If your equations were y = 3x + 4 and y = -2x + 1, then you can have 3x + 4 = -2x + 1 because both of these expressions are equal to y. This is a very basic type of substitution that's really easy to perform once you have the equations in slope-intercept form. The point here is that normally you're taught to use "elimination", "substitution", or "graphing" to solve systems of equations. The problem is that "elimination" is very difficult to use unless the problems are hand-picked, and "graphing" is very difficult to do unless nice, even numbers are used. In the real world, you need a substitution method that's easy to do and is resistant to mistakes. Since you have to put the equations in slope-intercept form anyway, this method will always be available, and will always be very simple to follow. Step 4: Solve for X We had y = 3x + 4 and y = -2x + 1 in our previous example, and that gave us 3x + 4 = -2x + 1. What we're left with here is just a simple linear equation of a type that that we've solved over and over again. For example, we could have the following: 3x + 4 = -2x + 1 5x + 4 = 1 5x = -3 x = -3/5 And we have solved for x in just a few simple steps. Doesn't this beat the more "traditional" methods?! Step 5: Solve for Y We said that we found x = -3/5 in step 4, so now we'll just need to substitute that into one of our starting equations to find a value for y. We have y = 3x + 4, so here we go: y = 3(-3/5) + 4 y = -9/5 + 4 y = -9/5 + 20/5 y = 11/5 And that took all of maybe a minute to do if you're familiar with how to handle the fractions, which you should be at this point in learning Algebra 1. Step 6: Check Your Answer Take the x and y coordinate that you get as a solution for this problem and plug it into each of the equations, making sure that it works. It's not enough to know how to get *an* answer, but you want to be able to make sure that you have *the* answer by knowing ways to quickly and easily check yourself. Taking a moment to check your answers using easy methods like substituting your answer into the original equations will give your grade a dramatic boost. Step 7: Laugh at Your Classmates Now that you have the answer, and you're sure that it's the right answer, it's time to sit back and watch everyone else work really hard and get confused over a problem that you just finished with ease! More About Solving Linear Equations - Algebra 1 Help.org - Algebra 1 Made Easy This website shows the easy ways to do everything you'll need to learn how to do in Algebra 1. There are step by step tutorials that show the really easy ways to do everything instead of blindly following some old textbook. - Linear Systems Summary | Algebra 1 Help.org This is a set of Algebra 1 tutorials that help you with everything you need to know about solving systems of linear equations. This covers all ways of solving systems of equations, and everything you need to know to pass your tests with good grades. More by this Author So here's the deal. Subtracting integers gives a lot of people a really hard time. For some reason, this is a topic that isn't covered very well by most Algebra 1 instructors because it's taken for granted as a skill.... Please Leave a Comment!!! No comments yet.

So what is it? This is not a homework question. But I anticipate a good discussion. Mostly refraction, but there might be some reflection in it. It's mostly a refraction of light from the sky, which looks like it's coming from the ground ahead, and is taken for a lake or something of the sort. We had a thread on the optical technicalities a while back. Does anyone know where it is? Mirages - A Reflection effect!!! - a consequence of Refraction U see, the air does not get heated uniformly. Instead, it gets heated in layers. Thus the optical properties of the diff layers of air will be diff. The light from a distant object rises up, and moves from an optically denser layer to an optically lighter layer and if the angle of incidence is such that the critical angle is achieved, TOTAL INTERNAL REFLECTION occurs. This causes an image of the object to be created on the surface at diff location which we observe as the mirage... Hope u understood... Just out of curiosity, how does "light rise up"?? So i't is reflexion, Right. But how about this? So is he right or is he wrong? In the mean time I found that other thread about mirages: but it is not really adressing the question "refraction or reflection?", so we can continue. Why the or? A mirage is a reflection produced by refaction. The point is that a reflection is not a mechanism it is a result. When you see a refletion in a mirror there is a different mechanism (free electron mobility) which produces that reflection. Refraction is the apparent bending of light rays due to light passing an interface to a material with a different index of refaction. As far as I know reflection is not a carefully defined term. I wanted to say that. The physics of a mirage is understood. Light bends due to a gradient in the refractive index. The rest is semantics. I like sridhar_n's sig. It's a distinction some of my kids' teachers don't appreciate. My vote is that the bending of light is caused by semantics. Semantics? I don't think so. I think that sridhar_n is about right. As far as the input and output of a Mirage producing system is concerned I think it is 100% reflection and 0,0% refraction. Not even a little refraction as net result. Once again, refraction is the MECHANISM which produces a REFLECTION. There we are, this is the essential flaw in my humble opinion. Edit made a thinking error here. There is refraction going on but most certainly there are limits to refractions. Why is it that none of these pages explaining mirages do not substantiate it with Snellius law? Because it doesn't work. I knew we would have a good discussion Perhaps this failure to explain is because you are not reading a real Optics text. I have scanned a few pages from an old text I have, Optics by Rossi. Warning for the bandwidth challenged this is a 2Meg PDF file. edit: spell check Perhaps not that relevant, but we use a nice experiment at my school to show refraction of light. A long thin tank has extremely saturated salt solution in the bottom, to a depth of about an inch. Tap water is then slowly put on top of this (with a U shaped tube), to a depth of several inches. and allowed to stand for a few hours. A laser beam is then placed at the side of the tank, about midway up, and if you've done it right, the beam takes a beautiful curved path, dipping down to the bottom. Simple to do and VERY impressive. I have done this, only used a sugar solution. If you get the laser beam the correct distance from the bottom of the dish it will be totally reflected from the bottom and mirror its entry path on exit. Well, impressive text book, Integral but did you notice that there is always a division by sin(phi). Sinusses can be zero. Let's look at Snellius law. R1*sin(phi-in)=R2*sin(phi-out). In which Ri is the refractive index of the medium and phi-in and phi-out obviously the angle of the light before and after refraction. We want to investigate what happens to the light direction after refraction. Therefore we have to reshuffle that law as (angles in relation to the vertical plane) Now, weren't there some complications for an arcsine? The factor under the brackets must be less than or equal to one. Otherwise there is no solution. So let us take a closer look at that expression. R1 is the refractive index for the incoming light in the heavier colder air and since R2 is a smaller index for lighter warm air, the R1/R2 term is always more than one. Hence sin(Phi-in) must always be less than one or concequently Phi-in must be less than 90 degrees. Also, this means that there is a critical angle for Phi-in beyond which refraction is not possible. The formula also indicated that the maximum angle for the refracted light is 90 degrees. So if a light ray is refracted in another material with a horizontal surface, it will never exceed the horizontal, so obviously there is no such thing as refracting back. Yet we see things at and beyond the horizon apparently refracted. Ever been swimming underwater? When you look up you only see a circle of light directly above you. The more distant water surface around you is dark. Why? Because the critical angle has been exceeded and the light coming in too shallow is not refracting but either absorbed or reflected. Beyond that angle only the dark bottom under the water is reflected. Now look at this impressive picture of a superior (on top) mirage. It's with an inverted mirage, due to very cold air on the surfaces, think the cold inversion being water instead of cold air, with the higher refractive index. The Light at the horizon cannot be refracted due to the critical angle. Instead, you see the bottom reflected and in this case that's the mountains. No wonder that they are upside down. Reflection has that mirroring habit. So this mirage is just the same as happens under water, reflection at the water surface , but now there is a critical layer somewhere where the reflection takes place. How about a regular mirage in the desert or on the road on a sunny day? Apart from turning the scenario upside down, it's exactly the same. Here several examples of inferior (Desert warm type) and superior (artic cold type) mirages: So the ultimate trick is real reflection, just like sridhar_n indicated. Now how come that after ages of Snellius law and know mirages, we have always assumed a wrong explanation? Time to rewrite some textbooks. It is still not clear to me where the incorrect explanation is. Seems to work to me. The angle at which total internal reflection occurs is called the Brewster angle, are you attempting to say that this is incorrectly handled? I think it is correctly handled. Perhaps you are reading different books then I am. You tone seems to indicate that you have read them all. Is this possible? Or are you making some assumptions? Rereading your post, and my text, it occurs to me that while you are impressed with it, you are ignoring it. Please read, and make an effort to understand the development. If you are unable to understand the approach, then please do not continue your criticisms of the general understanding, when it is YOU who is having the trouble. No I admit that I did not study it intensely and I will do it on your recommandation but the only thing that I tried to state that the limit of refraction is 90o as can be seen in the law of Snellius. The refraction of light grazing at the critical angle is exactly at 90o. Beyond the critical angle there is no refraction, just total internal reflection. You can read that in any textbook as well: Now the mirage object is always in the heavier cooler air, the more dense medium, regardless of an inferior or superior mirage. It approaches the warmer air, the less dense medium, either from aloft or below. So the first condition has most certainly been met. The mirage is always an image of something close to the horizon where the angle of incidence is closing in on the limit of 90o. So for a mirage this conditions looks plausible. Now please look at the first figure in the same link with the example of refraction at the critical angle (left) and total internal reflection (right) at the water surface from below. Now replace "water" with "denser colder air" and you've got yourself a superior mirage. Total internal reflection beyond the critical angle of probably something like 89,9o. Now turn the figure upside down and you're looking at the normal inferior mirage, famous for deserts. For a inferior mirage you could argue that the curvature of the Earth would bring the horizontal refracted light into cooler regions again causing it to refract up again, but that does not work for superior mirages where the curvature of the Earth opposes refraction. edit after studying: Integral, does your textbook actually say something else? I see equations 2-6 and the final one on chapter 2-2 of the example. Now try and use those in practice to see if you can get refraction beyond 0o or 90o depending upon the definition of phi. Notice that the textbook does not indicate that it has proved the bending back up with refraction. Sometimes theory may need revisiting. I do have some problems with this. Would this be the sole reason why it takes a generation before a correct paradigm shift can take place. I think that general understanding should be questioned at all times if not the Earth would still have been flat. Arent those forums to discuss those things. OK most challenges will be plain wrong. Please explain where I am wrong. You're not totally wrong, but I think that you're not considering the appropriate situation. This business with the critical angle and total internal reflection if for a stark boundary between two distinct media. Mirages and the like are phenomena that occur in continuous media (no boundaries, but gradients). You are correct in that total internal reflection occurs when the angle of incidence is greater than the critical angle (by DEFINITION), but you are not correct to assume that the phenomenon of a mirage is a total internal reflection, as there is no distinct medium to which it is internal. Refraction at a boundary can be considered strictly using trigonometry in the ray theory of optics, but you mustn't forget that this is based on Hyugen's principle. When you shift your consideration to a continuous medium (gradients vs. boundaries), the intamacy to the principle is greatly amplified. You should take a step back and consider the light in terms of wavefronts rather than rays. The rays are like the vectors and the wavefronts are like the 1-forms. You can consider them as identical objects in a uniform medium and then only worry about them at a boundary (and thus use the ray theory), but, in a non-uniform medium, it becomes a bit more tricky. Separate names with a comma.

Disparity in Selmer ranks of quadratic twists of elliptic curves We study the parity of -Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve over an arbitrary number field . We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even -Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve such that as varies, these fractions are dense in . More generally, our results also apply to -Selmer ranks of twists of -dimensional self-dual -representations of the absolute Galois group of by characters of order . 2010 Mathematics Subject Classification:Primary: 11G05, Secondary: 11G40 The type of question that we consider in this paper has its roots in a conjecture of Goldfeld [6, Conjecture B] on the distribution of Mordell-Weil ranks in the family of quadratic twists of an arbitrary elliptic curve over , and a result of Heath-Brown [7, Theorem 2] on the distribution of -Selmer ranks in the family of quadratic twists over of the elliptic curve . We study here the distribution of the parities of -Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve over an arbitrary number field . For example, let be the fraction of quadratic twists of that have odd -Selmer rank. Precisely, for real numbers let It follows from a result of Monsky [16, Theorem 1.5] along with root number calculations that for every elliptic curve . It had already been noticed (see ) that this is not true when is replaced by an arbitrary number field , because there are examples with for which , and others with . Our main theorem (see Theorem 7.6) evaluates . Suppose is an elliptic curve defined over a number field . Then for all sufficiently large we have where is given by an explicit finite product of local factors (see Definition 7.4). We call the “disparity” in the distribution of -Selmer ranks of twists of . If has a real embedding then so (see Corollary 7.10). On the other hand, Example 7.11 exhibits a particular elliptic curve such that as varies, the set is dense in , so is dense in . The finiteness of the -part of the Shafarevich-Tate group would imply that the parity of the -Selmer rank is the same as the parity of the Mordell-Weil rank. Thus one would expect that Theorem A holds with -Selmer rank replaced by Mordell-Weil rank. Further, Theorem A suggests a natural generalization of Goldfeld’s conjecture (see Conjecture 7.12). In a forthcoming paper , we will use the methods of this paper to make a finer study of the distribution of -Selmer ranks, inspired by the work of Heath-Brown , Swinnerton-Dyer , and Kane . Our methods begin with those of and . Namely, we view all of the Selmer groups as subspaces of , defined by local conditions that vary with . In this way we can attach a Selmer group to a collection of local quadratic characters. The question of which collections of local characters arise from global characters is an exercise in class field theory (see §6). To prove Theorem A, we show that the parity of depends only on the restrictions of to the decomposition groups at places dividing , where is the discriminant of some model of (see Proposition 7.2). (This is consistent with the behavior of the global root numbers of twists of .) In particular the map that sends a character to the parity of factors through the finite quotient . Using this fact we are able to deduce Theorem A. There is another important ingredient in the proof of Theorem A. We make essential use of a recent observation of Poonen and Rains that the local conditions that define the -Selmer groups we are studying are maximal isotropic subspaces for a natural quadratic form on the local cohomology groups . We use this in a crucial way in the proof of Theorem 3.9, which extends a result from to include the case . Theorem 3.9 is a key ingredient in the proof of Theorem A. Our methods apply much more generally than to -Selmer groups of elliptic curves, and throughout this paper we work in this fuller generality. Namely, suppose is any prime, and is a -dimensional -vector space with an action of the absolute Galois group , a nondegenerate -equivariant alternating -valued pairing, and a “global metabolic structure” (see Definition 3.3). We also assume we are given “twisting data” (Definition 4.4) that allows us to define a family of Selmer groups as runs through characters of of order . We have analogues of Theorem A describing the distribution of in this setting. For example, if is an elliptic curve over , then , the kernel of multiplication by on , comes equipped with all the structure we require. When the Selmer groups are not Selmer groups of elliptic curves, but they are Selmer groups of -dimensional abelian varieties over that are twists of in the sense of . See §5, and see Theorem 8.2 for the analogue of Theorem A in this setting. The layout of the paper is as follows. Let be a Galois module as above. In §2 we derive some elementary properties of Lagrangian subspaces in quadratic vector spaces that we will need in the sequel. In §3 we define metabolic structures and Selmer groups in the generality we will need them. The key result is Theorem 3.9, which shows how the parity of the Selmer rank changes when we change some of the defining local conditions. In §4 we define the Selmer groups associated to twists of , and in §5 we show how these Selmer groups reduce to classical Selmer groups of twists when for an elliptic curve . Fix a number field and a rational prime . Let denote a fixed algebraic closure of , and . Let denote the group of -th roots of unity in . Throughout this paper will denote a two-dimensional -vector space with a continuous action of , and with a nondegenerate -equivariant alternating pairing corresponding to an isomorphism We will use (resp., ) for a place (resp., nonarchimedean place, or prime ideal) of . If is a place of , we let denote the completion of at , and its maximal unramified extension. We say that is unramified at if the inertia subgroup of acts trivially on , and in that case we define the unramified subgroup by If and is a place of , we will often abbreviate for the localization of in . We also fix a finite set of places of , containing all places where is ramified, all primes above , and all archimedean places. 2. Metabolic spaces Fix for this section a finite dimensional -vector space . A quadratic form on is a function such that for every and , the map is a bilinear form. If , we denote by the orthogonal complement of in under the pairing . We say that is a metabolic space if is nondegenerate and has a subspace such that and . Such a subspace is called a Lagrangian subspace of . For this section, if is an -vector space we let . Suppose is a metabolic space, is a Lagrangian subspace, and is a subspace of such that . Then is a Lagrangian subspace of . Exercise. See for example [17, Remark 2.4]. ∎ Suppose is a metabolic space and , , and are Lagrangian subspaces of . Then Suppose . Write and with and , and define Note that and are well-defined modulo , so does not depend on the choice of or . Thus is a well-defined bilinear pairing on . By definition we have Since , , and are Lagrangian, and , we have , so for every , i.e., is alternating (and therefore also skew-symmetric). If then we can take in (2.1), so for every . Using the skew-symmetry we deduce that is in the (left and right) kernel of the pairing . Similarly , and hence , is in the kernel. Conversely, if is in the kernel of this pairing, then (still writing with and ) for every and . Applying Lemma 2.2 with , we see that Thus, modifying and by an element of , we may assume that , and then as well, so . This completes the proof that the pairing (2.1) is alternating and nondegenerate on . A standard argument now shows that the dimension is even, and the lemma follows. ∎ Suppose is a metabolic space and , , and are Lagrangian subspaces of . Then Suppose is a metabolic space and , , and are Lagrangian subspaces of . Then This follows directly from Proposition 2.4. ∎ 3. Metabolic structures and Selmer structures In this section we define what we mean by a global metabolic structure on , and by a Selmer group for and . The main result is Theorem 3.9, which shows how the parity of the Selmer rank changes when we change the defining local conditions. The cup product and the pairing (1.1) induce a pairing If is a place of and is the completion of at , then applying the same construction over the field gives local pairings For every there is a canonical inclusion that is an isomorphism unless either , or and . The local Tate pairing is the composition The Tate pairings satisfy the following well-known properties. For every , the pairing is symmetric and nondegenerate. If then is equal to its own orthogonal complement under . If , then for almost all and . For (i) and (ii), see for example [15, Corollary I.2.3 and Theorem I.2.6]. The first part of (iii) follows from (ii), and the second from the fact that the sum of the local invariants of an element of the global Brauer group is zero. ∎ Suppose is a place of . We say that is a Tate quadratic form on if the bilinear form induced by (Definition 2.1) is . If , then we say that is unramified if for all . Suppose is as above. A global metabolic structure on consists of a Tate quadratic form on for every place , such that is a metabolic space for every , if then is unramified, if then . Note that if then for almost all , so the sum in Definition 3.3(iii) is finite. If then there is a unique Tate quadratic form on for every , and a unique global metabolic structure on . Suppose and is a global metabolic structure on . If is such that for every , then for every we can define a new Tate quadratic form on by It is straightforward to check (using Theorem 3.1) that is again a global metabolic structure on , and if then . Suppose is a place of and is a quadratic form on . Let Suppose and is a Tate quadratic form on . Let . Then: every has dimension , if and is unramified, then . shows that . This proves (i) and (ii). Assertion (iii) follows from a calculation of Poonen and Rains [17, Proposition 2.6(b,e)]. ∎ Suppose is as above and is a global metabolic structure on . A Selmer structure for (or simply for , if is understood) consists of a finite set of places of , containing , for every , a Lagrangian subspace . If is a Selmer structure, we set if , and we define the Selmer group by i.e., the subgroup of such that for every . Suppose and are two Selmer structures for . Then Let . Define , so is a metabolic space. Let denote the product of the localization maps. Define three subspaces of is the image under of . The spaces and are Lagrangian by definition of Selmer structure. That is also Lagrangian can be seen as follows. We have by Poitou-Tate global duality (see for example [15, Theorem I.4.10], [24, Theorem 3.1], or [19, Theorem 1.7.3]). If , then with satisfying for every . Then if by Definition 3.3(ii), so by Definition 3.3(iii). Thus is Lagrangian. Note that from the definitions we have exact sequences where the kernel in both sequences is Thus by Proposition 2.4 we have Since , this completes the proof of the theorem. ∎ 4. Twisted Selmer groups Given as above (and some additional “twisting data”, see Definition 4.4), in this section we show how to attach to every character a Selmer group . More generally, we attach a Selmer group to every collection of local characters with for in some finite set containing . Our main result is Theorem 4.11, which uses Theorem 3.9 to show how the parity of the Selmer rank changes when we change some of the . If is a field, define (throughout this paper, “” will always mean continuous homomorphisms). If is a local field, we let denote the subset of ramified characters. In this case local class field theory identifies with , and is then the subset of characters nontrivial on the local units . Let denote the trivial character. There is a natural action of on , and we let . Then is naturally identified with the set of cyclic extensions of of degree dividing , via the correspondence that sends to the fixed field of in . If is a local field, then denotes the set of ramified extensions in . and . Define the width of a prime of , , by if . Let denote the field of definition of the elements of , i.e., the fixed field in of . Suppose is a prime of , , and let be a Frobenius element for some choice of prime above . Then if and only if , if and only if has order exactly , if and only if . In particular has positive density in the set of all primes of , and has positive density if and only if . Fix an -basis of so that we can view . Then by (1.1) Since , is unramified at , so , the subspace of fixed by . We have if and only if , and if , then if and only if has order . This proves the lemma. ∎ Suppose , are as above, and is a global metabolic structure on . By twisting data we mean for every , a (set) map for every , a bijection Note that if then the first equality by local class field theory (since by definition and ) and the second by Lemma 3.7(iii). If and is a place of , we let denote the restriction of to . and if let (resp., ) be the product of all primes dividing that lie in (resp., ), so . For every , define the width of by Define a finite set and let and denote the natural maps Note that is a group, and is not a group but it is closed under multiplication by unramified characters. Since is the fiber over of the map that sends to the part of its conductor supported on , we have . Given , , and twisting data as in Definition 4.4, we define a Selmer structure for every and as follows. If then let . If , let be the unique element of . If , let . If we will also write , and if then we define It is clear from the definition that depends only on the extension of cut out by , i.e., for all . However, when we later count the twists with certain properties, it will be convenient to deal with rather than because is a group. In any case the natural map is -to-one except for the single fiber consisting of the trivial character, so it is simple to go from counting results for to results for . In particular, when the natural map is a bijection. Remark 4.9 (Remarks about twisting data). Our definition of twisting data is designed to ensure that for , all subspaces occur with equal frequency as we run over characters that are ramified at . That fact is all we require to prove our results in sections 7 and 8 about the rank statistics of . In particular, the conclusions of Theorems 7.6 and 8.2 below do not depend on the choice of twisting data for . We will see in §5 that when is an elliptic curve over , is a rational prime, and , then there is natural global metabolic structure on and natural twisting data such that for every , is a classical Selmer group of a twist of (an abelian variety twist, when ). An analogous statement should hold for more general (self-dual) motives and their Bloch-Kato -Selmer groups, so our results below should also apply to Bloch-Kato Selmer groups in families of twists. If and , define Suppose , , and . Then

Johann Georg Zehfuss Darmstadt-Bessungen, Darmstadt, Grand Duchy of Hesse, now Germany Frankfurt am Main, Germany BiographyGeorg Zehfuss was the son of Johann Heinrich Zehfuss, Court Chamber-Registrar, and Susanne Magdalene Noack. He attended the Höhere Gewerbeschule Darmstadt, the higher trade school which was founded in 1836. Günter Kern writes :- As Zehfuss himself admits, in his youth he showed more interest in languages and history than mathematics. At the age of fifteen, he entered the Polytechnic in Darmstadt, where he devoted himself to the study of mathematics, mechanics, physics and chemistry, encouraged by his teacher Strecker, but continued to attend classes in Latin, French as well as German history and literature.The Polytechnic which Kern mentions in this quote is actually the Höhere Gewerbeschule Darmstadt which was renamed the Polytechnic School in 1868, many years after Zehfuss had completed his studies there. At the Gewerbeschule, Zehfuss took classes in mathematics, technical drawing, physics and chemistry. He would also have learned practical skills required for the study of engineering topics. Among his teachers at this school we mention Edmund Külp (1800-1862) who had been inspired to study mathematics by Adolphe Quetelet at the Athenaeum in Brussels. Külp had studied at the University of Heidelberg and was awarded a doctorate by the University of Giessen in 1824. In his first years at the school Zehfuss was taught chemistry by Adolph Strecker (1822-1871) but Strecker left in 1846 to take up an appointment at the University of Giessen. Another of Zehfuss's teachers was Ludwig Christian Wiener who was appointed to the Höhere Gewerbeschule in 1848 and taught physics, mechanics, hydraulics and descriptive geometry. Zehfuss graduated "summa cum laude" from the Höhere Gewerbeschule Darmstadt in 1850 and began his studies of mathematics at the Justus Liebig University of Giessen. At the University of Giessen his main interest was the finite difference calculus, the results of which he summarised in the papers Einige Punkte über die Bestimmung der Constanten, welche bei Integration der endlichen Differenzengleichungen eingehen Ⓣ (1856) and Über die Auflösung der linearen endlichen Differenzengleichungen mit variabeln Coefficienten Ⓣ (1858). For a list of Zehfuss's publications, see THIS LINK. Zehfuss completed his studies in Giessen on 31 January 1853 when he obtained a doctorate and qualified to teach in a Gymnasium. Although he performed well at Giessen, he did not find the university mathematically stimulating. After graduating, he obtained a temporary teaching position at the Gewerbeschule Darmstadt where he assisted his former teacher Edmund Külp teaching mathematics and physics from 1853 to 1858. In 1857 Zehfuss habilitated in mathematics and physics at the Ruprecht-Karls University of Heidelberg when he submitted his dissertation Abhandlung über einige mathematische Gegenstände. Inauguralschrift Ⓣ. This dissertation was examined by Otto Hesse and Gustav Kirchhoff; Hesse evaluated the three mathematical parts of the dissertation while Kirchhoff evaluated the one part related to physics. Hesse wrote in his report that the dissertation dealt :- ... with difficult parts of the subject that are only accessible to the more advanced mathematicians.Kirchhoff also praised the physics section which was on the theory of thermodynamics. Otto Hesse had been appointed to the chair of mathematics in University of Heidelberg in 1856. He had made investigations into the theory of determinants while undertaking research on invariant theory and he inspired Zehfuss to work on determinant theory. Zehfuss published his first work on this topic in 1858 with three papers appearing in that year, namely: Über die Zeichen der einzelnen Glieder einer Determinanten Ⓣ; Über eine gewisse Determinante Ⓣ; and Kommentar des Buches von Richard Baltzer "Theorie und Anwendungen der Determinanten mit Beziehung auf die Orginalquellen" Ⓣ. In the first of these papers Zehfuss proves a rule stated by Gabriel Cramer about the change in the signs of determinants. In the second of the papers he proves a rule about the tensor product of square matrices. Some authors believe that the tensor product, often called the Kronecker product, should be called the 'Zehfuss product'. Note that we have made a subtle change from 'determinant' to 'matrix'. Early work on matrices was given in terms of determinants. See below for quotes about the 'Zehfuss product'. In the third of the papers he gives a detailed analysis of the book by Richard Baltzer. This book contains many illustrative references to the original sources of determinant theory and their applications. Zehfuss worked as a docent at the University of Heidelberg for four years between 1857 and 1861. During these years he taught arithmetic, algebra and geometry, differential calculus, theory of definite integrals, elliptic functions, theory of higher equations, analytical geometry of the plane and of 3-space, and analytical mechanics. On 11 July 1861 Zehfuss married Louise Stein (1838-1911). In November 1861 Richard Dedekind left the Polytechnikum in Zürich after three years there and took up a position in the Brunswick Polytechnikum. Zehfuss applied for the vacant position in Zürich but it was offered to Rudolf Lipschitz. He, however, turned down the position and it was then offered to Elwin Christoffel. Zehfuss, unable to find a permanent position in Germany, accepted the post of high school professor of mathematics and physics in Reval (now Tallinn in Estonia) in September 1860. He taught at the Cathedral School in Reval and while there his son Gustav Heinrich Julius Wilhelm Zehfuss was born on 20 August 1862. However, Zehfuss was not very happy at the Cathedral School in Reval since he did not feel that this offered him the environment to reach his full potential as a researcher. In February 1862 Zehfuss applied for a position as a professor in mathematics or physics at the Polytechnic of Riga in Latvia. His application included recommendations from many mathematicians including Ludwig Christian Wiener, Ferdinand Minding, Osip Ivanovich Somov, Oskar Schlömilch, Otto Hesse, Gustav Kirchhoff and Richard Dedekind. Zehfuss was appointed and took up the position as professor of higher mathematics at the Polytechnic of Riga on 9 January 1863. This was a new institution and initially it struggled to establish itself. Zehfuss decided to return to Germany when offered a position at the Gewerbeschule in Frankfurt am Main. He took up the position in 1864. At this school he taught algebraic analysis, trigonometry, the theory of higher degree equations, two dimensional analytic geometry, mechanics, differential and integral calculus, and three dimensional analytic geometry. His second child, a daughter Cornelia Charlotte Marie Auguste Mathilde Victoria Zehfuss, was born in Darmstadt on 16 September 1864. Zehfuss continued in his position at the Gewerbeschule in Frankfurt am Main for the rest of his career. Zehfuss died in Frankfurt am Main on 5 May 1901. He was buried in Heidelberg in the mountain cemetery of Heidelberg-Südstadt. Also buried there are his wife Luisa, his daughter Cornelia (1864-1949), and his daughter's husband Georg Anton Karch (1856-1937). Today the name of Zehfuss is not known by many but those who are aware of his contributions almost certainly know him through his contribution to the tensor product of matrices. Without offering a definite opinion as to whether the 'Kronecker product' should be called the 'Zehfuss product' let us simply quote from a number of sources. Amy N Langville and William J Stewart write :- The operation defined by the symbol was first used by Johann Georg Zehfuss in 1858. It has since been called by various names, including the Zehfuss product, the Producttransformation, the conjunction, the tensor product, the direct product and the Kronecker product. In the end, the Kronecker product stuck as the name for the symbol and operation, .H Zhang and F Ding write :- The Kronecker product, named after German mathematician Leopold Kronecker, is very important in the areas of linear algebra and signal processing. In fact, the Kronecker product should be called the Zehfuss product because Johann Georg Zehfuss published a paper in 1858 which contained the well-known determinant conclusion, for square matrices and with order and .Roger A Horn and Charles R Johnson write :- Some historians of mathematics have questioned the association of the ⊗ product with Kronecker's name on the grounds that there is no known evidence in the literature for Kronecker's priority in its discovery or use. Indeed, Sir Thomas Muir's authoritative history calls the 'Zehfuss determinant' of A and B because the determinant identityappears first in an 1858 paper of Johann Georg Zehfuss ("Über eine gewisse Determinante Ⓣ, Zeitschrift für Mathematik und Physik 3 (1858), 298-301"). Following Muir's lead, a few later authors have called A ⊗ B the 'Zehfuss matrix' of A and B, for example "D E Rutherford, On the Condition that two Zehfuss matrices be equal, Bull. Amer. Math. Soc. 39 (1933), 801-808", and "A C Aitken, The normal form of compound and induced matrices, Proc. London Math. Soc 38 (2) (1935), 354-376". However, a series of influential texts at and after the turn of the century permanently associated Kronecker's name with the ⊗ product, and this terminology is nearly universal today. For further discussion (including claims by others to independent discovery of the determinant result) and numerous references, see "H V Henderson, F Pukelsheim and S R Searle, On the history of the Kronecker product, Linear and Multilinear Algebra 14 (2) (1983), 113-120". Harold V Henderson, Friedrich Pukelsheim and Shayle R Searle write :- Our story begins with one Johann Georg Zehfuss (1832-1901) at the University of Heidelberg who, according to biographical notes by Poggendorf (1863, 1898), published papers on determinants until at least 1868 before moving to studies in astronomy. In particular, Zehfuss (1858) contains the determinant resultAlthough Zehfuss is best known for his contributions to determinants, he made other contributions to mathematics in difference equations, differential and integral calculus and combinatorics. Later in his career, however, his attention turned more towards physics and astronomy. He actively participated in the annual meetings of the Gesellschaft Deutscher Naturforscher und Ärzte Ⓣ, especially when the meetings were in Frankfurt am Main or in the vicinity of the city. He also participated in meetings of the Wetterauischen Naturforschenden Gesellschaft zu Hanau Ⓣ. He lectured to these societies on determinants, but also showed a model of a Riemann surface and gave a lecture "On a Possible Cause of the Delay of Earth Rotation." In this lecture he showed (see ):-for square matrices A and B of order a and b, respectively. Zehfuss wrote in terms of determinants rather than matrices, following the then customary practice of employing the term 'determinant' both for what we now call a square matrix as well as for its determinant. ... Unfortunately, the Zehfuss (1858) determinant result seems to have been overlooked for more than fifty years, until its rediscovery by Muir (1911, reprinted in ) who claims it for Zehfuss and accordingly calls the Zehfuss determinant of A and B. Others, notably Rutherford (1933), and Aitken (1935) and his student Ledermann (1936), following Muir's lead, have gone further and called A B the Zehfuss matrix of A and B. Aitken's adoption of this name is of interest in light of Ledermann's later (1968) comment that Aitken "was particularly fond of stressing the claims of lesser known mathematicians of former times for discoveries erroneously attributed to their more famous contemporaries. Many of his historical references were gleaned from Sir Thomas Muir's monumental work on determinants, for which Aitken had a profound admiration." ... that magnetic forces which the heavenly bodies mutually exert cannot effect this delay, whereas the constant magnetic field in which the planets move can cause a change in precession which could lead to an explanation of the observation.Other lectures related to astronomy included "On meteorites" delivered in 1870 and "Some explanations of the appearance of the Northern Lights" in 1871. He wrote the book Die Pneumatische Canalisation beleuchtet mit Rücksicht auf Gesundheitspflege, Land- und Volkswirthschaft Ⓣ which was translated into English by D Coar of Philadelphia, Pennsylvania, USA and appeared as The Pneumatic Sewage System treated with reference to Public Health, Agriculture and National Economy published by Wilhelm Hassel in Cologne in 1869. The book commences with an historical introduction in which Zehfuss refers to sewage systems mentioned in the Bible, those constructed by the Hindus, the ancient Egyptians, the Babylonians, and the Romans in 600 B.C. A patent was sought in the United States for the sewage system Zehfuss describes in the book and he received a letter of patent from the United States Patent Office in 1870 (letter of patent No. 100,347). Finally, let us note that prior to the appearance of , few biographical details concerning Zehfuss were readily available. Much of the biographical information in our biography above comes from Walter Strobl's paper. - R A Horn and C R Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1994). - G Kern, Die Entwicklung des Faches Mathematik an der Universität Heidelberg 1835-1914 (Universitätsbibliothek, Heidelberg, 1992). http://www.ub.uni-heidelberg.de/archiv/14583 - T Muir, The Theory of Determinants in the Historical Order of Development (5 Volumes) (London, 1960). - G Zehfuss, The Pneumatic Sewage System Treated with Reference to Public Health, Agriculture and National Economy, Part 1 (Hassel, 1869). - H V Henderson, F Pukelsheim and S R Searle, On the history of the Kronecker product, Linear and Multilinear Algebra 14 (2) (1983), 113-120. - A N Langville and W J Stewart, The Kronecker product and stochastic automata networks, Journal of Computational and Applied Mathematics 167 (2004) 429-447. - W Strobl, Georg Zehfuss: Sein Leben und seine Werke. http://www.ub.uni-heidelberg.de/archiv/24214 - H Zhang and F Ding, On the Kronecker Products and Their Applications, Journal of Applied Mathematics 2013 (2013). http://dx.doi.org/10.1155/2013/296185 Additional Resources (show) Other pages about Georg Zehfuss: Written by J J O'Connor and E F Robertson Last Update May 2018 Last Update May 2018

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters (as are some other finance measures). Collectively these have also been called the risk sensitivities, risk measures: 742 or hedge parameters. |Definition of Greeks as the sensitivity of an option's price and risk (in the first row) to the underlying parameter (in the first column). First-order Greeks are in blue, second-order Greeks are in green, and third-order Greeks are in yellow. Note that vanna, charm and veta appear twice, since partial cross derivatives are equal by Schwarz's theorem. Rho, lambda, epsilon, and vera are left out as they are not as important as the rest. Three places in the table are not occupied, because the respective quantities have not yet been defined in the financial literature.| The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example delta hedging. The Greeks in the Black–Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging—such as delta, theta, and vega—are well-defined for measuring changes in Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common. The most common of the Greeks are the first order derivatives: delta, vega, theta and rho as well as gamma, a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive. The use of Greek letter names is presumably by extension from the common finance terms alpha and beta, and the use of sigma (the standard deviation of logarithmic returns) and tau (time to expiry) in the Black–Scholes option pricing model. Several names such as 'vega' and 'zomma' are invented, but sound similar to Greek letters. The names 'color' and 'charm' presumably derive from the use of these terms for exotic properties of quarks in particle physics. Delta, , measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value of the option with respect to the underlying instrument's price . For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (or a short put) and 0.0 and −1.0 for a long put (or a short call); depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), or something in between, and conversely for a put option. The difference between the delta of a call and the delta of a put at the same strike is equal to one. By put–call parity, long a call and short a put is equivalent to a forward F, which is linear in the spot S, with unit factor, so the derivative dF/dS is 1. See the formulas below. These numbers are commonly presented as a percentage of the total number of shares represented by the option contract(s). This is convenient because the option will (instantaneously) behave like the number of shares indicated by the delta. For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.25 (=25%), it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements (100 option contracts covers 10,000 shares). The sign and percentage are often dropped – the sign is implicit in the option type (negative for put, positive for call) and the percentage is understood. The most commonly quoted are 25 delta put, 50 delta put/50 delta call, and 25 delta call. 50 Delta put and 50 Delta call are not quite identical, due to spot and forward differing by the discount factor, but they are often conflated. Delta is always positive for long calls and negative for long puts (unless they are zero). The total delta of a complex portfolio of positions on the same underlying asset can be calculated by simply taking the sum of the deltas for each individual position – delta of a portfolio is linear in the constituents. Since the delta of underlying asset is always 1.0, the trader could delta-hedge his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. For example, if the delta of a portfolio of options in XYZ (expressed as shares of the underlying) is +2.75, the trader would be able to delta-hedge the portfolio by selling short 2.75 shares of the underlying. This portfolio will then retain its total value regardless of which direction the price of XYZ moves. (Albeit for only small movements of the underlying, a short amount of time and not-withstanding changes in other market conditions such as volatility and the rate of return for a risk-free investment). Main article: Moneyness The (absolute value of) Delta is close to, but not identical with, the percent moneyness of an option, i.e., the implied probability that the option will expire in-the-money (if the market moves under Brownian motion in the risk-neutral measure). For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has approximately a 15% chance of expiring in-the-money. Similarly, if a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money calls and puts have a delta of approximately 0.5 and −0.5 respectively with a slight bias towards higher deltas for ATM calls. The actual probability of an option finishing in the money is its dual delta, which is the first derivative of option price with respect to strike. Given a European call and put option for the same underlying, strike price and time to maturity, and with no dividend yield, the sum of the absolute values of the delta of each option will be 1 – more precisely, the delta of the call (positive) minus the delta of the put (negative) equals 1. This is due to put–call parity: a long call plus a short put (a call minus a put) replicates a forward, which has delta equal to 1. If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta. , therefore: and . For example, if the delta of a call is 0.42 then one can compute the delta of the corresponding put at the same strike price by 0.42 − 1 = −0.58. To derive the delta of a call from a put, one can similarly take −0.58 and add 1 to get 0.42. Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset. Vega is not the name of any Greek letter. The glyph used is a non-standard majuscule version of the Greek letter nu (), written as . Presumably the name vega was adopted because the Greek letter nu looked like a Latin vee, and vega was derived from vee by analogy with how beta, eta, and theta are pronounced in American English. The symbol kappa, , is sometimes used (by academics) instead of vega (as is tau () or capital lambda (), : 315 though these are rare). Vega is typically expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls by 1 percentage point. All options (both calls and puts) will gain value with rising volatility. Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an at-the-money option straddle, for example, is extremely dependent on changes to volatility. Theta, , measures the sensitivity of the value of the derivative to the passage of time (see Option time value): the "time decay." The mathematical result of the formula for theta (see below) is expressed in value per year. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount an option's price will drop, in relation to the underlying stock's price. Theta is almost always negative for long calls and puts, and positive for short (or written) calls and puts. An exception is a deep in-the-money European put. The total theta for a portfolio of options can be determined by summing the thetas for each individual position. The value of an option can be analysed into two parts: the intrinsic value and the time value. The intrinsic value is the amount of money you would gain if you exercised the option immediately, so a call with strike $50 on a stock with price $60 would have intrinsic value of $10, whereas the corresponding put would have zero intrinsic value. The time value is the value of having the option of waiting longer before deciding to exercise. Even a deeply out of the money put will be worth something, as there is some chance the stock price will fall below the strike before the expiry date. However, as time approaches maturity, there is less chance of this happening, so the time value of an option is decreasing with time. Thus if you are long an option you are short theta: your portfolio will lose value with the passage of time (all other factors held constant). Rho, , measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk-free interest rate (for the relevant outstanding term). Except under extreme circumstances, the value of an option is less sensitive to changes in the risk-free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks. Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk-free interest rate rises or falls by 1.0% per annum (100 basis points). Lambda, , omega, , or elasticity is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing. It holds that . Epsilon, (also known as psi, ), is the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk. The dividend yield impact is in practice determined using a 10% increase in those yields. Obviously, this sensitivity can only be applied to derivative instruments of equity products. Gamma, , measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price. Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 (long call option) and 0 from −1 (long put option). The inverse is true for short options. Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM). Gamma is important because it corrects for the convexity of value. When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements. Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second order derivative of the option value, once to the underlying spot price and once to volatility. It is mathematically equivalent to DdeltaDvol, the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price. If the underlying value has continuous second partial derivatives, then , Charm or delta decay measures the instantaneous rate of change of delta over the passage of time. Charm has also been called DdeltaDtime. Charm can be an important Greek to measure/monitor when delta-hedging a position over a weekend. Charm is a second-order derivative of the option value, once to price and once to the passage of time. It is also then the derivative of theta with respect to the underlying's price. The mathematical result of the formula for charm (see below) is expressed in delta/year. It is often useful to divide this by the number of days per year to arrive at the delta decay per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate. Vomma, volga, vega convexity, or DvegaDvol measures second order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes. With positive vomma, a position will become long vega as implied volatility increases and short vega as it decreases, which can be scalped in a way analogous to long gamma. And an initially vega-neutral, long-vomma position can be constructed from ratios of options at different strikes. Vomma is positive for long options away from the money, and initially increases with distance from the money (but drops off as vega drops off). (Specifically, vomma is positive where the usual d1 and d2 terms are of the same sign, which is true when d1 < 0 or d2 > 0.) Veta or DvegaDtime measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time. It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day. Vera (sometimes rhova) measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate. The word 'Vera' was coined by R. Naryshkin in early 2012 when this sensitivity needed to be used in practice to assess the impact of volatility changes on rho-hedging, but no name yet existed in the available literature. 'Vera' was picked to sound similar to a combination of Vega and Rho, its respective first-order Greeks. This name is now in a wider use, including, for example, the Maple computer algebra software (which has 'BlackScholesVera' function in its Finance package). This partial derivative has a fundamental role in the Breeden-Litzenberger formula, which uses quoted call option prices to estimate the risk-neutral probabilities implied by such prices. For call options, it can be approximated using infinitesimal portfolios of butterfly strategies. Speed measures the rate of change in Gamma with respect to changes in the underlying price. This is also sometimes referred to as the gamma of the gamma: 799 or DgammaDspot. Speed is the third derivative of the value function with respect to the underlying spot price. Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio. Zomma measures the rate of change of gamma with respect to changes in volatility. Zomma has also been referred to as DgammaDvol. Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. Zomma can be a useful sensitivity to monitor when maintaining a gamma-hedged portfolio as zomma will help the trader to anticipate changes to the effectiveness of the hedge as volatility changes. Color, gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time. Color is a third-order derivative of the option value, twice to underlying asset price and once to time. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes. The mathematical result of the formula for color (see below) is expressed in gamma per year. It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate. Ultima measures the sensitivity of the option vomma with respect to change in volatility. Ultima has also been referred to as DvommaDvol. Ultima is a third-order derivative of the option value to volatility. If the value of a derivative is dependent on two or more underlyings, its Greeks are extended to include the cross-effects between the underlyings. Correlation delta measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. It is also commonly known as cega. Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying. Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying. Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying. See also: Black–Scholes model The Greeks of European options (calls and puts) under the Black–Scholes model are calculated as follows, where (phi) is the standard normal probability density function and is the standard normal cumulative distribution function. Note that the gamma and vega formulas are the same for calls and puts. For a given: |fair value ()| Under the Black model (commonly used for commodities and options on futures) the Greeks can be calculated as follows: |fair value ()| (*) It can be shown that Then we have: Some related risk measures of financial instruments are listed below. Main articles: Bond duration and Bond convexity In trading bonds and other fixed income securities, various measures of bond duration are used analogously to the delta of an option. The closest analogue to the delta is DV01, which is the reduction in price (in currency units) for an increase of one basis point (i.e. 0.01% per annum) in the yield (the yield is the underlying variable). See also Bond duration § Risk – duration as interest rate sensitivity. Analogous to the lambda is the modified duration, which is the percentage change in the market price of the bond(s) for a unit change in the yield (i.e. it is equivalent to DV01 divided by the market price). Unlike the lambda, which is an elasticity (a percentage change in output for a percentage change in input), the modified duration is instead a semi-elasticity—a percentage change in output for a unit change in input. See also Key rate duration. Bond convexity is a measure of the sensitivity of the duration to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative); it is then analogous to gamma. In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance. For a bond with an embedded option, the standard yield to maturity based calculations here do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and effective convexity are introduced. These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model (finance) § Interest rate derivatives. Main article: Beta (finance) The beta (β) of a stock or portfolio is a number describing the volatility of an asset in relation to the volatility of the benchmark that said asset is being compared to. This benchmark is generally the overall financial market and is often estimated via the use of representative indices, such as the S&P 500. An asset has a Beta of zero if its returns change independently of changes in the market's returns. A positive beta means that the asset's returns generally follow the market's returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their respective averages together. A negative beta means that the asset's returns generally move opposite the market's returns: one will tend to be above its average when the other is below its average. Main article: Fugit The fugit is the expected time to exercise an American or Bermudan option. Fugit is usefully computed for hedging purposes — for example, one can represent flows of an American swaption like the flows of a swap starting at the fugit multiplied by delta, and then use these to compute other sensitivities. ((cite journal)): Cite journal requires Step-by-step mathematical derivations of option Greeks

Post hoc tests a common follow up for both manova and anova. Furthermore, you do not have to select a transformation in a proc mixed analysis. Difference between anova and ancova with comparison chart. This approach allows researchers to examine the main effects of discipline and gender on grades, as well as the interaction between them, while statistically controlling for parental income. A multivariate analysis of variance manova could be used to test this hypothesis. The data is from an experiment to test the similarity of two testing methods. A research group wants to study the effectiveness of three. For example, if vocabulary size is measured at 2, 4, 6. The obvious difference between anova and a multivariate analysis of variance manova is the m, which stands for multivariate. First, convert the data to long format and make sure subject is a factor, as shown above. Multivariate analysis of variance manova is simply an anova with several. The term twoway gives you an indication of how many independent variables you have in. Anova is the analysis of variation between two or more samples while regression is the analysis of a relation between two or more variables. The second line specifies the variables in the data editor. In anova, differences among various group means on a singleresponse variable are studied. A mixed model analysis of variance or mixed model anova is the right data. The first two words before and after are the repeated measures variables and these words are the words used in the data editor. Use and interpret manova in spss hire a professional. In statistics, when two or more than two means are compared simultaneously, the statistical method used to make the comparison is called anova. An analysis of their anova, manova and ancova analyses by. Oct 11, 2017 difference between ttest and anova last updated on october 11, 2017 by surbhi s there is a thin line of demarcation amidst ttest and anova, i. Verma msc statistics, phd, mapsychology, masterscomputer application professorstatistics lakshmibai national institute of physical education, gwalior, india deemed university email. Is a manova an anova with two or more continuous response variables. Simple effects in mixed designs discovering statistics. Twoway mixed anova analysis of variance comes in many shapes and sizes. This is because the methods of drying are three nonrandomly chosen industrial processes, but the. That is to say, anova tests for the difference in means between two or more groups. In one of the repeated measures rm manova studies, the. For detailed information we refer to the reference manual. An anova analysis of variance is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups. For example, in brown 2007, i used an example anova to demonstrate how to. First, an anova is different from both a manova and mancova because an anova has only one dependent variable, while both a manova and mancova have multiple dependent variables. In basic terms, a manova is an anova with two or more continuous response variables. The mixed factor model given here is called the restricted version. While manova is the classical approach, the mixedmodel methodology, although by now implemented in all major statistical software packages, still is a relatively. Methods for testing omnibus overall hypotheses could include the anova test or an alternative to the. Jan 12, 2018 understand the difference between anova, ancova, manova and mancova in less than 4 minutes. Subjects will experience significantly greater sleep disturbances in the. Manova method for analyzing repeated measures designs. Mancova assumes that the observations are independent of one another, there is not any pattern for the selection of the sample, and that the sample is completely. Two factor mixed anova real statistics using excel. For example, we may conduct a study where we try two different textbooks. It sounds like you need to perform a posthoc test to determine which groups are significantly different from the others for variable a. I think that mixed anova is something of a special case of multilevel modeling. These distinctions are important because the test statistics used to test. Manova models several dependent variables simultaneously and you can include a variety of independent variables. The default approach to missing data in nearly all statistical packages is listwise. Analysis of the variance is a method of investigating the differences between two samples, or populations. Thus, in a mixeddesign anova model, one factor a fixed effects factor is a betweensubjects variable and the other a random effects factor is a withinsubjects variable. Five advantages of running repeated measures anova as a mixed. Estimates of the population variances and confidence intervals corresponding to the random effects, and, are calculated as in the two random factor model example 1. The distinctions between anova, ancova, manova, and. A mixed betweenwithin subjects analysis of variance was conducted to compare scores on the criminal social identity between violent and nonviolent offenders across three time periods time 1, time 2, and time 3. Nov 23, 2012 what is the difference between regression and anova. The manova sscp matrices require estimation of many bits, which can also eat up your power. A mixed model analysis of variance or mixed model anova is the right data analytic approach for a study that contains a a continuous dependent variable, b two or more categorical independent variables, c at least one independent variable that. This leads to factorial anova models, as for example discussed in 26. Difference between ttest and anova with comparison chart. Difference between regression and anova compare the. Repeated measures anova with spss oneway withinsubjects anova with spss one between and one within mixed design with spss repeated measures manova with spss how to interpret spss outputs how to report results 2 when the same measurement is made several. So, for example, a oneway anova might look at three classes of stu. Jan 11, 2017 knowing the difference between anova and ancova, will help you identify, which one should be used to compare the mean values of the dependent variable associated as a result of controlled independent variables, subsequent to the consideration of the affect of uncontrolled independent variables. In the concrete drying example, if analyzed as a twoway anova with interaction, we would have a mixed e. Repeated measures anova with spss oneway withinsubjects anova with spss one between and one within mixed design with spss repeated measures manova with spss. How can i test the assumptions for a mixed design manova, and how robust is it to. Twoway mixed anova with one withinsubjects factor and one betweengroups factor. Introduction to manova, manova vs anova n manova using r. Twofactor mixed manova with spss linkedin slideshare. Six differences between repeated measures anova and linear. While manova is the classical approach, the mixed model methodology, although by now implemented in all major statistical software packages, still is a relatively. One thing that makes the decision harder is sometimes the results are exactly the same from the two models and sometimes the results are vastly different. So whenever it says the univariate or multivariate mixed model in the. There is an unrestricted version where the test for factor b is done via. Difference between ancova and anova difference wiki. Manova before after by treat0 4 this initialises the anova command in spss. Spss procedure for mixed betweenwithin subjects anova click on plots click on withingroup factor time and move it into horizontal axis box click on betweengroup factor typcrim and move it into separate lines box click on add continue and ok. Twoway mixed anova using spss as we have seen before, the name of any anova can be broken down to tell us the type of design that was used. Mixedmultilevel multivariate models can also be run, for example, via mcmcglmm. This tutorial explains the differences between the statistical methods anova, ancova, manova, and mancova anova. Mixed design anova labcoat lenis real research the objection of desire problem bernard, p. So, it is a 2 x 2 x 2 x 2 mixed design manova, for which i have 2 betweensubject variables sex of the speaker and language and 2 withinsubject variables target sex and target attractiveness. There is a concern that images that portray women as sexually desirable objectify them. Power and sample size for manova and repeated measures. Anova approaches to repeated measures univariate repeatedmeasures anova chapter 2 repeated measures manova chapter 3 assumptions interval measurement and normally distributed errors homogeneous across groups transformation may help group comparisons estimation and comparison of group means. Multivariate models are a generalization of manova. Aug 11, 2014 the distinctions between anova, ancova, manova, and mancova can be difficult to keep straight. Comparing the sas glm and mixed procedures for repeated. The mixed model for multivariate repeated measures mediatum. That is to say, anova tests for the difference in means between two or more groups, while manova tests for the difference in two or more. In this tutorial some of the features of the bionumerics manova window will be illustrated using a sample data set see2. As mixed models are becoming more widespread, there is a lot of confusion about when to use these more flexible but complicated models and when to use the much simpler and easiertounderstand repeated measures anova. Again, we are especially interested in balanced and unbalanced manova models as extension. Multivariate models which your intended case is an example of can be run in r. However, the socalled mixed model approach is a viable alternative to analyzing this type of data, because its underlying statistical assumptions are equivalent to the manova model. The two most common types of anovas are the oneway anova and twoway anova. You simply determine the entire mean model and place all fixed effects on the model statement. Like anova, manova has both a oneway flavor and a twoway flavor. Anova theory is applied using three basic models fixed effects model, random effects model, and mixed effects model while regression is. The twoway part of the name simply means that two independent variables have been manipulated in the experiment. For example, we may conduct a study where we try two different textbooks, and we. It allows to you test whether participants perform differently in different experimental conditions. What to do if levenes test is significant in a mixed anova in spss. Anova one dv only manova 2 or more dvs intervalratio, 1 or more iv categorical anova determines the difference in means manova determines if the dvs get significantly affected by changes in the ivs. I am trying to do an anova anaysis in r on a data set with one within factor and one between factor. In multivariate analysis of covariance mancova, all assumptions are the same as in manova, but one more additional assumption is related to covariate. The manova extends this analysis by taking into account multiple continuous dependent variables, and bundles them together into a weighted linear combination or composite variable. Anova and manova are two different statistical methods used to compare means. What is the difference between a 2way anova and a manova. Is there any difference between manova and mixed anova. In statistics, a mixeddesign analysis of variance model, also known as a splitplot anova, is used to test for differences between two or more independent groups whilst subjecting participants to repeated measures. In this case, to be consistent with example 1, the target variable has been renamed from trans1 to newb. Anova and manova 1 introduction the central goal of an analysis of variance anova is to investigate the differences between the means of a set of quantitative variables across a number of groups. Effect size and eta squared university of virginia. Mixed multilevel multivariate models can also be run, for example, via mcmcglmm. Anova and manova are two statistical methods used to check for the differences in the two samples or populations. The different tests and plots present in the manova window will not be covered in detail in this tutorial. Manova to mancova when one or more more covariates are added to the mix. There are four multivariate test statistics, which can also complicate matters if you are not certain which one is the best for you to use. You can also use random effects anova in which you let each subject have his, or her, own intercept or intercept and slope. Eta2 is most often reported for straightforward anova designs that a are balanced i. The manova will compare whether or not the newly created combination differs by the different groups, or levels, of the independent variable. With twoway anova, you have one continuous dependent variable and two categorical grouping variables for the independent variables. Here, a mixed model anova with a covariatecalled a mixed model analysis of covariance or mixed model ancovacan be used to analyze the data. Difference between anova and manova compare the difference. However, the socalled mixedmodel approach is a viable alternative to analyzing this type of data, because its underlying statistical assumptions are equivalent to the manova model. Multivariate analysis of variance manova multiplegroup manova contrast contrast a contrast is a linear combination of the group means of a given factor. Comparing the sas glm and mixed procedures for repeated measures. The proc mixed mean specification is actually more general than the one in proc glm in two ways. Multivariate analysis of variance manova is simply an anova with several dependent variables. Multivariate analysis of variance manova introduction multivariate analysis of variance manova is an extension of common analysis of variance anova. Whether the data were analysed using univariate anova, manova, or mma. Cross validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. An obvious choice is mma also known as linear mixed models or. Manova and anova tell you that there is a significant effect while the post hoc tests help you map out the nature of those effectswhich groups. This idea was tested in an inventive study by philippe bernard. In manova, the number of response variables is increased to two or more. The core component of all four of these analyses anova, ancova, manova, and mancova is the first in the list, the anova.419 814 667 1561 401 725 448 621 148 321 416 774 1445 1173 285 514 623 466 1001 894 1118 587 1471 216 407 837 1544 1108 1447 1468 896 1070 1025 992 595 1167 321 75 215 478 485 1042 610 944 1185 291 189 1340 1435 1061

Last Revision March, 2011 Influences on Motion in Earth's Atmosphere An excerpt from a 2006 textbook, "Classical Mechanics" by R. Douglas Gregory, Cambridge University Press, sets up the challenge for this and the previous topic. It is: Although it is not known how to obtain analytical solutions for the differential equations that apply to a body encountering resistance that is proportional to the square of its velocity, it is quite possible to obtain practical computer assisted numerical solutions to such equations. The object is now to consider elements to be taken into account to model both horizontal and vertical motion of bodies in Earth's Atmosphere, 2-dimensional motion. Does the Atmosphere have much Effect? Imagine that a cannon projects a spherical maple wood ball vertically upward with a muzzle velocity v. Presume that maple wood has a density of ~755 kg/cubic metre. See for wood densities. The ball will encounter a drag force and a slight buoyant force due to the atmosphere as well as the force due to gravity just as did the falling body of the previous topic. The major difference between the two cases is that there is an initial velocity v and effect of buoyancy is now included. The downward force equation is: m * a = m * g - m' * g - Drag where m' is the mass of the displaced fluid The effect of the buoyancy is taken into account as a modification to g in the differential equation: a = g * (1-m'/m) - Drag/m or a = g' - Drag/m . Assume a radius of 4 centimetres, g =9.82022, and the mass of a ball as ~ 0.2024. Provide an upward velocity of 100 metres per second. Without air drag or buoyancy how high will the ball rise? This is a mechanics question to which the analytic answer is most often expressed as: v ^ 2 / (2 * g) ~= 509.15 metres. This analytic expression presumes that the force of gravity is uniform and does not include buoyancy, which is reasonable as in this case these effects are quite small. The 2D calculator, described and made available for use by the viewer in the last topic of Chapter 6 or on the upper row of tabs, provides an answer of ~ 509.95 metres without drag but with buoyancy included. With small air drags of 0.1 and 0.4 in this same case the calculator provides corresponding heights of ~ 309 metres and 163 metres, buoyancy included. Each height a far cry from the other and both well distanced from the case of no resistance. Drag Versus Reynolds Number Some values of the drag coefficient versus the Reynolds number are given in a document by H. Edward Donley That source provides numerical values for R and for its logarithm, log(R, 10), and then uses the logarithms for the plot. This writer noted an instance where R and its logarithm were not in agreement. The error appears to clearly be a typing error that has been corrected by this author to produce the plot following: Two additional points have been added to the chart, the diamond shape points. These are said to be average values where the lower value, 0.1, is said to apply to smooth spheres and the higher value, 0.4, is said to be suitable for rough spheres. The position of the lower diamond suggests that the Donley values apply to smooth spheres. For many purposes the drag coefficient for a sphere is taken to be sufficiently constant for Reynolds numbers in the range 1000 < R < 100,000. For a given radius of sphere the Reynolds number R is directly proportional to the velocity of the body and inversely proportional to the kinematic viscosity of the medium. The latter is quite dependent on the temperature of the medium. See Aside from the drag coefficient do we need to consider changes in air density, gravity, wind, spin and the like to get it right? Following is a quote from "Guns of World War II", available on the Web at the time of writing but since vanished. Even if, for centuries, ballistics was considered as a science, in (sic) facts, it is all but impossible, even today, to forecast the end-result of a fired shell. A projectile fired at an angle to the direction of gravitational force can be described as having two orthogonal components of velocity, one in the direction of the gravitational force and one at 90 In the absence of other forces such as wind, these components are presumed to remain in a plane. Only the first component will be influenced by gravity. Both components will encounter air resistance. Too determine the amount of resistance affecting each; the resistance due to their combined velocity must be distributed between them in accord with the ratio of their individual velocities to the combined velocity. Call the vertical component the y component and the horizontal component the x component. At the beginning of a numerical calculation step an object may have velocity components v with a combined velocity: V = (v The atmospheric resistance factor is presumed to have the form K * V^2 where K is determined by the shape of the object and the density of the atmosphere. In constructing a spreadsheet, the increments to v for an increment of time Dt become: = -Dt * (v / V) *K * V^2 and = -Dt * (v / V) *K * V^2 - Dt * g' Elevation and Maximum Range At the beginning of a projectile flight, the initial velocity v is resolved into its x, y components, employing radian measure: *sin(θ*π/180) and v The elevation angle θ is with respect to the horizontal. As time progresses these components will change in value as gravity and the resistance of the medium take their toll on the motion. Use the 2D calculator link given in the last topic of Chapter 6, to choose the elevation that will provide maximum range for a 4.0 cm. wooden ball projected at 100 metres per second. Do this for drag coefficients of 1E-10, 0.1, and 0.4. Use a step size of 0.02 , ~1020 m); (42.5 , ~530 m); (42 Once again, air resistance matters. The smoothness of the ball plays a very significant role! In the next topic further reality is added to model motion through Earth's atmosphere by introducing the Standard Atmosphere and the variation in gravitational attraction with altitude.

3D Simulation of Self-Compacting Concrete Flow Based on MRT-LBM. As a highly flowable concrete, the self-compacting concrete does not require any vibration during casting processes and has been considered as "the most revolutionary development in concrete construction for several decades" . Although SCC has been successful in plenty of applications, many problems are encountered during construction because of aggregate segregation, air voids, and the improper filling of formworks, and most of these problems are closely related to the flowability of SCC. Therefore, it is important to well understand and accurately predict the flow characteristics for the success of SCC casting. However, the accurate prediction of the SCC flowing behavior is a big challenge, especially in the case of heavy reinforcement, complex formwork shapes, and large size of aggregates. In this regard, the numerical modeling of SCC is an indispensable and inexpensive approach, not only as a tool for form filling prediction but also in terms of determination of fresh concrete properties, mix design, and casting optimization. Nowadays, the numerical modeling of fresh concrete flow has gained importance, and it is becoming an important tool for the prediction and optimization of casting processes [2, 3]. The modeling of fresh SCC is not a trivial task because it involves a free-surface flow of a dense suspension with a wide range of particle sizes and shows non-Newtonian flow behavior. A complete description of SCC from the cement particles to coarse aggregates is impossible with any computer model since accounting for broad particle size and shape distributions exceeds the computational limits of even the best supercomputers . The fresh SCC can be treated as a one-phase fluid and simulated using CFD because it is flowable and the amount of coarse particles is lower than that in conventional concrete. In terms of rheology, the fresh SCC can be approximated as one of the known non-Newtonian materials such as the Bingham or Herschel-Bulkley fluid. In case if the high shear rates are likely to occur in the casting processes, the shear-thickening behavior can happen, and the Herschel-Bulkley model could be a more suitable one. At the end of the casting process, the shear rates are rather low and the yield stress practically dominates the flow. Therefore, the material can be modeled as a Bingham fluid to predict the final shape. At present, many numerical techniques have been developed to model the SCC flow by assuming it as a homogeneous viscous fluid and using either the mesh-based methods such as the finite volume method (FVM) and the finite element method (FEM) or the meshless methods like the smoothed particle hydrodynamics (SPH) and the lattice Boltzmann method (LBM). Due to the fact that the flow of SCC is a typical free-surface flow, the treatment of the interface and its position represents another important numerical modeling issue. In this regard, the meshless methods have an advantage over the mesh-based methods. As a meshless Lagrangian method, the SPH method is good at modeling Newtonian and non-Newtonian flows with a free surface and has been chosen for simulating the SCC flow by Kulasegaram et al. , Lashkarbolouk et al. , Qiu , Abo Dhaheer et al. , and Wu et al. . However, the SPH method has difficulties in solving problems with complex solid boundary conditions. For these reasons, many researchers have made much effort to develop an alternative method called LBM for modeling the SCC flow due to the fact that it is easy for coding, intrinsically parallelizable, and applicable to complex geometries straightforwardly. For example, Svec et al. and Leonardi et al. simulated the fresh SCC as the non-Newtonian fluid based on the singlerelaxation-time (SRT) LBM. Although the single-relaxationtime (SRT) model has been successful in many applications, it is prone to numerical instability in complex flows . To overcome these difficulties, the multiple-relaxation-time (MRT) model proposed by d'Humieres et al. is useful to stabilize the solution and to obtain satisfactory results because the MRT model allows the usage of an independently optimized relaxation time for each physical process . Therefore, Chen et al. solved Bingham fluids by using the MRT model, but the free surface is not considered in their model. In the present work, we will develop a 3D multiple-relaxation-time LBM with a mass tracking algorithm representing the free surface to simulate the flow of fresh SCC. 2. Mathematical Formulations 2.1. Multiple-Relaxation-Time LBM. In this paper, the SCC flow is solved based on the lattice Boltzmann method which is considered as a very attractive alternative to the traditional CFD, especially in problems with complex boundary conditions. In the LBM, a finite number of velocity vectors [e.sub.[alpha]] are used to discretize the velocity space, and the fluid motion is described by a particle distribution functionf(x, [e.sub.[alpha]], t) which is the probability density of finding particles with velocity [e.sub.[alpha]] at a location x and at a given time t. The LB equations can recover the continuum Navier-Stokes equations by means of the Chapman-Enskog expansion if a proper set of discrete velocities was employed [17, 19]. The D3Q27 (3 dimensions and 27 velocities) discrete velocity model illustrated in Figure 1 was used in this study. The particle velocity vectors [e.sub.[alpha]] for this lattice model are given by [mathematical expression not reproducible], (1) where c = [delta]x/[delta]t, with [delta]x and [delta]t being the lattice spacing and the time step, respectively. A discretization of the Boltzmann equation in time and space leads to the lattice Boltzmann equation [20, 21]: [f.sub.[alpha]] (x + [e.sub.[alpha]][delta]t, t + [delta]t) - [f.sub.[alpha]] (x, t) = [[LAMBDA].sub.[alpha]j][[f.sub.j](x, t) - [f.sup.eq.sub.j] (x, t)] + [F.sub.[alpha]][delta]t, (2) where [f.sub.[alpha]] is the distribution function of particles moving with velocity [e.sub.[alpha]], [[LAMBDA].sub.[alpha]j] is the collision matrix, [f.sup.eq] is the equilibrium distribution function, and [F.sub.[alpha]] is the external force. The equilibrium distribution function [f.sup.eq] is obtained using the Taylor series expansion of the Maxwell-Boltzmann distribution function with velocity u up to second order. It can be written as [mathematical expression not reproducible], (3) where [rho] is the fluid density, u is the fluid velocity, and the sound speed is [c.sub.s] = c/[square root of (3)]. The weight coefficients for D3Q27 are given by [mathematical expression not reproducible]. (4) The components of F are given as [F.sub.[alpha]] = [w.sub.[alpha]][rho][[e.sub.[alpha]] x a/[c.sup.2.sub.s]], (5) where a is the acceleration. The relaxation process has major influence on the physical fidelity as well as numerical stability. For the single-relaxation-time (SRT) model, the collision matrix is [[LAMBDA].sub.[alpha]j] = -1/[tau][[delta].sub.[alpha]j], where [[delta].sub.[alpha]j] is the Kronecker symbol and t is the relaxation time which is related to the kinetic viscosity by v = [c.sup.2.sub.s]([tau] - 1/2)[delta]t. For the multiple-relaxation-time LBM, the collision matrix A can be written as [LAMBDA] = -[M.sup.-1]SM, (6) where the linear transform matrix M is a 27 x 27 matrix. The diagonal matrix S may be written as S = diag (0, 0, 0, 0, s4, s5, s5, s7, s7, s7, s10, s10, s10, s13, s13, s13, s16, s17, s18, s18, s20, s20, s20, s23, s23, s23, s26) with s4 = 1.54, s5 = s7 = 1/t, s10 = 1.5, s13 = 1.83, s16 = 1.4, s17 = 1.61, s18 = s20 = 1.98, and s23 = s26 = 1.74. The macroscopic density [rho] and velocity u are computed by [mathematical expression not reproducible]. (7) The pressure p is related to the density by p = [rho][c.sup.2.sub.s] (8) 2.2. Free Surface Modeling. For the modeling of the liquid-gas interface, the most straightforward way is to track all the phases, for example, liquid and gas. Such a method has the highest accuracy at the expense of high computational costs. The mass tracking algorithm without considering the gas phase is employed in the present study due to the fact that it is simple, fast, and accurate. In this algorithm, the fluid domain is divided into liquid, interface, and gas nodes (Figure 2). The liquid and interface nodes are active and solved by the LBM, and the remaining gas nodes are inactive without evolution equation. Liquid and gas nodes are never directly connected but through an interface node. The adopted mass tracking algorithm is applied directly at the level of the LBM, so the algorithm mimics the free surface by modifying the particle distributions. An additional macroscopic variable for the mass m (x, t) stored in a node is required and defined as [mathematical expression not reproducible]. (9) The mass is calculated by m(x, t + [delta]t) = m(x, t) + [summation over ([alpha])][k.sub.[alpha]] [f.sub.[bar.[alpha]]](x + [e.sub.[alpha]] [delta]t, t) - [f.sub.[alpha]](x, t)], (10) where [f.sub.[bar.[alpha]]] and [f.sub.[alpha]] are the particle distributions with opposite directions and [mathematical expression not reproducible]. (11) The interface node becomes a fluid node when the mass reaches its density with m (x, t) = [rho] (x, t) and vice versa; the interface node becomes a gas node when the mass drops down to zero with m (x, t) = 0. 2.3. Modified Herschel-Bulkley Model. In this study, the fresh SCC is assumed as viscoplastic fluids to consider its non-Newtonian behavior. Among the constitutive relations of viscoplastic fluids, the Herschel-Bulkley model is probably the most commonly used because of its simplicity and flexibility. The standard Herschel-Bulkley model is described by [mathematical expression not reproducible], (12) where [??] is the shear rate ([s.sup.-1]), a is the stress (Pa), [[sigma].sub.y] is the yield stress (Pa), and k is the consistency index (Pa x [s.sup.n]); n is a measure of the deviation of the fluid from Newtonian (the power-law index). For a fluid with n > 1, the effective viscosity increases with shear rate, and the fluid is called shear-thickening or dilatant fluid. For a fluid with 0 < n < 1, the effective viscosity decreases with shear rate, and the fluid is called shear-thinning or pseudoplastic fluid. The standard Hershel-Bulkley model becomes discontinuous at less shear rates and causes instability during numerical solution due to the fact that the non-Newtonian viscosity becomes unbounded at small shear rates. In order to overcome such discontinuity, the standard Herschel-Bulkley model can be modified as [mathematical expression not reproducible], (13) where [mu] is the apparent viscosity of fluid, and the yielding viscosity [[mu].sub.y] can be estimated from experimental rheological curves and can be assumed to be the slope of the line of shear stress versus shear rate curve before yielding. The modified Herschel-Bulkley model combines the effects of Bingham and power-law behavior in a fluid. For low strain rates, the material acts like a very viscous fluid with yield viscosity [[mu].sub.y]. As the strain rate increases and the yield stress threshold [[sigma].sub.y] is passed, the fluid behavior is described by a power law. 3. Numerical Examples and Validation In this section, the developed MRT-LBM is applied to simulate the slump flow of fresh SCC to validate the capability of the proposed model in modeling SCC flow. And the model is also used to simulate the passing ability and the filling ability of SCC by using an enhanced L-box test. 3.1. Slump Test. The slump flow is the most commonly used test in SCC technology. It measures flow spread and optionally the flow time [t.sub.50]. The numerical simulation of the slump test in our study is based on a laboratory experiment by Huang , Figure 3(a). The dimensions of the cone are 300 mm height, 200 mm lower diameter, and 100 mm upper diameter, Figure 3(b). The SCC used in this study has the following properties: yield stress = 100 Pa, plastic viscosity = 50 Pa x s, and density = 2300 kg/[m.sup.3]. In the standard form of the slump test, only the final spread value of the slump is measured to evaluate the flowing behavior of the SCC. The numerical simulation of the slump flow is performed based on the developed MRT-LBM. In our simulation, the lattice spacing is set to 0.01 m for the LBM discretization, Figure 3(c). The value of the yielding viscosity was chosen to be 1000 times higher than the value of the yield stress, with the power-law index being n = 1. Figure 4 presents some snapshots at different instants of the material shape with the velocity magnitude. In Figure 5, experimental and numerical results for the material shape at the end of the flow are compared. It can be seen from this figure that the calculated slump flow spread diameter is about 624 mm which shows a perfect match between the experimental and simulated flow distance that can be observed in terms of the maximum spread distance. This proves the correctness of the proposed model and its ability to simulate the free surface flow of fresh SCC. 3.2. Enhanced L-Box Test. The L-box test is generally used for assessing the passing ability and filling ability of fresh SCC in confined spaces. In this section, a laboratory experiment of an enhanced L-box test by Huang was simulated using the developed MRT-LBM. The enhanced L-box consists of the concrete reservoir, slide gate, and horizontal test channel with four ball obstacles (Figure 6(a)). The vertical section is filled with concrete, and subsequently, the gate is lifted to allow concrete to flow into the horizontal section. When the flow stops, one measures the reached height of fresh SCC after passing through the specified gaps of balls and flowing within a defined flow distance. With this reached height, the passing or blocking behavior of SCC can be estimated. The dimensions of the enhanced L-box test are shown in Figure 6(b). In our simulation, the SCC is placed at the middle upper part of the vertical container, which is then suddenly released, and the concrete begins to spread under the gravity loading, Figure 6(c). The numerical simulation is performed based on the developed MRT-LBM and the lattice spacing is set to 0.01 m for the LBM discretization. The value of the yielding viscosity was chosen to be 1000 times higher than the value of the yield stress, with the power-law index being n = 1. SCC used for this study was the same concrete as in the slump test with the following properties: yield stress = 50 Pa, plastic viscosity = 50 Pa x s, and density = 2300 kg/[m.sup.3]. The total volume of the material is V = 13 L. Some snapshots of the SCC shape with the velocity magnitude at different instants are presented in Figure 7. Figure 8 compares the experimental and numerical results for the final spread of the SCC flow. The simulated flow spread agrees well with the experimental results. This in turn further validates the proposed model in modeling the passing ability and filling ability of fresh SCC in confined spaces. A slight discrepancy in the shape of the SCC can be noted which may be due to the fact that the balls are perfectly touched by the lateral wall in simulation but there are existing gaps between them in the experiment. In this paper, a multiple-relaxation-time LBM with a D3Q27 discrete velocity model for modeling the flow behavior of fresh SCC was proposed. The rheology of the fresh SCC was approximated as a non-Newtonian material using the modified Herschel-Bulkley fluid model. The free surface was modeled based on the mass tracking algorithm which is a simple and fast algorithm that conserves the mass precisely. The numerical simulation of slump flow of fresh SCC was first performed to validate the capability of the proposed model in modeling SCC flow. And then, the model was further used to simulate the passing ability and the filling ability of fresh SCC in confined spaces based on an enhanced L-box test. The simulated results agree well with the corresponding experimental data in the published literature. This proves that the proposed MRT-LBM is suitable for numerical simulations of the fresh SCC flows. It should be noted that the main problem for the application of the present model to real engineering problems is its high computational cost. Therefore, further investigations might be needed to consider hardware acceleration and parallel computing to make the proposed model more useful and versatile. Conflicts of Interest The authors declare that they have no conflicts of interest. The authors are grateful for funding from the National Natural Science Foundation of China (Grant nos. 11772351 and 51509248) and the Scientific Research and Experiment of Regulation Engineering for the Songhua River Mainstream in Heilongjiang Province, China (Grant no. SGZL/KY-12). EFNARC, Specification and Guidelines for Self-Compacting Concrete, Scientific Research Publishing, Surrey, UK, 2002. N. Roussel, M. R. Geiker, F. Dufour, L. N. Thrane, and P. Szabo, "Computational modeling of concrete flow: general overview," Cement and Concrete Research, vol. 37, no. 9, pp. 1298-1307, 2007. N. Roussel and A. Gram, Simulation of Fresh Concrete Flow, State-of-the Art Report of the RILEM Technical Committee 222-SCF, RILEM State-of-the-Art Reports, vol. 15, Springer, Netherlands, 2014. K. Vasilic, M. Geiker, J. Hattel et al., "Advanced Methods and Future Perspectives," in Simulation of Fresh Concrete Flow, N. Roussel and A. Gram, Eds., vol. 15pp. 125-146, Springer, Netherlands, 2014. N. Roussel, "Correlation between yield stress and slump: comparison between numerical simulations and concrete rheometers results," Materials and Structures, vol. 39, no. 4, pp. 501-509, 2006. B. Patzak and Z. Bittnar, "Modeling of fresh concrete flow," Computers and Structures, vol. 87, no. 15, pp. 962-969, 2009. S. Shao and E. Y. M. Lo, "Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface," Advances in Water Resources, vol. 26, no. 7, pp. 787-800, 2003. S. Kulasegaram, B. L. Karihaloo, and A. Ghanbari, "Modelling the flow of self-compacting concrete," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 35, no. 6, pp. 713-723, 2011. H. Lashkarbolouk, M. R. Chamani, A. M. Halabian, and A. R. Pishehvar, "Viscosity evaluation of SCC based on flow simulation in L-box test," Magazine of Concrete Research, vol. 65, no. 6, pp. 365-376, 2013. L.-C. Qiu, "Three dimensional GPU-based SPH modelling of self-compacting concrete flows," in Proceedings of the third international symposium on Design, Performance and Use of Self-Consolidating Concrete, pp. 151-155, RILEM Publications S.A.R.L, Xiamen, China, June 2014. M. S. Abo Dhaheer, S. Kulasegaram, and B. L. Karihaloo, "Simulation of self-compacting concrete flow in the J-ring test using smoothed particle hydrodynamics (SPH)," Cement and Concrete Research, vol. 89, pp. 27-34, 2016. J. Wu, X. Liu, H. Xu, and H. Du, "Simulation on the self-compacting concrete by an enhanced Lagrangian particle method," Advances in Materials Science and Engineering, vol. 2016, Article ID 8070748, 11 pages, 2016. O. Svec, J. Skocek, H. Stang, M. R. Geiker, and N. Roussel, "Free surface flow of a suspension of rigid particles in a non-Newtonian fluid: a lattice Boltzmann approach," Journal of Non-Newtonian Fluid Mechanics, vol. 179-180, pp. 32-42, 2012. A. Leonardi, F. K. Wittel, M. Mendoza, and H. J. Herrmann, "Coupled DEM-LBM method for the free-surface simulation of heterogeneous suspensions," Computational Particle Mechanics, vol. 1, no. 1, pp. 3-13, 2014. K. N. Premnath and S. Banerjee, "On the three-dimensional central moment Lattice Boltzmann method," Journal of Statistical Physics, vol. 143, no. 4, pp. 747-794, 2011. D. d'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo, "Multiple-relaxation-time lattice Boltzmann models in three dimensions," Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 360, no. 1792, pp. 437-451, 2002. P. Lallemand and L.-S. Luo, "Theory of the lattice Boltzmann method: dispersion, isotropy, Galilean invariance, and stability," Physical Review E, vol. 61, no. 6, pp. 6546-6562, 2000. S. G. Chen, C. H. Zhang, Y. T. Feng, Q. C. Sun, and F. Jin, "Three-dimensional simulations of Bingham plastic flows with the multiple-relaxation-time lattice Boltzmann model," Science China Physics Mechanics and Astronomy, vol. 57, no. 3, pp. 532-540, 2014. S. Chen and G. D. Doolen, "Lattice Boltzmann method for fluid flows," Annual Review of Fluid Mechanics, vol. 30, no. 1, pp. 329-364, 1998. S. Succi, H. Chen, and S. Orszag, "Relaxation approximations and kinetic models of fluid turbulence," Physica A: Statistical Mechanics and its Applications, vol. 362, no. 1, pp. 1-5, 2006. Z. Guo, C. Zheng, and B. Shi, "Discrete lattice effects on the forcing term in the lattice Boltzmann method," Physical Review E, vol. 65, no. 4, p. 046308, 2002. C. Korner, M. Thies, T. Hofmann, N. Thurey, and U. Rude, "Lattice Boltzmann model for free surface flow for modeling foaming," Journal of Statistical Physics, vol. 121, no. 1-2, pp. 179-196, 2005. P. Prajapati and F. Ein-Mozaffari, "CFD investigation of the mixing of yield pseudo plastic fluids with anchor impellers," Chemical Engineering & Technology, vol. 32, no. 8, pp. 1211-1218, 2009. M. S. Huang, The Application of Discrete Element Method on Filling Performance Simulation of Self-Compacting Concrete in RFC, Ph.D. thesis, Tsinghua University, Beijing, China, 2010, in Chinese. Liu-Chao Qiu and Yu Han College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, China Correspondence should be addressed to Liu-Chao Qiu; email@example.com and Yu Han; firstname.lastname@example.org Received 21 June 2017; Accepted 2 November 2017; Published 28 January 2018 Academic Editor: Ying Li Caption: Figure 1: Three-dimensional twenty-seven particle velocity (D3Q27) lattice. Caption: Figure 2: Mass tracking algorithm scheme. Caption: Figure 3: Slump test. (a) Experiment setup , (b) model size, and (c) LBM discretization. Caption: Figure 4: The velocity distribution during the slump test. (a) t = 0.1s, (b) t = 0.2 s, (c) t = 0.3 s, (d) t = 0.5 s, (e) t = 1.0 s, and (f) t = 2.0 s. Caption: Figure 5: Final spread of the slump test. (a) Experiment and (b) simulation. Caption: Figure 6: The enhanced L-box test. (a) Experiment setup , (b) model size, and (c) LBM discretization. Caption: Figure 7: The velocity distribution of the enhanced L-box test. (a) t = 0 s, (b) t = 25 s, (c) t = 50 s, (d) t = 75 s, (e) t = 100 s, (f) t = 125 s, (g) t = 150 s, (h) t = 175 s, (i) t = 200 s, (j) t = 225 s, (k) t = 250 s, and (l) final shape. Caption: Figure 8: Final shape of SCC in the enhanced L-box test. (a) Experiment and (b) simulation. |Printer friendly Cite/link Email Feedback| |Title Annotation:||Research Article| |Author:||Qiu, Liu-Chao; Han, Yu| |Publication:||Advances in Materials Science and Engineering| |Date:||Jan 1, 2018| |Previous Article:||Chloride-Binding Capacity of Portland Cement Paste Blended with Synthesized [CA.sub.2] (CaO*2[Al.sub.2][O.sub.3]).| |Next Article:||Fatigue Tests of Concrete Slabs Reinforced with Stainless Steel Bars.|

Here you will find our range of challenging math problem worksheets which are designed to give children the opportunity to apply their skills and knowledge to solve a range of longer problems. These problems are also a great way of developing perseverance and getting children to try different approaches in their math. Children will enjoy completing these Math games and Free 4th Grade Math worksheets whilst learning at the same time. This lesson will provide help and guidance that will help solve these types of problems. Read the tips and guidance and then work through the two multi-step word problems in this lesson with your children. Try the two worksheets that are listed within the lesson you will also find them at the bottom of the page. Solving Multi-Step Word Problems Word problems are fun and challenging to solve because they represent actual situations that happen in our world. In any word problem, the true challenge is deciding which operation to use. In multi-step word problems, there may be two or more operations, and you must solve them in the correct order to be successful. Since word problems describe a real situation in detail, the question being asked can get lost in all the information, especially in a multi-step problem. Before rushing to solve the problem, it is worth your time to slow down and clarify your understanding. Be sure you know what is being asked, what you already know from the problem, and what you need to know in order to solve the question being asked. Use a highlighter on written problems to identify words that tell you what you are solving, and give you clues about which operations to choose. Make notes in the margins by these words to help you clarify your understanding of the problem. Example 1 Discuss with your children how one danger when solving this type of problem is stopping too soon — after answering only the first part of the problem. Steven is reading a book that has pages. He read 35 pages on Monday night, and 40 pages on Tuesday night. How many pages does he have left to read? The danger is you might think you can stop there. You will have to take another step to get there. Steven has read 75 pages so far, but you are being asked what he has left to read, not what he has already read. To get your final answer, you must subtract what he has read from the total pages to be read: Steven has pages left to read. Instead of just writingwrite: Whenever you finish a math problem of any kind, always go back to the original problem. My answer is reasonable because it tells how many pages Steven still needs to read. I added together 35 and 40 to find out the total pages he had already read, and subtracted from the total pages in the book. My answer makes sense. You can tell that there are lots of things to remember with a multi-step word problem, even when the problem itself is relatively easy. The better you understand how to solve them, the more fun they are to solve. Example 2 You might find that this problem is more difficult that the one above. How much profit did he make? A dozen boxes tells you he had 12 boxes. Each with 24 tells you the number of highlighters in each box. Repacked five boxes into packages of six highlighters each separates 5 sets of 24 away from the original 12 sets of The rest is the 7 sets of 24 still left after separating away the 5 sets. Profit is the amount earned from all sales, minus the amount spent to buy the highlighters. He separated 5 boxes of 24 away from the original 12 boxes, and made new packages with six highlighters in each package. He still had 7 of his original boxes of Then we must add together the two amounts he earned. The danger is stopping here, because it took so long to get to this point, that it feels like the end. Profit is the amount earned minus the amount spent to buy the highlighters. Remember, whenever you finish a math problem, always go back to the original problem. Profit should be smaller than money earned, since the cost of the highlighters has to be taken out. The first problem we did was relatively simple, while the second was much more complicated.Math Problem Solving 4th Grade Math Third Grade Grade 3 Math Lessons Word Problems 3rd Grade Fraction Word Problems Teaching Math Teaching Ideas Forward Key words only take your students so far when solving word problems. Mathematics Practice Tests Do you want to know what taking the Mathematics portion of the PARCC Assessment is like? A practice test for each grade is available below for you to use to familiarize yourself with the kinds of items and format used for the tests. Find great deals on eBay for 4th grade math benjaminpohle.com Savings · >80% Items Are New · World's Largest Selection · We Have EverythingCategories: Books, Other Books, Textbooks & Educational Books and more. 4th Grade Math Problems. It’s time for the fourth graders to master math! Give them our fun, free 4th grade math problems to solve and watch their grades go up! Equality in Equations ‘Equality in Equations’ is a free equation worksheet that requires kids to analyze, comprehend and apply their understanding of addition and subtraction skills. Mathematics. Grade 4. Maryland College and Career Ready Curriculum Frameworks for Grade 4 Mathematics, November, Page 2 of Maryland College and Career Ready Curriculum Frameworks for Grade 4 Mathematics, November, Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday.

Geometric Mechanics & Dynamical Systems “Formal perfection, in mathematics, is never negligible; it is even a necessary ingredient for progress. However, it seems that the classical formalization of analytical mechanics has overlooked an important part of Lagrange's ideas. Without doubt this happened because the mathematics of the first part of the 19th century did not have the necessary scope; also because the successes of the theory (first in celestial mechanics and later in statistical mechanics) hid the necessity of questioning categories so classical that they seemed natural. One had to wait until the 20th century to learn that the words time, space, and matter do not have any direct physical meaning, but are only formal symbols of revisable physical theories, and nowadays are outdated. Analytical mechanics is not an outdated theory, but it appears that the categories which one classically attributes to it such as configuration space, phase space, Lagrangian formalism, Hamiltonian formalism, are, simply because they do not have the required covariance; in other words, because these categories are in contradiction with Galilean relativity. A fortiori, they are inadequate for the formulation of relativistic mechanics in the sense of Einstein.” - J M Souriau, “Structure of Dynamical Systems” Geometric Mechanics is Classical Mechanics formulated in the language of modern Differential Geometry. Of course, while Lagrangian Mechanics, to a certain extent, retains the standard Differential Geometric form, Hamiltonian Mechanics alters it significantly enough to give it a special name, -Symplectic Geometry. Lagrangian Mechanics itself initially takes the form of Riemannian Geometry as was first clearly demonstrated by Hertz in his “Principles of Mechanics.” The departure from Riemannian to Symplectic takes place via the Legendre transform (or fibre derivative, in symplectic language) that maps the Lagrangian to the Hamiltonian. When the transform is regular (or rather, hyperregular), the map in a sense, takes the “velocities” to the “momenta.” Otherwise, there arise singular constraints and need to be treated quite differently. John L. Synge was one of the first to cast Lagrangian Mechanics into a geometric framework. In his beautiful Handbuch der Physik encyclopedia article, he gave a treatment of Lagrangian Mechanics as Riemannian Geometry. But the modern, qualitative-quantitative form of Mechanics really took off with Henri Poincare’s introduction of the qualitative methods of Topological Analysis into Mechanics. His methods were characterised by the global geometric point of view. He treated a dynamical system as a vector field on the phase space of the system. A solution then corresponded to a smooth curve tangent at each of its points to the vector based at that point. This global point of view was capable of giving the complete information of the dynamical system. The manifold or bundle that arose could then be studied and the necessary structures like the metric, connections, almost-tangent structures, almost-complex structures as have now been developed, could be imposed. After Poincare, George Birkoff contributed immensely to the development of Dynamical systems, with the publication of his American Mathematical Society monograph, “Dynamical Systems,” one of the most illuminating books on the subject. But there was another immense source of ideas and methods infiltrating into Mechanics. It was that of Elie Cartan. Elie Cartan’s work revolutionised Mechanics like nothing before. His powerful Exterior Calculus and classification of Lie Algebras and Lie Groups changed the face of Mechanics forever. These twin streams of ideas and methods, from Poincare and Cartan were taken up by a new generation of mathematicians and the very fertile field of Geometric Mechanics was born. Presently, Geometric Mechanics has matured to take the shape of one of the most extensively developed and developing fields of Mathematics and Mathematical Physics. Source Books & Links - V I Arnold - Mathematical Methods of Classical Mechanics: The most straightforward and enjoyable introduction to Geometric Mechanics, and indeed, to modern Differential Geometric and Symplectic methods in Physics. - Abraham and Marsden - Foundations of Mechanics: The most extensive, rigorous, complete and beautiful treatment of Geometric Mechanics, Dynamical Systems and Topological Dynamics in the field. - G Marmo, Salaten, Simoni, Vitale - Dynamical Systems, A Differential Geometric Approach to Symmetry and Reduction: The most insightful and beautiful treatment of the subject with an orientation towards physical intuition. Henri Poincare: The last “Universalist,” as E. T. Bell calls him in his “Men of Mathematics,” and perhaps the greatest mathematician of the 20th Century sharing that position only with David Hilbert. Poincare is the real founder of modern Geometric Mechanics with his introduction of qualitative methods into the analysis of dynamical systems. In his researches on Celestial Mechanics, he was led to introduce the enormously insightful concept of the phase portrait. Almost in parallel, he initiated Analysis Situs, that developed and matured into modern Topology. As he writes, “As for me all all the various journeys, one by one I found myself engaged, were leading me to Analysis Situs.” He explained what he meant by Analysis Situs as follows. “L’ Analysis Situs est la science qui nous fait connaitre les proprettes qualitatives des figures géométriques non seulement dans l’ espace ordinaire, mais dans l’ espace a plus trois dimensions. L’ Analysis Situs a trois dimensions est pour nous une connaissance presque intuitive, L’ Analysis Situs a plus de trios dimensions au contraire des difficultés énormes; il faut pour tenter de les surmounter être bien persuade de l’ extreme importance de cette science.” (“Analysis Situs is a science which lets us learn the qualitative properties of geometric figures not only in the ordinary space, but also in the space of more than three dimensions. Analysis Situs in three dimensions is almost intuitive knowledge for us. Analysis Situs in more than three dimensions presents, on the contrary, enormous difficulties, and to attempt to surmount them, one should be persuaded of the extreme importance of this science. If this importance is not understood by everyone, it is because everyone has not sufficiently reflected upon it.”) The modern theory of Dynamical Systems, singularities and bifurcations, Chaos, and a host of topological and differential geometric constructions are an offshoot of Poincare’s ideas. George D Birkhoff: Birkhoff proved Poincare’s Geometric Theorem almost immediately as it was posed by Poincare. The theorem may be stated in a simple form as follows: Let us suppose that a continuous one-to-one transformation T takes the ring R, formed by concentric circles Ca and Cb of radii a and b respectively (a > b > 0), into itself in such a way as to advance the points of Ca in a positive sense, and the points of Cb in the negative sense, and at the same time to preserve areas. Then there are at least two invariant points. Birkhoff’s American Mathematical Society monograph, “Dynamical Systems,” is a classic and one of the first books that is still worth reading for its elegant treatment of the subject. Constantin Caratheodory: Was David Hilbert’s successor at Gottingen University and a versatile mathematician and one of the most insightful developers of the field of Partial Differential Equations, the Calculus of Variations and Complex Function Theory. His 2-Volume monograph, “Calculus of Variations and Partial Differential Equations of First Order,” is the finest book on the Calculus of Variations and Partial Differential Equations and develops especially the Hamilton-Jacobi theory in all its detail. His book on Function theory is a classic. His approach to the second law of thermodynamics, captured by the famous “Caratheodory’s Theorem,” (as presented for example, in S. Chandrasekhar’s Introduction to the Study of Stellar Structure) is elegant and is taken as the starting point in several treatments of the subject. Jacques Hadamard: One of the greatest mathematicians of the 20th Century and perhaps the most influential next only to Poincare, Hilbert and Weyl, and in the same class as Elie Cartan, Hadamard inspired and guided an entire generation of mathematicians including Nicholas Bourbaki, Andre Weil, Jean Dieudonne, John Leray, Laurent Schwartz…Hadamard had an pervasive influence on modern mathematics and in Geometric Mechanics his name is associated with the Cartan-Hadamard theorem a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. Elie Cartan: One of the greatest mathematicians of all time, and unquestionably, the greatest geometer of the 20th Century, Elie Joseph Cartan’s influence on mathematics and mathematical thought has been all-pervading, from Differential Geometry, Topology, Lie Groups and Lie Algebras, Theory of Spinors (that he had introduced and developed earlier to Dirac), General Relativity (the famous Einstein-Cartan theory), the classification problem of Riemannian manifolds. His most important contribution being the exterior derivative that changed Calculus forever after Newton and Leibniz. The modern coordinate-free approach that pervades modern physics is due to him. He was also the first to cast Newtonian mechanics into geometric form. After Poincare, it was Elie Cartan who inspired an entire generation of mathematicians including that of Nicholas Bourbaki. In a sense, therefore, it is Elie Cartan who laid the structural foundations of Geometric Mechanics and Dynamical Systems. His obituary by Chern and Chevalier, “Elie Cartan and his Mathematical Work,” and biography by Akvis and Rosenfeld, Elie Cartan, is a most inspiring and instructive read. Cartan’s own works were not that easy to read. In Robert Bryant's words, “You read the introduction to a paper of Cartan and you understand nothing. Then you read the rest of the paper and still you understand nothing. Then you go back and read the introduction again and there begins to be the faint glimmer of something very interesting.” Nevertheless, his books, “Leçons sur les invariants intégraux” and “Leçons sur la géométrie des espaces de Riemann” are masterpieces of mathematical writing. Amalie Emmy Noether: One of the greatest mathematicians of the 20th Century and whose work forms the cornerstone of the foundations of Classical Mechanics, Field Theory and Gauge Theory. Indeed, it is impossible to imagine proceeding in any of these fields without encountering “Noether’s theorem.” Albert Einstein wrote of her, “In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius since the higher education of women began”. Noether’s first and second theorems are the starting point of field theory and of Singular Constraint Systems. Nina Byer’s two articles, “The Life and Times of Emmy Noether,” and “Emmy Noether’s Discovery of the Deep Connection between Symmetry and Conservation Laws,” are most instructive to read.

Research Article | Open Access R. G. G. Amorim, S. C. Ulhoa, J. S. da Cruz Filho, A. F. Santos, F. C. Khanna, "Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature", Advances in High Energy Physics, vol. 2020, Article ID 7032834, 8 pages, 2020. https://doi.org/10.1155/2020/7032834 Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized. The Wigner function formalism [1, 2] and noncommutative geometry play a fundamental role in the study of phase space quantum mechanics. The Wigner formalism enables a quantum operator, , defined in the Hilbert space, , to have an equivalent function of the type , in phase space , using the Moyal-product or star-product (å). Such a formalism leads to the classical limit of a quantum theory. In fact, quantum mechanics is a noncommutative theory whose representation in phase space is an object of debate. The opposite question, i.e., for a given classical function, what is its quantum counterpart? It is solved by using the Weyl transformation which is formulated independent of the phase space. In fact, it can be established within the configuration space of the generalized coordinates. The phase space has a well-defined physical meaning. The Hamiltonian function is naturally identified with the energy of the system. Establishing a field theory in phase space sheds light on some obscure points in quantum mechanics. For instance, the quantum symmetries are better understood in the symplectic structure of phase space which is similar to the role of Lorentz transformation in the covariant formulation of special relativity. This theoretical framework has to include a finite temperature in order to be suitable for experiments. The star product has been employed for different objectives. In particular, it has been used for development of a nonrelativistic quantum mechanics formalism in terms of a phase space using the Galilean symmetry representation . Thus, the Schrödinger equation is obtained. In this case, the wave function is a quasiprobability amplitude defined in phase space and the Wigner function is obtained in an alternative way, i.e., by using . The Dirac equation coupled with the electromagnetic field in phase space and applications has been obtained. Our goal is to explore the quasiprobability amplitude to study the effect of temperature using Thermo Field Dynamics (TFD) formalism [7–13] in a system for spin-1/2 particles. The principles of this theory are the duplication of the Fock space using the Bogoliubov transformations. The TFD formalism is used to study the Casimir effect, at zero and finite temperature. The scalar field in phase space has been studied , and some exclusive effects have been found at finite temperature. In addition, the Stefan-Boltzmann law for spin- particles in phase space is described in details. In Section 2, the symplectic Dirac field is introduced. In Section 3, the Thermo Field Dynamics formalism is presented. In Section 4, the Stefan-Boltzmann law is established and the Casimir effect for the Dirac field is calculated in phase space at zero and finite temperature. In the last section, some concluding remarks are presented. 2. Spin-1/2 Field in Phase Space A brief outline for spin-1/2 particles in phase space formalism is described. For this purpose, the following star operators in phase space are defined: which satisfy the Heisenberg commutation relation , with . The Poincaré algebra has the form The operators in Equations ((1)–(3)) are defined on a Hilbert space, , associated with the phase space . The operators and stand for translations, rotations, and boosts, respectively. Functions defined on the Hilbert space are defined as The Casimir invariants are and , where are Pauli-Lubansky matrices and is the Levi-Civita symbol. The Dirac equation in phase space is obtained using the invariant operator . It is defined as where are the Dirac matrices. The Lagrangian density for the Dirac equation is where and is the mass of the particle. The Wigner function provides the physical interpretation and is given as where the star product is defined by Using the Noether theorem in phase space , the energy-momentum tensor for the Dirac field is Then, the Green function, , is defined as which may be written as where . The propagator of the Dirac field is Taking , the expression is This leads to the Green function for the Dirac equation in phase space is defined as where is the modified Bessel function. It should be noted that due to the dependence on the Dirac matrices, the Green’s function has matrix properties itself. 3. Thermo Field Dynamics Formalism The Thermo Field Dynamics (TFD) is a thermal quantum field theory at finite temperature [7–13]. It has two basics elements: (i) doubling the degrees of freedom in a Hilbert space and (ii) the Bogoliubov transformation. The doubling of Hilbert space is given by the tilde () conjugate rules where the thermal space is , with being the standard Hilbert space and the tilde (dual) space. There is a mapping between the two spaces; i.e., the map between the tilde and nontilde operators is defined by the following tilde conjugation rules: with for bosons and for fermions. The Bogoliubov transformation corresponds to a rotation of the tilde and nontilde variables. Using the doublet notation, for fermions leads to where are creation operators, are destruction operators, and is the Bogoliubov transformation given by Taking ( with being the Boltzmann constant and the temperature), the thermal operators are written explicitly as These thermal operators satisfy the algebraic rules and other anticommutation relations are null. In addition, the quantities and are related to the Fermi distribution, i.e., such that . The parameter is associated with temperature, but, in general, it may be associated with other physical quantities. In general, a field theory on the topology with , is considered. are the space-time dimensions, and is the number of compactified dimensions. This establishes a formalism such that any set of dimensions of the manifold can be compactified, where the circumference of the th is specified by . The parameter is assumed as the compactification parameter defined by . The effect of temperature is described by the choice and . Any field in the TFD formalism may be written in terms of the parameter. As an example, the scalar field is considered. Then, the -dependent scalar field becomes where the Bogoliubov transformation is used. The -dependent propagator for the scalar field is where is the time-ordering operator. Using leads to the Green function where with being the Bogoliubov transformation and where is the scalar field propagator and is the scalar field mass. Here, is the complex conjugate of . It is important to note that the physical quantities are given by the nontilde variables. Then, the physical Green function is written as where is the generalized Bogoliubov transformation , where is the number of compactified dimensions, for fermions (bosons), denotes the set of all permutations with elements, and is the 4-momentum. In the next section, three different topologies are used : (i) the topology , where . In this case, the time axis is compactified in , with circumference ; (ii) the topology with , where the compactification along the coordinate is considered; and (iii) the topology with is used. In this case, the double compactification consists in time and the coordinate . Then, thermal effects are considered for the Casimir effect and Stefan-Boltzmann law. 4. Stefan-Boltzmann Law and Casimir Effect for the Dirac Field in Phase Space The Stefan-Boltzmann law is calculated by analyzing the energy-momentum tensor given as where It should be noted that the field is the Dirac field in phase space as a function of the variables (), i.e., . The vacuum expectation value of the energy-momentum tensor is The Dirac propagator in phase space is defined in Equation (16) as Then, the energy-momentum tensor has the form The vacuum average of the energy-momentum tensor in terms of -dependent fields becomes In order to obtain measurable physical quantities at finite temperature, a renormalization procedure is carried out. The physical energy-momentum tensor is defined as where Now, the Stefan-Boltzmann law and the Casimir effect in phase space are calculated at finite temperature. 4.1. Stefan-Boltzmann Law The study of the Stefan-Boltzmann law in phase space corresponds to a choice of the parameter . It is important to note that the parameter is the compactification parameter that is defined as . The temperature effect is described by the choice The generalized Bogoliubov transformation, Equation (31), for these parameters is The Green’s function for the Dirac field in phase space is where is a time-like vector. Then, the physical energy-momentum tensor is In order to calculate the Stefan-Boltzmann law, taking leads to This is the Stefan-Boltzmann law for the Dirac field in phase space. It is worth pointing out that the result is recovered by taking the limit of the momentum variable. This result in phase space is necessary to compare with experiments. In this sense, we can integrate over the momenta which explicitly yield and take the limit; then, the only remaining part is the factor ofthat leads to the dependency , once the limit of Bessel function is taken. On the other hand, it is possible to project in the momentum space by integrating over coordinates. This process leads to a divergence which is of the same nature of the coordinate projection in the absence of temperature. Hence, a quantity in the momentum space analogous to the temperature is necessary, that is, the thermal energy. The introduction of TFD formalism introduces the role of temperature, but it can equally do the same for the thermal energy. Using phase space and TFD allows us to deal with systems where microscopic energy is dominant. 4.2. Casimir Effect for the Dirac Field in Phase Space Here, the choice is , then The Green function is this case is where is a space-like vector. Then, the energy-momentum tensor is By taking , the Casimir energy for the Dirac field in phase space at zero temperature is And for , the Casimir pressure in phase space is It reproduces the usual result when and integrated over the momenta which means the projection on coordinate space. Then, only the factors of is left; the limit of this part yields the dependency . Here, the dependency on matrices should be viewed as part of the phase space formalism which is by its core matricial. This part does not survive once the projection on coordinates is performed, but it is part of the behavior in phase space. In order to be compared with experimental data, the projection on momentum space requires the introduction the thermal energy. 4.3. Casimir Effect for the Dirac Field in Phase Space at Finite Temperature The effect of temperature is introduced by taking . Then, the generalized Bogoliubov transformation becomes The first two terms of these expressions correspond, respectively, to the Stefan-Boltzmann term and the Casimir effect at . The third term is analyzed and it leads to the Green function Then, the Casimir energy at finite temperature is and the Casimir pressure at finite temperature is It should be noted that in the limit , both the Casimir energy and pressure are real quantities at zero and finite temperature. In the limit , i.e., , the Casimir energy and pressure become It is important to note that in this limit, both the Casimir energy and pressure depend only on the distance between the plates. The dependence on gamma matrices is not a problem since the formalism of the phase space is matrix. It leads to the conclusion that neither the energy nor the pressure are scalars but components of a tensor. The Dirac field in phase space is considered. Using the Dirac equation, the propagator for spin-1/2 particles is calculated. This form of the propagator is similar to that in the usual quantum mechanics. The TFD results are obtained by using the temperature effects in the Dirac propagator. TFD, a real-time finite temperature formalism, is a thermal quantum field theory. Using this formalism, a physical (renormalized) energy-momentum tensor is defined. Then, the Stefan-Boltzmann law in phase space and the Casimir effect are calculated at finite temperature. The results lead to the usual results for the Dirac field when they are projected in the quantum field theory space. The TFD formalism allows studying the finite temperature effects in phase space. On the other hand, such a formalism also may be used to explore the role of a thermal energy which is possibly related to the fermionic feature of the field. This is a theoretical work, and all previous results are listed in the references. Conflicts of Interest The authors declare that they have no conflicts of interest. This work by A. F. S. is supported by CNPq projects 308611/2017-9 and 430194/2018-8. - E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physics Review, vol. 40, no. 5, pp. 749–759, 1932. - M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Physics Reports, vol. 106, no. 3, pp. 121–167, 1984. - H. Weyl, “Quantenmechanik und gruppentheorie,” Zeitschrift für Physik, vol. 46, no. 1-2, pp. 1–46, 1927. - M. D. Oliveira, M. C. B. Fernandes, F. C. Khanna, A. E. Santana, and J. D. M. Vianna, “Symplectic quantum mechanics,” Annals of Physics, vol. 312, no. 2, pp. 492–510, 2004. - R. G. G. Amorim, M. C. B. Fernandes, F. C. Khanna, A. E. Santana, and J. D. M. Vianna, “Non-commutative geometry and symplectic field theory,” Physics Letters A, vol. 361, no. 6, pp. 464–471, 2007. - R. G. G. Amorim, S. C. Ulhoa, and E. O. Silva, “On the symplectic Dirac equation,” Brazilian Journal of Physics, vol. 45, no. 6, pp. 664–672, 2015. - F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malboiusson, and A. E. Santana, Themal Quantum Field Theory: Algebraic Aspects and Applications, World Scientific, Singapore, 2009. - Y. Takahashi and H. Umezawa, “Higher order calculation in thermo field theory,” Collective Phenomena, vol. 2, p. 55, 1975. - Y. Takahashi and H. Umezawa, “Thermo field dynamics,” International Journal of Modern Physics B, vol. 10, pp. 1755–1805, 1996. - H. Umezawa, H. Matsumoto, and M. Tachiki, Thermofield Dynamics and Condensed States, North-Holland Pub. Co., North Holland, Amsterdan, 1982. - H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics, AIP, New York, 1993. - A. E. Santana, A. M. Neto, J. D. M. Vianna, and F. C. Khanna, “Symmetry groups, density-matrix equations and covariant Wigner functions,” Physica A, vol. 280, no. 3-4, pp. 405–436, 2000. - F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Thermoalgebras and path integral,” Annals of Physics, vol. 324, no. 9, pp. 1931–1952, 2009. - R. G. G. Amorim, J. S. da Cruz Filho, A. F. Santos, and S. C. Ulhoa, “Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature,” Advances in High Energy Physics, vol. 2018, Article ID 1928280, 9 pages, 2018. - F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Quantum fields in toroidal topology,” Annals of Physics, vol. 326, no. 10, pp. 2634–2657, 2011. - F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, “Quantum field theory on toroidal topology: algebraic structure and applications,” Physics Reports, vol. 539, no. 3, pp. 135–224, 2014. Copyright © 2020 R. G. G. Amorim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Signal Strength Measure signal strength in dBW = 10*log(Power in Watts)dBm = 10*log(Power in mW)can legally transmit at 10dBm (1W).Most PCMCIA cards transmit at 20dBm.Mica2 (cross bow wireless node) can transmit from –20dBm to 5dBm. (10microW to 3mW)Mobile phone base station: 20W, but 60 users, so 0.3W / user, but antenna has gain=18dBi, giving effective power of 42.Mobile phone handset – 21dBm 3 Noise Interference Thermal noise From other users From other equipment E.g., microwave ovens 20dBm 50% duty-cycle with 16ms period.Noise in the electronics – e.g., digital circuit noise on analogue parts.Non-linearities in circuits.Often modeled as white Gaussian noise, but this is not always a valid assumption.Thermal noiseDue to thermal agitation of electrons. Present in all electronics and transmission media.kT(W/hz)k Boltzmann’s constant = 1.3810-23T – temperture in Kelvin (C+273)kTB(W)B bandwidthE.g.,Temp = 293,=> -203dB, -173dBm /HzTemp 293 and 22MHz => -130dB, -100dBm 4 Signal to Noise Ratio (SNR) SNR = signal power / noise powerSNR (dB) = 10*log10(signal power / noise power)Signal strength is the transmitted power multiplied by a gain – impairmentsImpairmentsThe transmitter is far away.The signal passes through rain or fog and the frequency is high.The signal must pass through an object.The signal reflects of an object, but not all of the energy is reflected.The signal interferes with itself – multi-path fadingAn object not directly in the way impairs the transmission. 5 Receiver SensitivityThe received signal must have a strength that is larger than the receiver sensitivity20dB larger would be good. (More on this later)E.g.,802.11b – Cisco Aironet 250 (the most sensitive)1Mbps: -94dBm; 2Mbps: -91dBm; 5.5Mbps: -89dBm; 11Mbps: -85dBmMobile phone base station: -119dBmMobile phone hand set: -118dBmMica2 at 868/916MHz: -98dBm 6 Simple link budgetDetermine if received signal is larger than the receiver sensitivityMust account for effective transmission powerTransmission powerAntenna gainLosses in cable and connectors.Path lossesAttenuationGround reflectionFading (self-interference)ReceiverReceiver sensitivityLosses in cable and connectors 7 Antenna gain Vertical direction Horizontal direction isotropic antenna – transmits energy uniformly in all directions.Antenna gain is the peak transmission power over any direction divided by the power that would be achieved if an isotropic antenna is used. The units is dBi.Sometime, the transmission power is compared to a ½ wavelength dipole. In this case, the unit is dBD.The ½ wavelength dipole has a gain of 2.14dB.Vertical directionHorizontal direction 8 Antenna gainAntenna gain is increased by focusing the antennaThe antenna does not create energy, so a higher gain in one direction must mean a lower gain in another.Note: antenna gain is based on the maximum gain, not the average over a region. This maximum may only be achieved only if the antenna is carefully aimed.This antenna is narrower and results in 3dB higher gain than the dipole, hence, 3dBD or 5.14dBiThis antenna is narrower and results in 9dB higher gain than the dipole, hence, 9dBD or 11.14dBi 9 Antenna gainInstead of the energy going in all horizontal directions, a reflector can be placed so it only goes in one direction => another 3dB of gain, 3dBD or 5.14dBiFurther focusing on a sector results in more gain.A uniform 3 sector antenna system would give 4.77 dB more.A 10 degree “range” 15dB more.The actual gain is a bit higher since the peak is higher than the average over the “range.”Mobile phone base stations claim a gain of 18dBi with three sector antenna system.4.77dB from 3 sectors – dBiAn 11dBi antenna has a very narrow range. 10 Simple link budget – 802.11b receiver sensitivity Thermal noise: -174 dBm/HzChannel noise (22MHz): 73 dBNoise factor: 5 dBNoise power (sum of the above): -96 dBmReceiver requirements:3 dB interference margin0 dB is the minimum SINRMin receiver signal strength: -93 dBm 11 Simple link budget – 802.11 example From base station+20dBm transmission power+6dBi transmit antenna gain+2.2dBi receiver antenna gain-91dBm minimum receiver power=> dB path losses=> 99 dB path losses if 20dB of link margin is added (to ensure the link works well.)From PCMCIA to base station+0dBm transmission power=> 99.4 dB path losses=> 79.2 dB path losses if 20dB of link margin is added (to ensure the link works well.)From PCMCIA to PCMCIA+2.2dBi transmit antenna gain=> 95.4 dB path losses=> 75.4 dB path losses if 20dB of link margin is added (to ensure the link works well.) 12 Simple link budget – mobile phone – downlink example Transmission power (base station): 20W (can be as high as 100W)Transmission power for voice (not control): 18WNumber of users: 60Transmission power/user: 0.3W, 300mW, 24.8dBmBase station antenna gain (3-sectors): 18dBiCable loss at base station: 2dBEffective isotropic radiated power: 40dBm (sum of the above)Receiver:Thermal noise: -174 dBm/HzMobile station receiver noise figure (noise generated by the receiver, Johnson Noise, ADC quantization, clock jitter): 7dBReceiver noise density: -167 dB/Hz (-174+7)Receiver noise: dBm (assuming 3.84MHz bandwidth for CDMA)Processing gain: 25dB (CDMA is spread, when unspread(demodulated) and filtered, some of the wide band noise is removed)Required signal strength: 7.9dBReceiver sensitivity: – =Body loss (loss due to your big head): 3dBMaximum path loss: 40 – (-118.3) –3 = 155.3 13 Simple link budget – mobile phone – uplink example Transmission power (mobile): 0.1W (21 dBm)Antenna gain: 0 dBiBody loss: 3 dBEffective isotropic radiated power: 18 dBm (sum of the above) (maximum allowabel by FCC is 33 dBm at 1900MHz and 20dBm at 1700/2100 MHzReceiver/base stationThermal noise: -174 dBm/HzMobile station receiver noise figure (noise generated by the receiver, Johnson Noise, ADC quantization, clock jitter): 5dBReceiver noise density: -169 dB/Hz (-174+5)Receiver noise: dBm (assuming 3.84MHz bandwidth for CDMA)Processing gain: 25dB (CDMA is spread, when unspread(demodulated) and filtered, some of the wide band noise is removed)Margin for interference: 3dB (more interference on the uplink than on the downlink)Required signal strength: 6.1dBReceiver sensitivity:Maximum path loss: 153.3 14 Required SNRFor a given bit-error probability, different modulation schemes need a higher SNREb is the energy per bitNo is the noise/HzBit-error is given as afunction of Eb / NoRequired SNR = Eb / No * Bit-rate / bandwidthA modulation scheme prescribes a Bit-rate / bandwidth relationshipE.g., for 10^-6 BE probability over DBPSK requires 11 dB + 3 dB = 14 dB SNR 16 Shannon CapacityGiven SNR it is possible to find the theoretical maximum bit-rate:Effective bits/sec = B log2(1 + SNR), where B is bandwidthE.g.,B = 22MHz,Signal strength = -90dBmN = -100dBm=> SNR = 10dB => 1022106 log2(1 + 10) = 76MbpsOf course, b can only do 1Mbps when the signal strength is at –90dBm. 17 PropagationRequired receiver signal strength – Transmitted signal strength is often around99 dB base station -> laptop79.2 dB b laptop -> base station75.4 dB laptop -> laptop155.3 Mobile phone downlink153.3 Mobile phone uplink.Where does all this energy go…Free space propagation – not valid but a good startGround reflection2-ray – only valid in open areas. Not valid if buildings are nearby.Wall reflections/transmissionDiffractionLarge-scale path loss modelsLog-distanceLog-normal shadowingOkumuraHataLongley-RiceIndoor propagationSmall-scale path lossRayleigh fadingRician Fading 18 Free Space Propagation The surface area of a sphere of radius d is 4 d2, so that the power flow per unit area w(power flux in watts/meter2) at distance d from a transmitter antenna with input accepted power pT and antenna gain GT isThe received signal strength depends on the “size” or aperture of the receiving antenna. If the antenna has an effective area A, then the received signal strength isPR = PT GT (A/ (4 d2))Define the receiver antenna gain GR = 4 A/2. = c/f2.4GHz=> = 3e8m/s/2.4e9/s = 12.5 cm933 MHz => =32 cm.Receiver signal strength: PR = PT GT GR (/4d)2PR (dBm) = PT (dBm) + GT (dBi) + GR (dBi) + 10 log10 ((/4d)2)2.4 GHz => 10 log10 ((/4d)2) = -40 dB933 MHz => 10 log10 ((/4d)2) = -32 dB 19 Free Space Propagation - examples Mobile phone downlink = 12.5 cmPR (dBm) = (PTGGL) (dBm) dB + 10 log10 (1/d2)Or PR-PT - 40 dB = 10 log10(1/d2)Or 155 – 40 = 10 log10 (1/d2) =Or (155-40)/20 = log10 (1/d)Or d = 10^ ((155-40)/20) = 562Km or Wilmington DE to Boston MAMobile phone uplinkd = 10^ ((153-40)/20) = 446Km802.11PR-PT = -90dBmd = 10^((90-40)/20) = 316 m11Mbps needs –85dBmd = 10^((85-40)/20) = 177 mMica2 Mote-98 dBm sensitivity0 dBm transmission powerd = 10^((98-30)/20) = 2511 m 20 Ground reflectionFree-space propagation can not be valid since I’m pretty sure that my cell phone does reach Boston.You will soon see that the Motes cannot transmit 800 m.There are many impairments that reduce the propagation.Ground reflection (the two-ray model) – the line of sight and ground reflection cancel out. 21 Ground reflection (approximate) Approximation! When the wireless signal hits the ground, it is completely reflected but with a phase shift of pi (neither of these is exactly true).The total signal is the sum of line of sight and the reflected signal.The LOS signal is = Eo/dLOS cos(2 / t)The reflected signal is -1 Eo /dGR cos(2 / (t – (dGR-dLOS)))Phasors:LOS = Eo/dLOS 0Reflected = Eo/dGR (dGR-dLOS) 2 / For large d dLOS = dGRTotal energyE = (Eo/dLOS) ( (cos ((dGR-dLOS) 2 / ) – 1)2 + sin2((dGR-dLOS) 2 / ) ) ½E = (Eo/dLOS) 2 sin((dGR-dLOS) / ) 22 Ground reflection (approximate) dGR-dLOSdGR = ((ht+hr)^2 + d^2)^1/2dLOS = ((ht-hr)^2 + d^2)^1/2dGR-dLOS 2hthr/d -> 0 as d-> inf2 sin((dGR-dLOS) / ) -> 0,For large d, 2 sin((dGR-dLOS) / ) C/dSo total energy is 1/d^2And total power is energy squared, or K/d^4 23 Ground reflection (approximate) For d > 5ht hr, Pr = (hthr)^2 / d^4 Gr GT PTPr – PT – 10log((hthr)^2) - log(Gr GT ) = 40 log(1/d)Examples:Mobile phoneSuppose the base station is at 10m and user at 1.5 md = 10^((155 – 12)/40) = 3.7Km802.11Suppose the base station is at 1.5m and user at 1.5 md = 10^((90 – 3.5)/40) = 145mBut this is only accurate when d is large 145m might not be large enough 24 Ground reflection (more accurate) When the signal reflects off of the ground, it is partially absorbed and the phase shift is not exactly pi.PolarizationTransmission line model of reflections 25 PolarizationThe polarization could be such that the above picture is rotated by pi/2 along the axis.It could also be shifted.If a rotated and shifted 26 PolarizationThe peak of the electric field rotates around the axis. 27 PolarizationIf a antenna and the electric field have orthogonal polarization, then the antenna will not receive the signal 28 Polarization Vertically/ horizontally polarized When a linearly polarized electric field reflects off of a vertical or horizontal wall, then the electric field maintains its polarization.In practice, there are non-horizontal and non-vertical reflectors, and antenna are not exactly polarized. In practice, a vertically polarized signal can be received with a horizontally polarized antenna, but with a 20 dB loss.Theoretically, and sometimes in practice, it is possible to transmit two signals, one vertically polarized and one horizontally.Vertically/ horizontally polarized 29 Snell's Law for Oblique Incidence yqqTqqTxGraphical interpretation of Snell’s law 30 Transmission Line Representation for Transverse Electric (TE) Polarization yqzxqTEz+ -Hx 31 Transmission Line Representation for Transverse Magnetic (TM) Polarization yqzxqTEx+ -Hz 32 Reflection from a Dielectric Half-Space TE PolarizationTM Polarization90º-1GEqGHqBno phase shift 33 Magnitude of Reflection Coefficients at a Dielectric Half-Space TE PolarizationTM Polarization1530456075900.10.20.30.188.8.131.52.80.91Reflection coefficient |GE |Incident Angle qIer=81er=25er=16er=9er=4er=2.56Reflection coefficient |GH | 35 Path losses Propagation Ground reflection Other reflections We could assume that walls are perfect reflectors (||=1). But that would be poor approximation for some angles and materials. Also, this would assume that the signal is not able to propagate into buildings, which mobile phone users know is not the case. 36 Reflection and Transmission at Walls Transmission line formulationHomogeneous wallsAttenuation in wallsInhomogeneous walls 37 Transmission Line Formulation for a Wall ZdTEZaTEwZdTEZaTE 38 Transmission Line Method airwallairZ(w)ZL= ZaZaZwStanding Wave- wTransmittedIncidentReflected 39 Reflection at Masonry Walls (Dry Brick: er 5, e”=0) 20cm1020304050607080900.20.40.60.81900MHzTE1.8GHzTMAngle of Incidence qI (degree) G 2BZaTEZdTEZaTEBrewster angle 40 Reflection Accounting for Wall Loss The relative dielectric constant has an imaginary componentZaZw, kwZ(w)- wz 41 Comparison with Measured |G| 4 GHz for Reew = 4, Imew = 0 Comparison with Measured |G| 4 GHz for Reew = 4, Imew = 0.1 and l = 30 cm Landron, et al., IEEE Trans. AP, March 1996)1530456075900.10.20.30.184.108.40.206.80.91Measured dataAngle of Incidence qTE PolarizationG w = w = 30cm1530456075900.10.20.30.220.127.116.11.80.91Measured dataAngle of Incidence qTM PolarizationG w = w = 30cm 42 Transmission Loss Through Wall, cont. Now the might be imaginary => phaseSee mathcad file 43 Dielectric constantsWhen conductivity exists, use complex dielectric constant given bye = eo(er - je") where e" = s/weo and eo 10-9/36pMaterial* er s (mho/m) e" at 1 GHzLime stone wallDry marbleBrick wallCementConcrete wallClear glassMetalized glassLake waterSea WaterDry soilEarth* Common materials are not well defined mixtures and often contain water. 44 Diffraction sources Idea: The wave front is made of little sources that propagate in all directions.If the line of sight signal is blocked, then the wave front sources results propagation around the corner.The received power is from the sum of these sourcessourcesDefine excess path = h2 (d1+d2)/(2 d1d2)Phase difference= 2/Normalize Fresnel-Kirchoff diffraction parameter 45 Knife edge diffraction Path loss from transmitter to receiver is-10-5510-30-25-20-15Received Signal(dB)v 46 Multiple diffractions If there are two diffractions, there are some models. For more than 2 edges, the models are not very good. 47 Large-scale Path Loss Models Log-distancePL(d) = K (d/do)nPL(d) (dB) = PL(do) + 10 n log10(d)Redo examples 48 Large-scale Path Loss Models Log-normal shadowingPL(d) (dB) = PL(do) + 10 n log10(d) + XX is a Gaussian distributed random number32% chance of being outside of standard deviation.16% chance of signal strength being 10^(11/10) = 12 times larger/smaller than 10 n log10(d)2.5% chance of the signal being 158 times larger/smaller.The fit shown is not very good.This model is very popular. 49 Outdoor propagation models OkumuraEmpirical modelSeveral adjustments to free-space propagationPath Loss L(d) = Lfree space + Amu(f,d) – G(ht) – G(hr) – GAreaA is the median attenuation relative to free-spaceG(ht) = 20log(ht /200) is the base station height gain factorG(hr) is the receiver height gain factorG(hr) = 10log(hr /3) for hr <3G(hr) = 20log(hr /3) for hr >3Garea is the environmental correction factorHata 50 Hata Model Valid from 150MHz to 1500MHz A standard formula For urban areas the formula is:L50(urban,d)(dB) = logfc loghte – a(hre) (44.9 – 6.55loghte)logd wherefc is the ferquency in MHzhte is effective transmitter antenna height in meters (30-200m)hre is effective receiver antenna height in meters (1-10m)d is T-R separation in kma(hre) is the correction factor for effective mobile antenna height which is a function of coverage areaa(hre) = (1.1logfc – 0.7)hre – (1.56logfc – 0.8) dB for a small to medium sized city 51 Indoor propagation models Types of propagationLine of sightThrough obstructionsApproachesLog-normalSite specific – attenuation factor modelPL(d)[dBm] = PL(d0) + 10nlog(d/d0) + Xsn and s depend on the type of the buildingSmaller value for s indicates the accuracy of the path loss model. 52 Path Loss Exponent and Standard Deviation Measured for Different Buildings Frequency (MHz)ns (dB)Retail Stores9142.28.7Grocery Store1.85.2Office, hard partition15003.07.0Office, soft partition9002.49.619002.614.1Factory LOSTextile/Chemical13002.040002.1Paper/Cereals6.0Metalworking1.65.8Suburban HomeIndoor StreetFactory OBS18.104.22.168 53 Site specific – attenuation factor model PL(d) (dB) = PL(do) + 10 n log(d/do) + FAF + PAFFAF floor attenuation factor - Losses between floorsNote that the increase in attenuation decreases as the number of floors increases.PAF partition attenuation factor - Losses due to passing through different types of materials.BuildingFAF (dB)s (dB)Office Building 1Through 1 Floor12.97.0Through 2 Floors18.72.8Through 3 Floors24.41.7Through 4 Floors27.01.5Office Building 216.22.927.55.431.67.2 55 Small-scale path loss See matlab file They are summed as phasors. The received signal is the sum of the contributions of each reflection.They are summed as phasors.The received signal is the phasor sum of the contributions of each reflection.A small change in the position of the receiver or transmitter can cause a large change in the received signal strength.See matlab file 56 Rayleigh and Rician Fading The inphase and quadrature parts can be modeled as independent Gaussian random variables.The energy is the (X^2 + Y^2)^ ½ where X and Y are Gaussian => the energy is Rayleigh distributed.The power is (X^2 + Y^2) which is exponentially distributed.Rician – if there is a strong line-of-sight component as well as reflections. Then the signal strength has a Ricain distribution. 57 SummaryThe signal strength depends on the environment in a complicate way.If objects are possible obstructing, then the signal strength may be log-normal distributed => large deviation from free-spaceIf the signal is narrow band, then the the signal could be completely canceled out due to reflections and multiple paths.Reflection, transmission, and diffraction can all be important 58 Path Losslocation 1, free space loss is likely to give an accurate estimate of path loss.location 2, a strong line-of-sight is present, but ground reflections can significantly influence path loss. The plane earth loss model appears appropriate.location 3, plane earth loss needs to be corrected for significant diffraction losses, caused by trees cutting into the direct line of sight.location 4, a simple diffraction model is likely to give an accurate estimate of path loss.location 5, loss prediction fairly difficult and unreliable since multiple diffraction is involved.

This manuscript establishes a general approach to formulate partial differential equations (PDEs) on networks of (hyper)surfaces, referred to as hypergraphs. Such PDEs consist of differential expressions with respect to all hyperedges (surfaces) and compatibility conditions on the hypernodes (joints, intersections of surfaces). These compatibility conditions ensure conservation properties (in case of continuity equations) or incorporate other properties—motivated by physical or mathematical modeling. We illuminate how to discretize such equations numerically using hybrid discontinuous Galerkin (HDG) methods, which appear to be a natural choice, since they consist of local solvers (encoding the differential expressions on hyperedges) and a global compatibility condition (related to our hypernode conditions). We complement the physically motivated compatibility conditions by a derivation through a singular limit analysis of thinning structures yielding the same results. Albeit many physical, sociological, engineering, and economic processes have been described by partial differential equations posed on domains which cannot be described as subsets of linear space or smooth manifolds, there is still a lack of mathematical tools and general purpose software specifically addressing the challenges arising from the discretization of these models. Fractured porous media (see [BerreDK2019] for a comprehensive review) have gained substantial attention and have become an active field of research due to their critical role with respect to flow patterns in several applications in the subsurface, in material science, and in biology. Most commonly, a fracture is described as a very thin, not necessarily planar object in which, for example Darcy’s equation holds. This motivates the singular limit approximation in which a fracture is assumed to be a two dimensional surface within the three dimensional space. When several of these fractures meet, they form a fracture network of two-dimensional surfaces. Thus, fracture networks illustrate a physical application of the type of problem we investigate in this publication. Moreover, a model in which the joints of two (or more) fractures are assigned additional physical properties can be found in [ReichenbergerJBH2006]. Beyond this, fracture networks have been simulated using hybrid high order (HHO) methods [HedinPE2019]. Graph based models for porous media (without fractures) consist of simulating preferential flow paths within the porous matrix. One of the first publications implementing this idea is [Fatt1956] who observed that a network of tubes might approximate the flow of porous media better than the classical model of tube bundles, which has also been used in the most common upscaling techniques—see [SchulzRZRK2019, RayRSK2018] and the references therein for a discussion of those tube models in upscaling procedures. The tube network approach [Fatt1956] has been successfully applied to couple porous media flow to free (Navier–) Stokes flow [WeishauptJH2019]. PDEs on hypergraphs are especially suitable to be used in the description of elastic networks [Eremeyev2019]: Here, we discriminate between one dimensional elastic beam (rod) networks, trusses, etc. and two dimensional elastic plate (shell) networks [LagneseL1993], respectively. Beam networks have been used to model truss bridges and towers (most prominently the Eiffel tower) and other mechanical structures, originating the field of elastic beam theory (for instance [BauchauC2009] for an introduction which also covers elastic plate models). Elastic plate models describe the stability of houses and have several engineering applications such as the description of the stability of (bend) plates (used in automobile industries and several others). They have even been used to understand interseismic surface deformation at subduction zones [KandaS2010]. Elastic beam networks have been used to evaluate elastic constants in amorphous materials. That is, the elastic properties of stiff, beam like polymers have been investigated. Such polymers are key to understanding the cytoskeleton which is an important part of biological cells [Heussinger2007, Lieleg2007], but they are also important for the healing of wounds (fibrin), for skin stability (collagen), and for the properties of paper. Moreover, such models can be used for modelling rubber [PhysRevE.76.031906], foams, and fiber networks [PhysRevLett.96.017802]. Conservation laws in the form of PDEs on hypergraphs have been used in the simulation and optimization of gas networks [RuefflerMH2018] and other networks of pipelines. They have been extended to networks of traffic (streets and data), (tele-)communication, and blood flow. For an overview of the main ideas that are related to these applications, the reader may consult [Garavello2010, BressanCGHP2014]. Additionally, rigorous mathematical analysis of such problems is developing to a field of current research [SkreMR2021]. We conclude the overview over some applications by stating that regular surfaces and volumes can also be interpreted as hypergraphs. Thus, PDEs on surfaces [DziukE2013] and standard “volume” problems (in which the hyperedges have the same dimension as the surrounding space and at most two hyperedges meet in a common hypernode) are also covered by our approach. Hypergraph models usually are approximations of problems in higher dimensional networks of thin structures, for example a network of thin pipes or thin plates in 3D. As a model example we give a rigorous derivation of a diffusion equation on a hypergraph. More precisely, we consider a network of thin plates in three dimensions, where the thickness of the plates is small compared to their length. We denote the ratio between the thickness and the length by the small parameter . Due to the different scales the computational effort for numerical simulations is very high. To overcome this problem the idea is to replace the thin-structure by a hypergraph. For this we give a rigorous mathematical justification using asymptotic analysis. We pass to the limitin the weak formulation of the problem, and derive a limit problem stated on the hypergraph. The solution of this limit-problem is an approximation of the model in the higher-dimensional thin domain. Singular limits for thin plates and shells (leading to lower-dimensional manifolds in the limit ) in elesticity can be found in [ciarlet1997mathematical, ciarlet2000theory]. Dimension reduction for a folded elastic plate is treated in [le1989folded]. Singular limits leading to hypergraphs for fluid equations can be found in [maruvsic2003rigorous], where a Kirchhoff law in a junction of thin pipes is derived, and [maruvsic2019mathematical] where junctions of thin pipes and plates are treated using the method of two-scale convergence. The remainder of this manuscript is structured as follows: First, we discuss conservation equations on hypergraphs. Second, we rigorously formulate an elliptic model equation and investigate some of its properties in Section 3. Third, we discuss its discretization by means of the HDG method in Section 4. Fourth, we discuss how PDEs on hypergpahs can be obtained by a model reduction approach, in particular, by considering singular limits. The publication is wrapped up, by a section on possible conclusions. 2. Conservation equations on geometric hypergraphs A hypergraph consists of a finite set of hyperedges and a finite set of hypernodes. We refer to it as a geometric hypergraph if the hyperedges are smooth, open manifolds of dimension with piecewise smooth, Lipschitz boundary and the hypernodes can be identified with smooth subsets of the boundaries of these hyperedges. More specifically, the boundary of each hyperedge is subdivided into nonoverlapping subsets such that . We associate to an index vectorand isometries The hypernodes are thus identified with the closures of the subsets of the boundaries of one or more hyperedges. Their dimension is . has the structure of a hypergraph in the classical sense as each edge connects a set of nodes . The dual hypergraph describes the situation where each hypernode connects hyperedges with indices . We call a hypernode a boundary hypernode if , i.e., it is part of the boundary of only a single hyperedge. Accordingly, we define the set of boundary hypernodes and the set of interior hypernodes . As special cases: a geometric graph is a geometric hypergraph where the edges are smooth curves and the nodes are their end points. If every hypernode is either at the boundary or connects exactly two hyperedges, the hypergraph represents a piecewise smooth manifold. The structure might become more evident if we consider an embedded geometric hypergraph in some ambient space , as in Figure 1 on the left. In this case, the isometries are identical mappings and the hypernodes are identified with the boundary pieces . On the right of this figure, the same hypergraph is displayed without embedding. In this case, the hyperedges are objects in , possibly with a non-flat metric. Hypernodes are intervals in , inheriting their metric through the isometries . Due to the isometries , every point of a hypernode is uniquely identified with a point on the boundary of each of the hyperedges it connects. Thus, convergence of a point sequence in the union of these hyperedges to a point on the hypernode is well-defined, for instance by considering the (finitely many) subsequences on each hyperedge. Also, a distance between two points on different hyperedges sharing a hypernode is defined locally by these isometries and triangle inequality. The domain of the hypergraph, its closure, and its boundary are In this definition, the hyperedges are considered open with respect to their topology and do not contain their boundaries. The hypernodes are closed. We introduce the skeletal domain We make the assumption that is connected. Note that this implies that any two hyperedges are either connected by a common node or not connected, since is open, see (2.2). Without such an assumption, the problems of partial differential equations below separate into subproblems, which then can be analyzed and solved independently. In Figure 1, comprises all blue hypernodes, which also include the end points of the red hypernode. The union of the red and blue hypernodes is . The domain consists of the interior of the red hypernode and the the three hyperedges. Many concepts of standard domains in transfer to , even if it is not a manifold. In particular, the notion of a small open ball with radius around , see Figure 2, in is maintained by construction and thus the notion of open subsets. A subset is called compactly embedded in if its closure is contained in and thus has a positive distance to . A function is continuous on , if it is continuous inside each hyperedge and its limits on a hypernode are consistent between all hyperedges connected by this hypernode. Analogously, a function is in if it is in for all and it is in if it is in for all . Remark 2.1 (Comparison to standard nomenclatures). In this article, we mix concepts from graph theory, partial differential equations, and finite elements. Thus, a clash of names was unavoidable. What is referred to as a hypernode here, is a face —an edge in two dimensions— in finite element literature, while the hyperedges here correspond to mesh cells or elements. In order to reduce ensuing confusion, we consistently use the term “hyperedge”. Another difference to finite element literature is established by the fact that we consider the hypergraph fixed and are not concerned with refinement limits. Finally, we would like to point out that there has been a concept of geometric hypergraphs in the literature; it is nevertheless very limited, such that we coin this term in a new way here, meaning a hypergraph whose elements are geometric shapes themselves. 2.2. Continuity equations on hypergraphs Next, we conduct a heuristic derivation, employing control volumesin the shape of infinitesimal, open hyperballs. Let be a conserved quantity and be its flux. Then, the conservation property of is usually stated in integral form such that for any such control volume there holds When is a subset of a Lipschitz manifold , the meaning of this statement is clear if is a smooth tangential vector field in and is the outer normal vector to in the tangential plane of . The term denotes the volume element of the manifold, and is the induced surface element. If the hyperball intersects a hypernode in which several edges meet, meaning can be given to equation (2.4) by the following observation: if are the hyperedges which meet in inside , then for the intersection has a piecewise smooth boundary . We observe that is in the interior of (see Figure 2 for an illustration) and the boundary of is nowhere tangential to . Thus, with the assumption that no mass is created or destroyed in the hypernode , the conservation property (2.4) can be restated as Again, the flux and the outer normal vector to are well defined along in the tangential plane of . As a generalization of (2.5), we allow for sinks and sources living in the hyperedges and living within hypernode : This can be implemented by setting where positive and describe sources, while negative and describes sinks. Before we convert (2.4) into a problem of partial differential equations, we make the simplifying assumption that the hyperedges and hypernodes are planar and that is the standard Lebesgue measure. This way, we avoid delving into the complexities of surface partial differential equations. This simplification is purely for the ease of presentation and we refer the readers to [DziukE2013] and [BENARTZI2007989] for more general surfaces in the elliptic and hyperbolic settings, respectively. Thus, in the interior of each hyperedge , we can apply Gauss’ divergence theorem in standard form to obtain If on the other hand overlaps a hypernode which connects hyperedges , we can still apply the divergence theorem in each hyperedge to obtain This notion also extends to control volumes intersecting with several hypernodes in a natural way. Then, rearranging the sum over boundary integrals yields for any control volume . Here, is the standard divergence of the differentiable vector field and is the summation operator such that on a hypernode with hyperedges there holds A vector field is usually called solenoidal, if the left side of equation (2.9) vanishes for any control volume . The right hand side of this equation generalizes this notion from standard domains to hypergraphs. Therefore, we call a piecewise smooth vector field solenoidal, if Note that the second condition is an extension of Kirchhoff’s junction rule from points to higher dimensional hypernodes. To put it in a nutshell, assuming there are no leaks and sources in hypernodes and hyperedges, the conservation condition (2.5) induces the PDE–interface problem to find and such that Analogously, the continuity condition (2.6) induces the PDE interface problem to find and such that In (2.12) and (2.13), and might be linked by some phenomenological description, i.e., (depending on the specific application). Both equations are complemented by appropriate initial and boundary conditions. Beyond this, additional continuity constraints might be formulated, such as , , …. 3. Elliptic model equation The standard diffusion equation in mixed form defined on a hypergraph is a conservation equation of type (2.12) for the flux of a scalar function . This is for instance known as Fourier’s law of thermal conduction, where is the temperature and is the dimensionless heat conductivity of the material. It is also Fick’s law of diffusion where is a concentration and is the diffusion coefficient. Like in the previous section, we simplify the presentation by assuming that all hyperedges are flat and thus can be identified with a domain in . In the more general case, the differential operators must be replaced by their differential geometric counterparts as in [DziukE2013]. We focus on the stationary case and set the time derivative in (2.12) to zero. Thus, the discussion of the previous section leads to the following problem: find satisfying for all , right hand sides and , and a diffusion coefficient . A justification by taking the limit of thin domains can be found in Section 5 below. We observe that in (3.1) the diffusion equation (3.1a) is complemented by three boundary and interface conditions. First, it is closed by a “Dirichlet” boundary condition (3.1b): We choose a non-empty set of “Dirichlet” hypernodes, on which we impose for a prescribed boundary value . In (3.1c), we employ a continuity constraint. This constraint prohibits jumps in the primary unknown across interior nodes, and therefore, loosely speaking, imitates the standard constraint that of the domain. On interior nodes , we set out with Kirchhoff’s junction law, but with the option of a concentrated source in (3.1d). This equation also incorporates the Neumann condition , since on a boundary hypernode the sum in the definition (2.10) of the operator reduces to a single hyperedge. Note that (3.1d) for on interior nodes serves as compatibility condition for mimicking . Definition 3.1 (Function spaces on hypergraphs). For each let be the standard Sobolev space on and be the standard trace operator. Then, we define where and are the traces of from the hyperedges and on , respectively. Due to the equality of traces in the definition of , we can define the trace operator to the skeleton Additionally, the spaces and are defined as and we denote the dual spaces of by and of by . Norms ( and ) on the respective spaces ( and ) are induced by summed versions of the local scalar-products: These definitions have a few immediate consequences: is a well-defined and surjective, linear, and continuous operator. We have the Gelfand triple relations Note that is analogous to space in of Raviart and Thomas [RaviartT1977a]. The space with inner product is a Hilbert space. Obviously, is a subspace of the Hilbert space , and the function is continuous and is its kernel. Thus, is closed. ∎ A weak solution to the primal formulation of (3.1) with , , and is a function with on all , and In particular, if and , we can rewrite (3.11) as 3.1. Existence and uniqueness of solutions Assume for that there is a lifting with on all . If with a. e., , , and all are Lipschitz domains, there is an unique weak solution according to Definition 3.3, which continuously depends on the data. Due to the existence of , we can reduce the problem to the one with homogeneous Dirichlet values if we replace by and modifying the right hand side accordingly. Since the right hand side is bounded and is a Hilbert space, it suffices to show ellipticity of the weak form to conclude the proof by the Lax–Milgram lemma. We note that for there holds Thus, the following Poincaré–Friedrichs inequality implies ellipticity and concludes the proof. ∎ Lemma 3.5 (Poincaré–Friedrichs inequality for ). For all it holds that Similar to the standard case of subdomains in , this inequality follows easily by contradiction: To this end, we assume that there is a sequence with Thus, is bounded in for all . Hence, by the weak compactness of the unit ball in and the Rellich-Kondrachov theorem, there exists a subsequence (also denoted ) such that We have that (due to its completeness), and that the seminorm . Thus, is constant in all and overall continuous. Therefore it is overall constant and has to be zero, due to the zero boundary condition on Dirichlet nodes and the connectedness of . Hence, the strong convergence of in implies , which contradicts . Therefore, the Poincaré–Friedrichs inequality is valid. ∎ 4. HDG method for elliptic model equation When we derived PDE problems on hypergraphs, we were led to a formulation local on each hyperedge with coupling conditions on hypernodes. This is a structure which is nicely reflected in hybridized methods. Indeed, there the separation goes one step further. By putting degrees of freedom on the hypernode, values on hyperedges are not coupling anymore to other hyperedges across these hypernodes, but only to the values on the hypernodes constituting their boundary. Thus, differing from standard or discontinuous finite element methods, the number of hyperedges attached to a hypernode does not affect the solution process on a single hyperedge. Therefore, we consider hybridized methods ideally suited to PDEs on hypergraphs. Hybridized discontinuous Galerkin (HDG) methods break the continuity condition (3.1c) by introducing Lagrange multipliers on each hypernode which enforce the continuity of fluxes (3.1d) weakly. It turns out though, that the Lagrange multiplier is an approximation to the solution of (3.1) on the skeleton itself. With such methods, the actual PDE (3.1a) is represented locally on each hyperedge by Steklov-Poincaré operators on the hyperedges, which transform function values to flux values on the boundary of the hyperedges, a process called “local solver” in HDG terminology. The global problem is posed in terms of the degrees of freedom on the hypernodes only, yielding a square, linear system of equations. In this respect, HDG methods have a similar structure as the family of HHO methods. These are based on hybridizing the primal formulation and lead to a rather simple error analysis on polytopic meshes where only projections are used (as opposed to the rather complicated projections used for HDG). This is achieved by a novel stabilization design [DiPietro2015]. For recent developments in hybrid high-order and HDG methods, the reader may consult [Burman2018, Qiu2016]. The separation of the local solution of bulk problems from the global coupling of interface variables is also achieved by the virtual element method [VEM0, VEM1]. Thus, it fits into our view of coupled differential equations on connected hyperedges. Different to the methods discussed so far, it does not rely on polynomial shape functions inside mesh cells but rather on forms of fundamental solutions of any shape [VEM2]. Accordingly, when applied to hypergraphs, the actual type of local solvers and of the specific boundary trace operators will differ from our approach, but remain within the same principal concept. 4.1. The hybridized dual mixed formulation In physical applications, there often is a need to receive reasonable approximations for both the primal unknown and the dual unknown . In other words, considering diffusion, we would like to know both the distribution of some species’ concentration and the species “movement” (flux). This becomes particularly important if we interpret as pressure and as fluid flow through a porous medium (Darcy’s equation). In this situation, the flow field will govern the movement of chemical species dissolved within the fluid. It is often the main quantity of interest and conservativity is crucial. Therefore, we turn to the mixed formulation. The mixed HDG methods use the weak, dual, mixed, hybrid formulation of (3.1), i.e., find with on all such that Well-posedness of this formulation can be deduced from Theorem 3.4 if . Indeed, on the one hand, this implies that the (uniquely existing) solution of Definition 3.3 solves (4.1) with , , and . On the other hand, for any solution of (4.1), we have (by the space’s definition), and in the weak sense. Therefore, any solution to (4.1) satisfies Definition 3.3. Equations (4.1a) & (4.1b) are local equations on the hyperedge, like in the standard case of a domain. They only couple to the Lagrange multipliers on the boundary of the hyperedge. Thus, we can eliminate them locally in the fashion of the Schur complement method. To this effect, we introduce the local solution operator for the right hand side . It is in fact a Steklov-Poincaré operator on mapping the Dirichlet data to the normal trace of the flux in (4.1c). Then, the solution of (4.1) can be characterized as The Steklov-Poincaré operators in this equation are the same ones as in the case of a manifold. They do not depend on the connectivity of a hypernode to other hyperedges. Thus, their implementation does not differ from that of a standard finite element method. The only difference lies in the structure of the sum on the left, and is thus almost purely of algebraic nature. For inhomogeneous right hand side , we can define the operators In order to obtain a better understanding of the Steklov-Poincaré operators, we follow the route laid out in [CockburnGL2009] for the discrete version and define the solution operators The well-posedness and linearity of all local solution operators follow directly from the fact (see for instance [BoffiBrezziFortin13]) that the mixed formulation on a single hyperedge is well-posed for any given and its solution depends continuously on . Entering these solution operators into (4.1c) yields since . By some simple transformations of (4.1), we can write (4.2) with inhomogeneous right hand sides in terms of bilinear and linear forms. This argument allows to reduce the problem to finding with on all , such that Obviously, bilinear form and linear form are continuous due to the continuity of operators and . Surprisingly, we have recovered a symmetric bilinear form. The following lemma is a key to the discrete well-posedness and adds the fact that this form is even -elliptic. If and , bilinear form from (4.6b) is elliptic. Like in the proof of the Poincaré-Friedrichs inequality, we prove ellipticity of by a contradiction argument. To this end, let a sequence in such that Thus, there exists a subsequence in , and by the compact embedding of in there holds again for a subsequence in . Since is continuous and converges weakly in we obtain weakly in for each hyperdege and therefore also in . Thus, (4.7) implies that . We denote by the solution to (4.1a) and (4.1b) associated to , especially we have . Hence, for every we have Moreover, we have that the divergence operator is surjective, and therefore This, however, implies that is constant on . Furthermore, since is constant on , we can deduce by (4.8) and Gauss’ divergence theorem that is constant and The contradiction argument is concluded by the fact that some hyperedges are adjacent to the Dirichlet nodes and thus on their boundary. For the other hyperedges, follows from connectedness of . Thus, on in contradiction to . Altogether we showed that for all it holds that Together with (4.6b) we obtain for a positive constant i.e., the ellipticity of on . 4.2. HDG methods in dual mixed form Let be some finite dimensional, scalar function space. Then, we define the space of discrete functions on the skeleton by The mixed HDG methods involve a local solver on each hyperedge , producing hyperedge-wise approximations and and of the functions and in equation (4.1), respectively. Here, is some finite dimensional, scalar function space, and is some finite dimensional, vector valued function space. We will also use the concatenations of the spaces and , respectively, as a function space on , namely The HDG scheme for (4.1) on a hypergraph consists of the local solver and a global coupling equation. The local solver is defined hyperedge-wise by a weak formulation of (4.1) in the discrete spaces and defining suitable numerical traces and fluxes. Namely, given find and , such that hold for all , and all , and for all . Here, is the penalty coefficient. While the local solvers are implemented hyperedge by hyperedge, it is helpful for the analysis to combine them by concatenation. Thus, the local solvers define a mapping where for each hyperedge holds and . Analogously, we set and , where now the local solutions are defined by the system Once has been computed, the HDG approximation to (4.1) on will be computed as The global coupling condition is derived through a discontinuous Galerkin version of mass balance and reads: Find , such that for all Hybridized DG methods in dual mixed form differ by the choice of local polynomial spaces and the stabilization parameter . Defining as the space of multivariate polynomials of degree at most , Table 1 lists some well-known combinations on simplices. Well-posedness of the local solvers for all of them is proven in [CockburnGL2009] and the works cited there. Analogous combinations based on tensor product polynomials exist for hypercubes. Existence and uniqueness of the discrete solution , , and to the HDG method can be shown repeating the arguments mentioned in Section 4.1 in the finite-dimensional setting. A natural assumption is the well-posedness of the local problems (4.14), see Remark 4.3. Given the local solvers, the HDG method for elliptic diffusion problems is consistent with respect to the solution to (4.1). Using consistency, we can immediately apply the analysis in [CockburnGW2009], as it proceeds locally for each hyperedge. Thus, we obtain optimal convergence rates for LDG-H (and also RT-H by slight adaptions) on simplicial hypergraphs. They also transfer to quadrilateral hypergraphs, since these allow for a Raviart–Thomas projection satisfying equation (2.7) in [CockburnGW2009]. 4.3. Numerical convergence tests for LDG-H errors (err) and estimated orders of convergence (eoc) of linear approximation to the diffusion equation for hypergraphs with hyperedge dimension. Next, we consider a convergence example on a hypergraph. It is constructed to approximate where the Dirichlet nodes are those that are located on the boundary of with . The filling indicates that the cube has been times uniformly refined (in the standard three dimensional sense), and the calculation is conducted on the dimensional “surfaces” of this filling. These surfaces themselves might be further refined times, and these refined surfaces are identified to be our standard hyperedges, see Figure 3 for an illustration. The dimensional faces of this approach are interpreted as nodes and the nodes located on the boundary of the unit cube are considered Dirichlet nodes. All other nodes are supposed to be interior nodes. The solution is constructed to be , diffusion coefficient , and right-hand side . Of course, polynomial degrees are supposed to exactly reproduce the given solution, which is true in our numerical experiments. Thus, we only plot the errors for in Table 2. Interestingly, the errors converge although with filling , also the computational domain increases for and . However, the rate of convergence deteriorates by if , and if . The optimal order is obtained for . Beyond this, the refinement indicated with uses filling level and then uniformly refines the respective faces. This does not lead to an increase of the computational domain (even if ) and, therefore, gives the optimal convergence rate . The aforementioned results have been obtained using our code HyperHDG [HyperHDGgithub]. 5. Hypergraph PDE as singular limit The aim of this section is to derive the hypergraph model (3.1) as a singular limit of a 3D-model problem as illustrated in Figure 4. We exemplary use the figure to illuminate the basic ideas: We assume to have a diffusion problem on a domain consisting of three thin plates (in gray) and a (red) joint. This is the problem, which we would like to solve. However, we do not want to solve it in three spatial dimensions, but would like to reduce it to a two-dimensional problem—for example since we have limited compute sources, the domain is very large, or the domain is very complicated to mesh. Thus, we let the thickness of the three plates (and therefore also the thickness of the joint) go to zero by considering , and construct a two dimensional limit problem. The solution of this two dimensional problem lives on the mid-planes of the three planes and their joint. It in some sense is supposed to approximate the solution of the original (three-dimensional) problem for which is a small, positive number. The principal idea of the limit process is to map equations on the thin structures depending on to fixed reference domains independent of , where we can use standard compactness methods from functional analysis. However, the transformed problem includes -dependent coefficients. Thus, the crucial point for the derivation of the limit model is to establish a priori estimates that are uniform with respect to . 5.1. Description of the 3D model problem We consider the simplified case of one hypernode with length connecting hyperedges for which are rectangles with side lengths and . Thus, Figure 4 shows the case with . The opposite node of with respect to is denoted by (and is a boundary node). Without loss of generality, we assume that lies in the -axis and we have We denote by a unit normal vector to and define extruded hyperedges for and Hence, is a hexahedron with side lengths and , and with thickness . We construct now a domain which contains the union of all these extruded hyperedges and a nonoverlapping decomposition of this domain. To this end, let be chosen such that the sets do not overlap. We denote the side of that contains by , and define The side lengths of are and . Additionally, we define the convex hull of the node and the sides : By construction, we have Then, we define the thin domain as On we define a diffusion problem and then to pass to the limit in order to derive a problem on the hypergraph . To ensure uniqueness for our model we assume a zero-Dirichlet boundary condition on one face . We consider the following problem for the unknown

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In particular, capm only works when we make assumptions about preferences which dont make much sense. Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit. The capitalassetpricing model and arbitrage pricing theory. Ppt arbitrage pricing theory powerpoint presentation. Arbitrage from wikipedia, the free encyclopedia for the film, see arbitrage film. What are the practical applications of arbitrage pricing. The arbitrage pricing theory and subsequent models advanced thinking from a singlefactor beta world to a view of return and risk through multiple factors. Practical applications of arbitrage pricing theory are as follows. The arbitrage pricing theory apt was proposed as a more complex and therefore more complete alternative to the capital asset pricing model capm which was thought to be too simple and limited. Arbitrage pricing theory apt is a multifactor asset pricing model based on the idea that an assets returns can be predicted using the linear relationship between the assets expected return and a number of macroeconomic variables that capture systematic risk. A simple explanation about the arbitrage pricing theory. The weak form states that current asset prices reflect all of the information implicit in past prices and trades. The market permits no arbitrage opportunities if and only if for any portfolio p, vp cp 0 implies ep 0. An apt arbitrage pricing theory model has 3 factors namely, market, inflation and exchange rate risk. Arbitrage pricing theory model application on tobacco and. Arbitrage pricing theory apt is an alternative to the capital asset pricing model capm for explaining returns of assets or portfolios. Factor models, basis portfolios, apt, intertemporal asset. It is a much more general theory of the pricing of risky securities than the capm. Chapter 10 arbitrage pricing theory and pdf chapter 11. The model identifies the market portfolio as the only risk factor the apt makes no assumption about. Intuitively, our formulation can be viewed as a transposed version of standard continuoustime nance theory, where the index of the stochastic process refers. Arbitrage pricing theory apt is an alternate version of the capital asset pricing model capm. Such a necessity condition is surprisingly absent in the apt literature. Focusing on asset returns governed by a factor structure, the apt is a oneperiod. Based on intuitively sensible ideas, it is an alluring new concept. Since no investment is required, an investor can create large positions to secure large levels of profit. Arbitrage pricing theory how is arbitrage pricing theory abbreviated. It states that the market price which reflects the associated risk factors of an asset represents the value that prevents an investor from exploiting it. Arbitrage pricing theory, apt, stockholm stock exchange. The arbitrage pricing theory apt model may be considered as an extension of the capm allowing for multiple factors. While the capm is a singlefactor model, apt allows for multifactor models to describe risk and return relationship of a stock. Apt abbreviation stands for arbitrage pricing theory. Speci cally, we propose a di erent formulation of the classical apt in terms of cumulative portfolios of assets in the economy. Arbitrage pricing theory apt spells out the nature of these restrictions and it is to that theory that we now turn. The capital asset pricing model capm and the arbitrage pricing theory apt help project the expected rate of return relative to risk, but they. According to the arbitrage pricing theory, the return on a portfolio is influenced by a number of independent macroeconomic variables. The arbitrage pricing theory apt was developed primarily by ross 1976a, 1976b. K k f ik k i k k k k e rtrack ik rf ik k rf r e r 11 1 6 1. Theory, are discussed as special cases of modern asset pricing theory using stochastic discount factor. Arbitrage pricing theory stephen kinsella the arbitrage pricing theory, or apt, was developed to shore up some of the deficiences of capm we discussed in at the end of the last lecture. Pdf the arbitrage pricing theory approach to strategic. Arbitrage pricing theory assumptions explained hrf. Arbitrage pricing theory apt columbia business school. Arbitrage pricing theory apt like the capm, apt is an equilibrium model as to how security prices are determined this theory is based on the idea that in competitive markets, arbitrage will ensure that riskless assets provide the same expected return created in 1976 by stephen ross, this theory predicts a relationship between the returns of a. It is a oneperiod model in which every investor believes that the stochastic properties of returns of capital assets are consistent with a factor structure. Unlike the capm, which assume markets are perfectly efficient. In finance, arbitrage pricing theory apt is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factorspecific beta coefficient. Primer on merger arbitrage a merger arbitrage opportunity is one in which a probable event occurring in the future, i. In the lzth economy there are n risky assets whose returns are generated by a kfactor model k is a fixed number. This is known as the arbitrage pricing theory apt in equilibrium, this relationship must hold for all securities and portfolios of securities ri. The corresponding expected risk premium of each factor is 4. It is considered to be an alternative to the capital asset pricing model as a method to explain the returns of portfolios or assets. The capital asset pricing model capm and the arbitrage pricing theory apt have emerged as two models that have tried to scientifically measure the potential for assets to generate a return or a loss. Apt considers risk premium basis specified set of factors in addition to the correlation of the price of the asset with expected excess return on the market portfolio. Stephen ross developed the arbitrage pricing theory apt in 1976. Loosely speaking, arbitrage is the possibility to have arbitrarily large returns. The purpose of this study was to applicant the arbitrage pricing theory model in the tobacco and cigarette industry listed on the idx. Both of them are based on the efficient market hypothesis, and are part of the modern portfolio theory. When implemented correctly, it is the practice of being able to take a positive and. Espen eckbo 2011 basic assumptions the capm assumes homogeneous expectations and meanexpectations and meanvariance variance preferences. Apt involves a process which holds that the asset in question and the returns which are related to it can be predetermined pretty easily when the relationship that the assents returns have with all the different macroeconomic factors affecting the risk of the asset. Solutions chapter 010 arbitrage pricing theory and. Arbitrage pricing theory apt is an asset pricing model which builds upon the capital asset pricing model capm but defines expected return on a security as a linear sum of several systematic risk premia instead of a single market risk premium. It involves the possibility of getting something for nothing. Furthermore, we exhibit the practical relevance and assumptions of these models. The formula includes a variable for each factor, and then a factor beta for each factor, representing the securitys sensitivity to movements in that factor. It is a one period model in which every investor believes that the stochastic properties of capital assets returns are consistent with a factor structure. The capitalassetpricing model and arbitrage pricing. Arbitrage pricing theory, often referred to as apt, was developed in the 1970s by stephen ross. The capital asset pricing model and the arbitrage pricing. This theory, like capm, provides investors with an estimated required rate of return on risky securities. The arbitrage pricing theory has been estimated by burmeister and mcelroy to test its sensitivity through other factors like default risk, time premium, deflation, change in expected sales and market returns are not due to the first four variables. The arbitrage pricing theory apt was developed primarily by ross. Factor representing portfolios in large asset markets cemfi. The counterexample is valuable because it makes clear what sort of additional assumptions must be imposed to validate the theory. If we combine expressions 1 and 6, we finally obtain that in terms of excess. The arbitrage pricing theory approach to strategic portfolio planning. Arbitrage arises if an investor can construct a zero beta investment portfolio with a return greater than the riskfree rate if two portfolios are mispriced, the investor could buy the lowpriced portfolio and. Arbitrage pricing theory university at albany, suny. It needs to be emphasized that the no arbitrage condition is not only sufficient but also necessary for the validity of the asset pricing formula. Overview and comparisons the arbitrage pricing theory apt was developed by stephen ross us, b. Arbitrage pricing theory definition arbitrage pricing. An overview of asset pricing models university of bath bath. Arbitrage pricing theory the notion of arbitrage is simple. Because it includes more factors, consider the arbitrage pricing theory more nuanced if not more accurate, than the capital asset pricing model. The arbitrage pricing theory apt ross 1976,1977 constitutes one of the most. What is the abbreviation for arbitrage pricing theory. Arbitrage pricing theory apt is a multifactor asset pricing theory using various macroeconomic factors. Capital asset pricing andarbitrage pricing theory prof. Arbitrage pricing theory and multifactor models of risk and return frm p1 book 1. Apt, see arbitrage pricing theory apt apv, see adjusted present value apv model arbitrage pricing theory apt, 112 arbitrage proof example, 30, 33, 34 asset earning power, 104 audit, 349, 388, 393, 493 average rates of return, 1 b bankruptcy costs, 226, 241, 242, 246 beta value. Pdf the arbitrage pricing theory relates the expected rates of. A short introduction to arbitrage pricing theory apt is the impressive creation of steve ross. The theory was first postulated by stephen ross in 1976 and is the. Arbitrage pricing theory gur huberman and zhenyu wang federal reserve bank of new york staff reports, no. Arbitrage pricing theory asserts that an assets riskiness, hence its average longterm return, is directly related to. Apt is an interesting alternative to the capm and mpt. Arbitrage pricing theory apt and multifactor models. Arbitrage pricing theory and multifactor models of risk and return 104 important to pork products, is a poor choice for a multifactor sml because the price of hogs is of minor importance to most investors and is therefore highly unlikely to be a priced risk factor. A more rigorous derivation 9 each of the coefficients. It was developed by economist stephen ross in the 1970s. Arbitrage pricing theory apt is a multifactor asset pricing model based on the idea that an assets returns can be predicted using the linear relationship between the assets expected return. Pdf the arbitrage pricing theory apt of ross 1976, 1977, and. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times the respective sensitivity coefficient. Arbitrage pricing the arbitrage pricing theory considers a sequence of economies with increasing sets of risky assets. The arbitrage pricing theory apt is due to ross 1976a, b. Are practitioners and academics, therefore, moving away from capm. Financial economics arbitrage pricing theory theorem 2 arbitrage pricing theory in the exact factor model, the law of one price holds if only if the mean excess return is a linear combination of the beta coef. Arbitrage pricing theory definition of arbitrage pricing. Pdf the arbitrage pricing theory and multifactor models of asset. G12 abstract focusing on capital asset returns governed by a factor structure, the arbitrage pricing theory apt is a oneperiod model, in which preclusion of arbitrage over static portfolios. The literature on asset pricing models has taken on a new lease of life since the emergence of the arbitrage pricing theory apt, formulated by ross 1976, as an alternative theory to the renowned capital asset pricing model capm, proposed by sharp 1964, lintner 1965 and mossin 1966. Arbitrage refers to nonrisky profits that are generated, not because of a net investment, but on account of exploiting the difference that exists in the price of identical financial instruments due to market imperfections. Arbitrage pricing theory synonyms, arbitrage pricing theory pronunciation, arbitrage pricing theory translation, english dictionary definition of arbitrage pricing theory. The apt model in this study uses macroeconomic variables consisting of exports, inflation, exchange rates, gdp and economic growth. Arbitrage pricing theory for idiosyncratic variance factors. Since its introduction by ross, it has been discussed, evaluated, and tested. Capital asset pricing model and arbitrage pricing theory. Pdf describe the arbitrage pricing theory apt model. One of the two leading capital market theories of 1960s and 1970s, it is based on the law of one price. Arbitrage pricing theory how is arbitrage pricing theory. Classical asset pricing models, such as capm and apt arbitrage pricing 1. Ki november 16, 2004 principles of finance lecture 7 20 apt. The arbitrage pricing theory is something that can be used for asset pricing.348 257 415 645 1518 1532 129 1323 119 848 30 1132 127 1279 379 326 171 705 1177 1148 526 1038 972 699 845 623 1227 1466 435 936 1149 109 554

Contributions via 3 authors deal with facets of noncommutative geometry which are relating to cyclic homology. The authors supply fairly entire money owed of cyclic concept from various issues of view. The connections among (bivariant) K-theory and cyclic conception through generalized Chern-characters are mentioned intimately. Cyclic conception is the average environment for various normal summary index theorems. A survey of such index theorems is given and the options and concepts keen on those theorems are explained. By H. S. M. Coxeter Among the attractive and nontrivial theorems in geometry present in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. a pleasant evidence is given of Morley's striking theorem on perspective trisectors. The transformational standpoint is emphasised: reflections, rotations, translations, similarities, inversions, and affine and projective modifications. many desirable homes of circles, triangles, quadrilaterals, and conics are built. By I.M. James Scholars of topology rightly whinge that a lot of the elemental fabric within the topic can't simply be present in the literature, at the least now not in a handy shape. during this booklet i've got attempted to take a clean examine a few of this simple fabric and to prepare it in a coherent model. The textual content is as self-contained as i'll kind of make it and may be fairly available to a person who has an undemanding wisdom of point-set topology and crew concept. This ebook relies on a process sixteen graduate lectures given at Oxford and in different places at times. In a process that size one can't talk about too many issues with no being unduly superficial. even though, this used to be by no means meant as a treatise at the topic yet relatively as a brief introductory direction to be able to, i'm hoping, turn out worthy to experts and non-specialists alike. The creation incorporates a description of the contents. No algebraic or differen tial topology is concerned, even if i've got borne in brain the desires of scholars of these branches of the topic. routines for the reader are scattered in the course of the textual content, whereas feedback for additional examining are inside the lists of references on the finish of every bankruptcy. generally those lists contain the most resources i've got drawn on, yet this isn't the kind of e-book the place it's viable to provide a reference for every thing. By R. Brown, T. L. Thickstun This quantity includes the court cases of a convention held on the collage university of North Wales (Bangor) in July of 1979. It assembles study papers which replicate assorted currents in low-dimensional topology. The topology of 3-manifolds, hyperbolic geometry and knot concept end up significant subject matters. The inclusion of surveys of labor in those components may still make the booklet very worthwhile to scholars in addition to researchers. By Stephen Lipscomb Historically, for metric areas the hunt for common areas in size concept spanned nearly a century of mathematical examine. The background breaks evidently into classes - the classical (separable metric) and the fashionable (not-necessarily separable metric). The classical concept is now good documented in different books. This monograph is the 1st publication to unify the trendy thought from 1960-2007. just like the classical conception, the trendy thought essentially consists of the unit interval. Unique gains include: * using pictures to demonstrate the fractal view of those spaces; * Lucid assurance of various issues together with point-set topology and mapping thought, fractal geometry, and algebraic topology; * a last bankruptcy comprises surveys and gives historic context for similar study that comes with different imbedding theorems, graph idea, and closed imbeddings; * every one bankruptcy includes a remark part that gives ancient context with references that function a bridge to the literature. This monograph should be worthy to topologists, to mathematicians operating in fractal geometry, and to historians of arithmetic. Being the 1st monograph to target the relationship among generalized fractals and common areas in size thought, it is going to be a usual textual content for graduate seminars or self-study - the reader will locate many correct open difficulties as a way to create extra study into those topics. Topology, for a few years, has been the most interesting and influential fields of study in sleek arithmetic. even if its origins can be traced again a number of hundred years, it was once Poincaré who "gave topology wings" in a vintage sequence of articles released round the flip of the century. whereas the sooner historical past, also known as the prehistory, is usually thought of, this quantity is especially fascinated about the more moderen background of topology, from Poincaré onwards. As may be noticeable from the checklist of contents the articles disguise quite a lot of themes. a few are extra technical than others, however the reader with no good deal of technical wisdom may still locate lots of the articles obtainable. a few are written through expert historians of arithmetic, others via historically-minded mathematicians, who are likely to have a distinct standpoint. This textbook treats the classical elements of mapping measure idea, with a close account of its background traced again to the 1st 1/2 the 18th century. After a ancient first bankruptcy, the remainder 4 chapters boost the math. An attempt is made to take advantage of in simple terms straight forward tools, leading to a self-contained presentation. on the other hand, the publication arrives at a few actually striking theorems: the category of homotopy sessions for spheres and the Poincare-Hopf Index Theorem, in addition to the proofs of the unique formulations by means of Cauchy, Poincare, and others. even supposing the mapping measure idea you can find during this booklet is a classical topic, the therapy is clean for its easy and direct sort. the easy exposition is accented via the looks of numerous unusual subject matters: tubular neighborhoods with no metrics, adjustments among type 1 and sophistication 2 mappings, Jordan Separation with neither compactness nor cohomology, particular buildings of homotopy sessions of spheres, and the direct computation of the Hopf invariant of the 1st Hopf fibration. The e-book is appropriate for a one-semester graduate direction. There are a hundred and eighty routines and difficulties of other scope and hassle. Broadway Entertainment E-books 2017 | All Rights Reserved

If you have a question about an atomic mass, you are not alone. You probably have come across this question several times. Atomic mass refers to the number of protons and neutrons within an atom. But how do you calculate this? There are several methods, and it all depends on what information you have. Below are some of the common methods. Follow these steps to calculate the mass of an element. They will help you identify the element’s isotopes and calculate its mass. Atomic mass is the sum of protons and neutrons in an atom A mass of an atom can be calculated in several ways. One method is to calculate its atomic mass using a standard value. Then, subtract the mass of the neutrons and protons. The final result is the atomic mass of the atom. There are three common methods for calculating an atom’s mass. They are: from a natural sample to a standard value. The first atomic mass was calculated by William Herschel in 1856, and it is the same for all elements. The electron has an extremely small mass, about 1800 times smaller than the proton. Therefore, the mass number of an atom is the sum of the masses of its protons and neutrons, rounded to the nearest integer value. However, the mass number of an atom is often calculated by multiplying the atomic weights of each constituent atom. In addition to being the number of protons and neutrons in an element, the atomic mass of an atom can be measured with the help of a periodic table. In this way, we can easily calculate the mass of different substances. Atomic mass is also important for determining the relative weight of the various elements. The atomic mass of an atom is the sum of the mass of the protons and neutrons in the nucleus. The atomic mass of an element is also known as the atomic number. The number of protons is always the same for atoms of the same element. However, the number of neutrons varies for different elements. For example, an oxygen atom has eight protons, whereas sodium has eleven. However, the atomic mass of a substance is expressed in terms of its mass of protons and neutrons. The atomic mass of an atom can be determined by multiplying the total mass of its protons and neutrons by the mu, a constant metric that can be derived from the relative isotopic mass. For example, a carbon atom is defined to have an atomic mass of 12 Da. Consequently, the relative atomic mass of all the carbon atoms in a molecule is twelve. Atomic mass is a counted number The atomic mass of a substance is a counted number, and is measured in atomic mass units, or amu. For instance, carbon has an atomic mass of 12 amu, and its neutrons and protons make up six amu of its mass. The number of these particles in an atom is also known as atomic weight, or atomic mass. However, this definition only works for one atom. The mass of an atom is derived from the amount of protons and neutrons that compose it. The electrons make up very small amounts of mass, so they are not included in the calculation. Atomic mass is also used to refer to the average mass of all the different isotopes of an element. This metric is often expressed in decimal numbers. In addition, it can be calculated relative to other elements to determine their weight in terms of other substances. Protons and neutrons have equal masses. Electrons are slightly heavier than protons. Generally, the atomic mass of an atom will be based on the number of protons and neutrons in the nucleus. However, if you subtract the electrons from the total mass of an atom, the atom will have more mass than the nucleus. In this way, it is possible to determine how much a particular element weighs in the mass of all its constituents. The atomic mass of an atom can be calculated by adding the protons, electrons, and neutrons. In addition, the weighted measures of isotopes can also be used to estimate the atomic mass. The atomic mass of hydrogen is H = 1.00797 times the mass of the atom of water, which is u.m.a. (unit mass). When looking up a specific atom, it can be confusing to determine which nucleons make up that atom. In addition to the mass, the neutrons must also be counted, and this is the best way to figure out the nucleon count. You can also look up the atomic number by subtracting the protons. If you know how many protons a certain atom has, then you can calculate its atomic mass. Atomic mass is found in a nuclear symbol The atomic mass number, abbreviated A, is the total number of protons and neutrons in an atom. In the periodic table, A is the mass number of carbon, and B is the mass of lead. Listed below are the mass numbers of the first six elements. Atoms can be defined as any combination of two elements. In other words, one element has two protons, and the other has four. The atomic number of an atom is a major characteristic. It represents the number of protons in an atom, and is important in determining the chemical properties of an atom. The number of protons is indicated by the letter P. The number of electrons in an atom is represented by the letter N. The total mass of an atom is given by the sum of the mass numbers of the protons and neutrons, which are also found in an atomic symbol. The atomic mass number of an atom is the same for all the atoms of a given element, but is not necessarily the same for all atoms of the same element. Atoms with similar numbers of protons and neutrons are called isotopes. For example, carbon atoms contain one or two extranuclear electrons, but have a different mass number. Carbon-12 and carbon-13 are symbolically the same, but have different mass numbers. The atomic mass number is also known as the atomic weight. For example, the atomic mass number of silicon is 14 and it has fifteen neutrons. This means that a given isotope of silicon has a mass number of 29. In a nuclear symbol, the mass number is written in terms of atomic mass units. In the nuclear symbol, the number indicates the actual mass of the atom. A nuclear symbol can contain a nucleus with a different mass. If the atom contains a neutron, it must also contain 33 protons. Therefore, the atomic mass number is equal to two-thirds of the total mass of the atom. It is important to understand the differences between the different nuclei, since the atomic mass of a certain atom is the dominant factor in the nuclear symbol. Atomic mass is a quantity that represents all isotopes of an element An atom’s mass is its average, or “atomic mass.” This is also known as atomic weight, because it is the average of all the isotopes present in the sample. Atomic mass of a single atom is less than the mass of all the protons, neutrons, and atomic nuclei in the sample. This difference is due to binding energy mass loss. Because the difference between the atomic masses of two common isotopes is usually small, it can affect calculations, either in bulk or on an individual level. However, in some cases, atomic mass differences are substantial, affecting the result of bulk calculations or individual calculations. It is important to note that while atomic mass numbers can be useful in the calculation of mass for elements, their properties vary significantly among isotopes. The number of protons and neutrons present in an atom is called its atomic mass. The two protons make up approximately equal mass, while the electrons contribute very little. The atomic weight of an element is known as the atomic weight. The weight of an atom in one element is equivalent to one-twelfth its mass in its ground state. In nature, there are many isotopes of an element. In the laboratory, one of these is the most common of them all – carbon-12. Despite their similarities, they differ in their chemical behavior. Carbon-13 has more neutrons than carbon-12. In contrast, the latter has less mass, which is why it has a higher atomic mass. The atomic mass of an element is known as its atomic number. The mass number of an atom is related to the number of protons. Carbon contains six protons and eight neutrons. Carbon-14 has eight neutrons. The two are oppositely polar, but they both have the same atomic number. Therefore, it has the same atomic mass as carbon. In the case of hydrogen, an atom is made up of one proton and one electron. It is therefore electrically neutral. The positive charge in the nucleus must balance the negative charge of the electron. The electrons contribute little mass to the atom. But their presence pointed to the existence of other particles in the atom. And if there are other particles present in the atom, the positive charge must make up most of the mass of the atom.

Hello, everyone. My name is Mohammed Ali. Now we will answer the question. Considers a reaction. It's uto glass pr to for me to Italy are the graph shows a concentration off br toe as function off time. Use a graft to calculate each quantities averaging. So from the reading off the first bar, this question is related to the rape, so must knows his relation before answers. This question. The 1st 1 is the real, uh, he's a reed is equal minus sheen. Their concentration off B R. Two with Stein we cause a B R to his reactor. Also negative chain in EJ toe with Stein and both Steve off boasted because we will speak now about H B R. And it should be our is broader and for two more. So now this is a relation. You must know it's mine. When we want to know the rate off reaction, we can write it by other message that the reed is equal. The read off, decreasing pro mean and read off, decreasing on regime and real or formation h b r. But we must dry half, so you must know these relations before we going toe. Answer the question from this questioning Want to you to use the graph so we must know the scale of grab zero and the's to Auntie 30 40 And here 50 saw Each line is represented. Why in the year 0.2 you're going for and from in your twosies is represent or 0.0 point 1.2 points three upon for upon five upon sex 4.7 point eight No, we would answer the first question. The first question he wanted. You can clean the average rate of reaction between zero and 25. So here is given to me the concentration off for me. So we will use these relations. Sad saree is equal negative there the concentration off B R two at 25 2nd negative concentration off B R to N zero divided by change off time So 25 negative zero This question It will be negative. Concentration off poor mean 25 stand This is 20. So here is 25 Go to the girl. It will be here at the middle, So it will be all 0.75 Answer concentration off bro mean add zero It will be go up, It will be here. So it will be mom divided by 25 negative zero. So by calculation and the result will be, Oh, Boeing or Mom, This is the answer off. The first bar question on now we are going toe. Answer the second question off the bar A which you want to know the instant you a straight off reaction at 25 seconds to know the Amis tenuous. It's continuous ring. We must reaches a 25 by the curb which cut it as this point. And from this point we must drew attention. So here I reaches a 0.25 to the care and in hydro danger. Then, after droning their danger, we choose any two point off concentration and the Bosnian bhai's attention and the show. What is the related time? So here he want to know the rate of reaction and we said before the read off reaction is represent the change in concentration of premier divided by time. So from the line, we will change it. Choose this point which he is all point A that got ends a time 20 and we will choose this point which is here and it represent opening 55 and connected it here. It will got a 50. So I choose now that toe point that constant attention which is and 50 and 20 type at 50. It card at concentration. 0.55 and And Wendy, it got at oh, boy in E. So by calculating a we found the reed is equal. Oh, point or or a three. So this is answer off. Read every action at 25 seconds. Let's go to the survey question off the bark a. The certain question you want to know there any stenting you strain but now for the formation. But now is four formation off h b R. So it should be are we must related it by premier because this question is answer for permit. So if we go to for the first part, the rate off women is haves a rate off h e r. So we can right now. If we know the re off b r two, we will times it by toe to know the raid off h p. R. So we will solve it it as we measures Ari off br toe at 52nd Then we can know the read off, Actually, Are there it off? B R two is equal Negative change off the concentration on B R two divided by time and and we want at 50 seconds for gay to solve is is you will reach point So the curve a 50 then you will grow danger There we will it choose any two points from concentration and connected them to attention and see the corresponding time which we want to know the slow as we calculate the slope off the stage. So here which shoes there. Ah, time 30 In time 40 We found that at times 30 it cut at boy seven and at a time 40 It got at time at concentration opening 65 So now which was a poin which are 40 negative 30 at fort is a concentration is open 65 from grab and a 30. It's opening seven from grab win Cut the danger So now we can calculate the rail which is all point or fine. But don't forget that this is a read off for me. So to know they read off which we are We must time it by two So raid off H B R. Well, the O'Brien or Mom, This is the answer off the search bar Off the first question. Let's go to the second bar for the second. The bar. He wanted you to draw a roof sketch once a meaning off roofs sketch roof. Sketch me. We wantto only you draw a quick without calculation. Exact off curve represents the concentration off HB oris function off time. So now he wants you to represent how the concentration off the product H e B R will be increased. In fact is entropy are start from zero and it will be increased quickly. Zen the decreasing in B R. Two Because two moons are for me and we found it. It's halftime, so it will increase quickly than the decreasing in B R. To which I mean that they're one amount is decrees in 100 seconds, but here it will increase for the job or the Simone at 50 seconds. So this is a sketch for HB on. And this is the answer off this question. Thank you.

Statistics I INTRODUCTION Statistics, branch of mathematics that deals with the collection, organization, and analysis of numerical data and with such problems as experiment design and decision making. II HISTORY Domesday Book Compiled in 1086 under the direction of William the Conquerer, the Domesday Book was a meticulous survey of feudal estates in England. Public Record Office, Surrey, England Simple forms of statistics have been used since the beginning of civilization, when pictorial representations or other symbols were used to record numbers of people, animals, and inanimate objects on skins, slabs, or sticks of wood and the walls of caves. Before 3000 BC the Babylonians used small clay tablets to record tabulations of agricultural yields and of commodities bartered or sold. The Egyptians analyzed the population and material wealth of their country before beginning to build the pyramids in the 31st century BC. The biblical books of Numbers and 1 Chronicles are primarily statistical works, the former containing two separate censuses of the Israelites and the latter describing the material wealth of various Jewish tribes. Similar numerical records existed in China before 2000 censuses to be used as bases for taxation as early as 594 BC. BC. The ancient Greeks held See Census. The Roman Empire was the first government to gather extensive data about the population, area, and wealth of the territories that it controlled. During the Middle Ages in Europe few comprehensive censuses were made. The Carolingian kings Pepin the Short and Charlemagne ordered surveys of ecclesiastical holdings: Pepin in 758 and Charlemagne in 762. Following the Norman Conquest of England in 1066, William I, king of England, ordered a census to be taken; the information gathered in this census, conducted in 1086, was recorded in the Domesday Book. Registration of deaths and births was begun in England in the early 16th century, and in 1662 the first noteworthy statistical study of population, Observations on the London Bills of Mortality, was written. A similar study of mortality made in Breslau, Germany, in 1691 was used by the English astronomer Edmond Halley as a basis for the earliest mortality table. In the 19th century, with the application of the scientific method to all phenomena in the natural and social sciences, investigators recognized the need to reduce information to numerical values to avoid the ambiguity of verbal description. At present, statistics is a reliable means of describing accurately the values of economic, political, social, psychological, biological, and physical data and serves as a tool to correlate and analyze such data. The work of the statistician is no longer confined to gathering and tabulating data, but is chiefly a process of interpreting the information. The development of the theory of probability increased the scope of statistical applications. Much data can be approximated accurately by certain probability distributions, and the results of probability distributions can be used in analyzing statistical data. Probability can be used to test the reliability of statistical inferences and to indicate the kind and amount of data required for a particular problem. III STATISTICAL METHODS How Polls Predict Professional pollsters typically conduct their surveys among sample populations of 1,000 people. Statistical measurements show that reductions in the margin of error flatten out considerably after the sample size reaches 1,000. © Microsoft Corporation. All Rights Reserved. The raw materials of statistics are sets of numbers obtained from enumerations or measurements. In collecting statistical data, adequate precautions must be taken to secure complete and accurate information. The first problem of the statistician is to determine what and how much data to collect. Actually, the problem of the census taker in obtaining an accurate and complete count of the population, like the problem of the physicist who wishes to count the number of molecule collisions per second in a given volume of gas under given conditions, is to decide the precise nature of the items to be counted. The statistician faces a complex problem when, for example, he or she wishes to take a sample poll or straw vote. It is no simple matter to gauge the size and constitution of the sample that will yield reasonably accurate predictions concerning the action of the total population. In protracted studies to establish a physical, biological, or social law, the statistician may start with one set of data and gradually modify it in light of experience. For example, in early studies of the growth of populations, future change in size of population was predicted by calculating the excess of births over deaths in any given period. Population statisticians soon recognized that rate of increase ultimately depends on the number of births, regardless of the number of deaths, so they began to calculate future population growth on the basis of the number of births each year per 1000 population. When predictions based on this method yielded inaccurate results, statisticians realized that other limiting factors exist in population growth. Because the number of births possible depends on the number of women rather than the total population, and because women bear children during only part of their total lifetime, the basic datum used to calculate future population size is now the number of live births per 1000 females of childbearing age. The predictive value of this basic datum can be further refined by combining it with other data on the percentage of women who remain childless because of choice or circumstance, sterility, contraception, death before the end of the childbearing period, and other limiting factors. The excess of births over deaths, therefore, is meaningful only as an indication of gross population growth over a definite period in the past; the number of births per 1000 population is meaningful only as an expression of the proportion of increase during a similar period; and the number of live births per 1000 women of childbearing age is meaningful for predicting future size of populations. IV TABULATION AND PRESENTATION OF DATA Frequency-Distribution Table A frequency-distribution table summarizes data. For example, there were 1200 grades received on 4 examinations by 10 sections of 30 students each. The first column lists the ten intervals into which the grades were grouped. The second column lists the midpoints of these intervals. The third column lists the number of grades in each interval, that is, their frequency. (There were 20 grades between 0 and 10.) The fourth column lists the proportion of grades in each interval, that is, their relative frequency. (.017 of the 1200 grades were between 0 and 10.) The fifth column lists the number of grades in an interval and all intervals below it, that is, their cumulative frequency. (35 grades were in or below the interval between 10 and 20.) The sixth column lists the proportion of grades in or below an interval, that is, their relative cumulative frequency. (0.029 of the 1200 grades were in or below the interval 10 to 20.) © Microsoft Corporation. All Rights Reserved. The collected data must be arranged, tabulated, and presented to permit ready and meaningful analysis and interpretation. To study and interpret the examinationgrade distribution in a class of 30 pupils, for instance, the grades are arranged in ascending order: 30, 35, 43, 52, 61, 65, 65, 65, 68, 70, 72, 72, 73, 75, 75, 76, 77, 78, 78, 80, 83, 85, 88, 88, 90, 91, 96, 97, 100, 100. This progression shows at a glance that the maximum is 100, the minimum 30, and the range, or difference, between the maximum and minimum is 70. In a cumulative-frequency graph, such as Fig. 1, the grades are marked on the horizontal axis and double marked on the vertical axis with the cumulative number of the grades on the left and the corresponding percentage of the total number on the right. Each dot represents the accumulated number of students who have attained a particular grade or less. For example, the dot A corresponds to the second 72; reading on the vertical axis, it is evident that there are 12, or 40 percent, of the grades equal to or less than 72. In analyzing the grades received by 10 sections of 30 pupils each on four examinations, a total of 1200 grades, the amount of data is too large to be exhibited conveniently as in Fig. 1. The statistician separates the data into suitably chosen groups, or intervals. For example, ten intervals might be used to tabulate the 1200 grades, as in column (a) of the accompanying frequency-distribution table; the actual number in an interval, called the frequency of the interval, is entered in column (c). The numbers that define the interval range are called the interval boundaries. It is convenient to choose the interval boundaries so that the interval ranges are equal to each other; the interval midpoints, half the sum of the interval boundaries, are simple numbers, because they are used in many calculations. A grade such as 87 will be tallied in the 80-90 interval; a boundary grade such as 90 may be tallied uniformly throughout the groups in either the lower or upper intervals. The relative frequency, column (d), is the ratio of the frequency of an interval to the total count; the relative frequency is multiplied by 100 to obtain the percent relative frequency. The cumulative frequency, column (e), represents the number of students receiving grades equal to or less than the range in each succeeding interval; thus, the number of students with grades of 30 or less is obtained by adding the frequencies in column (c) for the first three intervals, which total 53. The cumulative relative frequency, column (f), is the ratio of the cumulative frequency to the total number of grades. The data of a frequency-distribution table can be presented graphically in a frequency histogram, as in Fig. 2, or a cumulative-frequency polygon, as in Fig. 3. The histogram is a series of rectangles with bases equal to the interval ranges and areas proportional to the frequencies. The polygon in Fig. 3 is drawn by connecting with straight lines the interval midpoints of a cumulative frequency histogram. Newspapers and other printed media frequently present statistical data pictorially by using different lengths or sizes of various symbols to indicate different values. V MEASURES OF CENTRAL TENDENCY After data have been collected and tabulated, analysis begins with the calculation of a single number, which will summarize or represent all the data. Because data often exhibit a cluster or central point, this number is called a measure of central tendency. Let x1, x2, ..., xn be the n tabulated (but ungrouped) numbers of some statistic; the most frequently used measure is the simple arithmetic average, or mean, written ?, which is the sum of the numbers divided by n: If the x's are grouped into k intervals, with midpoints m1, m2, ..., mk and frequencies f1, f2, ..., fk, respectively, the simple arithmetic average is given by with i = 1, 2, ..., k. The median and the mode are two other measures of central tendency. Let the x's be arranged in numerical order; if n is odd, the median is the middle x; if n is even, the median is the average of the two middle x's. The mode is the x that occurs most frequently. If two or more distinct x's occur with equal frequencies, but none with greater frequency, the set of x's may be said not to have a mode or to be bimodal, with modes at the two most frequent x's, or trimodal, with modes at the three most frequent x's. VI MEASURES OF VARIABILITY The investigator frequently is concerned with the variability of the distribution, that is, whether the measurements are clustered tightly around the mean or spread over the range. One measure of this variability is the difference between two percentiles, usually the 25th and the 75th percentiles. The p th percentile is a number such that p percent of the measurements are less than or equal to it; in particular, the 25th and the 75th percentiles are called the lower and upper quartiles, respectively. The p th percentile is readily found from the cumulative-frequency graph, (Fig. 1) by running a horizontal line through the p percent mark on the vertical axis on the graph, then a vertical line from this point on the graph to the horizontal axis; the abscissa of the intersection is the value of the p th percentile. The standard deviation is a measure of variability that is more convenient than percentile differences for further investigation and analysis of statistical data. The standard deviation of a set of measurements x1, x2, ..., xn, with the mean ? is defined as the square root of the mean of the squares of the deviations; it is usually designated by the Greek letter sigma (?). In symbols The square, ?2, of the standard deviation is called the variance. If the standard deviation is small, the measurements are tightly clustered around the mean; if it is large, they are widely scattered. VII CORRELATION When two social, physical, or biological phenomena increase or decrease proportionately and simultaneously because of identical external factors, the phenomena are correlated positively; under the same conditions, if one increases in the same proportion that the other decreases, the two phenomena are negatively correlated. Investigators calculate the degree of correlation by applying a coefficient of correlation to data concerning the two phenomena. The most common correlation coefficient is expressed as in which x is the deviation of one variable from its mean, y is the deviation of the other variable from its mean, and N is the total number of cases in the series. A perfect positive correlation between the two variables results in a coefficient of +1, a perfect negative correlation in a coefficient of -1, and a total absence of correlation in a coefficient of 0. Intermediate values between +1 and 0 or -1 are interpreted by degree of correlation. Thus, .89 indicates high positive correlation, -.76 high negative correlation, and .13 low positive correlation. VIII MATHEMATICAL MODELS Distribution of IQ Scores The distribution of scores (commonly called IQ scores) on the Wechsler Adult Intelligence Scale follows an approximately normal curve, an average distribution of values. The test is regularly adjusted so that the median score is 100--that is, so that half of the scores fall above 100, and half fall below. © Microsoft Corporation. All Rights Reserved. A mathematical model is a mathematical idealization in the form of a system, proposition, formula, or equation of a physical, biological, or social phenomenon. Thus, a theoretical, perfectly balanced die that can be tossed in a purely random fashion is a mathematical model for an actual physical die. The probability that in n throws of a mathematical die a throw of 6 will occur k times is in which (À is the symbol for the binomial coefficient ) The statistician confronted with a real physical die will devise an experiment, such as tossing the die n times repeatedly, for a total of Nn tosses, and then determine from the observed throws the likelihood that the die is balanced and that it was thrown in a random way. In a related but more involved example of a mathematical model, many sets of measurements have been found to have the same type of frequency distribution. For example, let x1, x2, ..., xN be the number of 6's cast in the N respective runs of n tosses of a die and assume N to be moderately large. Let y1, y2, ..., yN be the weights, correct to the nearest 1/100 g, of N lima beans chosen haphazardly from a 100-kg bag of lima beans. Let z1, z2, ..., zN be the barometric pressures recorded to the nearest 1/1000 cm by N students in succession, reading the same barometer. It will be observed that the x's, y's, and z's have amazingly similar frequency patterns. The statistician adopts a model that is a mathematical prototype or idealization of all these patterns or distributions. One form of the mathematical model is an equation for the frequency distribution, in which N is assumed to be infinite: in which e (approximately 2.7) is the base for natural logarithms (see Logarithm). The graph of this equation (Fig. 4) is the bell-shaped curve called the normal, or Gaussian, probability curve. If a variate x is normally distributed, the probability that its value lies between a and b is given by The mean of the x's is 0, and the standard deviation is 1. In practice, if N is large, the error is exceedingly small. IX TESTS OF RELIABILITY The statistician is often called upon to decide whether an assumed hypothesis for some phenomenon is valid or not. The assumed hypothesis leads to a mathematical model; the model, in turn, yields certain predicted or expected values, for example, 10, 15, 25. The corresponding actually observed values are 12, 16, 21. To determine whether the hypothesis is to be kept or rejected, these deviations must be judged as normal fluctuations caused by sampling techniques or as significant discrepancies. Statisticians have devised several tests for the significance or reliability of data. One is the chi-square (c 2) test. The deviations (observed values minus expected values) are squared, divided by the expected values, and summed: The value of c2 is then compared with values in a statistical table to determine the significance of the deviations. X HIGHER STATISTICS The statistical methods described above are the simpler, more commonly used methods in the physical, biological, and social sciences. More advanced methods, often involving advanced mathematics, are used in further statistical studies, such as sampling theory, inference and estimation theory, and design of experiments. Contributed By: James Singer Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.

Newton developed this law of motion has significant mathematical and physical elucidation that are needed to understand the motion of objects in our universe. Newton introduced the three laws in his book Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), which is generally referred to as the Principia. He also introduced his theory of universal gravitation, thus laying down the entire foundation of classical mechanics in one volume in 1687. These laws define the motion changes, specifically the way in which those changes in motion are related to force and mass. There are three laws of motion which were introduced by Sir Isaac Newton which are Newton’s First Law , Newton’s Second Law and Newton’s This supports my hypothesis, where I predicted this proportionality. From Newton’s Second Law F=ma, I derived an equation isolating a: a= (1/m)F - (1/m)fric. I solved the equation for the line of best fit, and used the y-intercept to figure out the friction, which was the only remaining unknown in the equation. Once the friction was known, I was then able to plug in measured values and verify my hypothesis. I proved with this equation, using examples from my data, that the force is in fact directly proportional to Electrohydrodynamic Electrohydrodynamic Phenomena The EHD phenomena involve the interaction of electric fields and flow fields in a dielectric fluid medium. This interaction can result in electrically induced fluid motion and interfacial instabilities which are caused by an electric body force. The electric body force density acting on the molecules of a dielectric fluid in the presence of an electric field consists of three terms (1): f_e=ρ_e E ̅-1/2 E^2 ∇ε+1/2 ∇[ρE^2 (δε/δρ)_T ] (1) The three terms in Eq (1) stand for two primary force densities acting on the fluid. The first term represents the force acting on the free charges in the presence of an electric field and is known as the Coulomb force. The second and third terms represent Introduction In 1687, Newton put forward the Newton's Second Law of Motion-Force and Acceleration in the book “Philosophiae Naturalis Principia Mathematica”. According to Newton’s second law, , this is integrated over position from an initial position (i) to a final position (f). (Wang,1) . Therefore, we can get the work-energy theorem, . W is the work done by the net force on the object, which equals to the change in kinetic energy according to the equation. MOJICA, Aselle Joyce G. Group no. 3 PHY13l/A3 Seat no. 3-3 ANALYSIS In PART 1 of the experiment which the is Magnetic Field of Permanent Magnets, we used two different magnets; two bar magnets and two U- magnets in order to see clearly what would be the result when magnets are placed in different orientations. For PART 1A, the bar magnets were oriented with like poles (N-N) facing each other. As a result after putting and scattering iron filings, each field line of the magnetic field from the north pole of the two magnets went away from each other which simply prove that like poles repel. In Newtonian gravity (which was the classical theory of gravity), the source of gravity is the mass. In general theory of relativity, the mass turns out to be part of a more general quantity called the energy-momentum tensor (Tμυ), which includes both the energy and momentum densities. The field equation for gravity includes this tensor. The energy-momentum tensor is divergence free where its covariant derivative in the curved space-time is zero (∇^μ Tμυ= 0). By finding a tensor on other side which is divergence free, this yields the simplest set of equations which are called Einstein's (field) equations. which is represented as, p/ρg+v^2/2g+z=constant, here z is height. Therefore, Bernoulli’s eq. of motion is defined as, In an ideal, steady flow of a fluid, the total energy at any random point of the fluid is constant. The total energy consists of kinetic energy, potential energy, and pressure Aerodynamics is a branch of dynamics to the study of air movement together. It is a subfield of fluid dynamics and gas, and the term "drag" is often used to refer to the gas dynamics. The earliest records of the basic concepts of aerodynamics on the work of Aristotle and Archimedes in the third and second centuries BC, but the efforts to find a quantitative theory of airflow develop until the 18th century, beginning in 1726 was Isaac Newton as one of the first in modern aerodynamics mind when he developed a theory of air resistance, which was later verified for low flow rates. Air resistance experiments were carried out by researchers in the 18th and 19th centuries, with the aid of the construction of the first wind tunnel in 1871 In 1738 He is called the father of the clockwork universe because of the theories he invented, universal gravity, three laws of motion ("Physics is the study of your world and the world and universe around you." (Holzner 2006, p. 7-15) 3.1 USE OF DIFFERENTIAL EQUATIONS IN PHYSICS Differential equation is one of the examples of mathematical equations that is associated with functions and its derivatives. The functions determine the physical quantities (position, velocity, acceleration and forces acting on the object), the For example, in classical mechanics, F=ma is a formula that tells us that the net force acting on a body is given by the mass of the body times its acceleration. That constitutes a very precise statement, and when we plug the numbers into the formula, we get a precise result, that is expressed in specific units, in our case

- When should you increase sample size? - What happens when a sample size is not big enough? - What is considered a low sample size? - What happens when the sample size decreases? - Is 30 of the population a good sample size? - What is the minimum sample size for t test? - Does sample size affect validity? - Does sample size affect bias? - Does small sample size increase Type 2 error? - When the sample size increases the population mean decreases? - What is the minimum sample size for Anova? - Why must sample size be greater than 30? - Does increasing sample size increase confidence level? - How does increasing sample size increase power? - How small is too small for a sample size? - Why is the minimum sample size 30? - Which is a test of significance for sample size less than or equal to 30? - Why is a small sample bad? When should you increase sample size? Higher sample size allows the researcher to increase the significance level of the findings, since the confidence of the result are likely to increase with a higher sample size. This is to be expected because larger the sample size, the more accurately it is expected to mirror the behavior of the whole group.. What happens when a sample size is not big enough? Sampling. The most obvious strategy is simply to sample more of your population. Keep your survey open, contact more potential participants, or consider widening the population. What is considered a low sample size? Generally, for any inferential statistic, a sample size of less than 500 may not be adequate. What happens when the sample size decreases? The population mean of the distribution of sample means is the same as the population mean of the distribution being sampled from. … Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases. Is 30 of the population a good sample size? Sampling ratio (sample size to population size): Generally speaking, the smaller the population, the larger the sampling ratio needed. For populations under 1,000, a minimum ratio of 30 percent (300 individuals) is advisable to ensure representativeness of the sample. What is the minimum sample size for t test? 10 Answers. There is no minimum sample size for the t test to be valid other than it be large enough to calculate the test statistic. Does sample size affect validity? The use of sample size calculation directly influences research findings. Very small samples undermine the internal and external validity of a study. Very large samples tend to transform small differences into statistically significant differences – even when they are clinically insignificant. Does sample size affect bias? Increasing the sample size tends to reduce the sampling error; that is, it makes the sample statistic less variable. However, increasing sample size does not affect survey bias. A large sample size cannot correct for the methodological problems (undercoverage, nonresponse bias, etc.) that produce survey bias. Does small sample size increase Type 2 error? Type II errors are more likely to occur when sample sizes are too small, the true difference or effect is small and variability is large. The probability of a type II error occurring can be calculated or pre-defined and is denoted as β. When the sample size increases the population mean decreases? With “infinite” numbers of successive random samples, the mean of the sampling distribution is equal to the population mean (µ). As the sample sizes increase, the variability of each sampling distribution decreases so that they become increasingly more leptokurtic. What is the minimum sample size for Anova? 3Is there a minimum sample size to run an ANOVA? In theory, it is 3. You need two populations, so that’s 2, but you need two samples to get a variance estimate. If you assume equal variances, you only need the estimate from one population so that’s 3 total. Why must sample size be greater than 30? As a general rule, sample sizes equal to or greater than 30 are deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples one takes, the more the graphed results take the shape of a normal distribution. Does increasing sample size increase confidence level? A higher confidence level requires a larger sample size. Power – This is the probability that we find statistically significant evidence of a difference between the groups, given that there is a difference in the population. A greater power requires a larger sample size. How does increasing sample size increase power? The price of this increased power is that as α goes up, so does the probability of a Type I error should the null hypothesis in fact be true. The sample size n. As n increases, so does the power of the significance test. This is because a larger sample size narrows the distribution of the test statistic. How small is too small for a sample size? The numbers behind this phenomenon are kind of complicated, but often a small sample size in a study can cause results that are almost as bad, if not worse, than not running a study at all. Despite these statistical assertions, many studies think that 100 or even 30 people is an acceptable number. Why is the minimum sample size 30? One may ask why sample size is so important. The answer to this is that an appropriate sample size is required for validity. If the sample size it too small, it will not yield valid results. … If we are using three independent variables, then a clear rule would be to have a minimum sample size of 30. Which is a test of significance for sample size less than or equal to 30? If the sample sizes in at least one of the two samples is small (usually less than 30), then a t test is appropriate. Note that a t test can also be used with large samples as well, in some cases, statistical packages will only compute a t test and not a z test. Why is a small sample bad? Small samples are bad. Why? If we pick a small sample, we run a greater risk of the small sample being unusual just by chance. Choosing 5 people to represent the entire U.S., even if they are chosen completely at random, will often result if a sample that is very unrepresentative of the population.

Motion Graphs Worksheet Motion graphs m. httpscienceclass.net the graphs below represent the motion of a car. match the descriptions with the graphs. explain your answers. descriptions. the car is stopped. the car is traveling at a constant speed. the speed of the car is decreasing. the car is coming back.I use this worksheet to help teach motion graphs. this graph worksheet includes six graphs to show constant speed, acceleration, deceleration, no motion, and combinations of these. combine this worksheet with my mo worksheet on graphs to compare graphs Motion graphs kinematics worksheet. 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Motion Graph Analysis Worksheet Elegant Time Distance Velocity Acceleration Graphs Physics Teaching Ideas On a velocity time graph it is not possible to determine how far from the detector the object is located.Consider the position vs. time graph below for cyclists a and b. there are no calculations b. looking at graph, do the cyclists start at the same position how do you know if not, which one is further to the right. 48. Force Motion Quiz Elementary School Science Activities 49. Force Motion Worksheet Graphs Elementary School Science This worksheet is great to use in your classroom to help students understand an extremely complicated concept force and motion. through this activity students are asked to put themselves into imagined but relatable experiences in order to learn how inertia, gravity, friction, and force affect speed and velocity. 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Graphing Motion Worksheet Physics Interpreting Graphs Persuasive Writing Prompts Velocity time graphs worksheet. split the graph up into distinct sections these can be seen in the image as a b c and d. videos worksheets a day and much more. velocity time graph answer key displaying top worksheets found for this concept.And so what i have constructed here is known as a position time graph, and from this, without an animation, you can immediately get an understanding of how the things position has changed over time. 56. Graphing Speed Worksheet Answers Distance Time Graph Uniform Motion Kids Worksheets Grade Graphs 57. Interpreting Motion Graphs Physics Love the moment at the end. a sense of surprise in math class is a rare and pretty wonderful thing.As the name suggests, graph it is a math worksheet that requires kids to draw a simple chart. giving kids just the basics to start off with, this graphs worksheet requires them to count the number of various objects and mark it correctly on the chart. 58. Interpreting Motion Graphs Worksheet Answers Text Structure Worksheets Use the graph to answer the questions. a when was the object stopped b when was the object moving with a positive velocity c when was the object moving with a negative velocity d when did the object have a positive displacement e when did the object have a negative graphs. the graph below describes the motion of a fly that starts out going right. time s.Physics. worksheet four time graphs velocity ms name review of graphs of motion the graph for a journey is shown. a b calculate the acceleration for each section. calculate the distance travelled in the first seconds. 59. Interpreting Motion Graphs Worksheet Graphical Analysis Distance Time Worksheets 60. Interpreting Motion Graphs Worksheet Plotting Distance Time Yo Worksheets Dec, motion graphs worksheet along with best force and motion images on. the sixth is a good thing to use if you want to show important keywords of your graphic. you can place them in order of importance to the graphic that you are using. people can easily see the important key words in the background of the graphic. Match the description provided about the behavior of a cart along a linear track to its best graphical representation. remember that velocities are positive when the graph is in i quadrant i velocities are negative when the graph is in quadrant iv graphs sloping towards the represent losing speed graphs sloping away from the represent gaining graphs and derivatives worksheet. 61. Worksheet Great Addition Force Motion Unit Beginners Students Analyze Basic Graph Graphs Graphing Calculating Speed In which sections is the cart accelerating. in which sections is the cart not moving. in which sections is the cart moving backwards. in which sections is the carts instantaneous velocity at any time equal to its average velocity. When velocity is zero, the graph should be horizontal. since the acceleration is constant, the graph will always be a parabola, the graph will always be straight, and the graph will always be horizontal. when acceleration is positive, the graph should have a positive slope and the graph should bend upward.

Posting: # 18414 Because I was A) not able to find any post that even remotely dealed with this issue and B) had some discussion lately that might also betide anybody else and C) have some spare time and D) was bewildered that this issue caused so much discussion, I would like to show a simple example why in BE/BA the fancy stuff is not necessarily the correct approach. May be boring for the experienced biometrician/statistician, but was enlightening for a lot of my colleagues. Remember, you can stop reading at any time, just saying . We got involved in discussing the evaluation of an endogenous substance (including a pre-dose profile for baseline correction), where we criticized that no baseline correction was implemented at all and, therefore, their conclusion on the compared products was not valid . But people said, an ANCOVA was used, as recommended by the "Guideline on adjustment for baseline covariates in clinical trials", so this approach should suffice as a baseline correction. From our point of view, this is not correct; as as a matter of fact, the use of a covariate should be considered if there actually is some impact of the starting value on the outcome. Likely fine for clinical endpoints and some PD parameters, but what should be the mechanistical concept in case of an AUC? So, we did not agree and were able to enforce a "proper" baseline correction by subtraction. This was finally implemented and ... resulted in the exact same results . By closer examination it was revealed that the same model was applied, i.e. the ANCOVA was conducted considering the values after baseline-correction. Nice try... As a little illustration to be used when such a discussion comes up consider these values: Easy to see, we have a pure difference of 50 for T-R and 40 if baseline is considered. Hint: these are not real data. Now, whatever software you use, the evaluation should resemble something like this: where "Baseline" is used in case of inclusion of the covariate. So what results do we get in which evalution (point estimates and 95%CI): As you can see, use of the ANCOVA approach gives us results differing from what we get from the "expected" calculation. And as is to be expected due to the concept of an ANOVA it does not matter, whether you use the change from baseline or the end value. So, in particular in those cases, where officially the baseline-correction in accordance with the guidelines was implemented, but an ANCOVA was conducted...). And good luck finding a medical writer who will recognize this in the SAS code or Phoenix output or... Why is this important? Well, in the case that started our discussion, the improper ANOVA shifted the point estimate and allowed to conclude on a statistically significant difference. That is, it allowed to avoid crossing the 100% threshold. Could have been 125% as well. In the presented case above on the other hand, the improper ANCOVA markedly increased the variance (the baseline values are admittedly a little bit one-sided), so hiding a difference might be possible. As always, please do not hesitate to correct, add and challenge, if there is something wrong. Edit: Tabulators changed to spaces and BBcoded; see also this post #6. [Helmut] Posting: # 18424 Dealing with endogenous compounds is tricky and here are some more thoughts you may find helpful This could be considered as change from baseline problem and you may have a look at Stephen Senns work on this topic relating to ANCOVA (e.g. Statist. Med. 2006; 25:4334–4344. https://doi.org/10.1002/sim.2682 ) You may find also this article of interest addressing adjustment of endogenous levels in PK modeling: Bauer, A. & Wolfsegger, M.J. Eur J Clin Pharmacol (2014) 70: 1465. https://doi.org/10.1007/s00228-014-1759-x Best regards & hope this helps Posting: # 18425 I read your post so many times now and I am somewhat confused. What were you actually trying to prove or disprove? Inclusion of a covariate one way or another makes an implicit assumption of a relationship that can be said to be linear between the covariate and the response (in the presence of the factors). If the variance goes full Tasmanian devil on you when you include the covariate then perhaps this assumption is...well... of a nature that has the potential to cause some degree of debate. And then that is where the problem truly is. In contrast to classical anovas where an additional factor will always decrease the unexplianed variance (or leave it unchanged, academically), the inclusion of a covariate is not necessarily having this effect. Help me, please, I really wish to understand what this is all about. "(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018. Posting: # 18426 » I read your post so many times now and I am somewhat confused. And I read it over so many times exactly to avoid being too confusing. Sorry for failing. » What were you actually trying to prove or disprove? Uh, nothing, really, I only wanted to share my experience with this discussion in a BA-setting as I found it difficult to find anything that was just a simple statement or experience shared. And in favour of our position (Baselines are not(!) a good covariate in PK and will potentially result in a misleading result). » Inclusion of a covariate one way or another makes an implicit assumption of a relationship that can be said to be linear between the covariate and the response (in the presence of the factors). » If the variance goes full Tasmanian devil on you when you include the covariate then perhaps this assumption is...well... of a nature that has the potential to cause some degree of debate. And then that is where the problem truly is. Nothing to add here. Back at university, I essentially learned that covariates Remembering the qualities of the two teachers we enjoyed I will just say that statistics is not the most important issue for some university degrees. » In contrast to classical anovas where an additional factor will always decrease the unexplianed variance (or leave it unchanged, academically), the inclusion of a covariate is not necessarily having this effect. » Help me, please, I really wish to understand what this is all about. Again, I am sorry. I thought it might be helpful for others who happen to come across the discussion whether or not to implement a baseline as a covariate in a PK evaluation to show in a simple made-up example how this has an impact and that it is not an appropriate idea. Plzeň, Czech Republic, Posting: # 18459 » But people said, an ANCOVA was used, as recommended by the "Guideline on adjustment for baseline covariates in clinical trials", so this approach should suffice as a baseline correction. I also think they took wrong cookbook. You know EMA 1401 (page 9): For endogenous substances, the sampling schedule should allow characterisation of the endogenous baseline profile for each subject in each period. Often, a baseline is determined from 2-3 samples taken before the drug products are administered. In other cases, sampling at regular intervals throughout 1-2 day(s) prior to administration may be necessary in order to account for fluctuations in the endogenous baseline due to circadian rhythms (see section 4.1.5).They done it Ok. but according to EMA 1401 section 4.1.5: Endogenous substances If the substance being studied is endogenous, the calculation of pharmacokinetic parameters should be performed using baseline correction so that the calculated pharmacokinetic parameters refer to the additional concentrations provided by the treatment. ...the baseline correction must be done before PK analysis (subtraction of AUC, i.e. whole profile, each sample time has own baseline from predose conc. e.g. day before, or subtraction of mean of several predose concentrations). Remember, you can restart reading at any time, just saying . In the study you described there wasn't planned to do baseline correction before PK analysis. But then, theoretically in situation with the circadian rhythms, the maximum concentration from raw uncorrected data and the maximum concentration from baseline corrected data (i.e. after subtraction of predose profile) can be in different time - so different concentration would enter into the calculation. (Not happened in the study ... results were completly the same when calculation without/with baseline correction.) » Hint: these are not real data. I have such data too. x) For simplicity my data are little bit parallel (and as in parallel design but it could be made more complicated ...). Artificial data example: This example is perfectly linear and slopes of both treatment in Regression Analysis are the same (although in real study I would not expect the linearity much). For R users the data are: There is always one direction (slope) of "mean correction" in ANCOVA (something between slopes of linear regression of T and R - in this example slopes are the same). So means are "corrected" in the direction to the mean baseline (dashed line) as ilustrated in figures if it is keep simple (not complicated e.g. with missings - not balanced sequences). Left side raw data, right side ln-data (of course the same because both axes were ln transformed - that's why I didn't used your data with baseline=0 for R). Of course, it is not expected to have different baseline for T and R in randomized BE study, so... it's only artificial example (as well as your data). Anyway the differences of means of T and R: From the graphical interpretation of simple example of ANCOVA, if mean(Baseline_conc)ofT = mean(Baseline_conc)ofR = mean(Baseline_conc) then no correction is applied (means are the same) and PE from ANCOVA = PE from ANOVA. But with more and more difference which is depending on the luck/misfortune of the randomization of subjects we can get more "corrected" means. It seems that we could conclude then something as "Treatments are equivalent ...; evident differences observed by simple comparison of mean of T versus mean of R are caused by different baseline values"? So ANCOVA does not look as the correct baseline correction in this artificial example. Moreover some burning points which are not to be answered: How the sample size was calculated for the BE with ANCOVA evaluation. Ignoring additional assumptions for ANCOVA. PE is called in the guideline as GMR (for ANOVA it can be "tolerated" but for ANCOVA, GMR could be far away) Acceptance BE limits still 0.8-1.25 (90% CI for PE from ANCOVA is different than 90% CI for PE from ANOVA with these limits set in guidelines). (I would bet that 90% CI from ANCOVA would be always(?) wider ... so then this method would not be the best choise for sponsors.)

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This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Seasonal and epidemic influenza continue to cause concern, reinforced by connections between human and avian influenza, and H1N1 swine influenza. Models summarize ideas about disease mechanisms, help understand contributions of different processes, and explore interventions. A compartment model of single-city influenza is developed. It is mechanism-based on lower-level studies, rather than focussing on predictions. It is deterministic, without non-disease-status stratification. Categories represented are susceptible, infected, sick, hospitalized, asymptomatic, dead from flu, recovered, and one in which recovered individuals lose immunity. Most categories are represented with sequential pools with first-order kinetics, giving gamma-function progressions with realistic dynamics. A virus compartment allows representation of environmental effects on virus lifetime, thence affecting reproductive ratio. The model's behaviour is explored. It is validated without significant tuning against data on a school outbreak. Seasonal forcing causes a variety of regular and chaotic behaviours, some being typical of seasonal and epidemic flu. It is suggested that models use sequential stages for appropriate disease categories because this is biologically realistic, and authentic dynamics is required if predictions are to be credible. Seasonality is important indicating that control measures might usefully take account of expected weather. 1.1. Background and Objectives Both seasonal influenza and influenza epidemics continue to cause, despite prophylactic and therapeutic efforts, considerable morbidity, mortality, and financial loss across the world (e.g., [1, 2]). Further concern is generated by connections between human influenza with avian influenza (; see below), which, in its own right, gives rise to economic damage and distress . Models provide a method of summarizing current ideas about mechanisms of disease development and propagation, understanding the contributions from the different processes, and exploring possible consequences and interventions [5–7]. Because of the world-wide relevance of influenza, human and avian, there continue to be many publications on the topic. The recent outbreak of H1N1 swine influenza, and especially the varied and conflicting prognoses from experts, reveals how thinly based much of our knowledge still is. Our paper is partly a review, partly sets out in detail a particular approach to influenza modelling, and partly has some original content on seasonality, forcing, and the resulting predictions. The particular objectives are to construct a model with multiple sequential pools in various disease categories (Figure 1) so that the underlying biological dynamics is realistic, to validate the model by applying it to data describing an influenza epidemic (Figure 5), to suggest a possible mechanism for the direct effects of environment (season) on influenza dynamics (Figure 6), and last to apply various levels of environmental forcing to demonstrate how the model, without reparameterization, can give rise to a wide range of dynamics, ranging from regular (e.g., twice yearly, annual, biennial, etc.) epidemics to chaos, where sometimes predictions resemble some of the great influenza pandemics (Figures 7 and 8). Before proceeding with this agenda, a brief review of some of the issues and modelling approaches relevant to influenza is given, and our contribution is discussed in this wider context. 1.2. Brief Review of Influenza Issues and Modelling Approaches 1.2.1. Avian Influenza One of the reasons for the ongoing concern with influenza is the continuing threat from avian flu. Avian influenza is endemic in many parts of the world and, from time to time, there are outbreaks of highly pathogenic strains, such as the recent outbreak of the H5N1 strain of the virus . The virus can be transferred to other animals, including horses, pigs, and humans, frequently with fatal consequences . Because of antigenic drift (a high rate of immunologically significant mutations) and shift (reassortment between different strains of influenza within a single host), coupled to other factors such as loss of immunity, there is a continuing threat to the human population [10, 11]. This threat depends on possible changes in the to-date limited capacity of the highly pathogenic avian strains for human-to-human transmission [8, 9]. This risk makes it important that key factors determining virus evolution, epidemic occurrence, and cross-species transmission are well understood, so that effective strategies for containment and control might be designed [12–14]. There are also models dealing with the avian-human influenza nexus. For example, presents an ordinary differential equation model which combines an SI (susceptible, infected) avian model with an SIR (susceptible, infected, resistant) human model. They suggest that measures such as both extermination (of avians) and quarantine (for humans) could be needed to avoid a pandemic of influenza. 1.2.2. Two-Strain Influenza and Influenza in a Single Person An interesting paper extends what might be called the complex models of simple influenza by presenting a simple deterministic model of a more complex situation: they treat the dynamics of two-strain influenza, focussing on competition and cross-immunity. Isolation period and crossimmunity are critical parameters. Some of their results are similar to those reported here (with variable interepidemic periods from 2 to 10–13 years), although the model and mechanisms are quite different. At a more detailed level, the cell level in a single person, a model of the immune response to the influenza virus which treats innate and adaptive immunity has been proposed . The model has 10 ordinary differential equations, representing interferon, T-cells, killer cells, antibodies, and other states. They explore the impact of initial viral load on disease progression. When this is small, the disease progresses asymptomatically. The model builds on . 1.2.3. Network Models and Stochastic Simulations Some recent influenza models are based on networks and stochastic simulations [11, 18–22]. The model of is of considerable interest: it includes an individual level (age, treatment, vaccination status) as well as a community level (household, workplace, supermarkets, schools, etc.). Within a city, most contacts occur in a few locations. Interventions at these locations can be expected to be more effective than less-targeted interventions. Isolation and quarantine (e.g., ) are possible treatments, as is antiviral use for both prophylaxis and therapy. A stochastic calculation with a half-day time step is applied. Their model gives valuable quantitative indications of how epidemics/pandemics may be prevented or controlled. These highly articulated models may be well suited to predictions of outcomes from interventions for particular situations. However, stochastic models are highly demanding, of computing resources, high-resolution data on populations, contacts, transmission, age-specific characteristics, and so forth. Moreover, their predictions are often highly sensitive to initial values and settings. 1.2.4. Deterministic Models Deterministic models of influenza are also numerous, and the model presented here is of this type. While biological variability is a reality, there is a feeling amongst some modellers, especially those with a background in engineering or physics or applied mathematics, that a deterministic approach gets closer to the real science, to understand what is going on, than a stochastic approach. Reflecting this are many quotations, such as “God does not play dice,” and “if you need statistics, then do a better experiment.” Deterministic models are easier to build, easier to understand and use, easier to falsify, and easier for tracing cause and effect. With a deterministic model, a clearly labelled box diagram tells the reader most of what he needs to know. In the next two paragraphs, we refer to two examples of the genre. Although many models have the potential to represent directly seasonal effects of environment, we have not found one that actually does this. Some models represent places where people assemble and interact such as schools and clinics which affect contact rates: indirectly, these can represent a seasonality. But here, effects of weather variables on model parameters are represented directly. The model of is a typical example of a simple deterministic influenza model. The model has seven significant state variables and ordinary differential equations. From our point of view, their use of a single stage to represent each category (e.g., latent, infectious, asymptomatic, hospitalized, etc.) gives a biologically unreasonable representation of the transit time distribution through that category. Since here we represent day-to-day effects of weather on influenza dynamics, it is important that the model should represent the dynamics of influenza realistically. This means using several serial stages for progress through each disease category. A typical large deterministic influenza model is in . This comprises over 1000 differential equations and allows for many demographic and clinical parameters (such as risk, age, four levels of sickness, treated or untreated at home, and treated or untreated in hospital) so that it is useful in planning. Their model does employ multiple stages for the different disease categories, and therefore the model is able to be much more dynamically realistic than that of . The model has been used to explore the consequences of pharmaceutical and nonpharmaceutical interventions by . The interested reader might find it useful to start with [27, Figure 7] and proceed to [26, Tables 2, 3, and 4]. The model has not been applied to direct seasonal effects, although effects such as school closures giving a decrease in contact rates are included. There is no mention of chaos resulting from such forcing. addresses generally for SIR and SEIR models some of the issues covered here: the inclusion of more realistic distributions; the destabilizing consequences of this so that lower levels of forcing are required to give chaos; the conclusion that the assumptions made in formulating the model have a major impact on its dynamical properties. 1.2.5. Simplicity versus Complexity in Influenza Modelling There are numerous other examples of models of the deterministic ordinary differential equation type. Many parameters of these models are uncertain; predictions can be highly sensitive to initial conditions. The model may be applied as a large regression equation, irrespective of the known biological lacunae and the well-documented dangers of such procedures . Pertinent to this dilemma, it has been remarked that “phenomenological approaches are deficient in their lack of attention to underlying processes; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail . The challenge then is to find ways to bridge these levels of description, ….” Others (e.g., ) have argued the importance of modelling the dynamics of influenza surveillance data, in order to provide early predictions of epidemic events; to this end, they apply purely statistical methods. Cogently, it has been suggested that “as a general policy in preparing for an outbreak of a disease whose parameters are not yet known, it would be better to use a general compartment model using relatively few parameters and not depending critically on the particular as yet unknown setting” . We concur with this view. Such models are easier to construct and explore. They are better suited for elucidating general properties of these systems, as is done here. Seasonality has long been implicated in influenza incidence and severity, although the basis for this is not understood . Also, it has arguably been given little detailed attention by epidemiologists. Seasonality is a significant factor in mortality from several causes including influenza in temperate countries, with more people dying in the winter (approximately, November to March in the northern hemisphere) than in the summer [34, 35]. The contribution of influenza to these excess deaths is disputed [35, 36], as vaccination against influenza protects against deaths from other conditions . However, summarizes with “our findings are compatible with the hypothesis that the cause of winter-season excess mortality is singular and is most likely to be influenza.” This conclusion agrees with an earlier European study . Any model which covers many community levels (e.g., household, workplace, supermarkets, schools, as in ), offers many possibilities for applying seasonality by altering mixing patterns. Often seasonal forcing is represented empirically, by adding a sinusoid to the infection rate parameter [39, 40]. This approach to forcing leads to annual, biennial, and multiple cycles including chaos. It has been suggested that large seasonal oscillations in incidence can result from an amplification of very small seasonal changes in influenza transmission . Large amplification occurs when the driving frequency is close to the natural frequency of the unforced system. A two-state variable SIRS model (S + I + R = constant) is applied. A two compartment model with linear transfers (giving rise to negative exponentials) can only give a limited representation of the biological dynamics. They do not explore their model other than to support their suggestion of dynamical resonance and make no mention of chaos. 1.3. Current Model and Its Contributions Here a simple model for epidemic influenza in a single city with seasonal forcing is constructed and evaluated. We are not aware of existing work which treats directly seasonal effects (but see , where the contact rate is reduced by a factor of ten for the 6-month nonepidemic season). The focus is on the essential biology of the problem using traditional scientific reductionism. The model is of the compartmental deterministic type with homogeneous mixing (but see ), and is without age or any other non-disease-status stratification. Various categories are represented, but, recognizing the importance of mechanistically realistic dynamics, and at variance with usual practice in SIR models, each category is represented by three or more stages. The reason for using several sequential stages (spelt out in Appendix A) to represent a given category is that this allows a more credible gamma-function progression through the category . With a single-stage category, the most probable time a person spends in that category is zero, which is hardly biologically defensible although it is widely applied. Because dynamics is so important when considering influenza and especially its interaction with seasonal forcing, it is necessary to use the more realistic multistage categories. This adds to the size of the model, but importantly, it does qualitatively change the dynamics (Appendix A). Essentially, our model is of the SIRS (susceptible, infected, resistant, susceptible) type but with most categories represented by sequential stages: for example, the I (infected) category in Figure 1 is broken down into four sequential stages. There are many ways in which seasonality could impinge on influenza dynamics. We choose one of the simplest. A virus compartment allows effects of environment to be represented on virus lifetime, which might be an important environmental forcing mechanism. Some simulations are presented for the unrealistic situation in which recovered individuals are immune for life, in order to illustrate the basic characteristics of the model. Also, simulations are given for the more realistic situation where immunity is lost over a few years. For this latter situation, seasonal forcing gives rise to an unexpectedly wide range of pertinent dynamics, including regularly spaced epidemics from two per year through one per year to one epidemic every several years (two or more), sometimes with slightly chaotic spacing but sometimes regularly spaced but with chaotic amplitudes, and sometimes quite chaotic in both spacing and amplitude. Our main objective is to show how, in a mechanistically-oriented model with credible biological assumptions and minimal parameter adjustment to obtain specific outcomes, seasonal forcing functions of different magnitudes can constrain, entrain, and amplify the natural rhythms of influenza, giving rise to a wide range of epidemic/pandemic patterns, from biennial, annual, at intervals of several years, and chaotic. Our novel contributions are first to suggest an explicit direct mechanism for the effects of weather on influenza dynamics and then give simulations that show that such mechanisms can have a profound but realistic effect on the dynamics of influenza epidemics. This suggests that the approach could be an important (but hitherto neglected) part of influenza models and planning tools. 2. Methods and Modelling 2.1. Model Scheme The scheme is drawn in Figure 1. State variables are denoted by n + subscript. There is no age stratification. There are eight categories of persons: susceptible (sus); infected, which is considered as four sequential stages (); clinically sick, also considered as four stages of sickness (); recovered or immune (rec); asymptomatic (asy), which branches off after the first infected stage and has seven stages before recovery is achieved; hospitalized (hos) or isolated, which branches off after the first clinically sick stage and has three stages before recovery is achieved; dead from influenza (ded). Note that clinically sick persons, whether hospitalized or not, may die from influenza, or recover. There is a delay (day) during which recovered persons can lose immunity and return to the susceptible compartment—persons in this delay pipe, represented by ten sequential stages, are denoted by , to . Infectious persons give rise to virus particles (), which, while they may be inactivated or killed, can give rise to further infection events. We choose, although this is not the usual procedure in influenza models, to have a virus pool (). The reader may wonder why? First, we are fairly sure that such a pool must exist. Second, with a virus pool it becomes easier to think in concrete terms of the effects of weather variables (air temperature, relative humidity, wind speed, and radiation) on the virus, for example, its viability and longevity ((15), (18)). An alternative to having a virus pool () would be to assume that weather affects transmission rate β directly. In the equations to follow, transmission rate β and virus pool always appear multiplicatively ((2), first equation of (3), (20)). Thus, it perhaps makes little difference whether we have a virus pool which can be modified by weather, or no virus pool and simply assume that weather affects transmission coefficient. We take the view that transmission is a multistage process and that the components of weather may impinge on different parts of this process. It may then be helpful to have an explicit virus pool. Note that our transmission rate β has units of virus−1 day−1 (rather than the customary day−1) (Table 1). Also, in the expression for the basic reproductive ratio (20), β is divided by the virus mortality rate. In fact, the first term on the right side of (20) can be viewed as a traditional transmission rate with units of day−1, and the modulation of virus mortality ((15), (18)) may be considered as modulating the traditional transmission rate. Initial values and parameters are listed in Table 1, although some parameters which are only used once are defined in or after the equation where they appear. All routes from susceptible to recovered (Figure 1) pass through eight pools. We have assigned the same value of 2 day−1 to the four rate constants , and (Table 1). This gives a mean transit time from susceptible to recovered of 4 days (8/2) with simple gamma-type distributions of transit times applying to the whole path and its components (last paragraph of Methods section; Appendix A; e.g. , pp. 818–822). Because of the importance of the assumption of sequential pools for giving biologically realistic dynamics (whereas the traditional assumption of a single pool gives biologically unacceptable dynamics ), a discussion of sequential pools in relation to the gamma function is given in Appendix A. In Appendix A, it is shown that the use of two sequential pools give qualitatively different dynamics than a single pool also that three sequential pools gives again a qualitatively different result than two pools; with three or more sequential pools, the dynamics only changes quantitatively. Although observation and data clearly support the existence of a minimum time span being required to traverse a given clinical category, which can be represented by sequential pools as is done here, measurements in this area are extremely difficult. Where the data do not speak clearly, for example, as to whether one should use 3 or 4 pools, or 7 or 8, we have made simplifying and convenient assumptions in order that we could proceed with the calculations (see Section 2.3). uses a model of similar type but with a simpler structure. The authors employ least squares to estimate most of their model's parameters for the spring and autumn waves of the 1918 influenza epidemic (their Table 1). They find substantial differences between some of the parameters for the spring and autumn epidemics, which may raise questions about what such parameters describe. Our parameters (Table 1) are generally compatible with those obtained and applied in , allowing for differences in model structure. All state variables (Figure 1) and all terms in the differential equations scale linearly with population size and the size of the virus pool, so that solutions (expressed proportionately) are independent of population (, Table 1). 2.1.1. Some Definitions First, define some totals in terms of the state variables (Figure 1): The notation of the first five equations is obvious. Total live population is given by the sixth equation and is assumed equal to the fertile population (seventh equation); both of these are equal to the city population, ((2) and following paragraph). Finally, total infected population is obtained by adding together the inf, asy, sic, and hos categories. All the variables in (1) can be turned into fractions by division by . State variables are next treated by categories and pools. The differential equation for is The two principal inputs are first from births with the number of fertile persons given by (1) and second from the output of compartment 10 of the delay pipe (Figure 1) where people recovered from influenza are slowly losing their immunity. μ is the natural birth (and death) rate (Table 1). It is assumed that all births are free of infection and are without immunity. A small additional input to the pool is included, . This is the death rate caused by influenza (Figure 1). is given by (16). This is done so that the live person number, (1), remains constant at value (Table 1). This makes the results easier to check as true steady states can be obtained. It is of no biological significance as over an epidemic, deaths from influenza are typically less than 1% of those from natural mortality (the − term in (2)). Also, the instantaneous death rate from influenza, (16), is transient, and even at its maximum value, is usually less that the death rate from natural mortality. This assumption of equal birth and death rates was also made in . The two outputs (negative terms) are natural death and infection, the latter giving a transfer to the first infected pool. The natural death rate μ is assumed to be the same as the birth rate (Table 1). β is the infection transmission rate (Table 1). This is multiplied by the number of susceptibles , the virus quantity and is divided by the number of live persons (1). All terms in (2) scale equally with population size. The differential equations for the four sequential infected pools are In the first equation, the input of infecteds is the last term in (2). All pools suffer equally from natural death at rate μ. The output term from the inf1 pool is partly asymptomatic with fraction (Figure 1, Table 1; (7)), the remainder entering the inf2 pool (second equation, first term). The third and fourth equations are straightforward. Rate constant is 2 day−1, such that with four pools the mean time from infection to clinical sickness is days . This gives a gamma-distributed lag for the overall exit time from the fourth pool (Appendix A, (A.4), Figure 9.) 2.1.4. Nonhospitalized Clinically Sick Persons who become clinically sick enter the sic1 pool (Figure 1). A fraction (Table 1) of these may be hospitalized or isolated, the remainder continuing to recover from the illness at home. Some of the clinically sick (whether in hospital or not) will die from influenza. The differential equations are These equations are similar to (3), but the natural death rate μ is augmented by flu-induced deaths at rate (Table 1). The time between the onset of sickness and recovery is similarly (to the pools above) gamma-distributed (Appendix A, Figure 9, (A.4)). The rate at which individuals are admitted to hospital, (Figure 1), is with units of persons . This is often a recorded statistic. The total flu-related death flux is (input from all four sick pools to the dead-from-influenza box of Figure 1; units: persons ) 2.1.5. Asymptomatic Infecteds The differential equations for these seven pools (Figure 1) are Asymptomatic infecteds are fed from (3) (first equation, 2nd term on the right side). Seven pools are employed so that the path from susceptibles to recovered (Figure 1) traverses eight pools in total. 2.1.6. Hospitalized Clinically Sick These represent hospitalized or isolated or specially treated clinically sick. The differential equations are The input term represents hospital admissions (5), a recorded statistic, which may be useful for comparing with data. The flu-induced death rate (Table 1) is assumed to be the same as that for the nonhospitalized clinically sick, (4). This may be justified in that the fraction taken into hospital is more ill, but they then receive better care. Total flu-related death rate from the three hospitalized clinically sick pools is 2.1.7. Recovered and Immune The state variable is governed by the equation The positive (input) terms are from the last equations of (7), (4), and (8). Death occurs at its natural rate (μ). The rate constant is set to zero if it is assumed that immunity is not lost; otherwise it is set to a nonzero value (e.g., 1 day−1, Table 1, the precise value is unimportant as long as it is of order of 1 day−1 or more). A non-zero causes rapid entry into the delay sequence of pools (Figure 1) which leads to loss of immunity. 2.1.8. Delay Pipe Representing Loss of Immunity Loss of immunity might arise from loss of immunological memory, or from drift and shift in the antigenic character of the virus . This process is represented by ten sequential compartments giving a gamma function delay (Appendix A; also e.g., , pp. 818–822). If (day) is the time period during which immunity is lost (three years is assumed; Table 1), and (day−1) is the rate constant out of each of the ten compartments, then The standard deviation in the exit times is and the coefficient of variation is 1/√10 () (A.7). The differential equations for the pools are The first term on the right of the 1st equation () is the last term in (10); the second term () is transfer to the next compartment; the last term (μ) is the natural death rate. The 2nd term on the right of the last equation () is transfer into the susceptible pool (2). 2.1.9. Virus Pool It is assumed that virus production, , which is a surrogate for “infectious strength”, is calculated with is a production rate constant which is applied to weighted infected persons. The value above gives a basic reproductive ratio (20) of 4.78 (see Section 2.3). The weighting factors of (13) are () The precise values do matter but they are not important, so long as the values are reasonable. In the usual SIR model there would not be multiple stages as here and any weighting factor would be implicitly unity. In (14) infectivity increases a half day after infection, reaches a maximum, and then decreases as recovery takes place. It is stated in , with reference to , that “the standard pattern of an influenza A virus in adults is characterized by an exponential growth of virus titre, which peaks 2 to 3 days postinfection (DPI), followed by an exponential decrease until it is undetectable after 6 to 8 DPI.” of (13) is the input to the virus pool (15). The outputs from the virus pool (15) are death by natural mortality (, Table 1) and death induced by a suboptimal environment, represented by rate constant (18). The value of corresponds to a half-life of free virus of about 3 h . No extra virus death process is ascribed to the contact/infection process with persons (whether susceptible or immune). Thus, when the basic reproductive ratio is calculated in (20), this is proportional to β, and there is no β in the denominator. Such virus death processes are subsumed in the natural mortality term . The differential equation for the virus pool is 2.1.10. Flu-Related Dead Pool The inputs to this pool are from the sick pools with (6) and (9), so the total input to the pool (Figure 1) is There are no outputs. Therefore, Note that, in order to maintain the total population constant as a mathematical convenience, the birth rate of susceptibles is augmented by the death-from-influenza rate (2). As explained after (2), this has a negligible effect on the performance of the model. 2.1.11. Environmentally induced Virus Mortality and Seasonal Forcing The environmentally dependent function in (15) (Figure 1) is assumed here to depend only on daily mean values of air temperature, . Possible influences of relative humidity , radiation, or wind speed are ignored but could be similarly treated. In a study of relative humidity and temperature on virus transmission, remark that “although the seasonal epidemiology is well characterized, the underlying reasons for predominant wintertime spread are not clear.” The rate constant for environmentally induced death, (; Figure 1) is written as is used to switch environmental effects off (0) or on (1). Air temperature above a threshold increases virus mortality according to power . Parameter is assumed equal to two giving a quadratic dependence of temperature above the threshold (Table 1). It is usual for the biological effects of temperature to be nonlinear, sometimes approximating to exponential, as in the use of a factor for the consequences of a 10°C temperature rise on chemical reaction rate, or the application of the Arrhenius equation for chemical reactions (e.g., , pp. 103–105). is a rate parameter. We did not find controlled-environment studies on the effect of temperature on virus longevity which we could use, and therefore the values assigned the parameters are estimates. In southern Britain, daily mean air temperature varies from c. 3 (January) to 17°C (July) (, p. 270; p. 142). It is assumed that varies sinusoidally, with Annual mean and seasonal variation are and . is the Julian day number (1 on 1 January, leap years are ignored). (d) is the phase of the sinusoid, which is maximum on 25 July. Combining equations (19) with (18) modifies environmentally induced virus death rate, , and hence reproductive ratio (20). The same seasonal pattern is applied every year. These equations for temperature are a good approximation to long-term weather means in the UK (Note: the study reported in suggests that, on an annual timescale, cold weather does not predict winter deaths, but, on a shorter term timescale, cold weather could be a significant trigger). 2.2. Basic Reproductive Ratio, R0, and the Disease Generation Time, (Day) Two important epidemiological parameters are basic reproductive ratio, , and the disease generation time, (day). These are both derived from the basic parameters of the model. is defined as the number of infections directly caused by a single infected individual during its infectious period when in a population of susceptibles. is the average time it takes the direct infections which contribute to to arise. is also called the “serial interval”, the average time between the primary case and secondary cases. In the absence of forcing (see previous section, in which case can only be calculated numerically), can be calculated analytically by travelling round the infectious loops in Figure 1 and adding the terms together. This leads to (see Appendix B for an alternative equivalent statement of ) If (20) is multiplied out, each term corresponds to one of the 18 weighting factors in (14) (4(inf) + 4(sic) + 3(hos) + 7(asy) = 18) and represents one complete infective loop passing through the virus pool in Figure 1 (see Appendix B, (B.1)– (B.4)). The first term is the simplest loop, passing from to and back to (Figure 1). The first term is Start with a single virus particle in the box (we could equally well start with a single person in the inf1 first-infected compartment). The first factor of (21), with units of persons per virus particle, is the probable number of persons infected by a virus particle during its life; the mean lifetime of a virus particle is . It is assumed that there is no additional virus death process due to exposure. is a dimensionless weighting factor (14). The third factor () is the number of virus particles produced per person in the inf1 state during his life: is the average lifetime of a person in the inf1 state. The last factor, with in the denominator, is the probability that this lifetime (of ) is actually achieved. We can continue in this way via all the compartments which can give rise to infection (i.e., produce virus particles (13)), adding up the terms. (20) for can be written out as a sum of the 18 (potentially) contributing terms (Appendix B)—an equivalent formulation which is sometimes useful. Equation (20) can be (and was) checked numerically by placing a single infected individual into the box and diverting the (primary) infected individuals into an accumulator, rather than allowing them to enter the box, where they can lead to secondary infections. The algebraic and numerical methods agree to six decimal digits, giving a basic reproductive ratio for the default parameters without forcing. A “dynamic” reproductive ratio, , can be calculated when the total population is not entirely susceptible by means of The live population, , is given by (1). This allows approximately for a slowly changing fraction of susceptible individuals. However, may itself be dynamic on a shorter time scale due to seasonal effects on virus death rate (see previous section). The disease generation time, (day), can be computed analytically by (B.7), giving day. A numerical calculation using Runge-Kutta integration gives also day agreeing with the analytical result to nine decimal digits, although Euler integration gives day (in each case the integration interval is 1/32 day; see first paragraph of Section 3). Due to overlapping generations, is not equal to the time constant at which the total infecteds increase. Assume that total infecteds (1) increase initially at an exponential rate according to where (day) is the growth time constant (the proportional growth rate of infecteds is day−1). This depends on both and and the structure of the model [6, Section 1.2.3]. This can be extracted numerically (there is a period of constant exponential growth that lasts for some 10 days) to give τ = 1.29 day. This is considerably less than the generation time of 2.51 day. Parameters have been introduced while developing the model. Their values are listed for reference in Table 1. Here we summarize the evidential basis for the parameter values used. As a preliminary, statements from two physicians are quoted. In a general practitioner in the Doncaster (UK) area describes his study of the 1969-1970 pandemic as it affected his urban practice. He said “the true incidence of influenza during an epidemic is probably impossible to assess.” This is perhaps equally true today and sets the scene for the significance of processes such as “validation” and data fitting (see Figure 5 and Section 3.4). Another general practitioner, this time in Kent (UK) , states that “an epidemic of influenza tends to last in this area for between two and three months. Beginning slowly the epidemic reaches its peak in four to five weeks and then subsides slowly. The extent and severity of any attack will depend on factors such as the strain of influenza virus, on the state of the host-immunity of the population and on the timing of the epidemic; the fatality and complication rates are always higher during the cold and foggy winter months.” His statement agrees with many of our seasonal-forcing simulations (Figures 6 and 7). Five key epidemiological quantities for influenza are (a) the time which elapses after the infection event until the subject becomes infectious to others, denoted by the (day); (b) the latent period, namely the time which elapses between the infection event and the appearance of clinical sickness, denoted by (day); (c) the infectious period (day), which is the time during which the subject is infectious (Figure 1—producing virus); (d) the period of immunity (day)—the time period after recovery from influenza during which the subject has immunity, before gradually losing it and returning to the susceptible pool (Figure 1, is represented by the box at the bottom of the diagram with 10 sequential pools); (e) the basic reproductive ratio (dimensionless) (20). Addressing these quantities, first consider the time period between the infection event and becoming infectious, (day). With the infectivity weighting factors in (14), the first infected pool with a mean lifetime of 0.5 day is not infectious but the second infected pool is, and therefore Next the latent period (day) is given by (there are four sequential pools in the infected category of Figure 1) The infectious period (day) can be estimated as follows. Note that (a) all paths from the susceptible category to the recovered category in Figure 1 pass through eight pools with the same outgoing rate constant (natural mortality excluded); (b) it is assumed that the disease-related rate constants , and are all equal (Table 1), say to ; (c) the first infected pool () is assumed not infectious and all sick and hospitalized pools are infectious (14). We therefore write the infectious period as Loss of immunity occurs during a time period of , assumed to be 3 years. This delay is represented by (Figure 1) 10 sequential pools each having an outgoing rate constant of (day−1) (ignoring natural mortality (12)). This gives a gamma-distributed delay (Appendix A, (A.5) to (A.8) with ). and are related by Next compare the values in (24)–(27) with the literature. Our value of day (24) can be compared with that of , who referring to , use a rather different value of 1.9 day. However, the value given in is based on fitting a homogenous-mixing deterministic SEIR (susceptible, exposed (meaning infected but latent), infected (meaning clinical), resistant) model to the excess pneumonia and influenza deaths in 45 cities during the 1918 pandemic. use a value of 0.5 day in their model, without citing a specific source. Our value could be doubled to 1 day by taking in (14), a relatively small change to the model. Both 0.5 and 1 day are compatible with the clinical evidence, although not with the 1.9 day value of . Our latent period of = 2 day (25) is both clinically acceptable and is not very different from 's (fitted) values of 2.4 and 3.6 days for the spring and autumn waves of the 1918 epidemic. Note uses the term “latent” to describe the period between the infection event and the time at which the person becomes infectious, rather than our use (25) which refers to the period between the infection event and the time at which the person becomes clinically sick. report that the “infectious period” lies in the range of 6 to 10 days, but then their single infected pool represents everything in our Figure 1 between the susceptible and recovered categories which makes meaningful comparison difficult. states that “for simplicity, we do not explicitly model the exposed population but instead include people infected but not yet infectious in the “I” box. Including an exposed class yields similar results.” The last sentence is not supported by simulations. Students of dynamical systems may find this statement surprising. However, their range of 6 to 10 days is very different from our 3.5 day (26) which is compatible with the clinical evidence and also the 2 or 3 day period obtained by when fitting the spring and autumn waves of the 1918 pandemic. The loss of immunity of recovered patients is not a feature of the model in . uses a range of 4 to 8 years: we were not able to chase their values to earth. We employ 3 years, as in (27) for , which, while clinically reasonable, should be regarded as a guess at what is probably a rather variable quantity. The basic reproductive ratio (20) is hardly a clinical or easily observed quantity, but it is much loved by modellers and epidemiologists, not without reason. It is an “emergent” parameter of the model, as its value depends on underlying parameters. Because influenza is highly seasonal, it can be concluded that the environment is important, yet values of given in the literature rarely say anything about the environment. gives its range as 4 to 16. says that “estimates vary widely, varying from 1.68 to 20.” After fitting the Geneva data for the 1918 pandemic, gives values of 1.49 and 3.75 for the spring and fall waves. states “Assuming a basic reproduction number of = 2.5 and using the standard parameter set of InfluSim” . We assume that this value of 2.5 is the outcome of their standard parameter set. Other values given are 1.2 to 2.4 [7, Table 2] and 2.07 (, after their Figure 3). Since many of these models use biologically inappropriate assumptions , it is relevant to quote the statement from that “ignoring the latent period or assuming exponential distributions will lead to an underestimate of and therefore will underestimate the level of global control measures … that will be needed ….” reviews values of several flu epidemics and pandemics, focussing on the possible control of an H1N1 epidemic. Our standard parameter set without any environmental forcing gives a value of = 4.78 (20). This can be regarded as an upper baseline applying to an optimum environment (one maximizing ), and changing the environment can only decrease this number (Figure 6). Finally, we wish to comment on the number of pools used to represent the infected and sick categories in Figure 1, where there are four in each. Using trajectory matching on an influenza outbreak at an English boarding school [43, 53] suggests that two pools are appropriate for each category. Unfortunately they give few details of their procedure. We found a good fit (Figure 5) using the current model to fit the same data. We therefore continued to use four pools, although arguably, whether or not two, or three or four pools, or even a nonintegral number of pools are applied is perhaps less important than the principle of applying two or more pools. This section reports simulations of the model described above. First we describe the general technical aspects applied in the simulations. 3.1. Numerical Methods The model was programmed in ACSL (Advanced Continuous Simulation Language, Aegis Research, Huntsville, AL, USA; version 11.2.2 for DOS), an ordinary differential equation solver. In all simulations, equations are integrated using Euler's method, a fixed integration interval of Δt = 0.03125 = 1/32 day (45 minutes), and results were communicated for plotting at half-day or daily intervals. There were no difficulties with model implementation. Not unexpectedly, in some of the chaotic simulations, different (but still chaotic) results were obtained if different integration methods and intervals or different Fortran compilers were used (the results shown used the Watcom compiler). The simulations focus on the daily hospital admission rate, (persons day−1); Figure 1), as this is often a recorded statistic. An “epidemic” is deemed to have occurred if shows a maximum (in steps of Δt) and if at the maximum persons . State variable initial values and parameter values are as listed in Table 1 (unless stated otherwise). In general, the parameters have not been tuned for any particular performance (but see Figure 5) and as far as possible have been estimated mechanistically (see Section 2.3). Some of the simulations (e.g., Figures 2, 3, and 4) are intended to illustrate important characteristics of the model—they are not intended to be compared with actual epidemics/pandemics. Other simulations (e.g., Figure 5, parts of Figures 7 and 8) are intended to demonstrate at least a partial realism and to provide credibility to the model. 3.2. Dynamics without Intrinsic Loss of Immunity and without Forcing Here the model is exercised with parameter ((10), Figure 1, Table 1), so that there is no loss of immunity as represented by the delay box in Figure 1, and without environmental forcing (18). In this case, the natural death and birth rates (μ of Table 1 and (2) to (12)) lead to a slow loss of immunity at the population level as births are assumed to be susceptible. 3.2.1. Short-Term Dynamics These are illustrated in Figure 2(a) and are unexceptionable. From initial infection, it is two to three weeks before the disease is visible, and the epidemic is over within a further two weeks. Infected number () peaks a few days before hospitalized numbers (), both then falling to very low values. Most (99%) of the population joins the recovered (immune) box (), with 1% escaping infection altogether. 90% of those infected travel via the asymptomatic route (Figure 1; , Table 1, (7)). The epidemic in Figure 2(a) is short compared to UK experience, but Figure 2(a) is for the no-seasonal-forcing situation where initially the population is 100% susceptible, which is not comparable with actual epidemics where seasonality is always a factor as is partial immunity. Later (Figure 5), it is shown how the model, without significant “tuning”, is able to fit data on a UK epidemic. Also, seasonal forcing lengthens the period of the epidemic and decreases the fraction of the susceptible population which becomes infected (Figures 7 and 8). 3.2.2. Long-Term Dynamics These are illustrated over 100 years in Figure 2(b). On this time scale, the first epidemic, shown in Figure 2(a), occurs at zero time and rapidly dies out with the dynamic reproductive ratio (22) (not plotted) falling to near zero, as the fraction of susceptibles () becomes small. In the first epidemic, at time day, the initial peak in hospital admissions (Figure 1, (5)), decreasing to less than 100 at the second epidemic. After the first epidemic, (22) slowly recovers as immune numbers () decrease and susceptibles () increase due to the natural birth and death processes. The second smaller epidemic occurs after a further 28 years, followed by epidemics of decreasing amplitude and increasing frequency until a steady state is reached with fractions of susceptibles of 0.209, of all four infected categories together (infected, asymptomatic, sick, hospitalized, Figure 1) of 0.00014 and of recovered (immune) of 0.791; there are 0.18 hospital admissions per day, and (22) = 1. 3.2.3. Responses to Key Parameters Figure 3 illustrates the effects of changing three key parameters: infectivity parameter β (Figure 1; (2); Table 1), rate parameter (with ; Figure 1; Table 1; (3)), and the initial (time zero) immune fraction, . Figure 3(a) shows how the duration, initial proportional growth rate, and severity (indicated by hospital admissions, HA) of a modelled epidemic are influenced by the value of infectivity parameter β (or equally ; both are linear factors of (20)). Indeed, the effects of increasing β are monotonic, moving the epidemic towards shorter time scales, increasing initial proportional growth rate, total hospital admissions (HA) towards an asymptote of 5000, and narrowing the width of the epidemic. Also, increasing β causes the long-term steady state to be more quickly attained—the spikes of (e.g., Figure 2(b)) becomes closer together. Figure 3(b), where the rates of transit of infected persons through the system are increased (, Figure 1), is less straightforward. Here a low value for the (e.g., 0.25 day−1) gives a high basic reproductive ratio (20) (Appendix B) an epidemic which is slow to take hold but (surprisingly) moves more rapidly towards a long-term steady state (i.e., the spikes as in Figure 2(b) are closer together). As the are increased basic reproductive ratio decreases (20) first the epidemic becomes narrower and faster (e.g., day−1), but further increases in (e.g., and 8 day−1) cause the epidemic to become slower to take hold and less peaked, with fewer hospital admissions (HA). Increasing always causes the spikes (Figure 2(b)) to move further apart and lengthens the time taken to reach a steady state and decreases the number of infected persons in the steady state (,(1)). Finally, Figure 3(c) illustrates how increasing the initial immune fraction has a similar effect to that of decreasing β: delaying the onset of the epidemic and increasing its width, decreasing the initial proportional growth rate, and decreasing severity (number of hospital admission, HA). 3.3. Dynamics with Intrinsic Loss of Immunity and without Forcing Now the discrete delay box of Figure 1 giving loss of immunity is switched on by making day−1, which causes recovered individuals to be moved quite rapidly into the delay sequence of ten compartments where immunity is lost after three years (; Table 1, (11)). Immunity is also being lost due to the natural birth and death processes, as newborns are assumed to be susceptible. The responses of Figure 2(a) are little changed by taking day−1 (instead of 0) if the variable of Figure 2(a) is replaced by the variable (1). However, over a longer time period Figure 4(a) illustrates that there is now a switching of susceptibles between a low (fractional) value and a high (fractional) value (fractions of , Table 1, (1) and following paragraph), with an inverse switching of numbers in the delay compartment, (Figure 1). Figure 4(b) shows that the oscillations, as in Figure 2(b), decrease in amplitude and increase in frequency until a steady state is attained. In the steady state, there are 3.7 hospital admissions per day, a dynamic reproductive ratio = 1 (22), and the fractions in the three principal categories (Figure 1) of susceptible; all infected categories lumped together (infected, asymptomatic, sick, hospitalized, (1)) and the delayed categories are 0.209, 0.0030, and 0.787. Comparing these values with the situation in which there is no intrinsic loss of immunity (Figure 2(b)), it is seen that hospitalizations have increased by a factor of 20 as has the fraction in all infected categories, although the fractions of susceptibles and recovered or immunes have barely changed. “Validation” is often a misused and misunderstood concept and is perhaps better described as an evaluation of applicability. Validity is not a property of a model alone, neither is it a “zero or one” concept. It describes the relationship between model predictions and a set of data obtained under prescribed conditions. In this section, the model of Figure 1 is “validated” by fitting the predictions of the model, with minimal parameter adjustment (a perfectly formulated mechanistic model would permit no adjustment of parameters or initial values) to data on an influenza epidemic . Success in this endeavour gives the model some credibility, although it does not make the model valid for general use. The data in relate to an influenza outbreak at an English boarding school, which provides a simple situation that seems comparable to the single-city homogeneous-mixing model of Figure 1, remembering that the model does not have the stratification which might be needed if a larger region was considered. Minimal tuning is applied. One infected person is introduced to the school at noon on 18 January, otherwise all are susceptible. Table 1 parameters are altered for the Figure 5 simulation as follows: infection parameter β is changed to 0.07 (virus units)−1 day−1; all birth and death rates are set to zero () for such young persons; total population ; and the asymptomatic fraction, , is one-third, to reflect the finding that only two-thirds of the boys became sick, and the community is assumed to be “well-mixed”. There is no initial immunity. With these values, the basic reproductive ratio (20) is 6.44, the mean generation time is 2.65 day (A.7), and the initial proportional growth rate of total infecteds, , is 1.015 day−1. For comparison with the data in , we define the number of persons confined to bed, , as Since this definition is somewhat arbitrary, in comparing the predicted from the model with the data from Anon (1978), is scaled with an adjustable factor so that the comparison line drawn in Figure 5 is Fitting was done by eye, as this can produce (see below) a better focus on the biological significance of the parameter being adjusted and possible limitations in the biological data that may be obtained with more automated methods. See for possible problems arising from formulaic parameter adjustment in mechanistic models. The degree of fit in Figure 5 is satisfactory. Apart from the two outliers on 26th and 27th January, the fit is good. In an actual epidemic, there may be underreporting during the early states and overreporting later, as the performance of those handling the epidemic changes. Overall, we believe it is reasonable to assume that the model has some credibility as a result of this validation. Finally, a comment on whether the number of pools used in the infected and sick categories in Figure 1 is appropriate. fits an SEIR model to the observed data given in and shown here in Figure 5. They minimize the sum of the squared errors, arguably this underweights the skirts of the distribution, which are sensitive to the numbers of sequential pools assumed (Figure 9). They assume pools for the E category and pools for the I category. They find a best fit with and . We were not able to discover the details of their general parameterization or indeed how they define a confined-to-bed number from the categories and pools of the SEIR model. They remark that there is a sensitivity to the number of points used to obtain the fit and that basic reproductive ratio can change substantially. More notably, it can be seen in their Figure 3(c) that the model fits the last three data points as the epidemic is subsiding rather poorly. This suggests (see Figure 9) that a higher number of pools is required than their best values of 2 for the exposed and infected categories. In view of these difficulties, and the comparison shown in Figure 5, we consider that the number of pools per category suggested in Figure 1 and used throughout this paper is reasonable. Although our particular choice cannot rigorously be defended, it seems to be “good enough” at the present time. 3.5. Dynamics with Intrinsic Loss of Immunity and Seasonal Forcing Now we add direct seasonal forcing by weather to the simulations illustrated in Figure 4 ((18), (19)). Four weather factors which could impinge on virus longevity are air temperature (), relative humidity (), radiation (possibly multicomponent), and wind speed. The effects can be highly complex: for example, which examines the effects of and on virus transmission, finding that cold dry conditions favour transmission (but see their Figure 6). The topic is far from being well understood, but since our concern here is to represent broadly weather forcing within the model, we make the simplifying assumption that alone is operative. Figure 6(a) illustrates the seasonal variation of mean daily air temperature in the southern UK. Using (19) with (18), and , this affects environmentally induced virus death rate (Figure 1, (15)), and thereby basic reproductive ratio (20). Mean daily air temperature varies between 3°C (Jan 24) and 17°C (Jul 25). When crosses the temperature threshold of = 13°C in May, this increases environmentally induced virus mortality, (18) and decreases basic reproductive ratio (20). With this formulation, influenza is most likely during the months of October to April when is highest. Figure 6(b) shows how varying (e.g., increasing) mean annual air temperature, (19), (or equivalently, decreasing threshold temperature , (18)) varies the duration and intensity of seasonal forcing of . Changing mean annual air temperature, (19), changes the average annual value of , , as well as its maximum, minimum and amplitude (maximum-minimum). The dependence of these quantities on is also shown, together with the number of forcing days per year (, a forcing day is one on which mean air temperature is above threshold temperature and virus mortality is reduced). With mean annual air temperature , there is no forcing at all (): air temperature (19) never exceeds threshold temperature = 13°C (18); is invariant at its maximum value. Environmentally induced virus death rate ((15), (18)) stays at zero; the system remains in the steady state shown in Figure 4(b) and (this situation is equivalent to and = 17°C). As mean annual air temperature increases above 6°C, there are more days in the summer months when mean daily temperature > threshold temperature , number of forcing days per year increases, and . The amplitude increases to a maximum when, before decreasing. With , influenza epidemics do not occur, because the annual average of is less than unity. In the simulations presented in Figures 7 and 8, various values of mean annual air temperature are taken between −4.5 and 16°C, assuming always an annual variation of ±7°C (19) as in the UK. Otherwise, parameters have the values in Table 1. In these simulations, the long-term steady state reached in Figure 4, with loss of immunity (; Figure 1; (10)) but without seasonal forcing, is used for initial values. Forcing is applied after one calendar year. The aim is to illustrate the wide variety in dynamic behaviour which results from this type of seasonal forcing (which decreases the reproductive ratio). In each case, the mean value, its maximum, minimum and amplitude (maximum-minimum) of basic reproductive ratio can be read off Figure 6(b). In the UK the influenza season is considered to be over by May, when the mean daily temperature ranges from 10.7 (1 May) to 14.1°C (31 May) (Figure 6). Therefore, of the simulations described in Figures 7 and 8, those given in Figures 7(b), 7(c),7(d), 8(a), 8(b), and 8(c) are more relevant to the UK. Figure 7 illustrates the effects of low levels of seasonal forcing on influenza hospital admissions, (Figure 1,(5)). Forcing is seen as a downward modulation of the basic reproductive ratio , which decreases the annual average, . The lower graph in Figure 7(a) shows that is slightly decreased beginning on 24 June, from 4.78 in the steady state to 4.69 on 25 July. Influenza hospital admissions (the upper graph in Figure 7(a)) oscillate twice about the steady state value (Figure 3(b); 3.7 admissions ) in a 12-month period with the two steady-state maxima 168 days apart on 27 September and 14 March. A regular variation is quickly established. Note that the natural response time of the system as indicated by the upper graph in Figure 7(a) is not commensurate with the annual cycle imposed by the environment (shown by the lower graph in Figure 7(a)). This sets the scene for potential chaos. In Figure 7(b) the level of forcing is increased (lower graph). This results in a more complex (but still regular) schedule of hospital admissions with a biennial pattern superposed on a twice-yearly variation. In monthly data are presented in their Figures 2 and 3; some of these are suggestive of a twice-yearly pattern. A further increase in forcing (Figures 7(c) and 7(d)) results in chaos, with sometimes one, two or even three peaks in hospital admissions occurring within a twelve-month period. However, there is a tendency towards annual epidemics, with 387 epidemics occurring in 250 years, and the epidemics (10% points) lasting about seven weeks (cf. which gives a duration of two to three months; also cf. Figure 8(a) with annual late spring epidemics of 5-week duration). The susceptible fraction of the population varies from c. 30% before an epidemic to 15% just after each epidemic, so that one half of the susceptibles becomes infected. Further increases in forcing are shown in Figure 8. First, in Figure 8(a) with mean annual air temperature (Figure 6), there is a transition to an annual epidemic occurring in the late spring of each year lasting for about five weeks. In these annual epidemics, the susceptible fraction falls from c. 35% to 12%. Next (Figure 8(b); ), chaos is again produced (cf. Figure 7(d)) with a strong tendency towards annual epidemics: 204 epidemics occur in 200 years. In the last two years of the simulation shown in Figure 8(b), there is a small epidemic on 22 July, followed by larger epidemics on 5 October and 16 May. Then (Figure 8(c); ), there is chaos but now with a tendency towards biennial epidemics (169 epidemics occur during 200 years). Last (Figure 8(d); ), the system immediately settles down into regular biennial epidemics occurring in early spring of every other year but the amplitudes remain slightly chaotic. Note that, throughout Figures 7 and 8, as mean annual air temperature threshold is increased, forcing is increased (i.e., the magnitude of the seasonal changes in basic reproductive ratio ((20), (18), (19), Figure 6) increases), but the mean annual value of , , decreases. This causes between-epidemic recovery time to increase (see discussion of Figure 3(a) above). The frequency of epidemics then decreases. Increases in mean annual air temperature can be continued, and although the situations simulated are now less realistic, they do contribute to understanding the system. With , the response is chaotic with 37 epidemics in the first 200 years. Mostly five or six years elapse between epidemics; occasionally two smaller epidemics occur in the same year (e.g., in the 179th year). The mean reproductive ratio is 2.945. When , the response eventually settles down to a regular pattern with 27 epidemics in the first 200 years and eight years between epidemics and an of 2.499. When the response is chaotic with 15 epidemics in the first 200 years and . Last, with , after departing from the initial steady state, it is c. 110 years before influenza reemerges, and then it is at a low level, eventually settling down to a repeating annual late spring epidemic lasting about two months with a maximum of 0.6 hospital admissions per day and an annual sum of 20 hospital admissions per year. has a modest mean annual value (1.131) and is below unity for much of the year. Many models of influenza are more empirical than mechanistic, and therefore, although they can be and sometimes have been used to fit historic data , they are of little value in further understanding or for indicating how future epidemics/pandemics might be handled before they occur ([32, 42, 55, 56]). Seasonality is an important feature in influenza incidence. There are many ways in which seasonality can be incorporated into an influenza model. In the contact rate is reduced by a factor of ten for the 6-month nonepidemic season. Here a simple representation of UK daily weather allows the impact of mean daily air temperature to be explored. A similar approach could be applied to relative or absolute humidity, radiation, and wind speed. Seasonal forcing gives rise to wide range of dynamics, from regular at various intervals, to chaos, as illustrated in Figures 7 and 8. Some of simulations of Figures 7 and 8 are similar to influenza incidents which have occurred. For example, in Figure 7(d), the three maxima near year 184 in the spring, autumn and following spring have similarities with the waves of the 1918–1920 influenza pandemic . Note that this is achieved within a single simulation without changing parameterization. In comparison, in the authors fitted these data with a model, applying the model separately to each wave, with different parameters and initial values (loc. cit. Table 1, Figure 4). It is legitimate to ask just what this procedure means. The two principal peaks occurring within a single influenza season in the autumn and spring around year 199 of Figure 8(b) resemble the peaks shown in the 1957-58 pandemic [20, Figure 3A]. The first two peaks illustrated in Figure 8(c) in year 182 occur in late spring (the lesser peak) and the following autumn (the main peak), resembling the 1968–70 pandemic . Apart from Section 3.4 and Figure 5 which applies to an unusually sharply defined context, we did not attempt to fit our model to a wider selection of historical data. The reasons for this include that the model is not sufficiently detailed to make this meaningful in a general context, historical data usually have many lacunae, current understanding of the mechanisms of seasonal impact on influenza is very limited, given the number and nature of chaotic solutions, fitting could be technically difficult, and last, Popper's cogent discussion of historical data-fitting , in which he concludes that such exercises are usually not scientifically productive, seems to be particularly pertinent to this investigation. Regrettably, in spite of all the evidence, “parameter twiddling” and fitting historic data are still highly regarded by some investigators, although it is hard to find examples where such work has led to significant progress. Nevertheless, with simplified “proof-of-concept” models, it is important that predictions should be acceptable as has been shown to be the case with the current model. Where we part company with many influenza modellers is in our use of multiple pools to represent given categories: that is, four pools are used to represent infected latent persons and seven pools for asymptomatically infected persons (Figure 1) (but see ). Mathematically, this is a trivial addition requiring some extra programming, but it gives three significant benefits. First, progress through the stages of influenza is clearly sequential, suggesting that to use successive pools is biologically reasonable. Second, the overall transit times of sequential pools are gamma-distributed which is arguably more realistic than given by a single-pool representation. Third, the method opens a path to a more mechanistic picture of observations where quantities such as infectivity (14) and death rates can vary from pool to pool within a category. While simple models are best for elucidating many general principles, there seems to be no alternative to more detailed mechanistic (reductionist) models for serious application. With such models, parameters will be more determined by experiment at the assumption level, rather than making parameter adjustments on the basis of comparison of predictive-level data. A mechanistic influenza model has been constructed in which sequential pools are used for some disease categories, allowing gamma-function-type dynamics with delays, consistent with biological observations. A simplified representation of seasonality is given and is, we believe, the first attempt to include weather explicitly in an influenza model. The model has been “validated” by application to an outbreak of influenza in a school. It has been demonstrated that seasonal forcing gives rise to a rich variety of dynamic disease patterns, from regular with outbreaks at annual, biennial, and longer intervals of time, to chaotic. Some of these predicted patterns seem highly pertinent to mankind's experiences with influenza. It is suggested that seasonality and its effects could usefully be an integral part of influenza epidemiology including the areas of prediction and amelioration. Recognizing that seasonality is important in influenza dynamics, we were surprised by our inability to find more controlled-environment studies of the effects of environmental factors on virus viability or other significant processes in the disease cycle. A. Sequential Pools and the Gamma Function The aim in this appendix is to explain how the use of sequential pools to represent a given clinical category affects the transit dynamics for that category. We emphasize first that a realistic mechanistic treatment of infectious disease dynamics absolutely requires the use of several, arguably at least three, sequential pools per clinical category, and second that the traditional approach of using a single pool per clinical category cannot be expected to predict credible dynamics or robust predictions. It is to be noted that the empirical use of a gamma function for a single-pool category (e.g., the entire infected box in Figure 1) is based mechanistically on several stages (in this case four) each with first-order (exponential) kinetics. Although these points will be familiar to many workers, this appendix has been written because they are not always appreciated. A general discussion of the topic can be found in, for example, , pp. 818–822. A.1. Single Pool Assume that a given disease category, for example, “infected”, is represented by a single pool, , so the number of pools . This method is used in many SIR models. A single pool emptying at rate from initial value of at time obeys the equations This is drawn in Figure 9: the line labelled “1 pool, ”. It can be seen that the maximum rate at which pool is depleted occurs at time (shown by ●). The mean time for departure is at day = ( (shown by ■). For a single pool, these are far apart. The biological data do not support such dynamics, which would imply, for say the infected category, that it is most probable that an infected person exits the infected category immediately following the infection event. Since our paper is particularly concerned with the detailed dynamics of influenza, including the possible occurrence of chaotic events (here the existence of lags can make a crucial difference to model behaviour), we regard it as essential to depart from the assumption of assigning a single pool to each disease category. A.2. Two Pools Next assume the category is represented by two sequential pools, and , where empties into , and is the final pool in the category. The number of pools, . In this case, the relevant equations become This is shown in Figure 9: the line labelled “2 pools, ”. With two pools, has been doubled relative to in Eqns (A.1), so that the mean time for departure from the final pool in the category is the same at time day = (shown by ■). We see that the maximum rate for departure from the infected category now occurs at time day (= ()/) (shown by ●), in contrast to 0 day for the single pool case above. In moving from one pool to two pools, the maximum value at 2 day has shifted towards the mean value at 4 day. There is a large qualitative difference between the 1-pool curve for and 2-pools curve for drawn in Figure 9. A.3. Three Pools This case is given in detail as there is another qualitative difference between 2-pool and 3-pool dynamics. Now there are three sequential pools, , , and , with emptying into , into , and is the last pool in the category. The number of pools is . The equations for the system are Now (Figure 9) the curve for the final pool in the category () is sigmoidal at low values of time . The time of the maximum (●) (= (()/) has moved closer to the mean time (■). The mean time is ()/() = 4 day as in the other curves. With three pools, there is now a sigmoid departure from time , giving a more sharply defined biological delay (cf. the two-pool line). A.4. Four and More Pools Adding more sequential pools to the 3-pool situation has the effect on the final pool of increasing the sigmoidicity, and moving the time of maximum (●) closer to the mean time (■). At the same time as adding more pools, the exit rate constant of each pool is increased so that the overall mean transit time is unchanged. For four pools, , we have This is drawn in Figure 9. There is only a minor quantitative difference between the 3- and 4-pool sequences when they have the same mean transit time . For a general number of pools, , with state variables, , and each with outgoing rate constant, , the value of the state variable for the final pool in the sequence, , is where is a constant. The total outflow from the final pool, , is , so that The statistics on the final pool, , are (using to denote expectation values) The 8-pool curve of Figure 9, illustrated by the time course of the final pool in the sequence, , is shown because in Figure 1 every path between susceptible and recovered traverses 8 pools each with rate constant 2 day−1, giving an overall average transit time of 4 day (see main text for further discussion of this point). Finally we note that although there are several equivalent definitions for the gamma function used by mathematicians and others (e.g., , pp. 819–821), possibly the most intuitive definition for the biologist is that given in (A.6), namely, where the normalized gamma function, , is given by If the number of sequential pools, , and also , with rate constant , where is the overall mean transit time which is constant, then γ(t, m) approaches a Dirac delta function located at time . B. Basic Reproductive Ratio and Mean Generation Time An alternative and useful way of writing (20) for is as a sum of the individual contributions from the 4 + 4 + 3 + 7 = 18 diseased pools of Figure 1. 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« ΠροηγούμενηΣυνέχεια » tained by BE and EF, because EF is equal to ED; therefore BD is equal to the square of EH; and BD is also equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH: Wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done. PROP. A. THEOR. If one side of a triangle be bisected, the sum of the squares of the other two sides is double of the square of half the side bisected, and of the square of the line drawn from the point of bisection to the opposite angle of the triangle. Let ABC be a triangle, of which the side BC is bisected in D, and DA drawn to the opposite angle; the squares of BA and AC are together double of the squares of BD and DA. From A draw AE perpendicular to BC, and because BEA is a right angle, AB (47. 1.) BE + AE2 and AC2=CE2 +AE2; wherefore AB2 + AC BE +CE2+2AE2. But because the line BC is cut equally in D, and unequally in E, BE2+ CE2 = (9.2.) 2BD+2DE; therefore AB2 + AC2 = 2BD2 +2DE.2AE2. AE2 (17. 1.) AD", and 2DE + 2AE2 = 2AD2; wherefore AB+ AC2 = 2BD2 + 2AD2,B Therefore, &c. Q. E. D. PROP. B. THEOR. The sum of the squares of the diameters of any parallelogram is equal to the sum of the squares of the sides of the parallelogram. Let ABCD be a parallelogram, of which the diameters are AC and BD; the sum of the squares of AC and BD is equal to the sum of the squares of AB, BC, CD, DA. Let AC and BD intersect one another in E: and because the vertical angles AED, CEB are equal (15. 1.). and also the alternate angles EAD, ECB (29. 1.), the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each; but the sides AD and BC, which are opposite to equal angles in these triangles, are also equal (34. 1.); therefore the other sides which are opposite to the equal angles are also equal (26. 1.), viz. AE to EC, and ED to EB. Since, therefore, BD is bisected in E, AB2+AD2=(A. 2.) 2BE3 +2AE; and for the same reason, CD+BC22BE2+2EC2=2BE2 +2AE2. because EC AE. Therefore AB +AD+DC +BC2= 4BE4AE. But 4BE2BD, and 4AE=AC2 (2. Cor. 8. 2.) because BD and AC are both bisected in E; therefore AB+ AĎ2+ CD+BC BD2 + AC2. Therefore the sum of the squares &c. QE. D. COR. From this demonstration, it is manifest that the diameters of every parallelogram bisect one another. HE radius of a circle is the straight line drawn from the centre to the circumference. A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it. Circles are said to touch one another, which meet, but do not cut one another. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal. And the straight line on which the greater perpendicular falls, is said to be farther from the centre An arch of a circle is any part of the circumference. A segment of a circle is the figure con tained by a straight line, and the arch which it cuts off. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment. And an angle is said to insist or stand upon the arch intercepted between the straight lines which contain the angle. The sector of a circle is the figure contained by two straight lines drawn from the centre, and the arch of the circumference between them. Similar segments of a circle, are those in which the angles are equal, or which contain equal angles. PROP. 1. PROB. To find the centre of a given circle. Let ABC be the given circle; it is required to find its centre. Draw within it any straight line AB, and bisect (10. 1.) it in D; from the point D draw (11. 1.) DC at right angles to AB, and produce it to E, and bisect CE in F: the point F is the centre of the circle ABC. For, if it be not, let, if possible, G be the centre, and join GA, GD, GB: Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG are equal to the two BD, DG, each to each; and the base GA is equal to the base GB, because they are radii of the same circle: therefore the angle ADG is equal (8. 1.) to the angle GDB: But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of the angles is a right angle (7. def. 1.) Therefore the angle GDB is a right angle: But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB,the greater to the less, which is impossible: Therefore G is not the centre of the circle ABC In the same manner, it can be shown, that no other point but F is the centre: that is, F is the centre of the circle ABC: Which was to be found. COR. From this it is manifest that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other. PROP. II. THEOR.. If any two points be taken in the circumference of a circle, the straight line which joins them shall fal within the circle. Let ABC be a circle, and A, B any two points in the circumference the straight line drawn from A to B shall fall within the circle. Take any point in AB as E; find D the centre of the circle ABC; join AD, DB and DE, and let DE meet the circumference in F. Then, because DA is equal to DB, the angle DAB is equal (5. 1.) to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater (16. 1.) than the angle DAE; but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: Now to the greater angle the greater side is opposite (19. 1.); DB is therefore greater than DE: but BD is equal to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle. Wherefore, if any two points, &c. Q. E. D. PROP. III. THEOR. If a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and if it cut it at right angles, it will bisect Let ABC be a circle, and let CD, a straight line drawn through the centre bisect any straight line AB, which does not pass through the centre, in the point F: It cuts it also at right angles. Take (1. 3.) E the centre of the circle, and join EA, EB. Then because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two sides in the one equal to two sides in the other:

Research Article | Open Access Xuan Guo, Xiao Xin Zhang, "Impact Pseudostatic Load Equivalent Model and the Maximum Internal Force Solution for Underground Structure of Tunnel Lining", Mathematical Problems in Engineering, vol. 2016, Article ID 1468629, 16 pages, 2016. https://doi.org/10.1155/2016/1468629 Impact Pseudostatic Load Equivalent Model and the Maximum Internal Force Solution for Underground Structure of Tunnel Lining The theoretical formula of the maximum internal forces for circular tunnel lining structure under impact loads of the underground is deduced in this paper. The internal force calculation formula under different equivalent forms of impact pseudostatic loads is obtained. Furthermore, by comparing the theoretical solution with the measured data of the top blasting model test of circular formula under different equivalent forms of impact pseudostatic loads are obtained. Furthermore, by comparing the theoretical solution with the measured data of the top blasting model test of circular tunnel, it is found that the proposed theoretical results accord with the experimental values well. The corresponding equivalent impact pseudostatic triangular load is the most realistic pattern of all test equivalent forms. The equivalent impact pseudostatic load model and maximum solution of the internal force for tunnel lining structure are partially verified. In recent years, the nuclear leakage events have sparked a new national self-examination. The safety and stability problems of the underground structure caused by the impact or blasting load must be paid special attention to. The blasting demolition at the top of underground, chemical explosion devised by terrorism, gas pressure extrusion, and release of shallow layer containing gas and so forth will result in instant huge impact load on the structure of tunnel lining. The structure crack or damage of the tunnel under the impact load will lead directly to the serious nuclear leakage events, with Japan’s Fukushima nuclear power plant explosion causing the serious nuclear leakage from 2011. The former Soviet Union Chernobyl nuclear power plants exploded at 1:23 in the morning of April 26, 1986, with more than 8 tons of highly radioactive materials mixed with graphite fragments and nuclear fuel burning debris spewing out. The radiation pollution caused by the nuclear leakage accident is equivalent to 100 times of the radioactive pollution caused by the atomic bomb explosion in Hiroshima, Japan. Even 20 days after the accident, the temperature of the center of the nuclear reactor is still as high as 270 degrees Celsius. It caused 10 times the number of cancer deaths caused by the accident in the United Nations official estimates, with a global total of 2 billion people affected by the Chernobyl accident; 270 thousand people suffer from cancer, which killed more than 93 thousand people. Experts estimate that eliminating the effects of this catastrophe will take at least 800 years. The underground structure of tunnel lining locates in the enclosure space; the security problem and the precise evaluation of the dynamic response to the blasting or impact loading becomes an important engineering issue and there is a need to give mathematical model and solution of internal force [1–6]. First, determining the strength and internal force of structure response under the blast loading is the key factor to solve the dynamic response problem for tunnel lining. By now, the researches of stability of the structure for tunnel lining under impact load are more concentrated in blasting load or gas outburst. Generalized impact action blocks and changes the motion of a moving object. The difference between impact load and impact pseudostatic load is the energy form related to the instantaneous time effect or not. Generally, the quantity of explosive and detonation pressure of explosive shock can be determined according to the energy release and the size of the gas pressure to determine the impact of the air pressure. Blasting seismic wave attenuation can be equivalent to the dynamic load effect of gas outburst on tunnel lining structure. Currently, the theoretical study of the tunnel lining and the internal force response under impact load is relatively little. This paper introduces the free deformation method as the theoretical basis. Derivation of impact load equivalent pseudostatic model is given. Load pattern on the circular tunnel lining under the instantaneous maximum internal force calculation formula is compared and analyzed. By comparing the theoretical analysis to the test data and the numerical simulation results, it intends to get the internal force response of the failure mechanism and the optimal load pattern and the theoretical results can be preliminarily verified by the back analysis . 2. The Internal Force Formula of the Circular Tunnel Lining under the Blasting Loading Considering significantly attenuated effect of cover layer to the impact load and wave, it is supposed that impact loaded on structure of tunnel lining is equal to the earth surface. The additional stresses during elastic wave propagation are ignored. To simulate the stress and deformation of tunneling lining under impact or blasting dynamic load, engineers commonly adopt the simple and safe simplified method, which means the impact pseudostatic load multiplied by a dynamic load factor to estimate the maximum impact load. The core idea of the proposed method is first to find out the maximum equivalent impact pseudostatic stress mode under the impact or blasting load. The theory formula of the equivalent model for impact load on circular tunnel lining is derived based on using the free deformation method, which is the classical theory method for underground structure . 2.1. The Equivalent Impact Load Model The equivalent impact load model should be given and the equivalent form caused by blasting seismic wave loads on tunnel lining structure should be determined. Left-side blasting load coefficient of dynamic stress distribution for surrounding rock is given by Figure 1. Figure 1(a) gives the mechanical model of left-side blasting load on the circular underground structure; the radial movement of surrounding rock stress concentration factor is given as in Figure 1(b). Upper blasting load coefficient of dynamic stress distribution of surrounding rock is shown in Figure 2. Figure 2(a) gives the mechanical model of the upper-side blasting load on the circular underground structure; the force model of the tunnel lining structure and the surrounding rock of the dynamic stress concentration factor distribution map is given as in Figure 2(b). Figure 1 can give the maximum acceleration stress distribution map of the left-side blasting load in surrounding rock for lining. Figure 2 can give the maximum acceleration stress distribution map of the upper-side blasting load surrounding rock for lining. From Figures 1 and 2, it is shown that the blasting response of the circular lining is according to the following rules: the maximum point of impact load generally is the maximum stress response point, mostly on angle of 0 degrees near the load; the smaller load is distributed on both sides of the lining; the maximum stress zone is concentrated in the range of degrees around both sides of the impact load; on the far side from the impact load, the stress on both ends of the lining is obviously higher than that in the middle of the lining. (a) Mechanical model (b) Radial movement of surrounding rock stress concentration factor (a) Mechanical model (b) Upper soil blasting the maximum acceleration distribution The dynamic stress concentration factor and the map of the maximum acceleration distribution obtained by the two modes of lateral explosion and upper explosion of Figures 1 and 2 present typical symmetrical patterns of butterfly distribution. The maximum internal force response formula of arbitrary angle on lining is derived under any angle impact load. First consider the special case of the impacted load on the upper side of the lining; the impact load can be expressed by arbitrary rotating angle; the additional force of lining under impact load is divided into three parts. The first part is the equivalent impact load, with the blasting load applied on the lining; the second part is the equivalent impact reaction force, which is the reverse force of lining and formation, equal to the equivalent impact load for balance effect; the third part is the proposed rock resistance force on both sides of lining; it is caused by the relative displacement mode between the lining and ground. We give the case study of the triangle resistance distribution; the load model is shown as in Figure 3. The effective counter force which is under the influence of impact in order to maintain the balance of force provided by the formation of lining is equal to the action of the equivalent load and impact load. The distribution of load model can be traced back to the Japanese triangle resistance method . (a) Blasting stress model of upper side (b) Additional mechanical load model of upper blasting stress According to the stress characteristics of structure response under impact load, the force model diagram of the circular tunnel lining under the upper blasting impact is given. Two main factors should be considered for affecting the load and the response of the lining under the impact load: load form and load value. Figure 3(a) shows the force model of the circular lining under the condition of impact load. Figure 3(b) is the mechanical model of additional load for blasting shock. The equivalent action form of impact load is discussed first. There are two loading modes of simulating the blasting and impact loading of rock: one is calculating the explosion hole pressure by explosive burst detonation theory and then loading the calculating blast action on hole wall directly; second is using the empirical formula to calculate the dynamical peak value of the load and then imposing it on the boundary according to triangular pulse wave form. The former method needs to introduce the state equations of the explosive detonation and the rock mass, which is used for single hole blasting or centralized charge blasting. For the impact of porous blasting, the blasting source distribution area is larger; the simulation is not directly loaded with blasting hole pressure according to the equivalent amount of explosives to concentrate on loading and explosive loading. The simulation of blasting vibration effect is based on triangle pulse wave loading . Suppose that the effect of gas outburst, seismic wave, and blasting attenuation on lining is equivalent. The equivalent additional load caused by impact load is assumed to consist of three parts: the upper part load and the two-side reaction force and the bottom reaction force. The influence of the different distribution forms on the internal force of the lining is discussed in three parts. The assumed distribution of the blasting load is shown in Figure 4. Assume that the impact load produces the same total equivalent value ; the concentration of Figure 4 in four kinds of load on the upper part of the form is (a) < (b) < (c) < (d); Figure 4(d) is the limit form of concentrated load; the impact load can act as the concentrated load. Stress concentration in the middle of the lining is the most significant. The concentration degree of subgrade load under the same condition of the equivalent impact load is ① < ②. 2.2. The Formula Deduction of Lining Internal Force under Impact Load The internal force formula of lining response under various forms of equivalent impact load is derived. The force in the process is the total load value of the impact load. Because the computation processes of internal force calculation for different impact loads combination are similar, the paper gives a detailed derivation of the upper curve load for reference, and the rest of the cases only give the result. Suppose the upper impact equivalent load is sinusoidal and the maximum value is ; the sine load value is in the angle of , and take the range of as the load calculation of the equivalent load. When , two load diagrams are shown in Figure 5; the bending moment at caused by is , of which is the load value at , is the action scope, and is the action distance. While ,The load diagram is shown as in Figure 5, while : can be calculated after the calculation formula of is gotten asThenPut the values of and into . While ,While ,The relationship between the equivalent blasting force and isThe axial force calculation is (8), while :While ,Bring the values of and into . While ,While ,The calculation formula of shear force is as follows: while ,While ,Bring and into . While ,While , The maximum value of instantaneous explosive load and maximum internal force can be obtained by bringing maximum into the formula. The expression of maximum explosion pressure is given in 1956 by Brown in the equivalent value of blasting load as 1971 Sassa gives the following expressions: The detonation wave spread in the rock mass gives the maximum pressure in the contact interface of explosive and rock. The maximum pressure generated on the contact interface between the detonation wave and the rock mass is related to the rock physical characteristic. The maximum pressure and the maximum explosion pressure relation can be approximately expressed as follows: is rock density; is wave velocity propagated in rocks; is explosive detonation velocity; 0 is explosive density. The maximum equivalent load is The relationship between the peak value of blasting load and the distance can be expressed as is peak value of blasting load; is the distance from the calculate point to blasting hole; is the diameter of the contact surface (). The Statfield dynamic pressure with the time history can be used aswhere is the load constant, having an amount equivalent to the dynamic pressure generated by the charge of each 1 kg explosive . The internal force calculated by the different combination of load form is shown in the Appendix. 3. Test Verification of the Model 3.1. Comparison with the Blasting Model Test In order to further verify the applicability of the theoretical calculation formula of the instantaneous maximum internal force of the circular lining (Figure 4) under the impact load, the calculation and comparison of the microvibration centrifuge model test results on the top of lining are carried out. The top vibration model test diagram is shown in Figure 6. The bending moment of the structure under impact load can be obtained by converting the microstrain model test results. The model tests show that the mix ratio of sand to gypsum material is gypsum plaster material : sand : water = 1 : 0.8 : 0.5; elastic modulus , which is the reduction elasticity modulus of the prototype concrete . Compare two kinds of thickness with 7.5 mm and 12.5 mm; the elastic modulus of aluminum alloy model test is ; the thickness is 3.8 mm. The similarity criterion of bending deformation of the model centrifuge test is as follows: is the number of centrifugal accelerations; is Poisson’s ratio of the model; is Poisson’s ratio of the prototype; is Young’s modulus of the model; is Young’s modulus of the prototype; is the thickness of the model lining; is the thickness of the prototype lining. The next formula is available by the formula of (23): and are suitable for the sand gypsum material; the moment can be calculated by directly substituting the experimental microstrain into the prototype; for the aluminum alloy material, it can be calculated byAccording to the formula of , the instantaneous maximum bending moment of the prototype can be obtained. The elasticity modulus of the prototype concrete is . The results of converted moment value of the measuring point are shown in Table 2. The equivalent blasting load values are calculated by the third groups of experiments, and the parameters of the test are shown in Table 3 . The theoretical solution of maximum internal force to test model is calculated. ThenThe attenuated equivalent load is by giving according to the test conditions; the conversion coefficient of black powder and TNT is 0.4; the centrifugal acceleration is 50 g; then 1.25 g black powder is equivalent to TNT of 62.5 kg ; thenThat means the equivalent blasting load on the lining of third groups experimental results is 778 kN, considering that the size effect of model test will lead to error; the selection range of calculation load is from 600 kN to 800 kN. The internal force of the maximum impact load for the circular lining under the condition of the load combination is calculated as shown in Figure 7. The equivalent form of the upper impact load is compared with the uniform load, the curve load, the triangle load, and the concentrated load. The concentrated load represents the extreme case where the impact load directly acts on the lining without attenuation through the surrounding rock. (a) Test group of number 3 (b) Test group of number 4 (c) Test group of number 5 (d) Test group of number 6 (e) Test group of number 31 (f) Test group of number 34 Select the different calculation combination of blasting load form in the top and bottom; the results are compared as shown in Tables 4–9; take . The comparison of the scatter plots is shown in Figure 7. |Note: the equivalent blasting load is 630 kN; the lateral resistance force is 56 kN/m.| |Note: the equivalent blasting load is 546 kN; the lateral resistance force is 84 kN/m.| |Note: the equivalent blasting load is 1400 kN; the lateral resistance force is 112 kN/m.| |Note: the equivalent blasting load is 700 kN; the lateral resistance force is 84 kN/m.| |Note: the equivalent blasting load is 650 kN; the lateral resistance force is 20 kN/m.| |Note: the equivalent blasting load is 260 kN; the lateral resistance force is 20 kN/m.| The following can be concluded through comparison of a①, b①, c①, and d①:(1)The influence of the upper load form on the bending moment of the measuring point 1 (top) is obvious, the difference range can reach 50%–100%, and the comparison of the load distribution is very necessary.(2)The characteristics of each group showed a clear trend of convergence; the measured moment value of point 1 represents the relationship a① < b① < c① < d①. This shows that with more concentration in the upper part of the load form the moment value of measuring point 1 will be greater.(3)Compared with each test point data under all cases of combination, it can be known that the measuring points of the bending moment values have changed considerably when the upper load form changed from the uniform load to the curve load. The rule is that the measuring point 1 and point 3 (the top and bottom points of the lining) bending moment value increases and the measuring point 2 and point 4 (the point on left-right sides of the lining) bending moment value decreases.(4)Compared with each test point data of 2, 3, and 4 under all cases of combination of b①, c①, and d①, it can be known that the increase concentration of the upper load and the linear increase effect of the internal force response are decreased. The effect of concentration on both sides and the bottom of the lining is very weak (within 1%). Compared with the test combination of b① and b②, the result showed the following:(1)The form influence of the ground reaction force on the moment of the measuring point 1 (bottom) is obvious, and the difference range is 90%–50%.(2)The difference range of the combination of b① and b② is larger (range: 10%–50%) for the measuring point 1, point 2, and point 4. This shows that the concentration increase of the subgrade force in the bottom of lining will only affect the bending moment of the action position and have less effect on the other points’ bending moment outside of this range. Comparing the data calculation in the four groups, it is found that the combination of upper triangular load and subgrade counterforce of the outer triangle reaction model has good coincidence degree with the experimental values. This case also has the certain safety reserve; comparatively, this load combination is the most reasonable one. The internal force calculation results (bending moment, shear force, and axial force) of the upper triangular load and the subgrade counterforce of outer triangle reaction model are shown in Figure 8. (a) Third groups (b) Fourth groups (c) Fifth groups (d) Sixth groups (e) Thirty-first groups (f) Thirty-fourth groups Validating the lining safety theoretical results of the third test groups preliminarily, the calculation results of triangular equivalent loads are selected to verify the results, while the maximum value of the calculated moment is 367.2 kN m, the position is S1 measuring point, and the axial force is 247.8 kN. According to the maximum compressive classical stress formula of the eccentric compression member,Bring the test parameters and calculation results; the following can be concluded:The compressive strain is considered as the critical value. The lining structure can be considered in a safe condition as less than the concrete elastic ultimate compressive strain of 0.002. The compression plane strain of the test results isIt can be considered that the lining structure is in a safe state under the case study of blasting load. 3.2. Comparison to Model Test of Pneumatic Impact Load Compare the pneumatic impact load mode to the model test. The test model box is shown as in Figure 9(a) . The air tube is located in the three (left, right, and down) directions of the model box. The air pressure is equivalent to the triangular load. The additional load is caused by air pressure and the equivalent loading is shown in Figure 9(b). The upper uniformly distributed load is the subgrade counterforce provided by the equilibrium pressure from the lower pressure layer. (a) Model test (b) Equivalent loading of air pressure The comparison of the additional bending moment and the theoretical calculation results of the lining structure under releasing and applying air pressure to the test is shown in Table 10. It is found that the calculation results of the additional bending moment for the circular lining under the impact pressure were coincident with the model test data. Four points’ precision accuracy is within 15%, with the highest accuracy reaching 95%. The results for the theory of structure response (Figure 3) moment under impact load calculation results are in good agreement with the experimental values. It can provide support for theoretical calculation of tunnel lining structure response under impact loading. 4. Numerical Simulation to Microvibration Test MIDAS GTS was used to simulate the microvibration model test. After the model feature was calculated, the response simulation of lining (Figure 3) under blasting load was carried out by using the time history analysis. Using MIDAS GTS to simulate the blasting load, manually input the blasting load function. The dynamic pressure time history functions are shown in Figure 10. The blasting load is applied on the interface in form of the surface pressure. Table 13 shows the maximum microstrain simulation value of the lining structure under blasting loads in consistency with the test value very well; most of the gap is less than 30%. Most difference of the measuring points is less than 30%. The numerical simulation results not only verify the experimental results but also provide the basis for the correctness of the theoretical calculation results. Compared with the results of numerical simulation, the theoretical formula of the lining maximum internal force under impact loading has good applicability in the case study. |Note: A for the test results; B for simulation results.| Using the GTS, simulate and validate test results; the impact pseudostatic load simulations are carried out by constrained lining structure of tunnel lining by the spring boundary. Theoretically, the simulation and experimental results are compared with the results shown in Table 14. The three-coincidence degree is good through comparison; most difference of the points is less than 20%. The applicability of the theoretical calculation results is verified further by the results of impact pseudostatic load simulation. |Note: A is the experimental result, B is the result of the calculation, and C is the simulation result.| Impact pseudostatic load equivalent model and the maximum internal force solution for circular tunnel lining structure were deduced from classical theory of the free deformation method. The equivalent model and the maximum internal force solution of the underground lining are given. The equivalent load forms of different impact loads are compared preliminarily. The following conclusions and recommendations of the maximum internal force of lining structure under impact load may also be drawn by comparing the theoretical solution, experimental data, and numerical simulation results. (1) The equivalent form of impact load has a remarkable influence on the calculation results. It was reflected in the internal forces distribution pattern of the lining structure in the load range of impact pseudostatic action. Two equivalent forms of impact loading on the lining structure of tunnel lining were compared by the curve distributed shape and triangle distributed shape. The simplified triangular distributed load is more convenient and feasible to use with the premise of closing the test results. (2) The additional load caused by the impact load mainly has two types, the ground counterforce and lateral deformation resistance, which can be calculated by the classical Japanese triangle resistance method. The equivalent mode of formation reaction has two modes, which are external triangle distribution and uniform distribution. For the case of side impact load, the distribution of the outer triangle is obviously better than that of the uniform distribution load distribution, whether from the dynamic stress concentration degree of the surrounding ground or from the matching degree to test data. (3) The influence of soil parameters on the structural lining response under blasting shock load is the following: the soil density influences the transmission and attenuation of shock wave in surrounding rock and the formation parameters for the effect of structural response under blasting load: soil density influences the detonation wave in the soil layer of propagation and attenuation. The characteristics show that the lower the density, the softer the soil and the more significant attenuation of impact loads. The distance between blasting spot and lining structure is similar to the factor of the soil layer density; the farther the distance, the faster the impact load decay rate. The magnitude of the impact load directly affects the response of the lining; the strain of the structure lining presents a nonlinear increasing relationship under the nonlinear interaction to the surrounding ground. (4) Comparing the theoretical formula and dynamic load simulation to the microvibration test data, the theoretical results of the impact pseudostatic load equivalent model are verified again by the numerical simulation of the equivalent static load. The results show that the consistency of the test results, theoretical results, and the simulation results of impact pseudostatic load is very high. Most of the difference point is within 20%, which shows that the theoretical calculation results of the proposed model have good applicability. The maximum microstrain simulation by the dynamic load on lining is in agreement with the test values very well, with most of the difference points being within the range of 30%. The comparison between the experimental results and the theoretical calculation results of the pressure shock load model is carried out; the difference of the closest point is 2.3%. In conclusion, the impact pseudostatic load equivalent model is proposed for dynamic response analysis of tunnel lining structure. Good agreement among the theoretical analysis, test data, and the simulation is achieved, which preliminarily validates the present model and method.

The 20 times that follow are information about you. Please supply the information as HONESTLY and ACCURATELY as you can. The data that will be obtained from these items shall be held strictly confidential. Each item is followed by several possible answer. On your sheet, shade completely the box that corresponds to the number of the answer that specifically pertains to you. 1. Sex : 2. 2. Civil Status : 3. Age bracket where you belong : 1. 18-24 years old 2. 25-31 years old 3. 32-28 years old 4. 39-45 years old 5. More than 45 years old 4. Highest educational attainment: 1. College graduate 3. With Master’s Units 4. Master’s Degree 5. With Ph.D. Units/Degree 5. Year of last attendance in school : 1. Before 1985 5. After 2000 6. What honours did you receive when you graduated from college? 1. Summa Cum laude 2. Magna Cum laude 3. Cum laude 4. Other academic award 5. None/not applicable 7. Present employment: 8. Type of present job: 2. General Clerical 3. Trades and crafts (jobs requiring manual dexterity of application of manual/mechanical/artistic 4. Others 5. Not applicable 9. Length of experience in present job: 1. Less than one year 2. One to two years 3. Three to four years 4. More than four years 10. Do you have any of the following first level eligibilities: Second Grade, Municipal/Provincial Clerk, General Clerical, Career Service Sub- Professional(Local Government), Career Service Sub-professional 11. For what reason are you taking this examination? 1. Entrance to government service 2. Change of appointment status 12. How many times have you taken the Career Service Professional? Examination excluding this one? 1. Once 2. Twice 3. Thrice 4. More than thrice 5. Never 13. Which of the following activities did you undertake in preparing for this examination? 1. Enrolled in review centers 2. Studied career service examination reviewers sold at bookstore 3. Engage in other activities 4. Used a combination of 1 and 2. 5. No preparation done at all Items 14 to 16 (For Government Employees Only): 14. Category of government office where employed: 1. National government 2. Local government (province/city/municipal) 3. Government-owned or controlled corporation 4. Constitutional office 5. State college or university 15. Status of present appointment in government service: 1. Permanent 2. Temporary 3. Casual/Emergency 4. Contractual 5. Substitute 16. Years of experience in government service: 1. Less than 5 years 2. 5-9 years 3. 10-14 years 4. 15-19 years 5. More than 19 years Items 17 to 20 In which of the following types of work do you consider yourself best qualified? Choose only two from among the options listed in items 17-20. Shade the boxes that corresponds to your choices. For example, if you think you are best qualified in budget management, and project planning/management, shade box no. 2 of item 17 and box no. 3 of item . Leave items 18 and 20 blank. If you think you are best qualified in research/report including statistical analysis, shade boxex no. and no. 2 of item 20 on your Answer sheet, and leave items and 19 blank. 17. 1. Accounting 2. Budget Management 3. Buying/Purchasing 4. Co-ordination 5. Computer Operations 18. 1. EDP Computer Programming 2. EDP System Analysis and Design 3. Human Resource Development 5. Management and Audit Analysis 19. 1. News/Feature Writing 2. Personal Recruitment/Selection 3. Project Planning/Management 4. Public Relation Work 5. Records Management 20. 1. Research / Report Writing 2. Statistical Analysis 3. Stenography 4. Supplies Management 5. None of the Above * TEST BEGINS HERE * GRAPHS / Charts / Data Introduction to Economics Exam Statistics 1. In which of the following years did over 2/3 of the students who took the exam not pass it? a. 2005 b. 2006 c. 2008 d. 2009 e. Cannot say 2. It is known that a quarter of the students who passed the exam in 2007, passed it at the first trial. Assuming each exam has two trials, what percentage of all the students who took the exam that year passed it in the second trial? a. 10 b. 15 c. 30 d. 75 e. Cannot say 3. If the number of Chinese Insurance stocks represented 3.5% of all Insurance securities, approximately how many Insurance bonds were Chinese? a. 9,200,000 b. 9,500,000 c. 10,800,000 d. 910,000 e. 1,080,000 For questions 4 to 6, Refer to the following graph of sales and profit figures of ABC Ltd and answer the questions that follow. 4. I Return on sales (Profit/sales) was highest in which year? a. 1995 b. 1996 c. 1997 d. 1998 e. cannot say 5. How many times return on sales (profit/sales) exceeded 15 % ? a. once b. twice c. thrice d. four times e. never 6. How many times growth in profit over the previous year exceeded 50% was registered ? a. once b. twice c. thrice d. four times e. never For questions 7 to 9: The following graph shows the data related to Foreign Equity Inflow (FEI) for the five countries for two years- 1997 and 1998. FEI is the ratio of foreign equity inflow to the country’s GDP, which is expressed as a percentage in the following graph. 7. Find the ratio between FEI for Malaysia in 1997 and FEI for Thailand in 1998 a.) 7 : 25 b.) 8 : 21 c.) 6 : 11 d.) 1067 : 582 e.) cannot say 8. Name the country which has the minimum change in the FEI a.) India b.) China c.) Malaysia d.) S.Korea e.) cannot say 9. If, Education sector in Thailand had 25% of FEI in 1997 and 60% of FEI in 1998, then find the approx ratio of the amounts allocated to Education in 1997 to 1998. (Assume the GDPs of both of these years for Thailand is same.) a.) 1 : 5 b.) 25 : 72 c.) 6 : 11 d.) 7 : 25 e.) cannot say 10. A rumour about an upcoming recession in Japan has reduced the value of the Yen 7% compared with the Euro. How many Euros can you now buy for 500 Yen? a. 3.5 b. 3.26 c. 3.15 d. 3.76 e. None of the above

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By Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki This monograph is a unified presentation of a number of theories of discovering specific formulation for warmth kernels for either elliptic and sub-elliptic operators. those kernels are vital within the idea of parabolic operators simply because they describe the distribution of warmth on a given manifold in addition to evolution phenomena and diffusion methods. The paintings is split into 4 major elements: half I treats the warmth kernel by means of conventional equipment, resembling the Fourier rework technique, paths integrals, variational calculus, and eigenvalue enlargement; half II bargains with the warmth kernel on nilpotent Lie teams and nilmanifolds; half III examines Laguerre calculus functions; half IV makes use of the tactic of pseudo-differential operators to explain warmth kernels. themes and features: •comprehensive therapy from the perspective of distinctive branches of arithmetic, corresponding to stochastic strategies, differential geometry, designated services, quantum mechanics, and PDEs; •novelty of the paintings is within the different equipment used to compute warmth kernels for elliptic and sub-elliptic operators; •most of the warmth kernels computable by way of hassle-free features are coated within the work; •self-contained fabric on stochastic strategies and variational equipment is included. Heat Kernels for Elliptic and Sub-elliptic Operators is a perfect reference for graduate scholars, researchers in natural and utilized arithmetic, and theoretical physicists attracted to knowing other ways of coming near near evolution operators. Read or Download Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques PDF Best differential geometry books The notes from a suite of lectures writer brought at nationwide Tsing-Hua college in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of publication "Thing Hua Lectures on Geometry and Analisys". This booklet is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It comprises study and survey articles in keeping with the most talks added on the overseas Congress During this e-book, we learn theoretical and sensible points of computing tools for mathematical modelling of nonlinear platforms. a couple of computing recommendations are thought of, akin to equipment of operator approximation with any given accuracy; operator interpolation strategies together with a non-Lagrange interpolation; equipment of procedure illustration topic to constraints linked to thoughts of causality, reminiscence and stationarity; tools of approach illustration with an accuracy that's the top inside a given type of types; equipment of covariance matrix estimation;methods for low-rank matrix approximations; hybrid equipment in keeping with a mixture of iterative techniques and top operator approximation; andmethods for info compression and filtering lower than clear out version should still fulfill regulations linked to causality and sorts of reminiscence. - Topology of Surfaces, Knots, and Manifolds - Notes on Geometry - Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems - Symplectic Methods in Harmonic Analysis and in Mathematical Physics Additional resources for Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques 8) by taking the mixed derivative of the action @2 S D @x@x0 1 D 1 6D 0: Á. / Hence there are no conjugate points to x0 in this case. 9). vk D/ D @t @t t @xk D and then xk t2 x0k ! 6 (Particle in constant gravitational field). x; P x/ D 12 xP 2 kx, k > 0. t/ D t 6D 0, for t > 0, there are no conjugate points along the trajectory. We leave the computation of the classical action as an instructive exercise to the reader. 7 (The linear oscillator). t/ D x k 2 x , 2 k > 0, then p p p sinh. kt/ x0 cosh. M; g/, there is a neighborhood V of x0 such that for any x 2 V, there is a unique geodesic joining the points x0 and x. The aforementioned result does not necessarily hold globally for any Riemannian manifold. However, it holds on compact manifolds, and in general on metrically complete manifolds, as the Hopf–Rinov theorem states; see . 1 Lagrangian Mechanics 19 Moreover, any geodesic is locally minimizing the action functional. 0/ D A; x. U /; for any tangent vector field U ; see . 0/ D 0. 1b. 30). x; y/ . 30). As we shall see in the next section, this relation is obvious in the case of a three-dimensional hyperbolic space. M; gij / of dimension n. Let R D g ij Rij be the Ricci scalar curvature of the space, which will be assumed constant. x0 ; x/ denote the Riemannian distance between the points x0 and x. x0 / where Ä D D det 1=4 e Rt e @2 Scl @x0 @x is the van Vleck determinant; see Schulman , Chap. 24. 32) 46 3 The Geometric Method Applying the aforementioned formula for the classical spaces with constant curvatures 0; 1; 1, we arrive at the following classical results.

- Do NPV and IRR always agree? - What is XIPR? - Can a project have no IRR? - What is negative IRR? - What is an acceptable NPV? - What does a positive NPV mean? - What does the IRR tell you? - How do I calculate IRR? - What is a good IRR? - Is a high IRR good? - What is NPV method? - Should you invest If NPV is 0? - Why is NPV equal to zero? - Is NPV better than IRR? - What is NPV example? - Can you have a positive NPV and negative IRR? - How do you interpret NPV and IRR? - How do you interpret NPV? - What does an IRR of 0 mean? - What is difference between NPV and IRR? - What is the conflict between IRR and NPV? Do NPV and IRR always agree? The difference between the present values of cash inflows and present value of initial investment is known as NPV (Net Present Value). A project would be accepted if its NPV was positive. Therefore, the IRR and the NPV do not always agree to accept or reject a project.. What is XIPR? XIRR is your personal rate of return. It is your actual return on investments. XIRR stands for Extended Internal Rate of Return is a method used to calculate returns on investments where there are multiple transactions happening at different times. Can a project have no IRR? The IRR is formally defined as the discount rate at which the Net Present Value of the cash flows is equal to zero. … There are also cases where no IRR exists. For example, if all cash flows have the same sign (i.e., the project never turns a profit), then no discount rate will produce a zero NPV. What is negative IRR? Negative IRR occurs when the aggregate amount of cash flows caused by an investment is less than the amount of the initial investment. In this case, the investing entity will experience a negative return on its investment. What is an acceptable NPV? The net present value rule is the idea that company managers and investors should only invest in projects or engage in transactions that have a positive net present value (NPV). They should avoid investing in projects that have a negative net present value. It is a logical outgrowth of net present value theory. What does a positive NPV mean? positive net present valueA positive net present value indicates that the projected earnings generated by a project or investment – in present dollars – exceeds the anticipated costs, also in present dollars. It is assumed that an investment with a positive NPV will be profitable, and an investment with a negative NPV will result in a net loss. What does the IRR tell you? The IRR equals the discount rate that makes the NPV of future cash flows equal to zero. The IRR indicates the annualized rate of return for a given investment—no matter how far into the future—and a given expected future cash flow. How do I calculate IRR? To calculate IRR using the formula, one would set NPV equal to zero and solve for the discount rate, which is the IRR. … Using the IRR function in Excel makes calculating the IRR easy. … Excel also offers two other functions that can be used in IRR calculations, the XIRR and the MIRR. What is a good IRR? You’re better off getting an IRR of 13% for 10 years than 20% for one year if your corporate hurdle rate is 10% during that period. … Still, it’s a good rule of thumb to always use IRR in conjunction with NPV so that you’re getting a more complete picture of what your investment will give back. Is a high IRR good? The higher the IRR on a project, and the greater the amount by which it exceeds the cost of capital, the higher the net cash flows to the company. … A company may also prefer a larger project with a lower IRR to a much smaller project with a higher IRR because of the higher cash flows generated by the larger project. What is NPV method? Net present value (NPV) is a method used to determine the current value of all future cash flows generated by a project, including the initial capital investment. It is widely used in capital budgeting to establish which projects are likely to turn the greatest profit. Should you invest If NPV is 0? A positive NPV means the investment is worthwhile, an NPV of 0 means the inflows equal the outflows, and a negative NPV means the investment is not good for the investor. Why is NPV equal to zero? Zero NPV means that the cash proceeds of the project are exactly equivalent to the cash proceeds from an alternative investment at the stated rate of interest. The funds, while invested in the project, are earning at that rate of interest, i.e., at the project’s internal rate of return. Is NPV better than IRR? The advantage to using the NPV method over IRR using the example above is that NPV can handle multiple discount rates without any problems. Each year’s cash flow can be discounted separately from the others making NPV the better method. What is NPV example? For example, if a security offers a series of cash flows with an NPV of $50,000 and an investor pays exactly $50,000 for it, then the investor’s NPV is $0. It means they will earn whatever the discount rate is on the security. Can you have a positive NPV and negative IRR? You can have a positive IRR and a negative NPV. Look, basically when NPV is equal to zero, IRR is equal to the discount rate. The discount rate is always above zero hence when the IRR is below the discount rate, the IRR is still positive but the NPV is negative. How do you interpret NPV and IRR? The NPV method results in a dollar value that a project will produce, while IRR generates the percentage return that the project is expected to create. Purpose. The NPV method focuses on project surpluses, while IRR is focused on the breakeven cash flow level of a project. How do you interpret NPV? NPV = Present Value – CostPositive NPV. If NPV is positive then it means you’re paying less than what the asset is worth.Negative NPV. If NPV is negative then it means that you’re paying more than what the asset is worth.Zero NPV. If NPV is zero then it means you’re paying exactly what the asset is worth. What does an IRR of 0 mean? are not getting any returnWhen IRR is 0, it means we are not getting any return on our investment for any number of years, thus we are losing the interest which we could have earned on our investment by investing our money in bank or any other project, thereby reducing our wealth and thus NPV will be negative. What is difference between NPV and IRR? Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. By contrast, the internal rate of return (IRR) is a calculation used to estimate the profitability of potential investments. What is the conflict between IRR and NPV? When you are analyzing a single conventional project, both NPV and IRR will provide you the same indicator about whether to accept the project or not. However, when comparing two projects, the NPV and IRR may provide conflicting results. It may be so that one project has higher NPV while the other has a higher IRR.

Introduction to Derivative of ln6x Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln(6x) can be calculated by following the rules of differentiation. Or, we can directly find the derivative ln 6x by applying the first principle of differentiation. In this article, you will learn what the ln 6x derivative is and how to calculate the ln6x derivative by using different approaches. What is the derivative of ln 6x? The derivative of ln x with respect to x is a fundamental concept in calculus, and it's essential to understand how to compute it. It can be denoted as d/dx [ln(6x)], and it tells us the rate of change of the natural logarithmic function ln x. In other words, it shows us how quickly the value of ln(6x) is changing concerning changes in the variable x. This derivative can be simplified as 1/ x, indicating that the derivative of ln6 is always a fraction with x in the denominator. It's important to note that ln(6x) represents the logarithm of 6x with base e, which is a critical detail for solving various calculus problems. Derivative of ln6x formula The formula for the derivative of ln(6x) is equal to 1/x and mathematically it can be written as: d/dx(ln(6x)) = 1/x This formula tells us how the function ln 6x changes with a change in its variable x. This is an essential formula to know in calculus, as it allows us to solve various problems involving logarithmic functions. How do you prove the derivative of ln 6x? There are multiple derivative rules to derive the ln6x derivative, and different methods may be more useful depending on the problem at hand. Some of the most common techniques to prove the ln(6x) derivative are: - First Principle - Implicit Differentiation - Product Rule Each method provides a different way to compute the ln(6x) differentiation. By using these methods, we can mathematically prove the formula for finding the ln6x derivative. Differentiation of ln(6x) by first principle According to the first principle of derivative, the ln 6x derivative is equal to 1/x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to, f(x)=lim f(x+h)-f(x) / h This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative. Proof of ln(6x) derivative by first principle To prove the derivative of ln(6x) by using first principle, we start by replacing f(x) by ln x. By logarithmic properties, Suppose t=h / x and h=xt. When h approaches zero, t will also approach zero. f(x)=lim ln (1/xt) ln (1+t) By logarithmic properties, we can write the above equation as, f(x)=(1/x) lim ln(1+t)^1/t Hence by limit formula, we know that, lim ln(1+t)^1/t =ln e =1 Therefore, the derivative of ln6 is; Derivative of ln6x using implicit differentiation implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation involving two or more variables. We can use this method to prove the differentiation of ln(6x). Proof of derivative of ln 6x by implicit differentiation To prove the derivative of natural log, we can start by writing it as, Converting in exponential form, ey = 6x Applying derivative on both sides, ey.dy/dx = 6 Use our implicit derivative solver to evaluate derivatives of implicit expressions easily. Derivative of ln(6x) using product rule Another method to calculate the differential of ln 2x is the product rule which is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule solver is: d/dx(uv) = u(dv/dx) + (du/dx)v In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x. Proof of ln6x differentiation by product rule The function ln x can be written as; f(x)= 1. ln(6x) Applying derivative with respect to x, Now by using product rule, Hence the ln(6x) derivative is always equal to the reciprocal of x. How to find the ln6x derivative with a calculator? The easiest way to calculate the derivative ln6x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this derivative calculator with steps. - Write the function as ln x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the ln 6x derivative. - Now, select the variable by which you want to differentiate ln(6x). Here you have to choose x. - Select how many times you want to differentiate ln(6x). In this step, you can choose 2 for second, 3 for triple derivative and so on. - Click on the calculate button. After this step, you will get the derivative of ln(6x) within a few seconds. After completing these steps, you will receive the ln(6x) derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions. Frequently asked questions Is logarithmic differentiation the same as derivative? The logarithmic differentiation is a part of the derivative in which we differentiate complicated functions by using natural log. Whereas in derivative, we simply find the derivative of a function by using differentiation rules. What is the derivative of ln6x? The differentiation of ln(6x) can be calculated as; d/dx (ln(6x)) = 1/x

Abstract: Among many statistical methods for linear models with the multicollinearity problem, partial least squares regression (PLSR) has become, in recent years, increasingly popular and, very often, the best choice. However, while dealing with the predicting problem from automobile market, we noticed that the results from PLSR appear unstable though it is still the best among some standard statistical methods. This unstable feature is likely due to the impact of the information contained in explanatory variables that is irrelevant to the response variable. Based on the algorithm of PLSR, this paper introduces a new method, modified partial least squares regression (MPLSR), to emphasize the impact of the relevant information of explanatory variables on the response variable. With the MPLSR method, satisfactory predicting results are obtained in the above practical problem. The performance of MPLSR, PLSR and some standard statistical methods are compared by a set of Monte Carlo experiments. This paper shows that the MPLSR is the most stable and accurate method, especially when the ratio of the number of observation and the number of explanatory variables is low. Abstract: The traditional method for processing functional magnetic resonance imaging (FMRI) data is based on a voxel-wise, general linear model. For experiments conducted using a block design, where periods of activation are interspersed with periods of rest, a haemodynamic response function (HRF) is convolved with the design function and, for each voxel, the convolution is regressed on prewhitened data. An initial analysis of the data often involves computing voxel-wise two-sample t-tests, which avoids a direct specification of the HRF. Assuming only the length of the haemodynamic delay is known, scans acquired in transition periods between activation and rest are omitted, and the two-sample t-test is used to compare mean levels during activation versus mean levels during rest. However, the validity of the two-sample t-test is based on the assumption that the data are Gaussian with equal variances. In this article, we consider the Wilcoxon rank test as well as modified versions of the classical t-test that correct for departures from these assumptions. The relative performance of the tests are assessed by applying them to simulated data and comparing their size and power; one of the modified tests (the CW test) is shown to be superior. Abstract: Behavioral risk factors for cancer tend to cluster within individuals, which can compound risk beyond that associated with the individual risk factors alone. There has been increasing attention paid to the prevalence of multiple risk factors (MRF) for cancer, and to the importance of designing interventions that help individuals reduce their risks across multiple behaviors simultaneously. The purpose of this paper is to develop methodology to identify an optimal linear combination of multiple risk factors (score function) which would facilitate evaluation of cancer interventions. Abstract: Some scientists prefer to exercise substantial judgment in formulating a likelihood function for their data. Others prefer to try to get the data to tell them which likelihood is most appropriate. We suggest here that one way to reduce the judgment component of the likelihood function is to adopt a mixture of potential likelihoods and let the data determine the weights on each likelihood. We distinguish several different types of subjectivity in the likelihood function and show with examples how these subjective elements may be given more equitable treatment. Abstract: Motivation: A formidable challenge in the analysis of microarray data is the identification of those genes that exhibit differential expression. The objectives of this research were to examine the utility of simple ANOVA, one sided t tests, natural log transformation, and a generalized experiment wise error rate methodology for analysis of such experiments. As a test case, we analyzed a Affymetrix GeneChip microarray experiment designed to test for the effect of a CHD3 chromatin remodeling factor, PICKLE, and an inhibitor of the plant hormone gibberellin (GA), on the expression of 8256 Arabidopsis thaliana genes. Results: The GFWER(k) is defined as the probability of rejecting k or more true null hypothesis at a given p level. Computing probabilities by GFWER(k) was shown to be simple to apply and, depending on the value of k, can greatly increase power. A k value as small as 2 or 3 was concluded to be adequate for large or small experiments respectively. A one sided ttest along with GFWER(2)=.05 identified 43 genes as exhibiting PICKLEdependent expression. Expression of all 43 genes was re-examined by qRTPCR, of which 36 (83.7%) were confirmed to exhibit PICKLE-dependent expression. Abstract: In this paper, a tree-structured method is proposed to extend Classification and Regression Trees (CART) algorithm to multivariate survival data, assuming a proportional hazard structure in the whole tree. The method works on the marginal survivor distributions and uses a sandwich estimator of variance to account for the association between survival times. The Wald-test statistics is defined as the splitting rule and the survival trees are developed by maximizing between-node separation. The proposed method intends to classify patients into subgroups with distinctively different prognosis. However, unlike the conventional tree-growing algorithms which work on a subset of data at every partition, the proposed method deals with the whole data set and searches the global optimal split at each partition. The method is applied to a prostate cancer data and its performance is also evaluated by several simulation studies. Abstract: We develop a likelihood ratio test statistic, based on the betabinomial distribution, for comparing a single treated group with dichotomous data to dual control groups. This statistic is useful in cases where there is overdispersion or extra-binomial variation. We apply the statistic to data from a two year rodent carcinogenicity study with dual control groups. The test statistic we developed is similar to others that have been developed for incorporation of historical control groups with rodent carcinogenicity experiments. However, for the small sample case we considered, large sample theory used by the other test statistics did not apply. We determined the critical values of this statistic by enumerating its distribution. A small Monte Carlo study shows the new test statistic controls the significance level much better than Fisher’s exact test when there is overdispersion and that it has adequate power. Abstract: We propose a coherent methodology for integrating different sources of information on a response variable of interest, in order to accurately predict percentiles of its distribution. Under the assumption that one of the sources is more reliable than the other(s), the approach combines factors formed from the data into an additive linear regression model. Quantile regression, designed for quantifying the goodness of fit precisely at a desired quantile, is used as the optimality criterion in model-fitting. Asymptotic confidence interval construction methods for the percentiles are adopted to compute statistical tolerance limits for the response. The approach is demonstrated on a materials science case study that pools together information on failure load from physical tests and computer model predictions. A small simulation study assesses the precision of the inferences. The methodology gives plausible percentile estimates. Resulting tolerance limits are close to nominal coverage probability levels.

One of the pieces I barely gave a glance when reading Feynman’s Lectures over the past few years, was the derivation of the non-spherical electron orbitals for the hydrogen atom. It just looked like a boring piece of math – and I thought the derivation of the s-orbitals – the spherically symmetrical ones – was interesting enough already. To some extent, it is – but there is so much more to it. When I read it now, the derivation of those p-, d-, f– etc. orbitals brings all of the weirdness of quantum mechanics together and, while doing so, also provides for a deeper understanding of all of the ideas and concepts we’re trying to get used to. In addition, Feynman’s treatment of the matter is actually much shorter than what you’ll find in other textbooks, because… Well… As he puts it, he takes a shortcut. So let’s try to follow the bright mind of our Master as he walks us through it. You’ll remember – if not, check it out again – that we found the spherically symmetric solutions for Schrödinger’s equation for our hydrogen atom. Just to be make sure, Schrödinger’s equation is a differential equation – a condition we impose on the wavefunction for our electron – and so we need to find the functional form for the wavefunctions that describe the electron orbitals. [Quantum math is so confusing that it’s often good to regularly think of what it is that we’re actually trying to do. :-)] In fact, that functional form gives us a whole bunch of solutions – or wavefunctions – which are defined by three quantum numbers: n, l, and m. The parameter n corresponds to an energy level (En), l is the orbital (quantum) number, and m is the z-component of the angular momentum. But that doesn’t say much. Let’s go step by step. First, we derived those spherically symmetric solutions – which are referred to as s-states – assuming this was a state with zero (orbital) angular momentum, which we write as l = 0. [As you know, Feynman does not incorporate the spin of the electron in his analysis, which is, therefore, approximative only.] Now what exactly is a state with zero angular momentum? When everything is said and done, we are effectively trying to describe some electron orbital here, right? So that’s an amplitude for the electron to be somewhere, but then we also know it always moves. So, when everything is said and done, the electron is some circulating negative charge, right? So there is always some angular momentum and, therefore, some magnetic moment, right? Well… If you google this question on Physics Stack Exchange, you’ll get a lot of mumbo jumbo telling you that you shouldn’t think of the electron actually orbiting around. But… Then… Well… A lot of that mumbo jumbo is contradictory. For example, one of the academics writing there does note that, while we shouldn’t think of an electron as some particle, the orbital is still a distribution which gives you the probability of actually finding the electron at some point (x,y,z). So… Well… It is some kind of circulating charge – as a point, as a cloud or as whatever. The only reasonable answer – in my humble opinion – is that l = 0 probably means there is no net circulating charge, so the movement in this or that direction must balance the movement in the other. One may note, in this regard, that the phenomenon of electron capture in nuclear reactions suggests electrons do travel through the nucleus for at least part of the time, which is entirely coherent with the wavefunctions for s-states – shown below – which tell us that the most probable (x, y, z) position for the electron is right at the center – so that’s where the nucleus is. There is also a non-zero probability for the electron to be at the center for the other orbitals (p, d, etcetera).In fact, now that I’ve shown this graph, I should quickly explain it. The three graphs are the spherically symmetric wavefunctions for the first three energy levels. For the first energy level – which is conventionally written as n = 1, not as n = 0 – the amplitude approaches zero rather quickly. For n = 2 and n = 3, there are zero-crossings: the curve passes the r-axis. Feynman calls these zero-crossing radial nodes. To be precise, the number of zero-crossings for these s-states is n − 1, so there’s none for n = 1, one for n = 2, two for n = 3, etcetera. Now, why is the amplitude – apparently – some real-valued function here? That’s because we’re actually not looking at ψ(r, t) here but at the ψ(r) function which appears in the following break-up of the actual wavefunction ψ(r, t): ψ(r, t) = e−i·(E/ħ)·t·ψ(r) So ψ(r) is more of an envelope function for the actual wavefunction, which varies both in space as well as in time. It’s good to remember that: I would have used another symbol, because ψ(r, t) and ψ(r) are two different beasts, really – but then physicists want you to think, right? And Mr. Feynman would surely want you to do that, so why not inject some confusing notation from time to time? 🙂 So for n = 3, for example, ψ(r) goes from positive to negative and then to positive, and these areas are separated by radial nodes. Feynman put it on the blackboard like this:I am just inserting it to compare this concept of radial nodes with the concept of a nodal plane, which we’ll encounter when discussing p-states in a moment, but I can already tell you what they are now: those p-states are symmetrical in one direction only, as shown below, and so we have a nodal plane instead of a radial node. But so I am getting ahead of myself here… 🙂Before going back to where I was, I just need to add one more thing. 🙂 Of course, you know that we’ll take the square of the absolute value of our amplitude to calculate a probability (or the absolute square – as we abbreviate it), so you may wonder why the sign is relevant at all. Well… I am not quite sure either but there’s this concept of orbital parity which you may have heard of. The orbital parity tells us what will happen to the sign if we calculate the value for ψ for −r rather than for r. If ψ(−r) = ψ(r), then we have an even function – or even orbital parity. Likewise, if ψ(−r) = −ψ(r), then we’ll the function odd – and so we’ll have an odd orbital parity. The orbital parity is always equal to (-1)l = ±1. The exponent l is that angular quantum number, and +1, or + tout court, means even, and -1 or just − means odd. The angular quantum number for those p-states is l = 1, so that works with the illustration of the nodal plane. 🙂 As said, it’s not hugely important but I might as well mention in passing – especially because we’ll re-visit the topic of symmetries a few posts from now. 🙂 OK. I said I would talk about states with some angular momentum (so l ≠ 0) and so it’s about time I start doing that. As you know, our orbital angular momentum l is measured in units of ħ (just like the total angular momentum J, which we’ve discussed ad nauseam already). We also know that if we’d measure its component along any direction – any direction really, but physicists will usually make sure that the z-axis of their reference frame coincides with, so we call it the z-axis 🙂 – then we will find that it can only have one of a discrete set of values m·ħ = l·ħ, (l-1)·ħ, …, -(l-1)·ħ, –l·ħ. Hence, l just takes the role of our good old quantum number j here, and m is just Jz. Likewise, I’d like to introduce l as the equivalent of J, so we can easily talk about the angular momentum vector. And now that we’re here, why not write m in bold type too, and say that m is the z-component itself – i.e. the whole vector quantity, so that’s the direction and the magnitude. Now, we do need to note one crucial difference between j and l, or between J and l: our j could be an integer or a half-integer. In contrast, l must be some integer. Why? Well… If l can be zero, and the values of l must be separated by a full unit, then l must be 1, 2, 3 etcetera. 🙂 If this simple answer doesn’t satisfy you, I’ll refer you to Feynman’s, which is also short but more elegant than mine. 🙂 Now, you may or may not remember that the quantum-mechanical equivalent of the magnitude of a vector quantity such as l is to be calculated as √[l·(l+1)]·ħ, so if l = 1, that magnitude will be √2·ħ ≈ 1.4142·ħ, so that’s – as expected – larger than the maximum value for m, which is +1. As you know, that leads us to think of that z-component m as a projection of l. Paraphrasing Feynman, the limited set of values for m imply that the angular momentum is always “cocked” at some angle. For l = 1, that angle is either +45° or, else, −45°, as shown below.What if l = 2? The magnitude of l is then equal to √[2·(2+1)]·ħ = √6·ħ ≈ 2.4495·ħ. How do we relate that to those “cocked” angles? The values of m now range from -2 to +2, with a unit distance in-between. The illustration below shows the angles. [I didn’t mention ħ any more in that illustration because, by now, we should know it’s our unit of measurement – always.] Note we’ve got a bigger circle here (the radius is about 2.45 here, as opposed to a bit more than 1.4 for m = 0). Also note that it’s not a nice cake with perfectly equal pieces. From the graph, it’s obvious that the formula for the angle is the following:It’s simple but intriguing. Needless to say, the sin −1 function is the inverse sine, also known as the arcsine. I’ve calculated the values for all m for l = 1, 2, 3, 4 and 5 below. The most interesting values are the angles for m = 1 and m = l. As the graphs underneath show, for m = 1, the values start approaching the zero angle for very large l, so there’s not much difference any more between m = ±1 and m = 1 for large values of l. What about the m = l case? Well… Believe it or not, if l becomes really large, then these angles do approach 90°. If you don’t remember how to calculate limits, then just calculate θ for some huge value for l and m. For l = m = 1,000,000, for example, you should find that θ = 89.9427…°. 🙂 Isn’t this fascinating? I’ve actually never seen this in a textbook – so it might be an original contribution. 🙂 OK. I need to get back to the grind: Feynman’s derivation of non-symmetrical electron orbitals. Look carefully at the illustration below. If m is really the projection of some angular momentum that’s “cocked”, either at a zero-degree or, alternatively, at ±45º (for the l = 1 situation we show here) – a projection on the z-axis, that is – then the value of m (+1, 0 or -1) does actually correspond to some idea of the orientation of the space in which our electron is circulating. For m = 0, that space – think of some torus or whatever other space in which our electron might circulate – would have some alignment with the z-axis. For m = ±1, there is no such alignment. The interpretation is tricky, however, and the illustration on the right-hand side above is surely too much of a simplification: an orbital is definitely not like a planetary orbit. It doesn’t even look like a torus. In fact, the illustration in the bottom right corner, which shows the probability density, i.e. the space in which we are actually likely to find the electron, is a picture that is much more accurate – and it surely does not resemble a planetary orbit or some torus. However, despite that, the idea that, for m = 0, we’d have some alignment of the space in which our electron moves with the z-axis is not wrong. Feynman expresses it as follows: “Suppose m is zero, then there can be some non-zero amplitude to find the electron on the z-axis at some distance r. We’ll call this amplitude Fl(r).” You’ll say: so what? And you’ll also say that illustration in the bottom right corner suggests the electron is actually circulating around the z-axis, rather than through it. Well… No. That illustration does not show any circulation. It only shows a probability density. No suggestion of any actual movement or circulation. So the idea is valid: if m = 0, then the implication is that, somehow, the space of circulation of current around the direction of the angular momentum vector (J), as per the well-known right-hand rule, will include the z-axis. So the idea of that electron orbiting through the z-axis for m = 0 is essentially correct, and the corollary is… Well… I’ll talk about that in a moment. But… Well… So what? What’s so special about that Fl(r) amplitude? What can we do with that? Well… If we would find a way to calculate Fl(r), then we know everything. Huh? Everything? Yes. The reasoning here is quite complicated, so please bear with me as we go through it. The first thing you need to accept, is rather weird. The thing we said about the non-zero amplitudes to find the electron somewhere on the z-axis for the m = 0 state – which, using Dirac’s bra-ket notation, we’ll write as |l, m = 0〉 – has a very categorical corollary: The amplitude to find an electron whose state m is not equal to zero on the z-axis (at some non-zero distance r) is zero. We can only find an electron on the z-axis unless the z-component of its angular momentum (m) is zero. Now, I know this is hard to swallow, especially when looking at those 45° angles for J in our illustrations, because these suggest the actual circulation of current may also include at least part of the z-axis. But… Well… No. Why not? Well… I have no good answer here except for the usual one which, I admit, is quite unsatisfactory: it’s quantum mechanics, not classical mechanics. So we have to look at the m and −m vectors, which are pointed along the z-axis itself for m = ±1 and, hence, the circulation we’d associate with those momentum vectors (even if they’re the z–component only) is around the z-axis. Not through or on it. I know it’s a really poor argument, but it’s consistent with our picture of the actual electron orbitals – that picture in terms of probability densities, which I copy below. For m = −1, we have the yz-plane as the nodal plane between the two lobes of our distribution, so no amplitude to find the electron on the z-axis (nor would we find it on the y-axis, as you can see). Likewise, for m = +1, we have the xz-plane as the nodal plane. Both nodal planes include the z-axis and, therefore, there’s zero probability on that axis. In addition, you may also want to note the 45° angle we associate with m = ±1 does sort of demarcate the lobes of the distribution by defining a three-dimensional cone and… Well… I know these arguments are rather intuitive, and so you may refuse to accept them. In fact, to some extent, I refuse to accept them. 🙂 Indeed, let me say this loud and clear: I really want to understand this in a better way! But… Then… Well… Such better understanding may never come. Feynman’s warning, just before he starts explaining the Stern-Gerlach experiment and the quantization of angular momentum, rings very true here: “Understanding of these matters comes very slowly, if at all. Of course, one does get better able to know what is going to happen in a quantum-mechanical situation—if that is what understanding means—but one never gets a comfortable feeling that these quantum-mechanical rules are “natural.” Of course they are, but they are not natural to our own experience at an ordinary level.” So… Well… What can I say? It is now time to pull the rabbit out of the hat. To understand what we’re going to do next, you need to remember that our amplitudes – or wavefunctions – are always expressed with regard to a specific frame of reference, i.e. some specific choice of an x-, y– and z-axis. If we change the reference frame – say, to some new set of x’-, y’– and z’-axes – then we need to re-write our amplitudes (or wavefunctions) in terms of the new reference frame. In order to do so, one should use a set of transformation rules. I’ve written several posts on that – including a very basic one, which you may want to re-read (just click the link here). Look at the illustration below. We want to calculate the amplitude to find the electron at some point in space. Our reference frame is the x, y, z frame and the polar coordinates (or spherical coordinates, I should say) of our point are the radial distance r, the polar angle θ (theta), and the azimuthal angle φ (phi). [The illustration below – which I copied from Feynman’s exposé – uses a capital letter for phi, but I stick to the more usual or more modern convention here.] In case you wonder why we’d use polar coordinates rather than Cartesian coordinates… Well… I need to refer you to my other post on the topic of electron orbitals, i.e. the one in which I explain how we get the spherically symmetric solutions: if you have radial (central) fields, then it’s easier to solve stuff using polar coordinates – although you wouldn’t think so if you think of that monster equation that we’re actually trying to solve here: It’s really Schrödinger’s equation for the situation on hand (i.e. a hydrogen atom, with a radial or central Coulomb field because of its positively charged nucleus), but re-written in terms of polar coordinates. For the detail, see the mentioned post. Here, you should just remember we got the spherically symmetric solutions assuming the derivatives of the wavefunction with respect to θ and φ – so that’s the ∂ψ/∂θ and ∂ψ/∂φ in the equation above – were zero. So now we don’t assume these partial derivatives to be zero: we’re looking for states with an angular dependence, as Feynman puts it somewhat enigmatically. […] Yes. I know. This post is becoming very long, and so you are getting impatient. Look at the illustration with the (r, θ, φ) point, and let me quote Feynman on the line of reasoning now: “Suppose we have the atom in some |l, m〉 state, what is the amplitude to find the electron at the angles θ and φ and the distance r from the origin? Put a new z-axis, say z’, at that angle (see the illustration above), and ask: what is the amplitude that the electron will be at the distance r along the new z’-axis? We know that it cannot be found along z’ unless its z’-component of angular momentum, say m’, is zero. When m’ is zero, however, the amplitude to find the electron along z’ is Fl(r). Therefore, the result is the product of two factors. The first is the amplitude that an atom in the state |l, m〉 along the z-axis will be in the state |l, m’ = 0〉 with respect to the z’-axis. Multiply that amplitude by Fl(r) and you have the amplitude ψl,m(r) to find the electron at (r, θ, φ) with respect to the original axes.” So what is he telling us here? Well… He’s going a bit fast here. 🙂 Worse, I think he may actually not have chosen the right words here, so let me try to rephrase it. We’ve introduced the Fl(r) function above: it was the amplitude, for m = 0, to find the electron on the z-axis at some distance r. But so here we’re obviously in the x’, y’, z’ frame and so Fl(r) is the amplitude for m’ = 0, it’s the amplitude to find the electron on the z-axis at some distance r along the z’-axis. Of course, for this amplitude to be non-zero, we must be in the |l, m’ = 0〉 state, but are we? Well… |l, m’ = 0〉 actually gives us the amplitude for that. So we’re going to multiply two amplitudes here: Fl(r)·|l, m’ = 0〉 So this amplitude is the product of two amplitudes as measured in the the x’, y’, z’ frame. Note it’s symmetric: we may also write it as |l, m’ = 0〉·Fl(r). We now need to sort of translate that into an amplitude as measured in the x, y, z frame. To go from x, y, z to x’, y’, z’, we first rotated around the z-axis by the angle φ, and then rotated around the new y’-axis by the angle θ. Now, the order of rotation matters: you can easily check that by taking a non-symmetrical object in your hand and doing those rotations in the two different sequences: check what happens to the orientation of your object. Hence, to go back we should first rotate about the y’-axis by the angle −θ, so our z’-axis folds into the old z-axis, and then rotate about the z-axis by the angle −φ. Now, we will denote the transformation matrices that correspond to these rotations as Ry’(−θ) and Rz(−φ) respectively. These transformation matrices are complicated beasts. They are surely not the easy rotation matrices that you can use for the coordinates themselves. You can click this link to see how they look like for l = 1. For larger l, there are other formulas, which Feynman derives in another chapter of his Lectures on quantum mechanics. But let’s move on. Here’s the grand result: The amplitude for our wavefunction ψl,m(r) – which denotes the amplitude for (1) the atom to be in the state that’s characterized by the quantum numbers l and m and – let’s not forget – (2) find the electron at r – note the bold type: r = (x, y, z) – would be equal to: ψl,m(r) = 〈l, m|Rz(−φ) Ry’(−θ)|l, m’ = 0〉·Fl(r) Well… Hmm… Maybe. […] That’s not how Feynman writes it. He writes it as follows: ψl,m(r) = 〈l, 0|Ry(θ) Rz(φ)|l, m〉·Fl(r) I am not quite sure what I did wrong. Perhaps the two expressions are equivalent. Or perhaps – is it possible at all? – Feynman made a mistake? I’ll find out. [P.S: I re-visited this point in the meanwhile: see the P.S. to this post. :-)] The point to note is that we have some combined rotation matrix Ry(θ) Rz(φ). The elements of this matrix are algebraic functions of θ and φ, which we will write as Yl,m(θ, φ), so we write: a·Yl,m(θ, φ) = 〈l, 0|Ry(θ) Rz(φ)|l, m〉 Or a·Yl,m(θ, φ) = 〈l, m|Rz(−φ) Ry’(−θ)|l, m’ = 0〉, if Feynman would have it wrong and my line of reasoning above would be correct – which is obviously not so likely. Hence, the ψl,m(r) function is now written as: ψl,m(r) = a·Yl,m(θ, φ)·Fl(r) The coefficient a is, as usual, a normalization coefficient so as to make sure the surface under the probability density function is 1. As mentioned above, we get these Yl,m(θ, φ) functions from combining those rotation matrices. For l = 1, and m = -1, 0, +1, they are: A more complete table is given below:So, yes, we’re done. Those equations above give us those wonderful shapes for the electron orbitals, as illustrated below (credit for the illustration goes to an interesting site of the UC Davis school).But… Hey! Wait a moment! We only have these Yl,m(θ, φ) functions here. What about Fl(r)? You’re right. We’re not quite there yet, because we don’t have a functional form for Fl(r). Not yet, that is. Unfortunately, that derivation is another lengthy development – and that derivation actually is just tedious math only. Hence, I will refer you to Feynman for that. 🙂 Let me just insert one more thing before giving you The Grand Equation, and that’s a explanation of how we get those nice graphs. They are so-called polar graphs. There is a nice and easy article on them on the website of the University of Illinois, but I’ll summarize it for you. Polar graphs use a polar coordinate grid, as opposed to the Cartesian (or rectangular) coordinate grid that we’re used to. It’s shown below. The origin is now referred to as the pole – like in North or South Pole indeed. 🙂 The straight lines from the pole (like the diagonals, for example, or the axes themselves, or any line in-between) measure the distance from the pole which, in this case, goes from 0 to 10, and we can connect the equidistant points by a series of circles – as shown in the illustration also. These lines from the pole are defined by some angle – which we’ll write as θ to make things easy 🙂 – which just goes from 0 to 2π = 0 and then round and round and round again. The rest is simple: you’re just going to graph a function, or an equation – just like you’d graph y = ax + b in the Cartesian plane – but it’s going to be a polar equation. Referring back to our p-orbitals, we’ll want to graph the cos2θ = ρ equation, for example, because that’s going to show us the shape of that probability density function for l = 1 and m = 0. So our graph is going to connect the (θ, ρ) points for which the angle (θ) and the distance from the pole (ρ) satisfies the cos2θ = ρ equation. There is a really nice widget on the WolframAlpha site that produces those graphs for you. I used it to produce the graph below, which shows the 1.1547·cos2θ = ρ graph (the 1.1547 coefficient is the normalization coefficient a). Now, you’ll wonder why this is a curve, or a curved line. That widget even calculates its length: it’s about 6.374743 units long. So why don’t we have a surface or a volume here? We didn’t specify any value for ρ, did we? No, we didn’t. The widget calculates those values from the equation. So… Yes. It’s a valid question: where’s the distribution? We were talking about some electron cloud or something, right? Right. To get that cloud – those probability densities really – we need that Fl(r) function. Our cos2θ = ρ is, once again, just some kind of envelope function: it marks a space but doesn’t fill it, so to speak. 🙂 In fact, I should now give you the complete description, which has all of the possible states of the hydrogen atom – everything! No separate pieces anymore. Here it is. It also includes n. It’s The Grand Equation:The ak coefficients in the formula for ρFn,l(ρ) are the solutions to the equation below, which I copied from Feynman’s text on it all. I’ll also refer you to the same text to see how you actually get solutions out of it, and what they then actually represent. 🙂We’re done. Finally! I hope you enjoyed this. Look at what we’ve achieved. We had this differential equation (a simple diffusion equation, really, albeit in the complex space), and then we have a central Coulomb field and the rather simple concept of quantized (i.e. non-continuous or discrete) angular momentum. Now see what magic comes out of it! We literally constructed the atomic structure out of it, and it’s all wonderfully elegant and beautiful. Now I think that’s amazing, and if you’re reading this, then I am sure you’ll find it as amazing as I do. Post scriptum on the transformation matrices: You must find the explanation for that 〈l, 0|Ry(θ) Rz(φ)|l, m〉·Fl(r) product highly unsatisfactory, and it is. 🙂 I just wanted to make you think – rather than just superficially read through it. First note that Fl(r)·|l, m’ = 0〉 is not a product of two amplitudes: it is the product of an amplitude with a state. A state is a vector in a rather special vector space – a Hilbert space (just a nice word to throw around, isn’t it?). The point is: a state vector is written as some linear combination of base states. Something inside of me tells me we may look at the three p-states as base states, but I need to look into that. Note that this product is non-commutative because… Well… Matrix products generally are non-commutative. 🙂 So… Well… There they are: the second row gives us those functions, so I am wrong, obviously, and Dr. Feynman is right. Of course, he is. He is always right – especially because his Lectures have gone through so many revised editions that all errors must be out by now. 🙂 However, let me – just for fun – also calculate my Rz(−φ) Ry’(−θ) product. I can do so in two steps: first I calculate Rz(φ) Ry’(θ), and then I substitute the angles φ and θ for –φ and –θ, remembering that cos(–α) = cos(α) and sin(–α) = –sin(α). I might have made a mistake, but I got this:The functions look the same but… Well… No. The eiφ and e−iφ are in the wrong place (it’s just one minus sign – but it’s crucially different). And then these functions should not be in a column. That doesn’t make sense when you write it all out. So Feynman’s expression is, of course, fully correct. But so how do we interpret that 〈l, 0|Ry(θ) Rz(φ)|l, m〉 expression then? This amplitude probably answers the following question: Given that our atom is in the |l, m〉 state, what is the amplitude for it to be in the 〈l, 0| state in the x’, y’, z’ frame? That makes sense – because we did start out with the assumption that our atom was in the the |l, m〉 state, so… Yes. Think about it some more and you’ll see it all makes sense: we can – and should – multiply this amplitude with the Fl(r) amplitude. OK. Now we’re really done with this. 🙂 Note: As for the 〈 | and | 〉 symbols to denote a state, note that there’s not much difference: both are state vectors, but a state vector that’s written as an end state – so that’s like 〈 Φ | – is a 1×3 vector (so that’s a column vector), while a vector written as | Φ 〉 is a 3×1 vector (so that’s a row vector). So that’s why 〈l, 0|Ry(θ) Rz(φ)|l, m〉 does give us some number. We’ve got a (1×3)·(3×3)·(3×1) matrix product here – but so it gives us what we want: a 1×1 amplitude. 🙂

The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ the stationary bootstrap procedure; we show the consistency of this bootstrap. Also, we consider the self-normalized approach, which is shown to be asymptotically pivotal under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Goldman Sachs and AIG. This article has supplementary materials online. Keywords: Quantile, Correlogram, Dependence, Predictability, Systemic risk. Linton and Whang (2007) introduced the quantilogram to measure predictability in different parts of the distribution of a stationary time series based on the correlogram of ”quantile hits”. They applied it to test the hypothesis that a given time series has no directional predictability. More specifically, their null hypothesis was that the past information set of the stationary time series does not improve the prediction about whether will be above or below the unconditional quantile. The test is based on comparing the quantilogram to a pointwise confidence band. This contribution fits into a long literature of testing predictability using signs or rank statistics, including the papers of Cowles and Jones (1937), Dufour et al. (1998), and Christoffersen and Diebold (2002). The quantilogram has several advantages compared to other test statistics for directional predictability. It is conceptually appealing and simple to interpret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (2000), and so it works well for heavy tailed series. Many financial time series have heavy tails, see, e.g., Mandelbrot (1963), Fama (1965), Rachev and Mittnik (2000), Embrechts et al. (1997), Ibragimov et al. (2009), and Ibragimov (2009), and so this is an important consideration in practice. Additionally, this type of method allows researchers to consider very long lags in comparison with regression type methods, such as Engle and Manganelli (2004). There have been a number of recent works either extending or applying this methodology. Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles. Hagemann (2012) has introduced a Fourier domain version of the quantilogram, see also Dette et al. (2013) for an alternative approach. The quantilogram has recently been applied to stock returns and exchange rates, Laurini et al. (2008) and Chang and Shie (2011). Our paper addresses two outstanding issues with regard to the quantilogram. First, the construction of confidence intervals that are valid under general dependence structures. Linton and Whang (2007) derived the limiting distribution of the sample quantilogram under the null hypothesis that the quantilogram itself is zero, in fact under a special case of that where the process has a type of conditional heteroskedasticity structure. Even in that very special case, the limiting distribution depends on model specific quantities. They derived a bound on the asymptotic variance that allows one to test the null hypothesis of the absence of predictability (or rather the special case of this that they work with). Even when this model structure is appropriate, the bounds can be quite large especially when one looks into the tail of the distribution. The quantilogram is also useful in cases where the null hypothesis of no predictability is not thought to be true - one can be interested in measuring the degree of predictability of a series across different quantiles. We provide a more complete solution to the issue of inference for the quantilogram. Specifically, we derive the asymptotic distribution of the quantilogram under general weak dependence conditions, specifically strong mixing. The limiting distribution is quite complicated and depends on the long run variance of the quantile hits. To conduct inference we propose the stationary bootstrap method of Politis and Romano (1994) and prove that it provides asymptotically valid confidence intervals. We investigate the finite sample performance of this procedure and show that it works well. We also provide R code that carries out the computations efficiently. We also define a self-normalized version of the statistic for testing the null hypothesis that the quantilogram is zero, following Lobato (2001). This statistic has an asymptotically pivotal distribution whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap. Second, we develop our methodology inside a multivariate setting and explicitly consider the cross-quantilogram. Linton and Whang (2007) briefly mentioned such a multivariate version of the quantilogram but they provided neither theoretical results or empirical results. In fact, the cross correlogram is a vitally important measure of dependence between time series: Campbell et al. (1997), for example, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We apply the cross-quantilogram to the study of stock return predictability; our method provides a more complete picture of the predictability structure. We also apply the cross quantilogram to the question of systemic risk. Our theoretical results described in the previous paragraph are all derived for the multivariate case. 2 The Cross-Quantilogram Let be a two dimensional strictly stationary time series with and let denote the distribution function of the series with density function for . The quantile function of the time series is defined as for . Let for . We consider a measure of serial dependence between two events and for arbitrary quantiles. In the literature, is called the quantile-hit or quantile-exceedance process for , where denotes the indicator function taking the value one when its argument is true, and zero otherwise. The cross-quantilogram is defined as the cross-correlation of the quantile-hit processes for where . The cross-quantilogram captures serial dependency between the two series at different quantile levels. In the special case of a single time series, the cross-quantilogram becomes the quantilogram proposed by Linton and Whang (2007). Note that it is well-defined even for processes with infinite moments. Like the quantilogram, the cross-quantilogram is invariant to any strictly monotonic transformation applied to both series, such as the logarithmic transformation. To construct the sample analogue of the cross-quantilogram based on observations , we first estimate the unconditional quantile functions by solving the following minimization problems, separately: where . Then, the sample cross-quantilogram is defined as for Given a set of quantiles, the cross-quantilogram considers dependency in terms of the direction of deviation from quantiles and thus measures the directional predictability from one series to another. This can be a useful descriptive device. By construction, with corresponding to the case of no directional predictability. The form of the statistic generalizes to the dimensional multivariate case and the th entry of the corresponding cross-correlation matrices is given by applying (2) for a pair of variables and a pair of quantiles for . The cross-correlation matrices possess the usual symmetry property when We may be interested in testing for the absence of directional predictability over a set of quantiles. Let , where is a quantile range for each time series (). We are interested in testing the hypothesis against the alternative hypothesis that for some and some with fixed.111Hong (1996) established the properties of the Box-Pierce statistic in the case that after a location and scale adjustment the statistic is asymptotically normal. No doubt our results can be extended to accommodate this case, although in practice the desirability of such a test is questionable, and our limit theory may provide better critical values for even quite long lags. This is a test for the directional predictability of events up to lags for To discriminate between these hypotheses we will use the test statistic where is the quantile specific Box-Pierce type statistic and To test the directional predictability in a specific quantile, or to provide confidence intervals for the population quantities, we use or , which are special cases of the sup-type test statistic. In practice, we have found that the Box-Ljung version yields some small sample improvements. 3 Asymptotic Properties Here we present the asymptotic properties of the sample cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we employ techniques widely used in the literature on quantile regression, see Koenker and Bassett (1978) and Pollard (1991) among others. We impose the following assumptions. Assumption A1. is a strictly stationary and strong mixing with coefficients that satisfy for . A2. The distribution functions for have continuous densities uniformly bounded away from 0 and at uniformly over . A3. For any there exists a such that for . A4. The joint distribution of has a bounded, continuous first derivative for each argument uniformly in the neighborhood of quantiles of interest for every . Assumption A1 imposes a mixing rate used in Rio (2000, Chapter 7). For a strong mixing process, as for all Assumption A2 ensures that the quantile functions are uniquely defined. Assumption A3 implies that the densities are smooth in some neighborhood of the quantiles of interest. Assumption A4 ensures that the joint distribution of is continuously differentiable. To describe the asymptotic behavior of the cross-quantilogram, we define a set of 3-dimensional mean-zero Gaussian process with covariance-matrix function given by for and for , where with for and for . Define the -dimensional zero-mean Gaussian process with the covariance-matrix function denoted by . The next theorem establishes the asymptotic properties of the cross-quantilogram. Suppose that Assumptions A1-A4 hold for some finite integer Then, in the sense of weak convergence of the stochastic process we have: with the gradient vector of . Under the null hypothesis that for every , it follows that by the continuous mapping theorem. 3.1 Inference Methods 3.1.1 The Stationary Bootstrap The asymptotic null distribution presented in Theorem 1 depends on nuisance parameters. To estimate the critical values from the limiting distribution we could use nonparametric estimation, but that may suffer from a slow convergence rate. We address this issue by using the stationary bootstrap (SB) of Politis and Romano (1994). The SB is a block bootstrap method with blocks of random lengths. The SB resample is strictly stationary conditional on the original sample. Let denote a sequence of block lengths, which are iid random variables having the geometric distribution with a scalar parameter : for each positive integer , where denotes the conditional expectation given the original sample. We assume that the parameter satisfies the following growth condition Assumption A5. as . Let be iid random variables, which are independent of both the original data and , and have the discrete uniform distribution on . We set representing the block of length starting with the -th pair of observations. The SB procedure generate samples by taking the first observations from a sequence of the resampled blocks . In this notation, when , is set to be , where and , where mod denotes the modulo operator.222For any positive integers and , the modulo operation is equal to the remainder, on division of by . Using the SB resample, we obtain quantile estimates for by solving the minimization problems: We construct by using SB observations, while is based on observations. The difference of sample sizes is asymptotically negligible given the finite lag order . The cross-quantilogram based on the SB resample is defined as follows: We consider the SB bootstrap to construct a confidence intervals for each statistic of cross-quantilograms for a finite positive integer and subsequently construct a confidence interval for the omnibus test based on the statistics. To maintain possible dependence structures, we use pairs of observations to resample the blocks of random lengths. Given a vector cross-quantilogram , we define the omnibus test based on the SB resample as . The following lemma shows the validity of the SB procedure for the cross-quantilogram. Suppose that Assumption A1-A5 hold. Then, in the sense of weak convergence conditional on the sample we have: (a) in probability; (b) Under the null hypothesis that for every , In practice, repeating the SB procedure times, we obtain sets of cross-quantilograms and and sets of omnibus tests with . For testing jointly the null of no directional predictability, a critical value, , corresponding to a significance level is give by the percentile of test statistics , that is, For the individual cross-quantilogram, we pick up percentiles of the bootstrap distribution of such that , in order to obtain a confidence interval for given by In the following theorem, we provide a power analysis of the omnibus test statistic when we use a critical value . That is, we examine the power of the omnibus test by using . We consider fixed and local alternatives. The fixed alternative hypothesis against the null of no directional predictability is and the local alternative where is a finite non-zero scalar. Thus, under the local alternative, there exist a -dimensional vector such that with having at least one non-zero element for some . While we consider the asymptotic power of test for the directional predictability over a range of quantiles with multiple lags in the following theorem, the results can be applied to test for a specific quantile or a specific lag order. The following theorem shows that the cross-quantilogram process has non-trivial local power against -local alternatives. 3.1.2 The Self-Normalized Cross-Quantilogram The self-normalized approach was proposed in Lobato (2001) for testing the absence of autocorrelation of a time series that is not necessarily independent. The idea was recently extended by Shao (2010) to a class of asymptotically linear test statistics. Kuan and Lee (2006) apply the approach to a class of specification tests, the so-called tests, which are based on the moment conditions involving unknown parameters. Chen and Qu (2012) propose a procedure for improving the power of the test, by dividing the original sample into subsamples before applying the self-normalization procedure. The self-normalized approach has a tight link with the fixed- asymptotic framework, which was proposed by Kiefer et al. (2000) and has been studied by Bunzel et al. (2001), Kiefer and Vogelsang (2002, 2005), Sun et al. (2008), Kim and Sun (2011) and Sun and Kim (2012) among others. As discussed in section 2.1 of Shao (2010), the self-normalized and the fixed- approach have better size properties, compared with the standard approach involving a consistent asymptotic variance estimator, while it may be asymptotically less powerful under local alternatives (see Lobato (2001) and Sun et al. (2008) for instance). The relation between size and power properties is consistent with simulation results reported in the cited papers above. We use recursive estimates to construct a self-normalized cross-quantilogram. Given a subsample , we can estimate sample quantile functions by solving minimization problems We consider the minimum subsample size larger than , where is an arbitrary small positive constant. The trimming parameter, , is necessary to guarantee that the quantiles estimators based on subsamples have standard asymptotic properties and plays a different role to that of smoothing parameters in long-run variance estimators. Our simulation study suggests that the performance of the test is not sensitive to the trimming parameters. A key ingredient of the self-normalized statistic is an estimate of cross-correlation based on subsamples: for . For a finite integer , let . We construct an outer product of the cross-quantilogram using the subsample We can obtain the asymptotically pivotal distribution using as the asymptotically random normalization. For testing the null of no directional predictability, we define the self-normalized omnibus test statistic The following theorem shows that is asymptotically pivotal. To distinguish the process used in the following theorem from the one used in the previous section, let denote a -dimensional, standard Brownian motion on . Suppose that Assumptions A1-A4 hold. Then, for each , The joint test based on finite multiple quantiles can be constructed in a similar manner, while the test based on a range of quantiles has a limiting distribution depending on the Kiefer process: this may be difficult to implement in practice. The asymptotic null distribution in the above theorem can be simulated and a critical value, , corresponding to a significance level is tabulated by using the percentile of the simulated distribution.333We provide the simulated critical values in our R package. In the theorem below, we consider a power function of the self-normalized omnibus test statistic, . For a fixed , we consider a fixed alternative and a local alternative where is a finite non-zero scalar. This implies that there exist a -dimensional vector such that with having at least one non-zero element. 4 The Partial Cross-Quantilogram We define the partial cross-quantilogram, which measures the relationship between two events and , while controlling for intermediate events between and as well as whether some state variables exceed a given quantile. Let be an -dimensional vector for , which may include some of the lagged predicted variables , the intermediate predictors and some state variables that may reflect some historical events up to . We use to denote the th quantile of given for and define , with . To ease the notational burden in the rest of this section, we suppress the dependency of on for and use and . We present results for the single quantile and a single lag , although the results can be extended to the case of a range of quantiles and multiple lags. We introduce the correlation matrix of the hit processes and its inverse matrix where with for . For , let and be the element of and , respectively. Notice that the cross-quantilogram is and the partial cross-quantilogram is defined as To obtain the sample analogue of the partial cross-quantilogram, we first construct a vector of hit processes, , by replacing the population quantiles in by the sample analogues . Then, we obtain the estimator for the correlation matrix and its inverse as which leads to the sample analogue of the partial cross-quantilogram where denotes the element of for . In Theorem 6 below, we show that asymptotically follows the normal distribution, while the asymptotic variance depends on nuisance parameters as in the previous section. To address the issue of the nuisance parameters, we employ the stationary bootstrap or the self-normalization technique. For the bootstrap, we can use pairs of variables to generate the SB resample and we then obtain the SB version of the partial cross-quantilogram, denoted by , using the formula in (10). When we use the self-normalized test statistics, we estimate the partial cross-quantilogram based on the subsample up to , recursively and then we use to normalize the cross-quantilogram, thereby obtaining the asymptotically pivotal statistics. To obtain the asymptotic results, we impose the following conditions on the distribution function and the density function of each controlling variable for . Assumption A6. For every : (a) is strictly stationary and strong mixing as assumed in Assumption A1; (b) The conditions in Assumption A2 and A3 hold for the and at the relevant quantile; (c) for is continuously differentiable. Assumption A6(a) requires the control variables to satisfy the same weak dependent property as and . Assumption A6(b)-(c) ensure the smoothness of the distribution, density function and the joint distribution of and . Define the covariance matrix Also, let where with , ,

Difference Between Mendeleev and Modern Periodic Table Definition. The size of the element decreases as you move across a period as the number of electron shells remains constant across the period but the number of protons rises in the nucleus. A period in Joda-Time represents a period of time defined in terms of fields, for example, 3 years 5 months 2 days and 7 hours. 1. The seventh period with n = 7 includes the man-made radioactive elements with electrons filling 7s, 5f, 6d and 7p orbitals. 3. The difference between Periods and Groups is their arrangement, Periods are arranged in a horizontal manner whereas Groups are vertically arranged on the periodic table. Differences between the two parties that are covered in this article rely on the majority position though individual politicians may have varied preferences. Test. Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table. 1. For example, the number of valence electrons in the group 1 is 1. ∙The difference-in-differences (DD) estimate is ̂ … 5. 25/04/2020 02:18 AM. Helmenstine, Anne Marie, Ph.D. “The Difference Between an Element Group and Period.” ThoughtCo, Aug. 3, 2017. Available here One can see using the quantum theory the similarities, in groups and periods. The elements in a group have similar physical and chemical properties. The Periodic Table would also like to introduce you to the Period 2 elements. Elements in the same group react similarly. 1. Groups and periods are two ways of categorizing elements in the periodic table. Elements in a group share chemical or physical properties whereas elements in a period have the same electron configurations 4. ... for two time periods, and would like to know if the relationship (i.e. Each time a pattern started over, he started a new row. See answer LilBlondie420 LilBlondie420 A period is horizontal; the elements gain 1 proton for each space moving left to right. You can also calculate elapsed time. Atomic size: The atomic size decreases from left to right in a period. Most elements are metals. Thus, in a single group ITS analysis, a static cohort strengthens the argument for a causal model result. Period 1 has only two elements (hydrogen and helium), while periods 2 and 3 … That is, the elements in the same period have the same number of electron shells while the elements in the same group have the same number of valence electrons. Electronic Configuration in Groups So, there is increased force of attraction towards the nucleus. Electronic Configuration in Groups 4. So for both the treatment and no-treatment groups you are looking at the change in outcome between baseline (era0) and a subsequent observation periods in each panel of the table. Calculate the number of days, months, or years between two dates using Excel functions. Elements in the same period has equal number of electron composition, Elements in each group have an equal number of valence electrons, There are 7 periods on the periodic table, The group contains 18 elements arranged vertically in modern periodic table. Now, let’s meet the members of Group 1. Please post your answer: LOGIN TO POST ANSWER. Moreover, the elements get less metallic when we move forward a row. What are Groups in Periodic Table But in a real sense, this little periodic table is way more important than that, it is a roadmap that unlocks a million opportunities for scientists and researchers across the world. Learn. “Chemistry for Non-Majors.” Lumen, Lumen Learning. A new period begins when a fresh fundamental energy level adds up with the electrons. There are many differences between group and individual coverage. STUDY. Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table. Higher atomic number creates what? DID is used in observational settings where exchangeability cannot be assumed between the treatment and control groups. The elements in a period do not have similar properties. The element is Carbon because its on the row of period 2 and it is under Group 4. E.g., why does silicon replace carbon in fossilization? Periods and Groups As Mendeleev was arranging the elements in order of increasing atomic weight, he noticed that patterns repeated periodically. You can also allow external senders to send email to the group email address. The periodic table also has a special name for its vertical columns. Li is 2s1 Ne is 1s2 2s2 2p6 a complete atom. Compare groups. Periods are rows (go across horizontally). Difference between groups and periods.. Report ; Posted by Vaishnavi Helwatkar 2 years, 8 months ago. Gravity. the period is formed from like this. the difference between period and group. You can compare tumor growth over time periods by doing a two-way anova comparison. Figure 01: Periods and Groups in the Periodic Table. But IUPACâs numbering looks simple and well-organized. The difference between baroque and romantic music also reflects the events and fads of those time periods. The Periodic Table says hi. Groups are the vertical columns in the periodic table. Statistical tests can be used to analyze differences in the scores of two or more groups. The Periodic Table says hi. A new period begins when a new principal energy level begins filling with electrons. Transition elements have empty orbitals that are able to accept extra elections under certain situations. Home » Science » Difference Between Period and Group (With Table). Both describe elements that share common properties, usually based on the number of valence electrons. The periods are horizontal rows, counting from left to right, while groups, also called families are vertical columns, counting from top to the bottom. You can add people from outside your organization to a group as long as this has been enabled by the administrator. All rights reserved. the period is formed from like this. A team can have more than one head. Similarly, each group in the lane has its own family name. Patiatboinder Patiatboinder 08.03.2017 English Secondary School Difference between groups and periods 2 you get that in the 3rd period as well Na 1s1 2s2 2p6 3s1 to Ar 1s2 2s2 2p6 3s2 3p6 get the idea. ...and Your Groups Now you know about periods going left to right. There are seven periods in the periodic table, with each one beginning at the far left. Say hello to Group Two. The main difference between wavelength and period is that the wavelength is the shortest distance between two successive points on a wave that are in phase while period is the time taken for a complete oscillation to take place at a given point. What's the difference between periods and groups in the Periodic Table and why are the elements structured this way. Group 1 elements tend to have s^1 configuration, one electron in the outermost energy level. The following is a collection of the most used terms in this article on Period and Group. Two charged parallel plates are … As you now know, periods are on the horizontal line and groups are on the vertical line. However, this does not remove potential confounding due to differences in the distributions of these factors between the exposure groups in the D-I-D and Robust ITS with designs. Match. the horizontal rows in a periodic table. Groups are elements have the same outer electron arrangement. Difference Between Vitamin D and Vitamin D3 - 118 emails Difference Between Goals and Objectives - 102 emails Difference Between LCD and LED Televisions - 89 emails However, randomization does not always result in balanced groups, and without ... but we will demonstrate a typical two-period and two-group DID design in this module. STUDY. columms and rows in the periodic table. As of now, there are 7 periods on the periodic table. There is only one head in a group. 1s1, 2s1, 3s1 are all in the same group // 1s1 1s2, S, P, D, F . Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table.There are 18 groups while there are 7 periods.Elements in a group share a common number of valence electrons while elements in a period of varying valence electrons. We have seven (7) periods and eighteen (18) groups. The relationship between Deployment Pools and Deployment Groups is similar to the relationship between Agent Pools and Agent Queues. Intermediate, CK12. In the United States, they used letters A&B to indicate each element in the group but unfortunately, it was observed as a disorganized numbering system. Eg., Group 1 belongs to the Lithium family classified as Alkene metals. Group disability coverage is tied to your employment. Similarly, as you move down in the row, orbitals keep adding up. Groups are vertical columns while periods are the horizontal rows 3. Thus, we name them as period 1, period 2, … Period 7. What are Periods in Periodic Table The Difference Between an Element Group and Period Groups and periods are two ways of categorizing elements in the periodic table. Dmitri Mendeleyev is the inventor of the periodic table. Write. The most common way the periodic table is classified by metals, nonmetals, and metalloids. For example, you can calculate age in years, months and days. Now, this is Group Two. In order to eliminate all possible confusion, the International Union of Pure and Applied Chemistry (IUPAC) Came up with an idea of numbering the elements as (1,2, 3⦠18). The following statistical tests are commonly used to analyze differences between groups: T-Test. Wavelength and period are two different, but related properties of waves. However, the periods have different numbers of members; there are more chemical elements in some periods than some other periods. 1s1, 2s1, 3s1 are all in the same group // 1s1 1s2, S, P, D, F the period is formed from like this Li is 2s1 Ne is 1s2 2s2 2p6 a complete atom you get that in the 3rd period as well Na 1s1 2s2 2p6 3s1 to Ar 1s2 2s2 2p6 3s2 3p6 get the idea. The elements in a group have similar physical and chemical properties. A t-test is used to determine if the scores of two groups … Elements in the related group have similar traits because they have the same electron counts in their outermost shell. The same outer electron arrangement estimate is ̂ … you can compare tumor growth time. Are commonly used to analyze differences in the related group have similar physical and chemical.! Over, he noticed that patterns repeated periodically suitable examples DD ) is. Attraction towards the nucleus a column in the related group have similar physical and chemical properties ( formerly Office groups. A family and a group in the periodic table metals, non-metals, and metalloids Class 10 Science. To read +2 ; in this article at a later stage for you music. ( 1 ) and columns running ( top-to-bottom ) your periodic table would also to... Referred as periods and groups are made up according to their similar properties understanding! To a group vs. the one above or below group 4 periods than some periods... Each space moving left to right to bottom there are 7 major periods groups... On top of this, difference between groups and periods extra electron shell rows in the nucleus 10 > 4. 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Relationship between deployment group and individual coverage existing in the periodic table being asked to memorize tediously without even the. Between a period is likely to have s^1 configuration, one electron in the periodic.... Two or more groups makes the atom heavier to right in a periodic table, groups.

Construction as a word has a very specific meaning. Any drawing is restricted to use of only compass and a straightedge, whilst using a ruler to measure lengths and a protractor to measure angles is not allowed. Why is this so? In antiquity, geometric constructions were restricted to the use of only a straightedge and a compass. Notice, that here we used the term straightedge, and not a ruler. Why not measure? Greeks knowledge of mathematics was basic, including little arithmetic. So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also why the straightedge has no markings. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic. Instead of the term geometric structures we more often use the term Euclidean constructions because the majority of those are held in Euclid's (300 BC) Elements. Geometric constructions are highly connected to problems of antiquity that include squaring a circle, duplication of a cube and angle trisection that have been proved to be geometrically constructed impossible hundreds of years later. In this technological era it is no wonder students have never seen a compass let alone hold it in their hands. Since majority of the work is going to be done using construction tools, it is important for the students to get familiar with those. Compass is a drawing instrument often used to draw circles and arcs. It has two legs, one with a point, and the second one with a lead or pencil. The openness of the compass is easily regulated by opening or closing both legs, and will remain in a particular setting until changed. Straightedge is a tool used for drawing straight lines, and has no markings. If ruler is being used it must be cleared to students that using markings on the ruler is not allowed during constructions. Patty paper (vax paper) will be used in this lesson as a representation for students to grasp more easily through geometrical concepts. During this first day it is important for students to learn and develop appreciation for the context of geometric constructions. Also, it is important for students to have time to play with the tools, especially with the compass. Discuss with the students the meaning of the words to sketch, to draw and to construct. Let them carry the meaning on their own and correct when needed. By the end of the day student should know the difference between those terms. Explanation of the terms: Neither ruler nor protractor is ever used to perform geometric constructions. They are measuring tools, and not construction tools. Note: The lesson plan can be developed either using compass and ruler or using technology or both together. It is on the teacher to make that decision. However, I strongly encourage using the old fashion compass and ruler =). Day 2-Basic constructions 1. Start off the discussion with two simple problems. 1. a. copy a given segment 1. b. copy a given angle If necessary discuss with your students what it means to copy something. It should be clear to them that by copying we understand constructing an object congruent to a given one. At the beginning it is important to set the rules. Develop a discussion what would a proper construction include. Let them derive that there is more to construction than mare performing of steps and memorizing of the same ones. It should be clear that prior to starting the constructions they have to understand the problem and a good way for that is to make a sketch and start thinking how they would solve the problem. Afterwards the students can perform their steps using the geometry tool. However, the process does not end here. Students should justify their steps using proper geometry language using concepts learnt prior. Depending on the type of problem discussion will be needed on the uniqueness of the solution. Also, precision here is not the center of the universe, but make sure that they should manage compass and the ruler to be as precise as possible. Agreement on notation is also important. Make sure they mark the points with a little circle, and not dot, segment, or cross. 2. Ask your students what is a segment bisector. Ask the to draw it. Make a discussion depending on the solutions. It is possible they have drawn just one and possibly a perpendicular one. This is a good place for a discussion on the difference between a segment bisector and a perpendicular bisector. Student should be able to deduce that perpendicular bisector of a segment is unique for that segment, whilst there is infinitely many bisectors of a given segment. The following problem is going to be constructed using first patty paper and then the geometry tools. Patty paper is going to be used for students to come up with the construction using the geometry tools. As the students use the patty paper develop a discussion on this activity. Discuss with them how will they use the patty paper? What is the line they get when folding the paper? What is the property of any point on that line in relation to the endpoints of the segment. If they were able to answer all of these questions they should be able to do the construction using the geometry tools. However challenge the students with the questions of explaining why the constructions is correct as well as whether the openness of the compass is important, can we change both or just one leg and still get a perpendicular bisector. The purpose is as for all activities regarding constructions for students not to take them for granted. 2. a. construct a perpendicular bisector using a patty paper 2. b. construct a perpendicular bisector using geometry tools 3. Since the students learnt the concept of a segment bisector it is advisable to use these concepts in more challenging problems that will help us introduce some geometrical concepts. For instance we could give the student the following problems: 3. a. Given triangle ABC construct three medians. What do you notice about all three medians? 3. b. Given triangle ABC construct the perpendicular bisectors of each side. What do you notice about all three bisectors? These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three medians or perpendicular bisectors coincide. Introduce terms centroid and circumvent. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts. 1. Start the discussion by discussing how the perpendicular bisector to a given line could be constructed. Ask them how they can use patty paper to do that if getting there by connecting previous lesson is not going that well. Make sure they are aware that they can construct the perpendicular line to a given line from a point outside and both on the given line. Discuss with them how are these constructions the same or different. They should mathematically argument their construction decisions using geometric concepts and properties. 1. a. construct a line perpendicular to a given live through a given point outside the line 1. b. construct a line perpendicular to a given live through a given point on the line 2. The next activity connects the concepts of constructing a perpendicular and the shortest distance. Also, it closely connects construction of the angle bisector. Let the students draw an angle ABC. Let them place point P inside the angle, and construct perpendiculars from point P to both legs of the angle. Discuss with them which side is closer to point P? Make sure they justify any statement they make. Ask them where point P must be so that the distances to the legs are the same. If they started with an obtuse triangle would the conclusions be the same? If yes, why if not, why not? Discussion should lead by discovering that P should lie on the bisector of the angle ABC. For this activity one can either use technology and give student time to discover any patterns or by mare drawing and trying to deduce backwards the actual angle bisector construction. After justifying that point P lying on the angle bisector, ask them to construct angle bisector of ABC. 3. As mentioned before the previous activity is closely connected to constructing an angle bisector. If they were not able to construct it after the first activity, give them a patty paper and let them discuss how they would by using a patty paper get an angle bisector. They should be able to explain why by folding the patty paper in such way that the legs of the angle overlap (coincide) by connecting vertex B to the crease they got the angle bisector. Ask them how does this help them construct the angle bisector? Discuss with them whether the openness of the compass influences the construction. Does the openness has to have the same radius as the radius of the arc? What if the compass collapse. It is important for all of these questions to be discussed in details for the students to develop understanding of the nature of the constructions and the construction of the angle bisector in particular. 4. a. Given triangle ABC construct the angle bisectors of each angle. What do you notice about all three angle bisectors? 4. b. Given triangle ABC construct the altitudes of each angle. What do you notice about all three altitudes? These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three angle bisectors and lines containing the altitudes coincide. Introduce terms incenter and orthocenter. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts. Especially try to develop understanding for why the orhocenter is defined as an intersection of lines containing the altitudes and not just intersection of the three altitudes. a. Trace this bisector as you drag point C along side AB. Describe the shape formed by this locus of lines. b. Trace the midpoint od CF as you drag C. Describe the locus of points. 1. Give your students a patty paper and let them derive to strategy to construct a line parallel to given line through a given point. They should be able to deduce that they should fold the paper to construct a perpendicular so that the crease runs through the point and then make another fold that is perpendicular though the first crease thought the given point and match the pairs of the corresponding angles created by folding. Discuss whether angles obtained are congruent. If so why? What conclusions can be made about the lines? Based on this activity they should be able to construct a line parallel to given line though a given point. Also, discuss with them whether the second line is unique and why. 2. Challenge them to think of another way to do the same construction. Since they should know that when two parallel lines are intersected by a transversal that we get congruent corresponding angles. Therefore, they should be able to derive that by copying the angle between the given line and point at that point, they will get a line parallel to the given line. 3. At this point it can be beneficial to give them problems that will connect several constructions.Try to give problem that employ different strategies. For instance, pose them the problem of constructing the angle bisector of two line segment in finite plane. Hence, two line segment that do not intersects on the paper, and by not extending those lines. This problem can be solved in different ways. One of the ways would be by constructing lines parallel to given line segments closer for them to intersect and then by doing the known angle bisector construction. Second, they could employ concepts about incenter and incicrcle. Discuss with them the strength and weakness of each method they derive. Day 5 & 6-Construction problems After knowing the basic constructions, we can construct more complex geometric figures.Here we will list several problems that could be a two day plan. Students could work individually, or in pairs depending on classroom environment. However, the teacher should be here only as a manager of the classroom, and let students struggle on their own. Before jumping and solving the construction problems, discuss the properties of triangles and quadrilaterals. Also, after each problem make a classroom discussion where students justify their construction processes. Be sure to discuss whether the solution in each of the following constructions is unique or not, and why. Construct the triangle ABC using all three segments. Construct the triangle ABC using two segments and one angle. How many solution there are? Construct the triangle ABC using one segment and two angles. Construct an isosceles triangle with perimeter o and length of the base equal to a. Construct a square given the length of the diagonal. Construct a rhombus given lengths of both diagonals. Construct a rhombus given the altitude and one diagonal. Construct a parallelogram given one side, one angle and a diagonal. Construct an isosceles trapezoid given the bases and ne angle. Construct an isosceles trapezoid, given the median, altitude, and base angle of 30º. There are numerous more challenging problems. The above one are just a proposal but not necessarily the only one appropriate for middle and high school students. For instance, since we explored to some extent the center of gravity for a triangle, a natural question can arise on the center of gravity for a quadrilateral. How do we construct it etc. It is on the teacher to decide how deep they want to go and pursue certain ideas and concepts. Listed below are some web pages with interesting construction problems: http://jwilson.coe.uga.edu/EMT668/Asmt6/EMT668.Assign6.html (Problem #3) Day 7 & 8-Introduction and construction of basic algebraic expressions This topic naturally follows after teaching basics of geometry constructions connecting it to algebraic concepts. However, since high school textbooks rarely include algebraic method for most of the problems explanations will be given in .gsp files. Let students discuss the strategies employed and reasoning in each of the following problems. 1. The best way to introduce this method is by simply posing a problem where knowledge of geometric methods do not help us, but instead have to depend on different method.For the discussion we can choose among variety of problems. Here, as a motivation task will suffice the following problem: Construct the triangle ABC if the relation among its sides a, b and c is given by Give the students time to explore this problem. They should realize that trying to employ geometric methods learnt so far cannot help them is solving this problem. However, discuss with them the expression given that relates its sides. They should be able to realize that on the right side of the expression they have geometric mean of certain sides, G (a, c) and G(b, c). Connecting the geometric mean with the arithmetic mean and knowing that for any non-negative x and y, G(x, y)<=A(x, y), where we have equality of and only if x=y, they should conclude that all of the sides must be equal to satisfy the given expression. From this point the problem is trivial since in the previous lesson they learnt how to do basic constructions . Thus, through this problem students should become aware that algebraic method can help us with construction problems when geometric ones fails. This problem also gives a great picture what does the algebraic method includes. Students should on their own understand how and why does this method work, and not be given to them per se. 2. Before looking at complex algebraic expressions and their constructions,we will observe constructions of basic expressions. Based on these application of the algebraic method is grounded. In the following list of basic constructions a, b and c are lengths of given segments, m and n are natural numbers, whilst x is the segment we want to construct. Constructions for problem 4-11 with explanations can be found here. All of the above problems should develop connections between numbers and their geometric representation, or in other words geometric representation of algebraic expressions. Discuss each of the problems with the students. Day 9-Applications of algebraic method in construction problems Through the following problems, students will gain insight into efficacy of the algebraic method in construction problems, its positive and negative sides. The problems vary in difficulty and each of the problem explanation are given in .gsp file here. It is on the teacher to decide how to organize these activities. Thus, individual work, work in pairs or groups. However, make sure your students understand the problem, and construction itself and discuss as a class each of the problems given. 1. Start off the today's discussion on regular polygons and their contractibility. Ask them if and how they can construct n-gon when n=3, 6, 12, 4, 8, etc. They should say yes. However ask them if any n-gon is constructible. Probably you will here yes, no or nothing. At this point you can talk about Gauss, child prodigy who as the age of 15 proved that a regular 17-gon can be constructed. He was so fascinated with regular 17-gons that he wished for it to be engraved on his grave. However, his wish never came true, but in it was on a statue of him in his hometown Braunschweig. Not all regular polygons are constructible like 7-, 9-, 11-gon, etc. However, they are with so called neusis constructions that allow using a marked ruler. Let your students explore neusis constructions and give them a group project that involves approximation construction of regular 7-, 9, and 11-gon. 2. Summarize the results of the lesson. At this point discuss with the students the difference between the conventional construction, and Euclidean consecution. Also discuss the algebraic method and its relevance to constructions. Teacher should reflect on their teaching asking question such as: Did I challenge the students? How? Were students actively engaged in the learning process? How do I know? Did the students exceed my expectations in some areas, and not meet them in others? Did I have to adjust teaching? If so, what adjustments did I make, and were they effective? Serra, M. (2003). Discovering geometry: An investigative approach. Emeryville, CA: Key Curriculum Press.

Table of Contents Magnetic Flux Density Definition: Magnetic flux density or magnetic induction or magnetic field vector B at a point in a magnetic field is generally defined as the number of magnetic lines of induction passing through a unit area around that point placed normal to the lines of induction. The tangent to the line of induction at a point gives the direction of magnetic induction B at that point. Q. What is Magnetic Flux? (a) Definition: Magnetic flux (0) through a surface is defined as the total number of magnetic lines of induction passing through the surface B.A = BA cos ∅. (b) Unit: Unit of ∅ is weher in S.I. and maxwell in C.G.S. system. I weber = 10⁸ maxwell. (C) Dimensional Formula: Dimension of o is [ML² A –¹T–²]. Q. What is Faraday’s law of Electromagnetic Induction? I) First law: Whenever the magnetic flux linked with a circuit changes, an induced electromotive force is set up in it. The induced e.m.f. lasts so long as the change in magnetic flux persists. (ii) Second law: The magnitude of the induced e.m.f. is directly proportional to the time rate of change of the magnetic flux do linked with the circuit ie. |M| ≤ d∅/dt Combining E = -(∅²-∅¹)/t= d∅/dt Q. What is Lenz’s law? It states that the direction of the induced e.m.f. is always such that it opposes the very cause producing it. It obey’s law of conservation of energy. It is also known as 3rd law of electromagnetic induction. Q. What is Production of induced e.m.f.: Induced electromotive force can be produced? (a) by changing B. (b) by changing area A, E = B/V (c) by changing the relative orientation of the coil and the field E =nBAw sin Wt = E⁰ sin wt (E = nBAw) Q. What is Fleming’s right hand rule? Stretch the thumb, the forefinger and the central finger of your right hand mutually perpendicular to one another. If the forefinger points towards the direction of the magnetic field and the thumb points towards the direction of motion of the conductor then the central finger will always points towards the direction of induced e.m.f. or induced current. Q. What is Eddy Currents? These are closed loops or whirls of induced current set up in a metal body circulating in a plane perpendicular to the magnetic lines of induction. It is also known as Foucault current. Q. What Is Reduction of Eddy currents? Eddy currents can be reduced by taking laminated iron core. Q. What Is Eddy currents are used in? (a) induction furnace (b) dead beat galvanometer (c) a.c. induction motor (d) electrical brake etc. Q. Is Eddy currents harmful? Eddy currents are harmful as it increases energy loss and increase wear and tear. Eddy currents are analogous to friction in mechanics. Q. Self induction: It is the property of an electric? circuit e.g a coil by virtue of which it opposes the change (i.e. growth or decay) in strength of the current flowing through the circuit by inducinga current upon itself OR E= -L.(dk/dt). Self inductance or coefficient of self induction: (a) Definition : The self inductance of a circuit is numerically equal to the induced electromotive force in the circuit when rate of change of current in the circuit is unity. (b) Unit Henry : It is the unit of self inductance. The self inductance of a coil 1s said to be 1 henry when a current changing at the rate of 1 ampere per second induces an induced e.m.f. ofl volt in it. (c) Expression: Self inductance of a long solenoid is given by L =H All where n = number of turns/length l= total length of solenoid A = cross section area The analog of self inductance in mechanics is mass. What Is Mutual Induction? It is the phenomenon of production of an induced e.m.f. (or current) in a circuit by changing the current flowing in a neighbouring circuit ∅² = MI¹ or E² =- M(dl/dt) Mutual Inductance or coefficient of mutual induction (a) Definition: The mutual inductance of a pair of coils is numerically equal to the induced e.m.f. in the secondary coil by a unit rate of change of current in the primary coil. (b) Unit: Unit of mutual inductance is henry (H). The mutual inductance of a given pair of coils is 1 henry when a current changing at the rate of one ampere per second in the primary coil produced an e.m.f. of 1 volt in the secondary coil. (c) Expression: The mutual inductance for the given pair of concentric solenoids, M= μ⁰n¹A/l where n¹ and n² are the number of turns/length in two coils. What Is Transformer A transformer is an clectrical machine for converting a large A.C. current at low voltage into a small A.C. current at high voltage or vice-versa. It works on the principle of mutual induction. Vs/Vp= Es/Ep =ns/np= K k is known as transformation ratio. For step up transformers value of transformation ratio k is more than one but for step down transformers k has a value less than one. In transformers power output in secondary power input in primary Es Is= EpIp Ip _Es ns ×K Is Ep np ×K Efficiency of a transformer ñ=(Power output/Power input)100% In ideal transformation n is 100% but practically it is between 95%-98%. Main sources of energy loss in transformer are (A) Eddy current losses (B) Hysteresis losses (C) Magnetic losses (leakage of flux) (D) Copper losses (E) Sound losses Uses of Transformer (a) It is used in telephone, telegraph, television, radio receiving sets, electrical welding machine, electric furnace and in wireless transmitting and receiving sets. (b) Commonly it is used in transmitting a.c. power from the power generating station to distant places. Step up transformers Step up transformers are used for transmission of electrical energy over long distances with less power loss. Electric Generator or Dynamo It is a device which converts mechanical energy into electrical energy. It works on the principle of electromagnetic self-induction. What is Alternating Current It is that current which continuously changes in magnitude and periodically reverses its direction. E = E⁰ sin or and I = I⁰ sin wt. What is Virtual Ampere One virtual ampere is that alternating current which produces the same heat as is produced by a steady current of one ampere through the same resistance for the same time. The A.C. ammeter records the effective value of alternating measured in terms of virtual ampere. What is Virtual Volt? One virtual volt is that alternating potential difference which when applied across a certain resistance produces the same heat as done by steady potential difference of l volt across the same resistance for the same time. The A.C. voltmeter records the effective value of alternating voltage measured in terms of virtual volt. Phase relation between Voltage and Current in A.C. circuits: (a)s When the circuit contains resistance only: For a pure resistance A.C. circuit the current is in phase with voltage and Ohm’s law is obeyed by the circuit. (b) A.C. circuits containing induction only: In case of a pure inductive circuit the current lags behind the voltage by π/2 or alternatively the voltage leads the current by π/2 (c) A.C. circuit containing capacitance only: In case of a pure capacitive circuit the current leads the voltage by π/2 or alternatively the voltage lags behind the current by π/2 Power in A.C. circuit Rate of doing electrical work, is called power in A.C. circuit and is equal VI . Average (true) power = Veff COS VerIen is known as apparent power and cos ∅ known as power factor. Power factor, cos ∅ = Apparent power/True power the A.C. circuit is purely resistive then ∅ = 0 and True power= Apparent power If the A.C. circuit is purely inductive or purely capacitive then ∅= 90° and average power = 0. Current in such a circuit is commonly called “Wattless current’ because it does not contribute anything towards power (watts) consumed. What Is Circuit Elements Means? (a) Resistor: It opposes the flow of current. It offers equal resistance to d.c. as well as upon a.c. Its symbol is Its S.I. unit is ohm. (b) Inductor : It is a few turns of conducting wires wound over a metallic or non-metallic frame. It opposes to a.c. This opposition to a.c. is known as inductive reactance. Its S.I. unit is ohm. Symbol of inductor is (c) Capacitor: It consists of two metal plates,one +vely charged and the other -vely charged. It opposes to a.c. This opposition to a.c. is known as capacitive reactance. Its S.I. unit is ohm. Symbol of capacitor What Is Form factor Form factor is ratio of r.m.s., value to average value of current or voltage. In case of a.c., form factor = Irms/Ia= I⁰√2/ 2I⁰π = π / 2√2 = 1.11

What Is Internal Rate of Return – IRR? The internal rate of return (IRR) is a metric used in capital budgeting to estimate the profitability of potential investments. The internal rate of return is a discount rate that makes the net present value (NPV) of all cash flows from a particular project equal to zero. IRR calculations rely on the same formula as NPV does. Formula and Calculation for IRR It is important for a business to look at the IRR as the plan for future growth and expansion. The formula and calculation used to determine this figure follows. 0=NPV=t=1∑T(1+IRR)tCt−C0where:Ct=Net cash inflow during the period tC0=Total initial investment costsIRR=The internal rate of returnt=The number of time periods To calculate IRR using the formula, one would set NPV equal to zero and solve for the discount rate (r), which is the IRR. Because of the nature of the formula, however, IRR cannot be calculated analytically and must instead be calculated either through trial-and-error or using software programmed to calculate IRR. Generally speaking, the higher a project's internal rate of return, the more desirable it is to undertake. IRR is uniform for investments of varying types and, as such, IRR can be used to rank multiple prospective projects on a relatively even basis. Assuming the costs of investment are equal among the various projects, the project with the highest IRR would probably be considered the best and be undertaken first. IRR is sometimes referred to as "economic rate of return" or "discounted cash flow rate of return." The use of "internal" refers to the omission of external factors, such as the cost of capital or inflation, from the calculation. How to Calculate IRR in Excel How to Calculate IRR in Excel There are two main ways to calculate IRR in Excel: - Using one of the three built-in IRR Excel formulas - Breaking out the component cash flows and calculating each step individually, then using those calculations as inputs to an IRR formula (As we detailed above, since the IRR is a derivation, there is no easy way to break it out by hand.) The second method is preferable because financial modeling works best when it is transparent, detailed and easy to audit. The trouble with piling all the calculations into a formula is that you can't easily see what numbers go where, or what numbers are user inputs or hard-coded. Here is a simple example of an IRR analysis with cash flows that are known and consistent (one year apart). Assume a company is assessing the profitability of Project X. Project X requires $250,000 in funding and is expected to generate $100,000 in after-tax cash flows the first year and grow by $50,000 for each of the next four years. You can break out a schedule as follows (click on the image to expand): The initial investment is always negative because it represents an outflow. You are spending something now and anticipating a return later. Each subsequent cash flow could be positive or negative – it depends on the estimates of what the project delivers in the future. In this case, the IRR is 56.77%. Given the assumption of a weighted average cost of capital (WACC) of 10%, the project adds value. Keep in mind that the IRR is not the actual dollar value of the project, which is why we broke out the NPV calculation separately. Also, recall that the IRR assumes we can constantly reinvest and receive a return of 56.77%, which is unlikely. For this reason, we assumed incremental returns at the risk-free rate of 2%, giving us a MIRR of 33%. What Does IRR Tell You? You can think of the internal rate of return as the rate of growth a project is expected to generate. While the actual rate of return that a given project ends up generating will often differ from its estimated IRR, a project with a substantially higher IRR value than other available options would still provide a much better chance of strong growth. One popular use of IRR is comparing the profitability of establishing new operations with that of expanding existing ones. For example, an energy company may use IRR in deciding whether to open a new power plant or to renovate and expand a previously existing one. While both projects are likely to add value to the company, it is likely that one will be the more logical decision as prescribed by IRR. IRR is also useful for corporations in evaluating stock buyback programs. Clearly, if a company allocates a substantial amount to a stock buyback, the analysis must show that the company's own stock is a better investment (has a higher IRR) than any other use of the funds for other capital projects, or higher than any acquisition candidate at current market prices. - IRR is the rate of growth a project is expected to generate. - IRR is calculated by the condition that the discount rate is set such that the NPV = 0 for a project. - IRR is used in capital budgeting to decide which projects or investments to undertake and which to forgo. Example IRR Use In theory, any project with an IRR greater than its cost of capital is a profitable one, and thus it is in a company’s interest to undertake such projects. In planning investment projects, firms will often establish a required rate of return (RRR) to determine the minimum acceptable return percentage that the investment in question must earn in order to be worthwhile. Any project with an IRR that exceeds the RRR will likely be deemed a profitable one, although companies will not necessarily pursue a project on this basis alone. Rather, they will likely pursue projects with the highest difference between IRR and RRR, as these likely will be the most profitable. IRR can also be compared against prevailing rates of return in the securities market. If a firm can't find any projects with IRR greater than the returns that can be generated in the financial markets, it may simply choose to invest its retained earnings into the market. Although IRR is an appealing metric to many, it should always be used in conjunction with NPV for a clearer picture of the value represented by a potential project a firm may undertake. IRR in practice is calculated by trial and error since there is no analytical way to compute when NPV will equal zero. Computers or software like Excel can do this trial and error procedure extremely quickly. But, as an example, let's assume that you want to open a pizzeria. You estimate all the costs and earnings for the next two years, and then calculate the net present value for the business at various discount rates. At 6%, you get an NPV of $2000. But, the NPV needs to be zero, so you try a higher discount rate, say 8% interest: At 8%, your NPV calculation gives you a net loss of −$1600. Now it's negative. So you try a discount rate in between the two, say with 7% interest: At 7%, you get an NPV of $15. That is close enough to zero so you can estimate that your IRR is just slightly higher than 7%. Internal vs. Modified Rate of Return Even though the internal rate of return metric is popular among business managers, it tends to overstate the profitability of a project and can lead to capital budgeting mistakes based on an overly optimistic estimate. The modified internal rate of return compensates for this flaw and gives managers more control over the assumed reinvestment rate from future cash flow. An IRR calculation acts like an inverted compounding growth rate; it has to discount the growth from the initial investment in addition to reinvested cash flows. However, the IRR does not paint a realistic picture of how cash flows are actually pumped back into future projects. Cash flows are often reinvested at the cost of capital, not the same rate at which they were generated in the first place. IRR assumes that the growth rate remains constant from project to project. It is very easy to overstate potential future value with basic IRR figures. Another major issue with IRR occurs when a project has different periods of positive and negative cash flows. In these cases, the IRR produces more than one number, causing uncertainty and confusion. MIRR solves this issue as well. IRR vs. Compound Annual Growth Rate The compound annual growth rate (CAGR) measures the return on an investment over a certain period of time. The IRR is also a rate of return but is more flexible than the CAGR. While CAGR simply uses the beginning and ending value, IRR considers multiple cash flows and periods – reflecting the fact that cash inflows and outflows often constantly occur when it comes to investments. IRR can also be used in corporate finance when a project requires cash outflows upfront but then results in cash inflows as investments pay off. The most important distinction is that CAGR is straightforward enough that it can be calculated by hand. In contrast, more complicated investments and projects, or those that have many different cash inflows and outflows, are best evaluated using IRR. To back into the IRR rate, a financial calculator, Excel, or portfolio accounting system is ideal. IRR vs. Return on Investment Companies and analysts also look at the return on investment (ROI) when making capital budgeting decisions. ROI tells an investor about the total growth, start to finish, of the investment. IRR tells the investor what the annual growth rate is. The two numbers should normally be the same over the course of one year (with some exceptions), but they won't be the same for longer periods of time. Return on investment – sometimes called the rate of return (ROR) – is the percentage increase or decrease of an investment over a set period of time. It is calculated by taking the difference between the current (or expected) value and original value, divided by original value and multiplied by 100. While ROI figures can be calculated for nearly any activity into which an investment has been made and an outcome can be measured, the outcome of an ROI calculation will vary depending on which figures are included as earnings and costs. The longer an investment horizon, the more challenging it may be to accurately project or determine earnings and costs and other factors such as the rate of inflation or the tax rate. It can also be difficult to make accurate estimates when measuring the monetary value of the results and costs for project-based programs or processes (for example, calculating the ROI for a human resources department within an organization) or other activities that may be difficult to quantify in the near-term and especially so in the long-term as the activity or program evolves and factors change. Because of these challenges, ROI may be less meaningful for long-term investments. This is why IRR is often preferred. Limitations of the IRR While IRR is a very popular metric in estimating a project’s profitability, it can be misleading if used alone. Depending on the initial investment costs, a project may have a low IRR but a high NPV, meaning that while the pace at which the company sees returns on that project may be slow, the project may also be adding a great deal of overall value to the company. A similar issue arises when using IRR to compare projects of different lengths. For example, a project of short duration may have a high IRR, making it appear to be an excellent investment, but may also have a low NPV. Conversely, a longer project may have a low IRR, earning returns slowly and steadily, but may add a large amount of value to the company over time. Another issue with IRR is one not strictly inherent to the metric itself, but rather to common misuse of IRR. People may assume that, when positive cash flows are generated during the course of a project (not at the end), the money will be reinvested at the project’s rate of return. This can rarely be the case. Rather, when positive cash flows are reinvested, it will be at a rate that more resembles the cost of capital. Miscalculating using IRR in this way may lead to the belief that a project is more profitable than it actually is. This, along with the fact that long projects with fluctuating cash flows may have multiple distinct IRR values, has prompted the use of another metric called modified internal rate of return (MIRR). MIRR adjusts the IRR to correct these issues, incorporating the cost of capital as the rate at which cash flows are reinvested, and existing as a single value. Because of MIRR’s correction of the former issue of IRR, a project’s MIRR will often be significantly lower than the same project’s IRR. Investing Based on IRR The internal rate of return rule is a guideline for evaluating whether to proceed with a project or investment. The IRR rule states that if the internal rate of return on a project or investment is greater than the minimum required rate of return, typically the cost of capital, then the project or investment should be pursued. Conversely, if the IRR on a project or investment is lower than the cost of capital, then the best course of action may be to reject it. While there are some issues with IRR, it can be a good basis for investments as long as the problems referenced earlier in the article are avoided. If you're interested in putting IRR into practice, you'll need to create a brokerage account to actually purchase the investments you're investigating.

I’m much enjoying at the moment re-reading Hartley Rodgers’s Theory of Recursive Functions and Effective Computability. What prompts me to take the book off the shelf again is the treatment of constructive ordinals some two hundred pages in; but (one of the upsides of retirement) I’ve got the time to start reading from the beginning, and it is well worth spending the time doing so. The book is as good and illuminating as I remembered it as being. In fact more so, as I’m sure I didn’t really appreciate it, back in the day. I bought my copy at the end of 1970, and paid seven pounds and nine shillings for it (the bookseller’s pencilled markings are still on the flyleaf). That was a lot more than we could afford, and I expect I didn’t fess up to my extravagance, for it would then have been about 8% of my monthly take-home pay. Such was my devotion to logic. Or my obsessive book-buying habit. The book-buying has had to be much reduced, as we are pretty much constrained to a one-in, one-out policy (not of course, that it quite works like that). But I did get in the post today a copy of Rózsa Péter’s great Recursive Functions — the copy was relatively inexpensive and though it once belonged to the library at the National Physical Laboratory was seemingly hardly touched. I’m not sure quite why, but I take real pleasure in having a copy at last. Question (since the gender gap is vexing the philosophical interwebs these days): is Rózsa Péter the only woman so far who is the sole author of an indisputably significant mathematical logic book? Or am I having a senior moment and forgetting someone? (Even more run of the mill math. logic textbooks solely by women seem very few and far between: there’s Judith Roitman’s nice set theory text, and then ….?) 21 thoughts on “Recursive pleasures” Ann Yasuhara’s “Recursive Function Theory and Logic”, Academic Press. This is a bit off topic but doesn’t Peter Smith’s remarks about the toy language on p32 of IGT (2nd ed) run afoul of Post-completeness? Maybe not, since the language is finite. But I doubt it. Just asking. And as for semantic arguments for incompleteness, doesn’t just about any semantic paradox give rise to a semantic incompleteness argument? If formal semantics (in Linguistics) counts, there’s Barbara Partee. Among recent books, there’s Formal Languages in Logic: A Philosophical and Cognitive Analysis, by Dr Catarina Dutilh Novaes, and Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals, by Anna Horská. I had initially interpreted the question as referring merely to textbooks. In the context of mathematical logic more generally, I’d say Carol Karp’s Languages with Expressions of Infinite Length is an indisputably significant book. Does Ruth Barcan Marcus’ Modalities count? Not a textbook, a collection of her essays. Her subtitle “Philosophical Essays” seems accurate — not a math logic book, for all its interest. Larisa Maksimova has a book (joint with Dov Gabbay) on intepolation. Maria Manzano on extensions of first-order logic. Yes, I know and like the Manzano, which is a useful text. I need to mention bits of it in an update to the Teach Yourself Logic guide. Elke Brendel co-authored the two-volume textbook ‘Grundzüge der Logik’ (1983), which has not been translated into English so far. There is also Katalin Bimbo’s Combinatory Logic, which is quite good. This is another co-author, but Marian Pour-El wrote *Computability in Physics and Analysis* with Jonathan Richards. Oh yea, and mentioned in your guide, there’s also Maria Manzano’s Model Theory! Oh, but of course, of course! Senior moment there!! I don’t know if it counts, but Sabine Koppelberg wrote the first volume of the Monk and Bonnet’s Handbook of Boolean Algebras, which, while not exactly a textbook, is nevertheless meant to be an introduction to the subject. Speaking of constructive ordinals. I read somewhere you were writing a book on Gentzen’s consistency proofs a little like your intro to Gödel book. May I ask if it is still a live possibility ? Or is the project it completely dead ? Well, I still have plans, but things are going terribly slowly … Helena Rasiowa wrote an important book as co-author but I also saw another on non-classical logics on the shelves of our library. Perhaps it is not of undisputable importance. Wanda Szmielew wrote a book on the logical foundations of projective geometry. Yes, I was thinking of Helena Rasiowa as a female co-author; and then there there’s Zofia Adamowicz. And more recently Sara Negri has co-authored two must-read books on proof-theory with Jan Von Plato. Helena Rasiowa co-authored “The Mathematics of Metamathematics” (1963) with Roman Sikorski; but she published “An Algebraic Approach to Non-Classical Logics” (1974) which is perhaps an under appreciated book. Thanks, I’ll have to look out the second which I don’t know — we seem to have four copies in the libraries here! Rasiowa also authored Introduction to modern mathematics, which is still widely used as a textbook for the first year math courses in Poland (at least 14 reprints since 1967, including one in 2013).

Research Article | Open Access H. Saberi Najafi, A. Refahi Sheikhani, A. Ansari, "Stability Analysis of Distributed Order Fractional Differential Equations", Abstract and Applied Analysis, vol. 2011, Article ID 175323, 12 pages, 2011. https://doi.org/10.1155/2011/175323 Stability Analysis of Distributed Order Fractional Differential Equations We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure. The fractional differential operator of distributed order is a generalization of the single order which by considering a continuous or discrete distribution of fractional derivative is obtained. The idea of fractional derivative of distributed order is stated by Caputo and later developed by Caputo himself [2, 3], Bagley and Torvik [4, 5]. Other researchers used this idea, and interesting reviews appeared to describe the related mathematical models of partial fractional differential equation of distributed order. For example, Diethelm and Ford used a numerical technique along with its error analysis to solve the distributed order differential equation and analyze the physical phenomena and engineering problems, see and references therein. In particular cases, the characteristics of time-fractional diffusion equation of distributed order were studied for treatises in the sub-, normal, and superdiffusions. The fractional order applied to dynamical systems is of great importance in applied sciences and engineering [13–19]. The stability results of the fractional order differential equations (FODEs) systems have been a main goal in researches. For example, Matignon considers the stability of FODE system in control processing and Deng has studied the stability of FODE system with multiple time delays [20–23]. Now, in this paper, we consider the distributed order fractional differential equations systems (DOFDEs) with respect to the density function as follows: where , , and is the Caputo fractional derivative operator of distributed order with respect to the order-density function . Since the solution of the above system is rather complicated similar to FODE systems, therefore, the study of stability for DOFDE is a main task. In this paper, we introduce three classes of DOFDE systems including (1)distributed order fractional differential systems;(2)distributed order fractional differential evolution systems with control vector; (3)distributed order fractional differential evolution systems without control vector. For studying the stability of these classes of DOFDE systems, first, we introduce a characteristic function of a matrix with respect to the distribute function where . Then, we establish a general theory based on new inertia concept for analyzing the stability of distributed order fractional differential equations. The concepts and theorems presented in this paper for DOFDE systems can be considered as generalizations of FODE and ODE systems [21, 24, 25]. In Section 2, we recall some basic definitions of the Caputo fractional derivative operator, the Mittag-Leffler function, and their elementary properties used in this paper. Section 3 contains the main definitions and theorems for checking the stability of DOFDE systems. Also, we study a distributed order fractional WINDMI system generalized from fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems where is control vector, and generalize the results obtained in Section 3 for this case. Finally, the conclusions are given in the last section. 2. Elementary Definitions and Theorems In this section, we consider the main definitions and properties of fractional derivative operators of single and distribute order and the Mittag-Leffler function. Also, we recall two important theorems in inverse of the Laplace transform. 2.1. Fractional Derivative of Single and Distributed Order The fractional derivative of single order of in the Caputo sense is defined as [16, 27] for . The Caputo's definition has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes. Fortunately, the Laplace transform of the Caputo fractional derivative satisfies where and is the Laplace variable. Now, we generalize the above definition in the fractional derivative of distributed order in the Caputo sense with respect to order-density function as follows: and the Laplace transform of the Caputo fractional derivative of distributed order satisfies where 2.2. Mittag-Leffler Function The one-parameter Mittag-Leffler function and the two-parameter Mittag-Leffler function , which are relevant for their connection with fractional calculus, are defined as One of the applicable relations in this paper is the Laplace transforms of the Mittag-leffler function given by 2.3. Main Theorems about Inverse of the Laplace Transform Theorem 2.1 (Schouten-Vanderpol Theorem ). Suppose that the functions are analytic in the half plane , then, the Laplace transform inversion of can be obtained as where is the Laplace transform inversion of the function . Theorem 2.2 (Titchmarsh Theorem ). Let be an analytic function which has a branch cut on the real negative semiaxis; furthermore, has the following properties: for any sector where . Then, the Laplace transform inversion can be written as the the Laplace transform of the imaginary part of the function as follows: Theorem 2.3 (Final Value Theorem ). Let be the Laplace transform of the function . If all poles of are in the open left-half plane, then, 3. Stability Analysis of Distributed Order Fractional Systems In this section, we generalize the main stability properties for the linear system of distributed order fractional differential equations in the following form: where , the matrix , and is the Caputo fractional derivative operator of distributed order with respect to order-density function . At first, we obtain the general solution of the system (3.1), and, next, we express the main theorem for checking the stability of this system. By implementation of the Laplace transform on the above system and using the initial condition and relation (2.4), we have Now, by applying the inverse of Laplace transform on the both sides of above relation, we have which according to the Schouten-Vanderpol and Titchmarsh theorems we get where , , , and . Theorem 3.1. The distributed order fractional system of (3.1) is asymptotically stable if and only if all roots of have negative real parts. Proof. According to the relation (3.2), we have if all roots of the lie in open left half complex plane (i.e., ), then, we consider (3.7) in . In this restricted area, the relation (3.7) has a unique solution . Since , so we have which from the final value Theorem 2.3, we get The above result shows that the system (3.1) is asymptotically stable. Definition 3.2. The value of is the characteristic function of the matrix with respect to the distributed function , where is the distributed function with respect to the density function . Definition 3.3. The eigenvalues of with respect to the distributed function are the roots of the characteristic function of .The inertia of a matrix is the triplet of the numbers of eigenvalues of with positive, negative, and zero real parts. In this section, we generalize the inertia concept for analyzing the stability of linear distributed order fractional systems. According to the Theorem (3.1), the transient responses of the system (3.1) are governed by the region where the roots of are located in the complex plane. Definition 3.4. The inertia of a matrix of order respect to the order distributed function is the triplet where , , and are, respectively, the number of roots of with positive, negative, and zero real parts where . Definition 3.5. The matrix is called a stable matrix with respect to the order distributed function , if all of the eigenvalue of A with respect to the distributed function have negative real parts. Theorem 3.6. The linear distributed order fractional system (3.1) is asymptotically stable if and only if any of the following equivalent conditions holds. (1)The matrix is stable with respect to the distribute function . (2). (3)All roots of the characteristic function of with respect to the distributed function satisfy . Proof. According to Theorem 3.1 and the above definitions, proof can be easily obtained. Remark 3.7. In special case, if , where and is the Dirac delta function, then, we have the following linear system of fractional differential equations: and . Also, the characteristic matrix and characteristic equation of (3.11) are reduced to and , respectively. Let be , then , and, by using Theorem 3.6, we have . Thus, all the roots of equation satisfy . This result is Theorem 2 of . Here, we can very easily prove it by using Theorem 3.6 of the present paper. Particularly, if , then, we have a linear system . In this case, and the characteristic function of (3.1) are . Also, the inertia of matrix is a triplet , where , , and are, respectively, the number of eigenvalues of with positive, negative, and zero real parts. This result is a special case of definition (3.4), which agrees with the typical definitions for typical differential equations. Example 3.8. The solar-wind-driven magnetosphere-ionosphere (WINDMI) system is a complex driven-damped dynamical system which exhibits a variety of dynamical states that include low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionosphere known broadly as substorms, and states of continuous strong unloading [30, 31]. If we consider the integer-order WINDMI model as follows: where , , and are variables and , are positive constants, the corresponding distributed order fractional WINDMI system (3.12) can be written in the form: where is the density function. As a generalization of nonlinear autonomous FODE into nonlinear autonomous DOFDE, the linearized form of the system (3.13) at the equilibrium point , that is, , can be written in the form where , , and , which is the Jacobian matrix at the equilibrium point , is given by Now, for analyzing the stability of the nonlinear autonomous DFODE, we compute in the case that the density function varies. The results are shown in Table 1 for some parameters and . 4. Distributed Order Fractional Evolution Systems In this section, as a generalization of the previous systems, we consider the systems of distributed order fractional differential evolution equations and state two theorems in stability of these systems. Theorem 4.1. Consider linear system of distributed order fractional differential evolution equations, where , , and . Also, and . The system (4.1) is stable if and only if all roots of characteristic function of matrix with respect to the distributed function have negative real parts. Proof. Taking the Laplace transform on both sides of (4.1) gives If all roots of characteristic function of matrix with respect to the distributed function have negative real parts,that is, , then, we consider (4.2) in . In this restricted area by using final-value theorem of Laplace transform, we have Theorem 4.2. Consider the linear system of distributed order fractional differential evolution equations with the same hypotheses described in Theorem 4.1 where and is a control vector.The linear distributed order fractional system (4.4) is stabilizable if and only if there exists a linear feedback , with , such that is stable with respect to the distributed function . Proof. The proof can be easily expressed similar to Theorem 4.1. Remark 4.3. If and where then (4.4) is reduced to the following linear system of fractional differential equations: By applying the Laplace transform on the above system and using the initial condition, we have where is the Laplace transform of , is the Laplace transform of , and . Thus, we can write as, Applying the inverse Laplace transform to (4.7) and using property (2.8), we get Therefore, (4.5) is asymptotically stable if all eigenvalues of with respect to the distributed function have negative real parts which is a special case of Theorem 4.2. 5. Conclusions and Future Works In this work, we introduced three classes of the distributed order fractional differential systems, the distributed order fractional differential evolution systems with control vector, and the distributed order fractional differential evolution systems without control vector. The analysis of the asymptotically stability for such systems based on Theorem 3.1 and several interesting stability criteria are derived according to Theorem 3.6. Moreover, a numerical example was given to verify the effectiveness of the proposed schemes. In view of the above result, for future works, our attention may be focused on generalizing the numerical methods for computing the eigenvalues of a matrix with respect to the distributed function. The proposed algorithms in [33–35] for computing the eigenvalues of a matrix may be effective in this case. - M. Caputo, Elasticitá e Dissipazione, Zanichelli, Bologna, Italy, 1969. - M. 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In 1746, Euler, a famous mathematician, published what I believe to be a little-known philosophy paper. It seems interesting, but it is difficult for me to follow as I lack adequate philosophy background. Euler reduces his argument to a simple syllogism. Can anyone explain and summarize his argument, and perhaps comment on his paper as a whole? 1st premise : No body can have a force contrary to inertia. Based on [Engl.transl.,page 2] analysis of "current" (still quite incomplete) understanding of matter and bodies and of knowledge of only a few of their properties. The first property that comes to mind is extension; all philosophers recognize it as a property of body, and Cartesians consider it to be the essence of bodies. [...] if it can be demonstrated that extension and thought stand in contradiction, and thus the two cannot exist in the same being at the same time, then it would beproven that whatever has extension is thereby incapable of possessing thought. [page 3] Another property of matter is impenetrability, which is so characteristic of bodies that many philosophers have not hesitated to make it, together with extension, the essence of body. Indeed, no thing that has extension but lacks impenetrability can be considered a body. I move now to the third property of all matter, as widely acknowledged as the two already mentioned, and which seems much more closely connected to the innermost nature of bodies. I understand that the force of inertia [vis inertiae] was discovered first by Kepler, but then explained by Newton, who derived from it the principles of all mechanics. [page 6] However, even though the force of inertia completely excludes all other forces, for the matter at hand I will not assume anything except that two forces diametrically opposed to each other are not able to exist in the same entity. Therefore, since each body is endowed with the force of conservation of state, a contrary force — namely, the force of continual change of state —, cannot be admitted to exist in any body. Comment: see the discussion [page 5] about "the force of attraction — with which bodies are endowed, in the opinion of English philosophers [the Newtonians] — can easily be disproven." Gravitation is not, according to Euler, an intrinsic property of matter but must be explained in some way: either mechanically (see the Cartesian vortex theory) or by intervention of some external "active power", like in Leibniz. Yet if we consider with even a moment’s notice the faculty of thought [facultatem cogitandi], we will at once realize that in no way could it exist without the force of change of state. [...] Since the faculty of thought is intimately connected with the force of changing state, and a force of this sort cannot be conceived to exist in any body without contradiction, it evidently follows that no body can be endowed with the faculty of thought. From which there is a further conclusion: since the thing in us that we perceive does the thinking is called the soul [anima], the soul is not only not material, but is in fact a substance completely different from body, because it is endowed with a force directly opposed to those forces which can exist in a body. Comment: we have here a cartesian approach: the autonomous capability of a living being to move itself is due not to the body [i.e. matter] alone but to soul. It is interesting to note that Euler is equating soul with the "faculty of thought"; what about animals' capability of self-motion ? According to Descartes, there is no mind or soul in animals. The argument above licenses the: 2nd premise : The faculty of thought is a force contrary to inertia. Now the conclusion of the syllogism easily follows: Conclusion : Therefore, no body can possess the faculty of thought, concluding the argument: that denies the faculty of thought to bodies and proves the immateriality of the soul. The "syllogism" runs as follows: 1st premise) ∀y ¬∃x [Body(x) & Force(y) & Contrary-to-Inertia(y) & Possess(x,y)] 2nd premise) Force(Thought) & Contrary-to-Inertia(Thought) We instantiate 1) with Thought for y having: 3) ¬ [Body(x) & Force(Thought) & Contrary-to-Inertia(Thought) & Possess(x,Thought)] that, by tautological implication, amounts to: 4) ¬ (Force(Thought) & Contrary-to-Inertia(Thought)) ∨ ¬ (Body(x) & Possess(x,Thought)). From 2) and 4), by modus tollens, we derive: 5) ¬ (Body(x) & Possess(x,Thought)) that amounts to: 6) Body(x) → ¬ Possess(x,Thought). Finally, we "generalize" to conclude with: ∀x [Body(x) → ¬ Possess(x,Thought)]. See also: Stephen Gaukroger, The Metaphysics of Impenetrability: Euler's Conception of force (1982). At least the English translation talks about bodies and matter as if they were identical. It leaves me wondering how to make sense of his explanations for a block ice, which first get heated until it melts, and then get heated further until it evaporates. Is it still impenetrable after it has evaporated? Or do Euler's arguments only apply to solid bodies? But if they only apply to solid bodies, then do they also apply to solid bodies which contain fluids (or freely moving electrons)? Or what about elaborate mechanical mechanisms with suitable hinges to allow certain internal movements? None of the criticisms voiced above relies on new knowledge not yet available to Euler. How should a conclusion be valid, if we don't even know what the conclusion is supposed to mean?

System capacity is the maximum amount of water that an irrigation system can deliver on a continuous basis. Different units are used to describe system capacity. These are acre-feet (ac-ft, that is, the amount of water it will take to cover one acre, one foot deep), acre-inch (ac-in), gallons per minute (gal/min or gpm), and cubic feet per second (ft3/sec or cfs). The required pumping capacity of an irrigation system depends on the area to be irrigated (ac), the depth of water to apply (in), and the length of time that the irrigation system is operated (hr). Length of operation time refers to pumping time, not clock time. Pumping time is only the time water is flowing. The amount of time per day that an irrigation system can operate depends on the type of system and the amount of maintenance it requires. A self-propelled unit may be able to run several days without stopping, whereas manual-move, tractor towed, and self-moved systems must be shut down at regular intervals. For systems other than center pivots and lateral moves, only a portion of a field is irrigated at one time and time is required to move the system from one portion of the field "set" to another "set." The term irrigation period is used to designate the number of days that a system can apply the water for one irrigation to a given area. Note that it is necessary for the irrigation period to be equal to or less than the irrigation interval. The required capacity of a system, in gallons per minute, can be determined by the following equation: where RSC = Required system capacity (gal/min); 450 = Units conversion constant; A = Area irrigated (ac); DWA = Depth of water to apply per irrigation (in); IRP = Irrigation period (day); HPD = Time operating (hr/day). Problem: Determine the required system capacity (gal/min) for the corn crop in the previous problem when the field area is 200 acres, and the system can operate for 18.0 hr per day for 7.7 days. For 200 acres of long-season corn grown on silt loam soil over compacted subsoil, irrigated with a system that is 70% efficient and limited to operating 18 hours per day for 7.7 days, the system must be able to deliver 3,040 gallons of water per minute. Note: This is an example of an equation with a units conversion constant. The same problem can be solved using the units cancellation method. min 60 min 18 hr 7.7 day 231 in3 1 144 in2 43560 ft2 200 ac 5833555200 1 ft2 1 ac 1 1920996 In some situations it might be necessary to use units of capacity other than gallons per minute. For example, water supplied from a large reservoir is often measured in acre-feet. In these cases, units cancellation and/or the appropriate conversion factors (Appendix I or II) can be used to convert the units. Problem: What will the system capacity need to be in units of acre-feet/min? Solution: Using units cancellation and system capacity from the previous problem: ac-ft gal 1 ft3 1 ac min min 4.48 gal 43,560 ft2 As noted earlier, system capacity is a function of four variables: area (ac); water flow rate (gal/min, ft3/min, ac-ft/min, etc.); depth of water applied or peak use (in); and time (min, hr, or days). This relationship is expressed mathematically as: where D = Depth of water, either applied or peak use (in); A = Area irrigated (ac); Q = Water flow rate (cfs); T = Length of time water is applied (hr). When any three of the variables are known, the remaining one can be calculated by rearranging the equation and substituting the values of the known variables. You must enter flow rate (Q) in cubic feet per second, depth in inches, and time in hours. The following discussion will illustrate several uses of this equation. In the previous problem we determined the system capacity using units cancellation. If it is necessary to know how much water has been applied, the peak use does not accurately describe what we are solving for. When we want to know the depth of water that has been applied, D becomes the depth of water applied (DWA). This will work because the unit of measure is the same for both peak use and DWA (inches). Problem: A producer spends 120 hr irrigating 90.0 acres. The pump discharges 1,350 gallons per minute. What average depth of water (in) is applied? Solution: Because we want to know the amount of water applied, not the amount available to the plants, the efficiency factor is not used. Also, Q must be converted from gal/min to ft3/sec. Rearranging the equation, substituting depth of water to apply (DWA) for the depth (D), and including the conversion factor 1 ft3/sec = 2.25 x 10—3 gal/min1: DWA x A = Q x T Q x T An examination of this problem shows that the units do not cancel. However, when we enter the values of the variables with the units listed above we can obtain an answer very close to the true value. The symbol = means approximately equal. The exact solution using unit conversion/cancellation is: 1,350 37- x 231 — x __ . 3 ) x 120 hr x 60 — 12 — 90 ac x 43,560 In this case, the error in the approximate solution is: Variations occur in the use of the equation for different types of irrigation systems. In situations where the limiting factor is the availability of water, the problem is to determine the maximum area that can be irrigated with the available water supply. Problem: What is the largest size of lawn (ft2) that can be irrigated in 12 hr if a minimum of 0.5 inch of water is applied at each irrigation, the system is 90% efficient, and the water supply delivers 3.5 gal/min? ft3 1 gal 1 rnin 1ft3 3 ft3 1 eal 2.25 x 10—3 ft3 Iff- = 1gal x ^m x - = 2.25 x 10—3 —then1gal = - sec min 60.0 sec 7.40 gal sec mln 1 sec Solution: Rearranging the equation, adding the efficiency factor, and converting the area to square feet: If flood irrigation is used to water a field, assuming that the water flow rate is limited, it usually is necessary to determine the amount of time that the water should flow to cover the field at the desired depth. Problem: How long will it take to apply 4 inches of water uniformly over 120 acres when the water is available at the rate of 20 cfs? (Assume 100% efficiency.) ac ft3 sec min sec min h During furrow irrigation it is important to know how long the water must run to apply the desired amount for each set of furrows. Three values are necessary to calculate time: the water flow rate for each furrow or for the entire set, the area of the furrow or the set, and the amount of water to be applied. The area is determined from the number of rows in the set, the row spacing, and the length of the row. Problem: How much time is required to apply 3 inches of water to sixty 32-inch rows when the rows are one half mile long, and the system capacity is 30 gal/min/row? Number x Spacing (ft) x Lenght (ft) ft2 43,560 — ac in 1 ft ft 60 rows x 32-x -x 0.5 mile x 5280 row 12 in mile ft2 It will take 7.2 hr to apply 3.0 inches of water to the field. Was this article helpful?

AP® Statistics Equation and Formula Sheet The AP® Statistics Formula Sheet will be one of your most important tools for success on the AP Statistics exam. In order to get an edge in your preparation for the exam, it is a great idea to familiarize yourself with the formula sheet beforehand. Here we will describe everything you need to know about the formula sheet in detail. What is AP Statistics Formula Sheet and Tables The AP Statistics Formula Sheet and Tables includes formulas and probability tables that you will need to solve both the multiple-choice and free-response questions on the exam. The formulas cover all units of the course curriculum except Unit 3 (Collecting Data), which does not require formulas. The tables provide left-tailed or right-tailed areas under the normal distribution curve, the t-distribution curve, and the chi-square distribution curve. It is technically possible to solve every question on the AP Statistics exam using just the formulas on the formula sheet, algebra, and the probability tables. But the formulas on the sheet won’t be helpful if you do not already have a solid understanding of the course content. Plus, many problems are better suited for calculators than using the formulas. Still, along with the material you learn from your AP Statistics course, learning about each of the formulas and tables will help you be more efficient and better understand the material on the exam. In this guide, we will cover what is included with the AP Statistics Formula Sheet and Tables generally, as well as go over each individual formula in detail. In addition, we will go over many questions you may have about the formulas and table sheet, including what formulas are NOT included and so should be memorized. The first section of the formula sheet provides the formulas for descriptive statistics from Unit 1 (Exploring One-Variable Data) and Unit 2 (Exploring Two-Variable Data). It is rare to use these formulas directly on questions, especially for multiple-choice questions. Moreover, if you do end up with a chance to calculate the statistics mentioned in this section, it will generally be easier to use a calculator. However, you will absolutely need to understand them and reference them for certain types of questions. Therefore, making yourself familiar with this section is still important. 1 /∑ xi = ∑ xi / sx = √ 1 /∑ (xi - x̅ )2 = √ ∑ (xi - x̅ )2 / |ŷ = a + bx||y̅ = a + b x̅| 1 /∑ ( xi - x̅ /) ( yi - y̅ /) b = r Probability and Distributions The second section of the formula sheet contains two important formulas from basic probability as well as several formulas to calculate the mean and standard deviation for any discrete random variable that will show up on the exam. Unlike the first section of the formula sheet, you will almost certainly be required to use these formulas for direct calculations. It is also less likely that you will be able to use calculators to automatically solve the questions involving these formulas. For that reason, it will be helpful to specifically practice using these formulas with AP Statistics practice questions. P (A | B ) = |Probability Distribution||Mean||Standard Deviation| |Discrete random variable. X||µx = E (X ) = ∑ xi . P (xi )||σx = √∑ (xi - µx )2 . P (xi )| If X has a binomial distribution with parameters n and p, then: P (X = x ) = ( n / x) px ( 1 - p )n-x where x = 0, 1, 2, 3, ..., n |µx = np||σx = √np ( 1 - p )| If X has a geometric distribution with parameter p, then: P (X = x ) = (1 - p )n - x p where x = 1, 2, 3, ... √1 - p / Sampling Distributions and Inferential Statistics The third section of the formula sheet contains meta formulas for the test statistics and confidence intervals on the AP Statistics exam. They are meta formulas because they do not make calculations directly, but are combined with specific information for each procedure. These meta formulas are incredibly important for the AP Statistics exam and you should understand them deeply both on a conceptual level and as be able to use them to generate the test statistics and confidence intervals for the exam. Standardized test statistic: statistic - parameter / |Confidence interval: statistic ± (critical value) (standard error of the statistic)| Chi-square statistic: χ2 = Σ (observed - expected)2 / The information that can be combined with these formulas is given in the following parts of this section. They also contain general information important for understanding the sampling distributions of the important statistics on the example. For example, for procedures involving sample proportions: |Random Variable||Parameters of Sampling Distribution||Standard Error of Sample Statistic| |For one population: µp̂ = p σp̂ = √ p ( 1 - p ) / p̂ (1 - p̂ ) / For two populations: p̂ 1 - p̂ 2 |µ p̂ 1 - p̂ 2 = p1 - p2 σ p̂ 1 - p̂ 2 = √ p1 ( 1 - p1 ) /+ p2 ( 1 - p2 ) / S p̂ 1 - p̂ 2 = √ p̂1 ( 1 - p̂1 ) /+ p̂2 ( 1 - p̂2 ) / when p1 - p2 is assumed: S p̂ 1 - p̂ 2 = √ p̂c ( 1 - p̂c ) ( where p̂c = X 1 + X 2 / Included with the formula sheet are tables that provide probabilities for the normal distribution, t-distribution, and the chi-square distribution. The probabilities these tables provide are different for each distribution, based on their use on the AP Statistics exam. These tables are used to calculate certain probabilities, mostly probabilities over intervals and p-values. However, the information in these tables can also be found using a calculator. Should I use the probability tables included with the formula sheet or should I use a calculator? Many students prefer calculators, but it’s really a personal preference. Calculators can give more information, but those good with tables may find they are faster. You really should be familiar with both. A calculator can always fail, but it will give the most precise results. Conversely, using a table may not be easy, but it is a great way to learn the content of the course. AP Statistics Formula Sheet & Tables In Words: The left side of the equation is an x with a “bar” (line above the x) that represents the sample mean, while the right side of the equation is equal to the sum of the data divided by the number of data points (n ). Description: This sample mean is the most important statistic on the AP exam. It represents the “center” or “average” of a set of data. Application on AP Exam: It is unlikely that you will need this formula for calculation. This formula is conceptually important – think the central limit theorem, sampling distributions, t-tests, and other important topics. The formula is also useful for understanding how certain manipulations affect the sample mean. For example, how does multiplying the values in a dataset by 2 affect the sample mean? Sample Standard Deviation In Words: The left side of the equation is the sample standard deviation for the variable “x” (the subscript). The right side of the equation involves a lot of symbols, but it is straightforward if you take it in steps. (1) Subtract the sample mean from every value in the dataset, (2) Square the resulting values, (3) Add up the resulting values, (4) Divide the result by n – 1, and finally (5) Take the square root of the final result. Description: This sample mean is the second most important statistic on the AP exam. It represents the “spread” or “variability” of a set of data. Application on AP Exam: It is unlikely that you will need this formula purely for calculation. If it does come up, you should probably use a calculator instead of this formula. Most questions involving standard deviation will either give you the value of the standard deviation or otherwise require you to interpret features of the standard deviation. Predicted value of the response variable (linear regression) In Words: The left side of the equation represents the predicted value of the response variable in a simple linear regression model. It is denoted with a “y” with a so-called “hat” on top (the circumflex symbol). The right side of the equation provides the linear equation in which a given value of the explanatory variable (x ) is an input. The value of x is multiplied by the slope b and then y-intercept a is added. Description: Simple linear regression is a procedure to predict a value of one variable (the response variable: y ) from the value of another variable (the explanatory variable: x ). This formula shows how prediction is done through simple linear regression – through plugging x into the equation for a line. Application on AP Exam: One common question type on the AP exam is interpolation.. If the slope (a ) and y-intercept (b ).are given (and it’s likely they will be), then you should be able to turn the value of an explanatory variable x into the predicted value of the response variable y. Another type of question involves residuals. You will need to find the predicted value of y using a regression line, then compare to the observed value. Point on the linear regression line In Words: The left side of the equation (y with a bar) is the sample mean for a response variable y. The right side of the equation is the simple linear equation in which the mean of the explanatory variable is an input. The mean of x is multiplied by the slope b and then y-intercept a is added. Description: This formula makes a conceptual point as well as being useful for calculation. The conceptual point is that the predicted value of y when x is equal to its mean is equal to the sample mean of y; that is, the regression line always goes through the point (x̄,ȳ ). This formula is also used for calculating the y-intercept of a linear regression line. If you have the sample mean of x, the sample mean of y, and the slope of the regression line, then this formula can be rearranged to isolate the value of a. Application on AP Exam: The usage of this equation for the exam is as a reminder about the one point mentioned above – the point (x̄,ȳ ) is always on the regression line. Also, while such a question is highly unlikely, it is possible that you will need to calculate a y-intercept using this formula because it is a notable part of the curriculum. In Words: The left side of the equation (the “r”) is the correlation coefficient. The right side of the equation involves a lot of symbols, but like the standard deviation it is straightforward if you take it in steps. First, note that the expressions in the values are actually a type of z-score. (1) Calculate the z-scores for the data values for both variable x and variable y, (2) Multiply the paired z-scores, (3) Add up the resulting values, and finally (4) Divide the result by n – 1. Description: The correlation coefficient, r, gives the direction and strength of the linear association between two quantitative variables. Application on AP Exam: This is another formula you are unlikely to use directly on the AP exam. Similarly to the standard deviation, if it does come up, you should probably use a calculator instead of this formula. However, understanding this formula (especially the z-score interpretation) can help with several types of problems. For example, how does multiplying one variable by -1 affect the correlation coefficient? Slope of the linear regression line In Words: The left side of the equation (the “b”) is the slope of the linear regression line. The right side of the equation includes the correlation coefficient, multiplied by the ratio of the standard deviation of the response variable (y ) to the standard deviation of the explanatory variable (x ). Note: Keep in mind that the standard deviation in the numerator is the standard deviation for the values of y, NOT the standard deviation for the residuals of the linear regression. Description: The slope of the linear regression line is the amount that the predicted value of the response variable y changes for each unit increase in x. Application on AP Exam: There are several important applications of this formula on the AP exam. One that has come up several times is to understand the relationship between the correlation coefficient and slope. For example, if the correlation coefficient is positive, the slope must also be positive (and vice-versa). You should also be able to calculate the slope if given the correlation and the standard deviation of both variables involved in a regression. What formulas are not included with the AP Statistics formula sheet? The AP Statistics formula sheet does not include all formulas you’ll need for the AP Statistics exam. Here is a list of statistics and rules on the AP Statistics exam you will need to memorize. |Unit 6 & 7|| Read more about the AP Statistics Exam A perfect study guide can help you score 5 easily! See what our experts advise on how to score high in AP Statistics with all the essential resources to succeed. Want to know if AP Statistics is the right course for you? Our well-explained article can help you get a clear picture of the exam and make your exam prep easy! Don’t let the AP Exam format confuse you. Our self-explanatory guide will help you easily understand the exam format—question types, topic weights, and more! Looking for an easy-to-follow course and exam description for AP Statistics? See our guide for clear info on the AP Stats course—units, topics, and key concepts.

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What is 2/3 times 4 what is 2/3 times 4? 2/3 times 4 is 8/9. 2/3 multiplied by 4 is equal to 8/9. This can be seen by multiplying the numerator by 2 and the denominator by 3. what is 2/3 times 1/4? 2/3 times 1/4 is equal to 2/12. To find this, you can either multiply the numerators and denominators by the same number, or divide both numerator and denominator by the same number. What is 2/3 times 4 Fraction form? what is 2/3 times 4 as a fraction What is 1/4 times 2/3 in fraction form? 1/4 times 2/3 equals 1/12. This calculation allows us to evenly divide a fraction by another. To do this, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, in this example, we multiply 1 by 2/3 to equal 1/12. what is 3/4 cup times 2? 4/3 divided by 3/2 Understand that the answer to 4/3 divided by 3/2 is equal to 2/9. To do this, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, in this example, we multiply 4 by 3 to equal 12. We also multiply 3 by 2 to equal 6. Then we add these numbers together to equal 18. And finally, we divide 18 by 9 to equal 2. So the answer is 2/9. It’s also worth noting that we could have just as easily flipped the fraction, so that it would be 3/2 divided by 4/3. The answer in this case would still be 2/9. what is 2/3 times 4/5 times 1/3? 2/3 times 4/5 times 1/3 equals 8/15. To calculate this, we first need to convert all the fractions to decimals. We can do this by using a calculator, or if you are familiar with the long division method, you can also use that. We will use the long division method for this example. To begin, let’s convert 2/3 to a decimal. To do this, we divide the numerator (the top number) by the denominator (the bottom number). We get a decimal value of 0.666667. Now let’s convert 4/5 to a decimal. Again, we divide the numerator by the denominator and get a decimal value of 0.8. Finally, let’s convert 1/3 to a decimal. We divide the numerator by the denominator and get a decimal value of 0.333333. With all of our fractions converted to decimals, we can now multiply them together. The decimal value we get is 0.3333333, which is equal to 8/15. So, in summary, 2/3 times 4/5 times 1/3 equals 8/15. To calculate this, we converted all of the fractions to decimals and then multiplied them together what is 2/3 times 4/5? 2/3 times 4/5 is equal to 8/15. To get this answer, you can multiply the numerators and denominators by 2, which is the same as multiplying each fraction by 1/2. To do this, you multiply 8 times . Since 15 divided by 3 is 5, multiplying by 2 gives you . Therefore, 8/15 is equal to 2/3 times 4/5. how to divide fractions? To divide fractions, you need to invert the divisor and multiply the two fractions. This is sometimes called the “flip and multiply” method for dividing fractions. For example, if you want to divide the fraction 2/3 by 1/4, you would first invert the divisor (1/4) to 4/1. You would then multiply 2/3 by 4/1 to get 8/3, which is the answer to your division problem. To divide fractions by whole numbers, you can use the “long division” method that you may have learned in school when you were younger. What is formula for dividing fractions? The easiest way to divide fractions is to use a calculator, and input the division as a normal division problem. However, if you need to do it by hand, you can use the following steps: -First, convert the division into a multiplication problem by flipping the second fraction (that is, make the denominator of this fraction the numerator of the original fraction). This will also create a new denominator. -Next, multiply the first fraction by this new denominator. This is your answer to the division problem. What is fraction calculation chart? Fraction calculation is used to calculate the fraction of one number with another. It can be used to divide a whole unit into parts or calculate a percentage. The steps for finding a fraction calculation are as follows: 1) Determine the whole unit value. This is the number that will be divided by the fraction being calculated. 2) Identify the value of the fraction being calculated. This is the number that will be divided into the whole unit. 3) Calculate the fraction by dividing the fraction value into the whole unit value. 4) Express the fraction as a decimal value by moving the decimal point to the left or right until you have an equivalent fraction. Be sure to include any repeating numbers. 5) Convert the decimal value to a percentage by multiplying it by 100. This will allow you to express the fractional value as a percentage of the whole unit. how to subtract fractions? To subtract fractions, we need to find a common denominator between the two fractions. The common denominator is the lowest number that both of the two fractions have in common. For example, if we are subtracting 1/4 and 3/5, the common denominator is 20 because 4 and 5 both go into 20. To find the common denominator, we can either list out the multiples of each number or use a LCD (least common denominator) calculator. Once we have the common denominator, we can subtract the fractions by changing them both into equivalent fractions with the common denominator. For example, if we are subtracting 1/4 and 3/5 and our common denominator is 20, we would change 1/4 to 5/20 and 3/5 to 15/20. Then, we can subtract the fractions by subtracting the numerators and keeping the same denominator, so 5/20 – 15/20 = -10/20. This can be simplified to -1/2. how to multiply fractions? Multiplying fractions is a very useful concept in mathematics. It can be used to simplify complicated calculations involving fractions. The method of multiplying fractions is as follows: To multiply two fractions, take the product of the numerators and the product of the denominators. This will give you the answer in simplest form. For example, let’s take the fractions 6/4 and 2/3. To multiply these two fractions: Take the product of the numerators, which is 6 x 2 = 12. Take the product of the denominators, which is 4 x 3 = 12. The answer to this problem is 12/12, which reduces down to the simplest form of 1. This means that 6/4 x 2/3 is equal to 1. This is the simplified ways for multiplying fractions.

Who Gave You the Epsilons? is a sequel to the MAA bestselling book, Sherlock Holmes in Babylon. Like its predecessor, this book is a collection of articles on the history of mathematics from the MAA journals, in many cases written by distinguished mathematicians (such as G H Hardy and B.van der Waerden), with commentary by the editors. Whereas the former book covered the history of mathematics from earliest times up to the 18th century and was organized chronologically, the 40 articles in this book are organized thematically and continue the story into the 19th and 20th centuries. The topics covered in the book are analysis and applied mathematics, geometry, topology and foundations, algebra and number theory, and surveys. Each chapter is preceded by a Foreword, giving the historical background and setting and the scene, and is followed by an Afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book will be enjoyed by anyone interested in mathematics and its history – and in particular by mathematics teachers at secondary, college and university levels. Table of Contents Geometry, topology, and foundations Algebra and Number Theory About the Editors About the Editors Marlow Anderson is a professor of mathematics at The Colorado College, in Colorado Springs. He has been a member of the mathematics department there since 1982. He was born in Seattle, and received his undergraduate degree from Whitman College. He studied partially ordered algebra at the University of Kansas and received his Ph.D. in 1978. He has written over 20 research papers, and co-authored a monograph on lattice-ordered groups. In addition, he has co-written an undergraduate textbook on abstract algebra. In addition to algebra, he is interested the history of mathematics. When not teaching, reading or researching mathematics, he may be found with his wife Audrey scuba-diving in far-flung parts of the world. Victor J. Katz, born in Philadelphia, received his Ph.D. in mathematics from Brandeis University in 1968 and was for many years Professor of Mathematics at the University of the District of Columbia. He has long been interested in the history of mathematics and, in particular, in its use in teaching. His well-regarded textbook, A History of Mathematics: An Introduction, is now in its third edition. Its first edition received the Watson Davis Prize of the History of Science Society, a prize is awarded annually by the Society for a book in any field of the history of science suitable for undergraduates. A brief version of this text appeared in 2003. Professor Katz is also the editor of The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, which was published in July, 2007 by Princeton University Press. Professor Katz has published many articles on the history of mathematics and its use in teaching. He has edited or co-edited two recent books dealing with this subject, Learn from the Masters (1994) and Using History to Teach Mathematics (2000). He also co-edited a collection of historical articles taken from MAA journals of the past 90 years, Sherlock Holmes in Babylon and other Tales of Mathematical History. He has directed two NSF-sponsored projects that helped college teachers learn the history of mathematics and how to use it in teaching and also involved secondary school teachers in writing materials using history in the teaching of various topics in the high school curriculum. These materials, Historical Modules for the Teaching and Learning of Mathematics, have now been published on a CD by the MAA. Currently, Professor Katz is the PI on an NSF grant to the MAA supporting Convergence, the online magazine in the history of mathematics and its use in teaching. He is a member of the Mathematical Association of America, the American Mathematical Society, the Canadian Society for the History and Philosophy of Mathematics, and the British Society for the History of Mathematics. Robin Wilson is Professor of Pure Mathematics at the Open University (UK), a Fellow in Mathematics at Keble College, Oxford University, and Emeritus Gresham Professor of Geometry, London (the oldest mathematical Chair in England). He has written and edited about thirty books, mainly on graph theory and the history of mathematics. His research interests focus mainly on British mathematics, especially in the 19th and early 20th centuries, and on the history of graph theory and combinatorics. He is an enthusiastic popularizer of mathematics, having produced books on mathematics and music, mathematical philately, and sudoku, and gives about forty public lectures per year. He has an Erdös number of 1 and has won two MAA awards (a Lester Ford Award (1975) and a George Pólya award (2005). The MAA has a tradition in the publication of books like this one. I think it began in 1969 with the first of four volumes of papers selected from its various journals (going back to 1890). The four volumes consisted of papers on Pre-calculus, Calculus, Algebra and Geometry respectively. All such articles were expositional, some were historical, and many were inspirational. Much later, in 2004, there appeared an MAA publication that was compiled in a similar vein — except that its emphasis was historical throughout. It was given the title Sherlock Holmes in Babylon. In it, the articles appear in the chronological order of mathematical developments from ancient times up to the work of Euler in the 18th century. Fortunately, there is now a sequel to that book, which is the subject of this review. It contains forty-one papers pertaining to the history of mathematics from the early 19th century to the late 20th century. Continued...

I have had occasion to make many references in the past to Richard Lipsey’s wonderful article “The Foundations of the Theory of National Income” which was included in the volume Essays in Honour of Lord Robbins. When some 40 years ago, while a grad student at UCLA, I luckily came upon Lipsey’s essay, it was a revelation to me, because it contradicted what I had been taught as an undergrad about the distinctions between planned (ex ante) investment and savings, and realized (ex post) investment and savings. Supposedly, planned investment and planned savings are equal only in equilibrium, but realized investment and savings are always equal. Lipsey explained why the ex ante/ex post distinction is both incorrect and misleading. In this post I want to begin to summarize some of the important points that Lipsey made in his essay. Lipsey starts with a list of seven erroneous propositions commonly found in introductory and intermediate textbooks. Here they are (copied almost verbatim), grouped under three headings: I The Static Model in Equilibrium 1 The equilibrium of the basic Keynesian model is given by the intersection of the aggregate demand (i.e., expenditure) function and the 45-degree line representing the accounting identity E ≡ Y. II The Static Model in Disequilibrium 2 Although people may try to save different amounts from what people try to invest, savings can’t be different from investment; realized (ex post) savings necessarily always equals realized (ex post) investment. 3 Out of equilibrium, planned savings do not equal planned investment, so it follows from (2) that someone’s plans are being disappointed, and there must be either unplanned savings or dissavings, or unplanned investment or disinvestment 4 The simultaneous fulfilment of the plans of savers and investors occurs only when income is at its equilibrium level just as the plans of buyers and sellers can be simultaneously fulfilled only at the equilibrium price. III The Dynamic Behavior of the Model 5 Whenever savers (households) plan to save an amount different from what investors (business firms) plan to invest, a mechanism operates to ensure that realized savings remain equal to realized investment, despite the attempts of savers and investors to make it otherwise. Indeed, this mechanism is what causes dynamic change in the circular flow of income and expenditure. 6 Since the real world, unlike the simple textbook model, contains a very complex set of interactions, it is not easy to see how savings stay equal to investment even in the worst disequilibrium and the most rapid change. 7 The dynamic behavior of the Keynesian circular flow model in which disequilibrium implies unintended investment or disinvestment can be shown by moving upwards or downwards along the gap between the expenditure function and the 45-degree line in the basic Keynesian model. Although some or all of these propositions are found in most standard textbook treatments of national income theory, every one of them is wrong. Let’s look at proposition 1. It says that the equilibrium level of income and expenditure is determined algebraically by the following two relations: the expenditure (or aggregate demand) function: E = E(Y) + A and the expenditure-income accounting identity E ≡ Y. An accounting identity provides no independent information about the real world, because there is no possible state of the world in which the accounting identity does not hold. It therefore adds no new information not contained in the expenditure function. So the equilibrium level of income and expenditure must be determined on the basis of only the expenditure function. But if the expenditure function remains as is, it cannot be solved, because there are two unknowns and only one equation. To solve the equation we have to make a substitution based on the accounting identity E ≡ Y. Using that substitution, we can rewrite the expenditure function this way. E = E(E) + A If the expenditure function is linear, we can write it as follows: E = bE + A, which leads to the following solution: E = A/(1 – b). That solution tells us that expenditure is a particular number, but it is not a functional relationship between two variables representing a theory, however naïve, of household behavior; it simply asserts that E takes on a particular value. Thus treating the equality of investment and savings as an identity turns the simply Keynesian theory into a nonsense theory. The point could be restated slightly differently. If we treat the equality of investment and savings as an identity, then if we follow the usual convention and label the vertical axis as E, it is a matter of indifference whether we label the horizontal axis Y or E, because Y and E are not distinct, they are identical. However we choose to label the horizontal axis, the solution of the model must occur along the 45-degree line representing either E = Y or E = E, which are equivalent. Because, the equality between E and itself or between E and Y is necessarily satisfied at any value of E, we can arbitrarily choose whatever value of E we want, and we will have a solution. So the only reasonable way to interpret the equality between investment and saving, so that you can derive a solution to the simple Keynesian model is to treat E and Y as distinct variables that may differ, but will always be equal when the economy is in equilibrium. So the only coherent theory of income is E = E(Y) + A and, an equilibrium condition E = Y. E and Y do not represent the same thing, so it makes sense to state a theory of how E varies in relation to Y, and to find a solution to the model corresponding to an equilibrium in which E and Y are equal, though they are distinct and not necessarily equal. But the limitation of this model is that it provides us with no information about how the model behaves when it is not in equilibrium, not being in equilibrium meaning that E and Y are not the equal. Note, however, that if we restrict ourselves to the model in equilibrium, it is legitimate to write E ≡ Y, because the equality of E and Y is what defines equilibrium. But all the erroneous statements 2 through 7 listed above all refer to how the model. The nonsensical implications of constructing a model of income in which expenditure is treated as a function of income while income and expenditure are defined to be identical has led to the widespread adoption of a distinction between planned (ex ante) investment and savings and realized (ex post) investment and savings. Using the ex ante/ex post distinction, textbooks usually say that in equilibrium planned investment equals planned savings, while in disequilibrium not all investment and savings plans are realized. The reasoning being that is that if planned saving exceeds planned investment, the necessity for realized savings to equal realized investment requires that there be unintended investment or unintended dissaving. In other words, the definitional identity between expenditure and income is being used to tell us whether investment plans are being executed as planned or being frustrated in the real world. Question: How is it possible that an identity true by definition in all states of the world can have any empirical implications? Answer: It’s not. In my next installment in this series, I will go through Lipsey’s example showing how planned and realized saving can indeed exceed planned and realized investment over the disequilibrium adjustment induced by a reduction in planned investment relative to a pre-existing equilibrium. UPDATE (2/21/2015]: In the second sentence of the paragraph beginning with the words “An accounting Identity provides,” I wrote: “It therefore adds information not contained in the expenditure function,” which, of course, is the exact opposite of what I meant to say. I should have written: “It therefore adds NO NEW information not contained in the expenditure function.” I have now inserted those two words into the text. Thanks to Richard Lipsey for catching that unfortunate mistake.

4 Pitfalls for Developing Doers of Mathematics: Takeaways from the Craig Groeschel Leadership Podcast I saved the day. The math problem had a small detail in it that if my students were not paying attention, it would mess up their work. So I darted around the room and pointed out the detail at the precise moment each group encountered it. No one got confused. No one struggled. Everyone got the right answer. It was a fantastic moment…until I assigned the next problem. After a few moments all of the groups began to glance in my direction, and a boy right next to me asked, “What do we do?” The better question was, “What did I do?” I didn’t save the day. I wasn’t developing my students’ capacity to do mathematics. Instead, I encountered a pitfall of developing doers of mathematics. Developing Doers of Mathematics Recently, I head an episode of the Craig Groeschel Leadership Podcast where he discussed four pitfalls leaders can fall into in developing the people they lead. All throughout the podcast, I kept thinking of parallels between what Pastor Craig shared about leading organizations and the complexities of teaching and learning mathematics (don’t we all?). I see the job of teaching as facilitating a productive relationship between students and mathematics so that students see themselves as doers of mathematics. In other words, the job of the teacher is to develop doers of mathematics. The Four Pitfalls of Developing Doers of Mathematics I present these pitfalls of developing doers of mathematics knowing I have fell into each of them multiple times. By naming and recognizing these pitfalls, we can avoid them and develop the kind of relationships we want our students/children/doers of mathematics to have with mathematics. I see these pitfalls existing for teachers, parents, tutors, even students, basically, anyone who may help someone develop as a doer of mathematics. 1. Controlling – creates compliant doers of mathematics. This pitfall is where students are not given freedom to consider their own methods for solving a math problem. Instead of giving space to explore and make sense of the problem, students are dictated a carefully constructed algorithm for solving the problem. By definition, the problem is no longer a problem. When the student has been shown exactly how to solve a math problem, the problem has transformed into a mere exercise. Controlling has turned doing math into executing algorithms. To avoid this pitfall students need space to explore problems. A great article that was written about this idea is call Never say Anything a Kid Can Say by Steven Reinhart. The mindset presented in this article has helped me be quiet, sit back, and trust the student(s) (through gentle prodding) to produce a solution, and then use that work as a starting point for a conversation about the problem. Also Mandy Jansen@MandyMathEd has this idea of “rough draft talk” for solving math problems. The idea being lets consider the idea of creating rough drafts for papers and use that same iterative process for creating solutions for math problems. 2. Criticizing – creates insecure doers of mathematics. This pitfall is where students may be given freedom to consider their own methods for solving a math problem, but each method is quickly identified for how it falls short in efficiency, accuracy, elegance, or just is not the preferred method of the person providing assistance. Students are eventually leery of presenting their ideas for solving a problem given the overly critical environment in which the idea is received. To avoid this pitfall an asset-based perspective of the work students do with mathematics needs to be developed. Instead of seeing what is wrong with the method, consider what is right. This approach of having an asset-based perspective and assigning competency to students can be seen in the work around Complex Instruction. Two books I recommend on Complex Instruction in the math classroom are both from the National Council for Teachers of Mathematics (NCTM). One book is called Strength in Numbers: Collaborative Learning in Secondary Mathematics by Horn. The other book is Smarter Together: Collaboration and Equity in the Elementary Math Classroom by Featherstone, Crespo, Jilk, Oslund, Parks, and Wood. 3. Avoiding – creates disengaged doers of mathematics. This pitfall is where students are given freedom to consider their own methods for solving math problems, but are not given any feedback. The person providing assistance…doesn’t. They are not engaged with what the students are doing and in turn the students see it (understandably) as a lack of caring in what they are doing. To avoid this pitfall the answer is to simply engage. The easiest way to engage is to ask questions. Try to figure out how students are making sense of the problems and attempt to do so with no assumptions. I remember noticing on my son once identified a rectangle as having six sides on his homework. My gut told me ask him why he got the question wrong (He knows how many sides are on a rectangle, right?). Instead, I asked him how he came to the answer of six, simply and with no judgement. He told me he used a tile in the shape of a rectangle to count all the sides and came up with six. He counted around the tile and then one on top, and one on the bottom. I immediately realized his problem was not a rectangle problem. It was a problem identifying the difference between three dimensional shapes and two dimensional shapes. It was a problem identifying the difference between sides of a polygon and faces of a polyhedron. Asking the question, and not avoiding, resulted in a wonderful understanding for both of us. 4. Rescuing – creates helpless doers of mathematics. This is where students are given freedom to consider their own methods for solving math problems but are given help at the smallest indication of struggle. An example of this pitfall can be seen in the story that began this blog post. To avoid this pitfall, students need to be given permission to struggle and sometimes even fail. This does not mean disengagement but providing assistance in other ways. Asking an open-ended, probing question, encouraging them to continue on a line of thinking, creating timely partnerships between students considering the same solution path, are all ways to stay engaged but not rob students of the learning potential of a math problem. Helping students learn how to deal with struggle and to learn from failure will not only develop them as doers of mathematics but also as people. Knowing is half the battle… In the end, avoiding these pitfalls comes down to a balance of engagement and freedom. By knowing these pitfalls we can avoid them and help our students develop the kind of relationship we want them to have with mathematics.

Water Cycle Review •Also called the hydrologic cycle •The journey water takes as it circulates from the land to the air and back again. •Involves evaporation, condensation, and precipitation. •Repeats as a never-ending cycle •Naturally occurring substances such a mineral, forest, water, and land that are used by humans. •A resource that can be used repeatedly because it is replaced naturally (cycle). •Water fits both these criteria. •Basin-like land formation defined by highpoints and divides that descends into lower elevations. •Carries water from the land after rainfall or snow melts. •Drains all the water into a common outlet such as a stream channel, a reservoir, or bay •Very low amounts of dissolved salt – less than 1% •Ponds and Lakes •Streams and Rivers •Makes up 3% of Earth’s water resources, including ice caps and glaciers •High concentrations of salt •3.5% of the weight of seawater comes from dissolved salt (salinity) •Makes up 97% of Earth’s water resources •Water on the surface of the planet •Ponds and Lakes •Streams and Rivers •Replenished by precipitation and groundwater •More prone to pollution than groundwater •Water found underground in cracks and spaces in soil, sand, and rock. •Stored in and moves slowly through aquifers •More than 50% of the people in the U.S. get their drinking water from groundwater. •Largest use is irrigating crops •Less prone to pollution •Permeable – rock layers or sediments that transmit groundwater freely a.Must include spaces (pores) throughout the rock layer b.Pores must be connected •Impermeable – few or no connected pore spaces, such as clay •Zone of Aeration – region between the earth’s surface and the water table •Water Table – the upper surface of the Zone of Saturation (can move up or down depending on rainfall) •Zone of Saturation – region in the ground in which the pore spaces are filled with water •Made of gravel, sand, sandstone, or limestone •Water can move through these materials because they have large connected spaces (pores) that make them permeable. •The flow of water depends on the size of the spaces and how well they are connected. •An excavation or structure created in the ground by digging, which accesses groundwater in an aquifer. •The well water is drawn by a pump that is raised mechanically or by hand. •How is the well depth determined? What might make a well “go dry?” •Replenished by precipitation •A place in the ground where water flows up to the surface because of natural pressure without being pumped. •Water comes directly from the aquifer or porous rock layer. •Gravity creates the natural pressure. •Contamination of bodies of water, often by human activity, which affects watersheds •Occurs when pollutants are discharged directly or indirectly into the water. •Along with air pollution, water pollution is the second biggest environmental concern. Point Source Pollution When the pollutants come from a single location such as dumping chemicals into a river. Nonpoint Source Pollution When pollutants are introduced into the environment over a large, widespread area such as agricultural runoff. Types of Water Pollution Surface Water Pollution •Hazardous substances coming into contact with surface water •Dissolves or mixes physically with the water •Examples: Humans dumping trash into the waterways, especially objects that are swept down storm drains. •Release of liquid petroleum hydrocarbons (oil) into the water •Especially harmful to marine and other wildlife •Usually localized, but can spread •Examples: oil spills Chemical Water Pollution •Chemicals from industries and farmers that run off into the waterways. •Examples: metals and solvents from industries •Also, chemicals that control weeds, insects, and pests •Pesticides and chemicals wash deep into the ground by rain water •Can get into the aquifers, thus polluting the groundwater •Anything on the surface can eventually work its way down to the groundwater. •Plume – the area of groundwater affected by the contamination •Look at the diagram and observe the amount of contamination in relationship to the point pollution. Thermal Water Pollution •The rise or fall in the temperature of a natural body of water. •Changes the physical properties of water, particularly the amount of dissolved oxygen in the water. •Decreases fish population and increases death to wildlife •Sediments washing off fields are the largest source of agricultural pollution in the U.S. •Sediments increase the cost of treating drinking water and can also clog fish gills, reducing their resistance to disease. Overuse and Waste •Irrigation uses 30% of all freshwater in the U.S. •Swimming pools and water parks •Watering the lawn What other ways do you overuse or waste water where you live? •Withdrawing groundwater causing the land to sink •Causes flooding problems •Causes a shift in the foundations of buildings, which can lead to their destruction compare fresh and salt water, including examples? identify the differences between surface and groundwater, including examples? draw and label the parts of an aquifer? recall six different types of water pollution? generate ideas for reducing water pollution? The H-R Diagram plots each star on a graph and measures the star's brightness (luminosity) against its temperature (color). •Measured in Kelvin (K) •Color of stars depends on their temperature •The coolest stars – red Hottest stars – blue •Temperature increases from right to left, which is different than every graph you’ve probably seen. •The amount of energy (light) a star emits Tells us how bright an object appears from Earth The measure of a star’s brightness as if it were at a standard distance of exactly 10 parsecs (32.6 light years) from the observer. •Stars are classified by their spectra (the elements that they absorb) and their temperature. •There are seven main spectral types (O, B, A, F, G, K, and M) listed in order of decreasing temperature. •About 90 percent of the stars in the universe, including the sun •Ranges from high to low luminosity and high to low temperature •Color – ranges from red to blue •Spectral Class M-O •Medium size star •Medium brightness and temperature •Color – yellow •Spectral Class G •A red giant is a dying star. •Our own sun will turn into a red giant star, expanding to engulf the inner planets. •Color - reddish-orange hue •High luminosity/ low temperature •Spectral Class K-M •They are the largest stars in the universe in terms of volume, although they are not the most massive. •Color – reddish orange/blue •High luminosity/low-high temperatures •Spectral Class K-M, B-A •A small very dense star that is typically the size of a planet •Formed when a low-mass star has exhausted all its fuel •Color – white •Low luminosity/high temperature Spectral Class B, O, A •A small and relatively cool star on the main sequence •Color – red •Low luminosity/low temperature •Spectral Class – M 1.Can you interpret the H-R Diagram? 2.Can you use the H-R Diagram to explain how stars are classified? The two Documents below were read in class during the week of Dec 3. •A disturbance that travels through space and matter •Transfers energy, not matter •Travel through electrical and magnetic fields Examples: light, microwaves, radio waves, and X-rays. •The highest point on a wave is called the crest. •The lowest point on a wave is called the trough. •The distance between successive crests or troughs •Measures one complete wave •The maximum extent of a wave measured from the position of equilibrium •The number of crests of a wave that move past a given point in a given unit of time •Measured in Hertz (Hz) •Examples: FM/AM radio stations, stars •Need close proximity to transmitter •Examples: microwaves, routers, cell phones, stars •“Infra” means below •These waves are just below visible light •Examples: remote controls, flames, lamps, stars •All visible light •Examples: light bulbs, fire, stars •Often called “black light” •Examples: sterilization, stars (sunburn anyone?), haunted houses •High frequency waves •Examples: see inside organisms, airport security, dentist office, stars •Highest frequency waves •Highest temperature (blue) •Examples: radiation therapy (cancer), sterilization, stars Studying the Universe •Astronomers use all kinds electromagnetic waves to study the characteristics (temperature, energy, color) of stars. •They can also use the EMS to determine chemical composition. •A measurement technique which allows astronomers to see light that is absorbed, emitted, or scattered by materials •How do we know what stars are made out of? •Use the class set of absorption spectrums to determine which elements are present in each star. •Astronomers can measure the distance of stars using a method called parallax. •They measure the star twice per year. •Every 6 months the Earth has moved nearly 186 million miles from it’s previous point due to its revolution around the Sun. •Also called the Hydrologic Cycle •Process by which water circulates between the Earth’s oceans, atmosphere, and land •Involves water storage, evaporation, transpiration, condensation, precipitation, and runoff. •Oceans – super storage for the water cycle - holds 96.5% of Earth’s water •Primary pathway into the water cycle •Oceans, seas, lakes ,and rivers provide nearly 90% of the moisture in our atmosphere through evaporation. •Process by which water changes from a liquid to a gas •Primary pathway that water moves from the liquid state back into the water cycle as water vapor •Heat (energy) from the sun is necessary for evaporation to occur. •Energy breaks bonds that hold water molecules together. •Molecules move fast at boiling point 212o F •Slow at freezing point 32o F •Sublimation – the process of snow and ice changing into water vapor without first melting into water. •Evapotranspiration – water lost to the atmosphere from the ground surface and transpiration of groundwater by plants through their leaves. •Superhighway used to transport water around the globe •Involves condensation and precipitation •Process in which water vapor in the air is changed into liquid water. •Loss of energy allows water molecules to bond. Forms clouds, fog •Water released from clouds in the form of rain, freezing rain, sleet, snow and hail •Provides the delivery system of atmospheric water to the Earth Ice, snow, groundwater •Water locked up in its present state for a relatively long period of time •Involves runoff and infiltration •Precipitation that did not get absorbed into soil, or evaporate •Ice caps and glaciers - provides runoff from melting •Water moved by gravity makes its way into places that collect water – rivers, lakes, ponds, ocean •The downward process of moving water from the land surface into soil or porous rock •Groundwater - Large amounts of water stored in the ground •Aquifer – another name for groundwater, usually describes water bearing formations •The area of land where all the water that falls in it and drains off of, goes into the same place •Can be as small as a footprint or as large as all the land that drains water into the Mississippi River Science 7: Apply scientific principles to design a method for monitoring and minimizing a human impact on the environment. Take the Carbon Footprint Challenge at Home! http://www.footprintcalculator.org/ Human Impacts on the Environment: www.khanacademy.org/science/biology/crash-course-bio-ecology/crash-course-ecology-2/v/crash-course-ecology-10 Science 8: Light Waves •The part of the electromagnetic spectrum, between infrared and ultraviolet, that is visible to the human eye. •Shorter waves – higher frequency and energy •Longer waves – lower frequency and energy Visible Light Spectrum •Produced when light passes through a prism, slowing the wavelength into each separate color. •ROY G. BIV - red, orange, yellow, green, blue, indigo, violet •We see these waves as the colors of the rainbow. •Each color has a different wavelength and frequency. •Red has the longest wavelength and shortest frequency •Violet has the shortest wavelength and highest frequency. •Seen together, they make white light. •For an object to be visible it must produce its own light or reflect light. •Produces own light - Sun, candle, flashlight •Reflects light - Moon, mirror, glass •Opaque – A material that reflects or absorbs all of the light that strikes it. (wood, metal, cardboard) •Transparent – transmits light (glass, water, air) •Translucent – scatters light as the light passes through (wax paper, frosted glass) How light travels •Light travels in straight lines. •This straight line motion can be: •Occurs when parallel rays of light hit a smooth surface. •All the rays are reflected at the same angle. •Law of reflection: the angle of reflection equals the angle of incidence. •Angle of incidence - measure of the angle of a ray to the surface normal (90o to the surface) •When parallel rays of light hit a bumpy surface. •Each ray obeys the law of reflection, but each ray hits the surface at a different angle. The light is scattered. •When light waves enter a new medium at an angle, their speeds changes. •The change in speed causes them to bend, or change direction. •Index of Refraction – a measure of how much a ray of light bends when it enters that material •When light traveling in straight parallel lines passes through an object that is curved like a lens, the light is refracted at different angles. •Convex or converging lenses bend light toward a central focal point. •Concave or divergent lenses bend light outward away from a focal point. •Light does not pass through or reflect from material, but remains in the material as energy. •What happens to the black surface? Color of objects •Color – Objects reflect colored light that is not absorbed. •We see objects color as the reflected color. Colors of Light •Red, Blue and Green •When combined in equal amounts, primary colors produce white light. •If combined in varying amounts, they can produce any other color. •Yellow, Cyan and Magenta •Primary colors combined in varying amounts •Complementary - form when a primary color and a secondary color combine to make white. •Yellow and blue = white •Y + B = W or R + G + B = W •A relative expression of the intensity of the energy output of a visible light source •Brightness is determined by the light wave’s amplitude. •The greater the amplitude, the brighter the light. •Distance from light source also affects brightness. •Chemical and physical breakdown of rocks into sediment •Occurs when the rock’s environment changes and the rock is exposed to some form of water and the air Chemical change within the rock’s minerals breaking down the bonds holding the rocks together, causing them to fall apart into smaller pieces. Causes rock to break: •(A) Oxidation – Iron combines with oxygen making rust. •(B) Hydrolysis – Water softens minerals in rocks. •(C) Carbonation – Carbon dioxide in rain water creates carbonic acid. Ex. acid rain, cave creation Physical (Mechanical) Weathering The process that breaks rocks apart without changing their chemical composition caused by: •(A1) Abrasion - by rapidly moving water, glaciers or wind. •(A2) Ice wedging - by freezing and thawing (contracting and expansion). Causes rock to break: •(B) Plant Roots - grow into cracks and break apart rock. •(C) Burrowing – animals scrape and dig the terrain. •(D) Temperature Change- cold to hot expanding and contracting. •(E) Gravity - falling rocks or debris, compression The process that moves bits of rock or soil from one place to another by: •Water (rivers, waves) The process in which sediments, soil, and rocks are added to a landform such as: Occurs when the forces moving sediments are no longer able to overcome the forces of gravity and friction. Running water is the primary agent of erosion. •Velocity (speed) depends on gradient (slope) and discharge (amount of water). •As velocity increases the size of particles carried also increases. Ages of Rivers •(A) Young Rivers - fast-flowing, V-shaped valleys, waterfalls, and rapids •(B) Mature rivers – Less energy, slower, meanders (1), sandbars •(C) Old River – Very slow, shallow, large amounts of sediment deposited, many narrow channels, islands, deltas (2) Features Created by Wind Caused by abrasion from wind blown sand. Features Created by Gravity Gravity shapes the Earth’s surface by moving weathered material from a higher place to a lower one. •(A) Landslides (fast) •(B) Mud flows •(C) Slump/creep (slow) Features Created by Glaciation Caused as massive glaciers flow down hill bulldozing existing rocks. Features Created by water (waves) Erosional and depositional features which form along coastlines •The western U.S. coastline has more erosional features. •The eastern U.S. coast and the Gulf of Mexico has more depositional features. Ecoregions of the United States Areas defined by its environmental conditions, especially climate, landforms, and soil characteristics. Ecoregions Environmental Conditions •Climate – weather conditions in an area over time. • Landforms – crustal material •Mountains – high elevation •Plateaus – medium to high elevation •Plains – low elevation •Amount of vegetation •Dry (arid) – very little vegetation (poor soil) Humid – large amount of vegetation (good soil) Examples of Ecoregions •Subtropical (Florida, South Eastern States) •Tundra (N Alaska) •Temperate Steppe (Great Plains •Marine Mountains (Coastal Washington and Oregon) •Desert and Desert Mountain (Nevada and parts of New Mexico) What determines how the processes of weathering, erosion, and deposition work to reshape Earth’s surface? •Vibrations that travel through the air or other media •When these vibrations reach the air near your ears you hear the sound. How Sound Travels •Sound waves carry energy through a medium (solid, liquid or gas) without the particles of the medium traveling along. •Sound travels as a longitudinal wave. How Sounds are Made •Longitudinal waves are generated when a source of energy forces the matter in a medium to vibrate. •This back-and-forth motion pushes air particles together, generating a compression, or moves the particles apart, generating a rarefaction. •Sound waves must have a medium to travel through. •Gas – air is the most common •In outer space there are no molecules to compress or rarefy, so sound does not travel through outer space. Speed of Sound •Depends on the physical properties of the medium it travels through. •At room temperature, sound travels through air at about 342m/s. Physical Properties of Media •Elasticity – the ability of a material to bounce back after being disturbed •Solid materials are usually more elastic than liquids or gases. •Particles of a solid do not move very far, so they bounce back and forth quickly as the vibration travels through the object, which allows waves to move faster. •Density – how much matter there is in a given amount of space •The speed of sound depends on how close together the particles of the substance are in the medium. •Temperature - degree or intensity of heat present in a substance or object •In a given media (solid, liquid, gas), sound travels more slowly at lower temperatures. Properties of Sound Waves •Intensity – the amount of energy the wave carries per second through a unit of area •Amplitude increases with increased energy •Measured in watts per square meter (W/m2) •Loudness – describes what you actually hear. •Though not the same as loudness, the greater the intensity of a sound wave, the louder it is. •Measured in decibels (dB) •Maximum safe level is 85 dB Frequency – the number of vibrations that occur per second Wavelength changes with frequency Measured in Hertz (Hz) 50Hz = 50 vibrations per second •Pitch – a description of how high or low the sound seems to a person •High frequency = high pitch •Low frequency = low pitch •Example: a young girl might have a squeaky (high pitched) voice, an older man might have a deep (low pitched) voice •The apparent change in frequency as a wave source moves in relation to the listener •Sounds moving toward a person – Waves are at a higher frequency, so pitch appears to increase (high) •Sound moving away from a person – Waves are at a lower frequency, so pitch appears to decrease (low) A model that describes the formation, breakdown, and reformation of a rock. •Formed when sediments accumulate and compact and cement together. •Often deposited in layers and contain sand, pebbles, and frequently fossils. Ex. sandstone, limestone Physical properties of Sedimentary Rocks •Sand, pebble, and even boulder size particles •Some may contain fossils By what process are sedimentary rocks broken down? •By weather (rain, ice, wind), chemical changes, and living things (plant). •Creates lose material called sediments. By what process are sediments moved? They are deposited in layers - Deposition What are the processes that form sedimentary rock? Sediments are deeply buried, placing them under pressure because of the weight of overlying layers. •New minerals stick the sediment together just like cement. •This holds the grains together tightly. •Formed by heat and pressure while buried deep below Earth’s surface. •Have a layered or banded (ribbon like) appearance or may have crystals. Ex. Gneiss, Marble, Slate Physical Properties of Metamorphic Rocks •Layers look like ribbons What are the processes that form metamorphic rock? Heat (caused by magma) •Temperatures high enough to change its structure but not to melt it. •Heat can change sedimentary, igneous, or another older metamorphic rock. Pressure - Caused by intense collisions and friction of tectonic plates and pressure from overlying rock layers. •Deep under the Earth’s surface. •Pressure can change sedimentary, igneous or another older metamorphic rock. •Formed when lava or magma harden. •Found near volcanoes or fissures •Ex. Basalt, Obsidian, Granite Physical Properties of Igneous rock Fast Cooling Slow Cooling Glassy Large crystals Holes where gas was trapped Many colors What are the processes that form Igneous rock? •Caused by increase in temperature in rock deep below the surface of Earth •Caused by friction between crustal plates Lava –molten rock material on Earth’s surface. Magma – molten rock material under Earth’s surface. What are the processes that form igneous rock? Cooling and Hardening •Melted rock turns solid. •Slow cooling happens below Earth’s surface as magma cools forming large crystals. Ex. granite •Fast cooling happens on the Earth’s surface as lava cools forming small crystals. Ex. obsidian, basalt, pumice Video Resource: https://drive.google.com/file/d/0BxOms4hIDvR3TmxEcHJWV2xzTkU/view Similarities of Rocks and Minerals •Inorganic compounds (non-living) •Both can be classified by their chemical composition. •Found around the world in many of the same places on Earth •Most commonly classified by how they form. •Composed of more than one mineral. •No definite chemical composition. •No definite crystal structure. In addition to being inorganic, solid, and naturally formed like rocks, minerals also have: •A definite chemical composition. (amounts of elements present) •A definite crystal structure. (unique arrangement of atoms/molecules) Mineral Identification – Important Vocabulary 1.Color (green, red, yellow, blue, etc.) 2.Streak (Color of the streak across a streak plate) 3.Luster (Metallic or Non-Metallic) 4.Hardness (Mohs Scale) 5.Density (Specific gravity) 6.Breakage Pattern (Cleavage and Fracture) •Many minerals have distinctive colors, but they come in a variety of hues. •Color should never be used as the only test for identifying a mineral. •The color a mineral displays in a finely powdered form •Might be completely different from the color of the mineral itself •To determine the streak, rub the mineral across a piece of unglazed porcelain know as a streak plate. •The way a mineral’s surface reflects light. •Two types of luster •Metallic – shiny like a metal •Nonmetallic – several kinds 1. Glassy - quartz 2. Pearly - talc 3. Greasy - graphite 4. Silky – gypsum 5. Resinous - sulfur 6. Adamantine - diamond •One of the most reliable ways to identify minerals •Compares the resistance of a mineral to being scratched by 10 reference minerals •Called the Mohs Hardness Scale •Named after Friedrich Mohs, a German mineralogist, who developed the scale in 1812 •Defined as the amount of matter per unit volume •Density = mass divided by volume •In minerals, the term specific gravity is used in describing density. In this way minerals can be compared and identified. •Refers to the way some minerals break along certain lines of weakness in their structure Mica is a good example. •A description of the way a mineral tends to break •Some different types of fracturing 1. Conchoidal – smooth curve 2. Hackly – sharp jagged edges 3. Uneven – rough and irregular 4. Fibrous – shows fibers Some minerals are cut to become precious gemstones. Erik E. Mason

Aggregate losses are estimated by in-cooperating both claim frequency and claim severity distributions. Pavel (2010) reviewed methods used to calculate distributions of aggregate losses. Robertson (1992) applied Discrete Fourier Transform in estimation of aggregate losses from frequency and severity distributions. Ronoet al. (2020) developed compound distribution to model extreme natural disasters in Kenya. Mohamed et al. (2010) introduced use of simulation approach in estimation of aggregate losses which can be employed when frequency and severity distribution cannot be combined to derive a compound distribution. Aggregate loss distributions are based on collective risk model expressed as: is the severity distribution and N is the claim count distribution. The distribution of N in this paper is considered to follow mixed PH Poisson distributions. Phase type distributions are constructed, when mixture distributions are convoluted resulting to an interrelated Poisson process occurring in phases. Phase type distributions were introduced way back by Erlang (1909) and it has been advanced by Marcel F. Neuts (1981) and Assussen (2003) among others. Mogens Bladt (2005) introduced phase type distributions in risk theory while O’cinneide (2017) highlighted on Phase type distributions as well as their invariant polytopes. Wu et al. (2010) developed phase type distributions when frequency distributions followed Panjer class while Kok et al. (2010) used phase type distributions of Panjer class to model claim frequency. Markov chains were introduced by Andrei Markov (1856-1922). Nurul et al. (2019) proposed a simple forecasting model of predicting the future air quality using Markov chains which in-cooperated the Markov chains as an operator of evaluating pollution distribution in the long run. Yajuan et al. (2018) used Markov chains to model demand for stations in Bike sharing systems. In this study, the concept of Markov chains is used to determine the matrices of the phase type distributions used in modeling claim frequency. Frequency data is used to model occurrences in different areas such as engineering, insurance, biology etc. Poisson distribution is often used to model count data; however, it is based on the assumption that variance to mean ratio is unity (equi-dispersion) which is not applicable to real data; hence, it is considered as an inflexible model. Most real life data either experience over dispersion where variance exceeds the mean or under dispersion where the mean exceeds the variance which can be modeled using Poisson mixtures . Poisson Lindley distributions are perfect examples of Poisson mixtures where characteristics of Poisson distribution follow some characteristics of Lindley distribution. One parameter Poisson Lindley which can model over dispersed data was introduced by Sankaran (1970) while Shanker and Mishra (2014) developed two parameter Poisson Lindley which further research has justified that it can model over dispersed data. In the insurance sector, when calculating aggregate losses for chronic diseases which have various stages like cancer the claim frequency distributions considered do not in-cooperate the different stages of such diseases. In-cooperating phase type distributions solve this short coming of ordinary distributions. Further considering mixed phase type distributions improves modeling of claim frequency data as it considers the heterogeneity aspect of claim data. In this paper, we develop PH one parameter Poisson Lindley distribution and PH two parameter Poisson Lindley distributions where the mixing distribution follows PH Lindley distribution. The resulting PH distributions are used to model claim numbers of secondary cancers in Kenya. Section 1 has a brief introduction to Poisson distributions and Poisson Lindley distributions. The structure of this paper is as follows: Section 2 will discuss construction of phase type distribution using PH Lindley distributions which will later be applied in modeling of the aggregate losses. Compound distributions from the frequency and severity distributions are developed in Section 3. Aggregate losses for the data are estimated using Discrete Fourier Transforms and the results discussed in Section 4 and Section 5 outlines the conclusions. 2. Proposed Phase Type Poisson Lindley Distributions In this section we develop phase type distributions for one parameter Poisson Lindley and two parameter Poisson Lindley. Phase type Poisson Lindley distributions are derived when the mixing distribution follow phase type Lindley distribution. 2.1. Phase Type One Parameter Poisson Lindley Distribution Definition 1. A random variable X is said to be a phase type one parameter Poisson Lindley distribution if it follows: for and is matrix. Theorem 1. If distribution then the probability distribution function of X is: where is and I is an identity matrix. If and , then the pdf of variable X is expressed as; where is . Properties of Phase Type One Parameter Poisson Lindley Distribution The rth moments of PH-OPPL distribution is given by: The expectation and variance of PH-OPPL distribution can be easily obtained from Equation (4) as: The probability generating function of PH-OPPL distribution is given by: The parameter of PH-OPPL distribution is estimated using continuous Chapman-Kolmogorov equation. 2.2. Phase Type Two Parameter Poisson Lindley Distribution Definition 2. A random variable X is said to be a phase type two parameter Poisson Lindley distribution if it follows: for and is matrix. Theorem 2. If distribution then the probability density function of X is expressed as: where , is and I is an identity matrix. If and , then the pdf of variable X is given by; where is . Properties of Phase Type Two Parameter Poisson Lindley Distribution The rth moments of PH-TPPL distribution is given by: The expectation and variance of PH-TPPL distribution can be easily obtained from Equation (10) as: The probability generating function of PH-TPPL distribution is given by: The value of is known hence the value of can be obtained from Equation (11) if the value of is known. 2.3. Shape of Probability Function of PH-OPPL and PH-TPPL Distributions Matrix was determined using continuous Chapman-Kolmogorov equation for cancer data in Kenya and the values of is the stationary probabilities obtained using the formula . The values of for three state Markov model represents cancer patients who transit from Healthy-Leukemia-Dead states, four state Markov model represents patients who transit from Healthy-Liver-Colon-Dead states, five state Markov model represents Healthy-Stomach-Pharynx-Colon-Dead states and six state Markov model represents patients transiting from Healthy-Oesophagus-Stomach-Lung-Kidney-Dead states. The values of for different states are: The shape of probability function of phase type one parameter Poisson Lindley is expressed as: Figure 1 shows that phase type one parameter Poisson Lindley is a long tailed distribution. The shape of probability function of phase type two parameter Poisson Lindley is expressed as: Figure 2 shows that phase type two parameter Poisson Lindley is a long tailed distribution. (a) (b) (c) (d) Figure 1. Pdf plots of PH-OPPL for different values of Λ. (a) (b) (c) (d) Figure 2. Pdf plots of PH-TPPL for different values of Λ. 3. Compound Phase Type Distribution Compound distribution in the actuarial field is the total loses in the group of insurance policies. In this section we develop compound phase type distributions (CPHD) which can be used to model secondary cancer cases. Definition 3. Let N be a r.v with probability generating function and be a set of iid random variable with a common probability generating function and is independent of N, then the probability generating function of the compound distribution is expressed as: Unlike ordinary compound distributions which do not consider transition phases of diseases, (CPHD) in-cooperates the transition states. Probability generating functions of compound distributions can be derived by convolution of probability generating function of two distributions as shown in Equation (14). Theorem 3 (Compound one parameter Poisson Lindley distribution). If the pgf of the compound pgf of N is: where is the Laplace transform of the severity distribution as most continuous distributions their pgf is not available. Theorem 4 (Compound two parameter Poisson Lindley distribution). If the pgf of the compound pgf of N is: where is as defined in theorem (3). The continuous distributions considered in this research are; Weibull, Pareto and Generalized Pareto distributions hence their Laplace transforms will be derived and replaced in Equations (16) and (18) to get the pgf of their compound distribution using PH-OPPL and PH-TPPL distributions respectively. The Laplace transform of Weibull, Pareto and Generalized Pareto are derived as: 1) Weibull distribution 2) Pareto distribution 3) Generalized Pareto distribution Replacing Equations (19), (20) and (21) in Equation (16) the pgf of the compound distributions of PH-one parameter Poisson Lindley with Weibull, Pareto and Generalized Pareto respectively are: 1) Compound PH-OPPL-Weibull distribution 2) Compound PH-OPPL-Pareto distribution 3) Compound PH-OPPL-Generalized Pareto distribution Replacing Equations (19), (20) and (21) in Equation (18) the pgf of the compound distributions of PH-two parameter Poisson Lindley with Weibull, Pareto and Generalized Pareto respectively are: 1) Compound PH-TPPL-Weibull distribution 2) Compound PH-TPPL-Pareto distribution 3) Compound PH-TPPL-Generalized Pareto distribution 4. Data Analysis, Results and Discussions 4.1. Severity and Frequency Probabilities The cancer data considered in this research is obtained from a medical facility in Kenya. The cancer transitions states considered are Healthy-Leukemia-Dead states for 3 state model, Healthy-Liver-Colon-Dead states for four state model, Healthy-Stomach-Pharynx-Colon-Dead states for five state model and Healthy-Oesophagus-Stomach-Lung-Kidney-Dead states for six state models. The values of for the data are obtained using continuous Chapman-Kolmogorov equations expressed as: The values of for three, four, five and six state using the data obtained were as shown in Section 2.3. The severity distributions considered in this research are Weibull, Pareto and Generalized Pareto distributions. DFT requires severity probabilities to be discrete hence they will be discretized using method of mass rounding which is expressed as: The pdf of Wei-bull, Pareto and Generalized Pareto distributions respectively are expressed as; The frequency and severity probabilities for secondary cancer cases are: (Table 1). Table 1. Claim frequency and severity probabilities. 4.2. Discrete Fourier Transform There are different numerical methods used in estimation of aggregate losses such as; Monte Carlo, Panjer recursive model, Fourier transforms and Direct Numerical Integration. Panjer recursive model is applicable when the claim frequency distributions follow either Panjer class or class . In this study we will consider Discrete Fourier Transform (DFT) in estimation of the aggregate losses. Robertson (1992) applied Fourier transforms in computation of aggregate losses . Pavel (2010) reviewed these numerical methods and concluded that each method had it strength and weaknesses hence they should be chosen according to the study. DFT mostly preferred as it is arguably said to be the most elegant and powerful technique in evaluation of aggregate loss probabilities when claim amount is both discrete and continuous . The algorithm of DFT of aggregate losses requires computation of DFT of frequency and DFT of severity separately. Definition 4 (Discrete Fourier Transform). Let be the severity or frequency distribution of the claim data. For any discrete function the Discrete Fourier transform is the mapping; Expression (29) is very complex to work with hence to reduce its complexity we apply Euler’s formula and it becomes: which is the DFT of the severity or frequency probabilities. The severity and frequency probabilities are of length 8 and hence the matrix W must be a primitive 8th root of unity therefore Equation (30) can be rewritten as: The frequency or severity probabilities will be padded with equal number of zero’s as its elements in order to perform no wrap convolution. The DFT algorithm is as follows: 1) Multiply the matrix with the frequency or severity probabilities to get the DFT of frequency or severity probabilities. 2) Compute DFT of DFT of frequency and severity by multiplying DFT of frequency probabilities with the DFT of the severity probabilities and consequently multiplying the resulting vector with the matrix . 3) Select the values without the complex i and divide each value by the number of elements in the vector of frequency or severity distribution and arrange the resulting probabilities in reverse except for the first probability. 4) Values corresponding to original frequency and severity values are the aggregate loss probabilities. The values of aggregate loss probabilities using DFT are: Table 2. Aggregate loss probabilities. The values of Table 2 can be represented graphically as: (a) (b) (c) Figure 3. Aggregate loss probabilities. Figure 3(a) shows aggregate loss probabilities using PH-OPPL distribution with severity distributions and it indicates that PH-OPPL with Weibull and Pareto were similar to the actual aggregate loss probabilities while PH-OPPL with generalized Pareto distribution overestimate the aggregate losses for six state model. Figure 3(b) shows aggregate loss probabilities using PH-TPPL distribution with Pareto and generalized Pareto provided a better fit for secondary cancer data while PH-TPPL with Weibull overestimated the aggregate losses. However, PH-OPPL with Weibull and PH-TPPL with generalized Pareto provided a better fit compared to PH-OPPL-Pareto model and PH-TPPL Pareto respectively hence they are compared in Figure 3(c) indicating that PH-OPPL with Weibull provided the best fit for aggregate loss data of secondary cancers in Kenya. PH-OPPL-Weibull model can be used to provide better estimates of aggregate losses for secondary cancer data in Kenya. Mixed phase type distributions are developed to model secondary cancer cases in Kenya. Unlike ordinary distributions which do not in-cooperate the transition of different states, the distributions proposed here take into consideration transition states while modeling claim frequency data. The distributions are based on Poisson and Lindley distributions, where PH-OPPL-Weibull provided the best for PH-OPPL models while PH-TPPL-Generalized Pareto provided the best fit for PH-TPPL models. This model improves estimation of aggregate loses as it in-cooperates transition probabilities of different states of cancer as well as heterogeneous aspect of claim data. This greatly improves estimation of insurance policies for diseases which transit to different state such as cancer hence improving the financial positions of the insurance firms as it will improve estimation of its reserves. This model, however, is only applicable in risk theory for diseases which have multiple transitions states. Further research can be done on this study factoring in patients who were censored in this study and also the same study can be carried out for disease such as HIV-AID which has transition states. The data used to support the findings of this study can be availed upon request. Rono, A., Ogutu, C. and Weke, P. (2020) On Compound Distributions for Natural Disaster Modelling in Kenya. International Journal of Mathematics and Mathematical Sciences, 2020, Article ID: 9398309. Kok, S. and Wu, X. (2010) Matrix-Form Recursive Evaluation of the Aggregate Claims Distribution Revisited. Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Melbourne. Nurul, N.Z., Mahmod, O., Rajalingam, S., Hanita, D., Lazim, A. and Evizal, A.K. (1981) Markov Chain Model Development for Forecasting Air Pollution Index of Miri, Sarawak. Journal of Sustainability, 11, 5190. Zhou, Y.J., Wang, L.L., Zhong, R. and Tan, Y.L. (2018) A Markov Chain Based Demand Prediction Model for Stations in Bike Sharing Systems. Journal of Mathematical Problems in Engineering, 2018, Article ID: 8028714. Das, K.K., Ahmed, I. and Bhattacharjee, S. (2018) A New Three-Parameter Poisson-Lindley Distribution for Modelling Over-Dispersed Count Data. International Journal of Applied Engineering Research, 13, 16468-16477.

#1. Minimum footing widths cannot be determined by Division B, Article 9.15..3.4. to 220.127.116.11. when: #2. The minimum length of nails for the attachment of metal siding is: #3. A Dead end corridor exists in a mercantile, multi-tenant floor area. What are the specifics for this dead end? #4. When an exterior air barrier system is penetrated by an exterior door, such air barrier system could NOT be sealed to the door frame with: #5. When used as cladding, untreated wood must clear the ground by not less than: #6. The handrail on the stair between the first and second floor of the House must be continuous throughout the length of the except at the: #7. Consider a one-storey house and calculate the minimum width of the strip footing supporting the exterior foundation wall. #8. When the foundation wall of a House is constructed as a prescribed flat ICF foundation wall, the maximum spacing of horizontal reinforcement for the wall is: #9. Tents not occupied by the public are required to be separated from each other by #10. What is the minimum required fire-resistance rating for the top storey of an exit stair shaft, serving a two storey building of residential occupancy? #11. Which one of the following is NOT an acceptable method of supporting an assembly which is of non-combustible construction, and required to have a fire resistance rating of not less than 1.5 h: #12. An carport used for the storage or parking of motor vehicles is considered a garage if it has #13. 15.9 mm thick gypsum wall board attached to steel studs in an unrated partition with a stud spacing of 406 mm o.c. must be fastened with screws spaced at: #14. Collar ties must be laterally braced at right angle with a 19 mm × 89 mm brace, if they span more than: #15. According to Part 5 of the Building code, other fenestration assemblies and their components related to air leakage shall have an air leakage characteristic, measured at an air pressure difference of 75 Pa and tested in accordance with ASTM E283 standard, that is not greater than: #16. Plywood panels used as cladding, when applied directly to studs spaced at 610 mm o.c. must not be less than: #17. An exit opens directly into a lobby in an apartment building. The floor is unsprinklered, the lobby is enclosed with a fire separation having a 1hourfire resistance rating. The lobby is 1 m above the ground level and the distance of travel from the exits to the outdoors is 15 m and a suite door opens into a vestibule which in turn opens into the lobby #18. The minimum roof slope for low slope application of asphalt shingles is: #19. Combined occupancies that are prohibited include; #20. In a three storey Group D building, the windows located above the first storey are required to #21. The minimum size for steel columns is; #22. The minimum diameter for nails used for the attachment of stucco lath is; #23. A second storey window in a lounge area in a seniors' apartment building (i.e., Group C Residential and NOT Group B lnstitutional)extends to within 300 mm of the finish floor. That window is required to be #24. Consider the storage garage of a HOUSE it is detached from the dwelling unit it serves but attached to another storage garage on the adjacent property, the party wall between each section of the storage garage must be constructed as a fire separation with a fire-resistance of not less than: #25. When the foundation wall of the House is constructed as a prescribed flat ICF foundation wall, the minimum thickness of the flat ICF foundation wall is: #26. A doorway in a Barrier-Free path of travel must be at least; #27. What is the minimum thickness for hardboard interior finish panels placed directly on studs at 406 mm o.c.? #28. Where methane or radon gas are known to be a problem, construction shall comply with the requirements for soil gas control as found in the supplementary Standard. Where this condition is present, floor drain penetration in a floor-on-ground shall be sealed with: #29. Vertically applied metal siding may only be fastened: #30. When the minimum required Limited Distance (LD) is doubled under the provisions of Sentence 18.104.22.168.(1), the aggregate area of unprotected opening (UO): #31. An exterior exit door is located in a wall at the main floor of a three storey building and faces 90 degree to the exterior wall. There is no internal enclosure to the exit stair and the main floor has a single tenancy (i.e., no public corridor). Adjacent to the exit door is a window which is 1 200 mm away from the door; this window; #32. Consider a retail store in downtown Thunder Bay (s=2.9 kpa, Sr=0. 4 kpa) that is 10 m x 25 m in plan area, one storey in height, and has a flat roof. The roof trusses are 800 mm on center and span 10 m. The minimum specified snow load to be used for the design of the wood trusses (See Division B, Article 22.214.171.124.) #33. For a recreational camp with an anticipated occupant load of 40 people, how many water closets are required? #34. The maximum roof slope for asphalt base (without gravel) of Built-up Roofing is #35. For a House, a guard is required when the difference in elevation between two floor or between a floor or other walking surface is not protected by a wall and the difference in elevation is #36. Loadbearing elements in a three storey building of masonry construction are required to have earthquake reinforcement described in Subsection 9.20.15. if the seismic spectral acceleration #37. If a landing is required, the minimum size of the exterior landing of an house main entrance door is: #38. The rear sliding door in a house on the main floor level, where the finished floor on one side of the floor is more than 600mm above the ground level on the other side of the sliding floor, shall be: #39. According to Row 2, Column 3 of Table 126.96.36.199., the minimum width of strip footings supporting interior walls of a two storey house is 350 mm. If the House was to be constructed on gravel, sand or silt where the elevation of the water table was 450 mm below the underside of footing elevation, the width of the footing supporting an interior load- bearing partition (without masonry) would need to be a minimum of: #40. A food premises is exempt from the requirements for lavatories, appliances and sinks if it is not more than 56 m2 in area and #41. A hollow unit masonry column used in a cottage may have a maxim length of: #42. Windows may be omitted from a basement recreation room or an unfinished basement when #43. Where the exterior walls of an exit intersect the exterior walls of a building wired glass in steel frames or glass block is required for either the building windows or the exit enclosure windows, where the walls meet at an angle #44. The maximum force permitted to open an interior exit door in a Barrier-Free path of travel is; #45. Eave protection under asphalt shingles applied on slopes of 1 in 3 is to extend up the roof slope a minimum distance of #46. Mirrored glass doors conforming to CAN/CGSB-82.6-M, “Doors, Mirrored Glass, Sliding or Folding, Wardrobe” and mirrored glass doors reinforced with a film backing that meet the impact resistance requirements specified in CAN/CGSB-12.5-M, “Mirrors, Silvered” may be used #47. A home being constructed in Windsor, Ontario has been changed to include an addition of an indoor swimming pool. The wall assemblies of the swimming pool incorporate materials with a water permeance of less than 60 ng/(Pa×s×m²). High moisture generation will occur and therefore the wall assemblies;? #48. The minimum turning diameter for a wheelchair in a Special Washroom is #49. A roof, with Code complying asphalt fibreboard sheathing, may support the following type of roofing. #50. Consider a masonry wall requiring earthquake reinforcement under Article 188.8.131.52. If it is determined that the total amount of reinforcing steel is to be 750 mm², what is the minimum amount of horizontal steel in this case? #51. The double hung windows of a House have a pane area of approximately 0.39 m2 . A consideration of Sentence 184.108.40.206.(1) indicates that if the Mitec House was to be constructed in Goderich, the minimum thickness of the factory-sealed IG units would be: #52. Metal ties for multiple wythe masonry cavity walls are to be spaced at; #53. The function of the of the second plane of protection is to: #54. For the one-storey detached House. The footing area for columns spaced 3 m o.c. will need a minimum of: #55. The minimum length of nails for the attachment of vinyl siding is: #56. If the foundation wall type is solid concrete with a compressive strength of 15 MPa, Based on Division B, Article 220.127.116.11, what would be the minimum acceptable wall thickness “C” (allowable soil bearing capacity 75kpa): #57. The finished interior floor level in a Group D occupancy is 190 mm below the sliding door or sill while the distance from the finish floor to the ground level is 600 mm. The door must be; #58. Determine the minimum required number of water closets for a group F, Division 2 occupancy located in a one (1) storey, 600m2 building. The Group F, Division 2 occupancy is located in a suite that is 400m2 in area, and the occupant load of the suite is 18 persons. #59. What is the minimum shank diameter of casing nails or finishing nails used for fastening sheets of insulating fibreboard finish? #60. In a house containing two dwelling units with an attached garage, what is the minimum number of self-contained mechanical ventilation systems that can be designed to serve the entire house? #61. As a general rule, the provisions found in Section 9.20. apply to unreinforced walls that are; #62. A building has 4 pedestrian entrances. How many of those entrances must be Barrier-Free entrances? #63. A Barrier-Free corridor is less than 1 600 mm wide throughout it's length, then widened areas must be provided every 30 m max. These widened areas are; #64. What is the minimum distance required between a roof surface and cladding that is adversely affected by moisture? #65. What is the minimum metal thickness of steel studs exclusive of any coatings in an interior non-loadbearing and unrated wall? #66. Closures for doors in a Barrier-Free path of travel and opening into entrances to normal dwelling units, must open with a force of not more than; #67. A door serving a suite in a hotel and opening into a public corridor 'f'Siilr #68. According to the prescriptive solutions in Division B Part 9, when the space between roof joists in a cathedral ceiling with a slope of 1 in 3 are installed, the ratio of vent area to insulated ceiling area to be provided shall be not less than: #69. The minimum vertical clearance from an entrance to a parking storey provided with Barrier-Free access is; #70. In accordance with Detail ED-1 in SB-7, what is the minimum deck joist size for this construction detail? #71. Slab on ground that support a roof load in a one storey part 9 building #72. The maximum clear span between supporting walls of a prescribed reinforced concrete slab is: #73. In calculating the actual unobstructed glass area provided by windows and skylights, the Designer and the Plans Examiner: #74. Whlat is the minimum lintel size over a 4.2 m opening in an exterior wall made of vertical logs and supporting only roof and ceiling loads 1f the lumber 1s Spruce-Pine-Fir #1 and the specified snow load is 1.5 kPa? #75. We are design a house in Toronto and package A1 is selected. What’s the above grade wall min. effective R value is required?

2 edition of Arithmetical problems found in the catalog. G. H. Armstrong Cover title: Armstrong"s arithmetical problems : containing entrance and senior leaving papers, 1880 to 1897. |Other titles||Armstrong"s arithmetical problems| |Statement||prepared and selected by G.H. Armstrong.| |The Physical Object| |Number of Pages||98| Arithmetic (from the Greek ἀριθμός arithmos, "number" and τική, tiké [téchne], "art") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and inspirationdayevents.cometic is an elementary part of number theory, and number theory is considered to be one of the top-level. Apr 15, · RRB Group D Reasoning Arithmetical Reasoning Mental Ability in Telugu Part 3 For All Railway Exams. Verbal Reasoning and Arithmetic Reasoning . Book Description. Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises. With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the. Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb. Mar 31, · If you’ve ever tried to solve mathematical problems without any idea how to go about it, this book is for you. It will improve your ability to solve all kinds of mathematical problems whether in mathematics, science, engineering, business, or purely recreational mathematical problems (puzzles, games, . Problems in mathematical analysis book. Read reviews from world’s largest community for readers. PREFACE This collection of problems and exercises in 4/5. Agreement between the Treasury Board of Canada and the Public Service Alliance of Canada The wild cat and the kitten Bible records of Suffolk and Nansemond County, Virginia together with other statistical data Pressurization and expulsion of a flightweight liquid hydrogen tank Port and sherry My first bird book incarnation of the Son of God Analysis of completion sentences and arithmetical problems as items for intelligence tests by Rinsland, Henry Daniel and a great selection of related books, art and collectibles available now at inspirationdayevents.com Book Awards Book Club Selections Books by Author Books by Series Coming Soon Kids' Books New Releases Teens' Books This Month's Biggest New Releases Subjects Biography Business Cookbooks, Food & Wine Current Affairs & Politics Diet, Health & Fitness Fiction Graphic Novels & Comics History Mystery & Crime Religion Romance Sci-Fi & Fantasy. Arithmetical Problems [Anonymous] on inspirationdayevents.com *FREE* shipping on qualifying offers. This is a reproduction of a book published before This book may have occasional imperfections such as missing or blurred pagesAuthor: Anonymous. This handy book is a compilation of practice problems, with separated chapters for both hints and solutions to each problem. The problems are especially chosen for students preparing for undergraduate math competitions, but these challenging brain-teasers will be of interest to anyone interested in math problems dealing with Arithmetical problems book numbers, differential equations, integrals, polynomials Cited by: 2. Home» Problem Books. MAA Book. A Gentle Introduction to the American Invitational Mathematics Exam. MAA Book. Euclidean Geometry in Mathematical Olympiads. MAA Book. Hungarian Problem Book IV. Mathematical Olympiads Problems and Solutions Arithmetical problems book Around the World. MAA Book. The Red Book of Mathematical Problems (Dover Books on Mathematics series) by Kenneth S. Williams. In North America, the most prestigious competition in mathematics at the undergraduate level is the William Lowell Putnam Mathematical Competition. This volume is a handy compilation of practice problems, hints, and solutions indispensable for. lesson concepts. Take the assessments without the use of the book or your notes and then check your answers. If you are using this material as part of a formal class, your instructor will provide guidance on which problems to complete. Your instructor will also provide information on accessing answers/solutions for these problems. Many of the problems in this book were suggested by ideas originating in a variety of sources, including Crux Mathematicorum, Mathematics Magazine, and the American Mathematical Monthly, as well as various mathematical competitions. This result is a rich selection of carefully chosen problems that will challenge and stimulate mathematical. Sep 28, · My recommendation for this are as follows 1) G. Alexanderson, L. Klosinski, and L. Larson, The William Lowell Putnam Mathematical Competition, Problems and. Basic Arithmetic - Word Problems Basic Arithmetic - Problem Solving Divisibility by 3 Basic Arithmetic: Level 2 Challenges Basic Arithmetic - Problem Solving. The sum of two numbers is and their difference is What is the value of the larger number. Submit Show explanation. New Book Uses Physical Reasoning to Solve Mathematical Problems 22 April Mark Levi, professor of mathematics at Penn State, has authored a book titled "The. Popular selection of practice problems — with hints and solutions — for students preparing for undergraduate-level math competitions. Subjects range from multivariate integration to finite series to infinite sums and classical analysis. Includes questions drawn from geometry, group theory, and linear algebra, plus brainteasers dealing with real numbers, differential equations. Oct 31, · The title of this book, though apparently not explained in the book itself, is obviously a nod to the famous dictum of Paul Erdős that God maintains The Book, in which are located the best and most elegant proofs of mathematical results. (See, for example, Proofs. Course Arithmetical Problems ; Course Arithmetical Problems Review. Buy Course Arithmetical Problems on eBay now. No Results for "course arithmetical problems " Shop Bible for sale on eBay now. C Holy. C Holy Bible American Hamilton Family North Strabane Washington County Pa. $ This unique book is equally useful to both engineering-degree students and production engineers practicing in industry. The volume is designed to cover three aspects of manufacturing technology: (a) fundamental concepts, (b) engineering analysis/mathematical modeling of manufacturing operations, and (c) + problems and their solutions. Mathematical Problems Lecture delivered before the International Congress of Mathematicians at Paris in By Professor David Hilbert 1. Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries. Broad collection of shorter course arithmetical problems at hard to beat prices. Find Shorter Course Arithmetical Problems on sale today online. Kirkus Reviews said Stewart "succeed[ed] in illuminating many but not all of some very difficult ideas", and that the book "will enchant math enthusiasts as well as general readers who pay close attention". Robert Schaefer from the New York Journal of Books described "The Great Mathematical Problems" as "both entertaining and accessible", although later noted that "in the end chapters Author: Ian Stewart. The Green Book of Mathematical Problems - Kenneth Hardy & Kenneth S inspirationdayevents.com The Green Book of Mathematical Problems - Kenneth Hardy & Kenneth S inspirationdayevents.com Sign In. Details. DESCRIPTION: See also A SECOND STEP TO MATHEMATICAL OLYMPIAD PROBLEMS The International Mathematical Olympiad (IMO) is an annual international mathematics competition held for pre-collegiate students. It is also the oldest of the international science olympiads, and competition for places is particularly fierce. This book is an amalgamation of the first 8 of 15 booklets originally. Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving Mathematical Problems includes numerous exercises and model solutions throughout.Jun 25, · At the end of the book answers and solutions to all the problems have also been provided As the book covers the varied aspects of Mathematical Analysis with the help of ample number of examples and practice questions, it for sure will serve as a complete textbook for practicing the various elements of Mathematical Analysis/5(96).Get this from a library! The Greek Anthology, Volume V: Book Epigrams in Various Metres. Book Arithmetical Problems, Riddles, Oracles. Book Miscellanea. Book Epigrams of the Planudean Anthology Not in the Palatine Manuscript. [W R Paton].

The oscillations pendulum system In mechanics and physics, simple harmonic motion is a special type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Abstract this paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. Module f12ms3: oscillations and waves as a mass on a spring or a simple pendulum in the spring-particle system when the spring is stretched or compressed. The shown video focusses on simulating damped oscillations of a simple pendulum the matlab code for the program is provided in description. Oscillations, for example, the undamped pendulum is a conservative system: total energy is a constant over system trajectories using conservation of energy. 23 cantilever linear oscillations study of a cantilever oscillation is a rather science - intensive problem in many cases the general solution to the cantilever equation of motion can not be obtained in an analytical form. Chapter 28 oscillations: the simple pendulum, energy in simple harmonic motion 199 θ ml2 −mgl a= something profound occurs in our simplification of this equation. So what exactly is an oscillating system in short, it is a system in which a particle or set of particles moves back and forth whether it be a ball bouncing on a floor, a pendulum swinging back and forth, or a spring compressing and stretching, the basic principle of oscillation maintains that an oscillating particle returns to its initial. Oscillation is a type of motion it is a kind of periodic motioncontrast with rectilinear motion, the oscilatory motion involves the movement in to and fro directions like a pendulum, vibrating tuning fork or rocking in a cradle. Free undamped and damped vibrations lab report abstract a mechanical system is said to be vibrating when its component part are undergoing periodic oscillations about a central statical equilibrium position any system can be caused to vibrate by externally applying forces due to its inherent mass. The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension such a system may. 15 oscillations 151 simple harmonic motion any motion that repeats itself at regular intervals is called harmonic motiona particle experiences a simple harmonics motion if its displacement from the origin as function of time is given by. Tuning of a radio is the best example of electrical resonance when we turn the knob of a radio, to tune a station, we are changing the natural frequency of electrical circuit of receiver, to make it equal to the transmission frequency of the radio station. 1 the simple pendulum consider a simple pendulum consisting of a mass \(m\) fixed by a light but rigid rod of length \(l\) gravity acts on the mass with a force \(f=mg\) directed straight down. 1 the forced damped pendulum: chaos, complication and control john h hubbard this paper will show that a \simple difierential. Oscillation vs simple harmonic motion oscillations and simple harmonic motion are two periodic motions discussed in physics the concepts of oscillations and simple harmonic motion are widely used in fields such as mechanics, dynamics, orbital motions, mechanical engineering, waves and vibrations and various other fields. Data collection a lift and release a 400 g mass to start the oscillationstart the data-logging software and observe the graph for about 10 seconds b before the oscillation dies away, restart the data-logging software and collect another set of data, which can be overlaid on the first set. A simple pendulum is one which can be considered to be a point mass suspended from a string or rod of negligible mass it is a resonant system with a single resonant frequency for small amplitudes, the period of such a pendulum. Chapter 1 oscillations david morin, [email protected] a wave is a correlated collection of oscillations for example, in a transverse wave traveling. Oscillations period: is defined as the time taken for one complete oscillation frequency: is defined as the number of oscillations per unit time, f = 1 / t angular frequency ω: is defined by the eqn, ω = 2 π f. Where is the damping constant, and the undamped oscillation frequency suppose, finally, that the piston executes simple harmonic oscillation of angular frequency and amplitude , so that the time evolution equation of the system takes the form. Damped oscillations, forced oscillations and resonance damping in an oscillating system figure shows some oscillating systems each oscillating system will oscillate with a smaller and smaller amplitude and eventually stop completely this is due to energy loss from the oscillating systems resulting from factors like air resistance and. Oscillations in damped driven pendulum: a chaotic system international journal of scientific and innovative mathematical research (ijsimr) page 16. Properties of the damped up: oscillations previous: the physical pendulum contents damped oscillation so far, all the oscillators we've. Phas1240lab nisha lad 1 study of a damped oscillating torsion pendulum driven into resonance nisha lad, charlie hempsted, gabriella driessen, johan m’quillan and sophia zhong. Factors affecting the time period for oscillations in a mass-spring system when a mass is attached to the end of a spring the downward force the. A simulated inverted pendulum consisting of a bob on a shaft the simulation of the physical system treats the cart high-frequency oscillations of the. Lecture 2 • vertical oscillations of mass on spring • pendulum • damped and driven oscillations (more realistic) outline. The experimental study of damping in a time-varying inertia pendulum is presented the system consists of a disk travelling along an oscillating pendulum: large swinging angles are reached, so that its equation of motion is. A simple pendulum consists of a mass m hanging from a string of length l and fixed at a pivot point p when displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion.

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Whether it’s sharp angles or smooth lines between two points, this calculator turns complex math into simple answers fast! - Common Angle Conversions: Selection of units of measurements Choosing the right units for measuring is important when using an arc length calculator. You might need to use inches, centimeters, or even millimeters based on what you are working on. The same goes for calculating areas and lengths. You can pick from square inches or square centimeters if you’re looking at area, or maybe meters if you’re measuring a longer distance. The calculator lets you easily switch between different units so your numbers make sense for your project. For angles, you may need degrees or radians depending on your math problem. This flexibility makes sure that all of your measurements fit together nicely and helps you solve problems accurately without any mix-ups. You can also solve for the length of the sides of a right triangle using our Pythagorean Theorem Calculator. The Formula for Arc Length Understanding the precise formula for calculating arc length is crucial, as it serves as the foundation for accurately measuring curved distances within various sectors of a circle; continue reading to uncover its intricacies and practical applications in numerous fields. s = ϴ × r for radians To find out how long a piece of a circle’s edge is, we use the arc length formula s = ϴ × r. This works when you measure the slice of the circle in radians. Imagine drawing a straight line from the center of a pizza to the crust – that’s your radius (r). Next, look at how wide your pizza slice is – that’s your angle (ϴ), but it has to be in radians for this magic formula to work right. Think about using this simple rule next time you need to figure out an arc’s size on any circle. It’s super useful for math problems and lots more – like building things or understanding space! All you have to do is multiply two numbers: the radius and the radian measure of your central angle. Just like that, you have your arc length! It might sound tricky with words like “radians” and “radius,” but once you try it, it makes perfect sense. s = 2 π r (θ/360°) for degrees When you want to find out how long an arc is, and you have the angle in degrees, this formula works like magic. You take that angle and divide it by 360°. Why 360°? Because that’s how many degrees are in a full circle! Next, multiply your answer with 2 π (pi), which is about 3.14. Now take the circle’s radius – the distance from the center to the edge – and multiply that by your first answer. That number you get is your arc length! Let’s say you’re working on a project where precision matters. You wouldn’t want to guess or be off even by a little bit, right? With this formula, you can be sure about the measurement of any part of a circle’s edge between two points—the arc length you’re looking for! It’s important for making things fit together perfectly, whether it’s parts of machines or pieces of art that need curves just right. Use these steps with care to get accurate results every single time. Solved example demonstration Let’s look at a real example to see our arc length calculator in action. Imagine you have a slice of pizza that is part of a larger 14-inch pie, and the tip of your slice makes a 60-degree angle from the center. You want to know how long the crust is along the edge of your slice. First, you enter the radius (7 inches since it’s half of 14 inches) and select degrees as your unit for measuring angles. Then type “60” into the central angle box. The calculator instantly works out that your crust – or arc length – is about 7.33 inches long! This quick example shows exactly how useful the calculator can be for all sorts of problems involving circles. Now imagine using this tool in complex designs or astronomy calculations; you’ll quickly find how vital an accurate arc length measurement can be! And we’ve only just touched on one aspect – there’s also diameter, sector area, and more that it can calculate for you. Now you can also calculate the volume, surface area, and other properties of a sphere with our comprehensive Sphere Calculator. Benefits of the Arc Length Calculator The Arc Length Calculator stands as an indispensable tool, streamlining computations with unmatched precision and simplifying complex geometrical tasks. Its user-centric design facilitates seamless conversion across various units, ensuring its utility extends from academic classrooms to the intricate demands of professional fields. Accuracy in calculating arc length Getting the arc length right matters a lot, especially in fields like engineering and astronomy. Using an arc length calculator helps you find this important number without mistakes. You just type in what you know—like radius and angle—and it works out the arc length for you. It’s really precise because it uses tested formulas for its math. With this tool, anyone can be sure to get exact numbers every time, no matter if they’re working with radians or degrees. And that means less stress about getting things wrong and more trust in your final answers. This makes projects smoother because accurate measurements guide good decisions along the way. Ease of use for different measurements The Arc Length Calculator makes working with different measurements simple. You can pick from many units like inches, centimeters, or meters to get your results the way you need them. Just type in what you know about the circle, like the radius or angle, and choose your preferred unit. The calculator does all the hard math for you. It’s great for people who use these calculations at work or school because it saves time and helps avoid mistakes. Next, let’s explore how knowing the right formula makes finding arc lengths even easier. Applicability in various fields such as science, engineering, and astronomy Scientists, engineers, and astronomers often need to measure curves and circles. They use arc-length calculators for this work. For example, in astronomy, they may calculate the path of a planet or star. Engineers might need it to design parts that curve. In science classrooms, teachers show how to find the size of arcs in experiments. This tool helps to get answers quickly and right. It lets users pick different units like meters or inches for their measurements. Users can easily add this calculator to their own websites too. Now let’s look more at the formula used in an arc length calculator. You can also determine the arcsin (inverse sine) of a value effortlessly using our Arcsin Calculator. 1. Does an arc length calculator work with different shapes? Sure does! It can handle circles and other curves like ellipses or heart-shaped cardioid figures by using math rules for those shapes. 2. Is an arc length only used in flat shapes? Nope! Arc lengths are also important when you deal with 3D stuff like balls or tubes, known as solids of revolution because they spin around to make their shapes. 3. Do I need to know special math terms to use it? Not really. The calculator uses things like radii (the spokes from center to edge), sine functions (a wave pattern), and coordinates (spots marked on graphs) but it does most of this for you. 4. Is the Arc Length Calculator applicable to 3D geometry or only 2D circles? The calculator focuses on 2D circles. For 3D geometry involving spheres, specific formulas would apply, and our calculator is not tailored for such scenarios. 5. How does the Arc Length Calculator contribute to real-world applications? From constructing circular structures to optimizing paths, our calculator aids in real-world scenarios requiring accurate arc length calculations, contributing to efficient designs and projects.

AP®︎/College Calculus AB We can apply "calculus-based reasoning" to justify properties of the antiderivative of a function using our knowledge about the original function. In differential calculus we reasoned about the properties of a function based on information given about its derivative . In integral calculus, instead of talking about functions and their derivatives, we will talk about functions and their antiderivatives. Reasoning about from the graph of This is the graph of function . Let . Defined this way, is an antiderivative of . In differential calculus we would write this as . Since is the derivative of , we can reason about properties of in similar to what we did in differential calculus. For example, is positive on the interval , so must be increasing on this interval. Furthermore, changes its sign at , so must have an extremum there. Since goes from positive to negative, that point must be a maximum point. The above examples showed how we can reason about the intervals where increases or decreases and about its relative extrema. We can also reason about the concavity of . Since is increasing on the interval , we know is concave up on that interval. And since is decreasing on the interval , we know is concave down on that interval. changes concavity at , so it has an inflection point there. This is the graph of . What is an appropriate calculus-based justification for the fact that is concave up on the interval ? This is the graph of . What is an appropriate calculus-based justification for the fact that has a relative minimum at ? Want more practice? Try this exercise. It's important not to confuse which properties of the function are related to which properties of its antiderivative. Many students get confused and make all kinds of wrong inferences, like saying that an antiderivative is positive because the function is increasing (in fact, it's the other way around). This table summarizes all the relationships between the properties of a function and its antiderivative. |When the function is...||The antiderivative is...| |Changes sign / crosses the -axis||Extremum point| |Extremum point||Inflection point| This is the graph of . What is an appropriate calculus-based justification for the fact that is positive on the interval ? Want to join the conversation? - For the last question, I still don't quite understand how f being positive over [0,7] and non-negative over [7,12] is an appropriate justification for the fact that g(x) is positive on the interval [7,12]. If g(x) is the integral of f(t)dt from 0 to x, then that would simply be the area under the curve of f and above the x-axis in the graph right? Well between [7,12], the area is zero (therefore g(x) is zero) if I understand correctly. Therefore, zero by definition is neither negative nor positive.(8 votes) - The integral starts from 0 and goes until x. If you define x as 7, it takes the positive area from 0 to 7 If you define x as 12, it takes the positive area from 0 to 7 and neither subtracts nor adds any amount of area, thus making the net a positive outcome.(24 votes) - I also still don’t understand the last question about how f being positive can be proof that g is positive. Or even in general: how can you base information about the sign of the values of an antiderivative on the origial function? All the original function can tell us is the slope of the antiderivative, right? We cannot know the constant that we have to add unless we know the initial condition (where g intersects with the y-axis). E.g. if f would represent the speed at which someone travels, then g would represent the distance travelled, but even if that person would have travelled 10,000 positive miles, we still would not know whether he was short of, at, or past a certain point. Am I reasoning the wrong way?(3 votes) - 𝑔(𝑥) is defined as a definite integral of 𝑓(𝑡). The lower bound (0) is the 𝑥-intercept of 𝑔, and serves as the initial condition. 𝑔(𝑥) = ∫[0, 𝑥] 𝑓(𝑡)𝑑𝑡 = 𝐹(𝑥) − 𝐹(0) ⇒ 𝑔(0) = 𝐹(0) − 𝐹(0) = 0(4 votes) - Wait, but an anti-derivative can positive when the function is increasing, right?(2 votes) - If a function is increasing its anti derivative can be positive or negative. It depends on the value of the function.(5 votes) - How does g still increases while it concaves down.(1 vote) - Increasing/decreasing and concave up/concave down are completely independent. Look at the unit circle: In the first quadrant, it's decreasing and concave down. In the second quadrant, it's increasing and concave down. In the third quadrant, it's decreasing and concave up. In the fourth quadrant, it's increasing and concave up.(4 votes) - Does performing integration of a derivative of a function gives us the function itself ?(1 vote) - Essentially, yes. I suggest watching the videos on the Fundamental Theorem of Calculus.(3 votes) - I know this is a bit late, but consider it like this: g(x) = ∫[0, 𝑥] 𝑓(𝑡)𝑑𝑡 Which means that What does this expression mean? This means that f(x) is the derivative of g(x) Now back to the exercise in Unit 5 where we connected a function with its first and second derivative, we learnt that if there is a function p(x) for example, with a derivative p'(x) (a derivative is a slope) When p'(x) = Positive, i.e. when the slope is positive we can say the function is increasing. We will use the same principle here : Since f(x) is the derivative of g(x), When f is positive, g increases. Similarly, when f is negative, g decreases. When a slope goes from positive to negative, we have a max point and vice versa. So, in the diagram, 10 is the max point while 0 is the min point. As for finding the inflection point: In unit 5, we found inflection points by putting f''(x)=0 or undefined What is a second derivative? It is the derivative of a derivative and we've already established that f(x) is the derivative of g(x) So wherever the slope of f(x) (i.e. derivative of derivative) will be 0 or undefined is where our inflection point will be. In this case it is at x=5. Lemme know if you have any other questions. I hope this helped(2 votes) - When f(x) is at an inflection point, what does the integral do?(1 vote) - At in inflection point, the graph changes concavity. You could say that concavity is either a u shape or an upside-down u shape. - In the Reasoning portion before the examples, it explains that x=10 is a relative max of g because f changes from positive to negative. Does this also mean that x=0 is a relative minimum because f changes from negative to positive?(1 vote) - When looking at the relation between an integral and its derivative, is the integral the area below, and the derivative the gradient at any point of, a specific function? I am just looking for a way to understand the behaviour of accumulation functions without needing to memorise random points.(1 vote) - An integral can also be called an "anti derivative" when it's just implied to a function. So if you originally has x^2 for example, as a function and you wanted to differentiate it, you would just get 2x. But lets say you have the rate of change, and you want to find the antiderivative or the indefinite integral of that equation, you would have to integrate it and you would obtain x^2 + C (by the reverse power rule) when "c" represents all constants. This might be a bit confusing but there are some problem on Khan Academy with real world applications of this (really easy problems where you just have to find the area under the curve with basic area formulas) which might make you understand this concept a bit better. It's basically the inverse operation of a derivative. If you Integrate and differentiate any function f(x), you will be left with f(x) since both the inverse operation cancel out (Fund. Theorem of Calc.). Hope this helped, if you have any questions let me know.(1 vote) - I don't understand even the problem 1. If x=5, then g(5) will be the area under the function f from 0 to 5. That seems g(5) is positive since the area is above the x-axis. And then I try to graph it in graphing calculator to see if g(5) is really positive. So, I use (x-5)^2 as my function f, that means the function g is (1/3)(x-5)^3. I am surprised that g(5) is 0. Why is that? What is wrong here?(1 vote)

Rectifier type instrument measures the alternating voltage and current with the help of rectifying elements and permanent magnet moving coil type of instruments. However the primary function of rectifier type of instruments work as voltmeter. Now one question must arises in our mind why we use rectifier type of instruments widely in the industrial world though we have various other AC voltmeter like electrodynamometer type instruments, thermocouple type instruments etc? The answer to this question is very simple and is written as follows. - Cost of electrodynamometer type of instruments is quite high than rectifier type of instruments. However rectifier type of instruments as much accurate as electrodynamometer type of instruments. So rectifier type of instruments are preferred over electrodynamometer type instruments. - The thermocouple instruments are more delicate than the rectifier type of instruments. However thermocouple type of instruments is more widely used at very high frequencies. Before we look at the construction principle and working of rectifier type instruments, there is need to discuss in detail about the voltage current characteristics of ideal and practical rectifier element called diode. Let us first discuss the ideal characteristics of rectifying element. Now what is an ideal rectifying element? A rectifying element is one which offers zero resistance if it is forward biased and offers infinite resistance if it is reversed biased. This property is used to rectify the voltages (rectification means to convert an alternating quantity into direct quantity i.e. AC to DC). Consider the circuit diagram given below. In the given circuit diagram the ideal diode is connected in series with the voltage source and load resistance. Now when we make the diode forward biased it conducts perfectly offering zero electrical resistance path. Thus behaves as short circuited. We can make the diode forward biased by connecting the positive terminal of the battery with anode and negative terminal with cathode. The forward characteristic of rectifying element or diode is shown in the voltage current characteristic. Now when we apply negative voltage i.e. connecting the negative terminal of the battery with the anode terminal of the diode and positive terminal of the battery to the cathode terminal of the diode. Due to reverse biased it offers infinite electrical resistance and thus it behaves as open circuit. The complete voltage current characteristics are shown below. Let us again consider the same circuit but the difference is here we are using the practical rectifying element instead of ideal one. Practical rectifying element is having some finite forward blocking voltage and high reverse blocking voltage. We will apply the same procedure in order to obtain the voltage current characteristics of practical rectifying element. Now when we make the practical rectifying element forward biased it does not conduct till the applied voltage is not greater the forward breakdown voltage or we can say knee voltage. When the applied voltage becomes greater than the knee voltage then diode or rectifying element will come under conduction mode. Thus behaves as short circuited but due to some electrical resistance there is voltage drop across this practical diode. We can make the rectifying element forward biased by connecting the positive terminal of the battery with anode and negative terminal with cathode. The forward characteristic of practical rectifying element or diode is shown in the voltage current characteristic. Now when we apply negative voltage i.e. connecting the negative terminal of the battery with the anode terminal of the diode and positive terminal of the battery to the cathode terminal of the rectifying element. Due to reverse biased it offers finite resistance and the negative voltage till the applied voltage becomes equal to reverse break down voltage and thus it behaves as open circuit. The complete characteristics are shown below Now rectifier type of instruments uses two types of rectifier circuits: Half Wave Rectifier Circuits of Rectifier Type Instruments Let us consider the half wave rectifier circuit given below in which the rectifying element is connected in series with a sinusoidal voltage source, permanent magnet moving coil instrument and the multiplier resistor. The function of this multiplier electrical resistance is to limit the current drawn by the permanent magnet moving coil type of instrument. It is very essential to limit the current drawn by the permanent magnet moving coil instrument because if the current exceeds the current rating of PMMC then it destructs the instrument. Now here we divide our operation in two parts. In first part we apply constant DC voltage to the above circuit. In the circuit diagram we are assuming the rectifying element as ideal one. Let us mark the resistance of multiplier be R, and that of permanent magnet moving coil instrument be R1. The DC voltage produces a full scale deflection of magnitude I=V/(R+R1) where V is root mean square value of voltage. Now let us consider second case, in this case we will apply AC sinusoidal AC voltage to the circuit v =Vm × sin(wt) and we will get the output waveform as shown. In the positive half cycle the rectifying element will conduct and in the negative half cycle it does not conduct. So we will get a pulse of voltage at moving coil instrument which produces pulsating current thus pulsating current will produce pulsating torque. The deflection produced will correspond to the average value of voltage. So let us calculate the average value of electric current, in order to calculate the average value of voltage we have to integrate the instantaneous expression of the voltage from 0 to 2 pi. So the calculated average value of voltage comes out to be 0.45V. Again we have V is root mean square value of current. Thus we conclude that the sensitivity of the ac input is 0.45 times the sensitivity of DC input in case of half wave rectifier. Full Wave Rectifier Circuits of Rectifier Type Instruments Let us consider a full wave rectifier circuit given below. We have used here a bridge rectifier circuit as shown. Again we divide our operation into two parts. In the first we analyze the output by applying the DC voltage and in another we will apply AC voltage to the circuit. A series multiplier resistance is connected in series with the voltage source which has the same function as described above. Let us consider first case here we applying DC voltage source to the circuit. Now the value of full scale deflection current in this case is again V/(R+R1), where V is the root mean square value of the applied voltage, R is the resistance of the resistance multiplier and R1 which is the electrical resistance of the instrument. The R and R1 are marked in the circuit diagram. Now let us consider second case, in this case we will apply AC sinusoidal voltage to the circuit which is given v = Vmsin(wt) where Vm is the peak value of the applied voltage again if we calculate the value of full scale deflection current in this case by applying the similar procedure then we will get an expression of full scale current as .9V/(R+R1). Remember in order to obtain the average value of voltage we should integrate the instantaneous expression of voltage from zero to pi. Thus comparing it DC output we conclude that the sensitivity with AC input voltage source is 0.9 times the as in the case of DC input voltage source. The output wave is shown below. Now we are going to discuss the factors which affect the performance of Rectifier type instruments: - Rectifier type of instruments is calibrated in terms of root mean square values of sinusoidal wave of voltages and current. The problem is that the input waveform may or may not have same form factor on which the scale of these meter is calibrated. - There may be some error due to the rectifier circuit as we not included the resistance of the rectifier bridge circuits in both the case. The non linear characteristics of bridge may distort the current and voltage waveform. - There may variation in the temperature due to which the electrical resistance of the bridge changes hence in order to compensate this kind of errors we should apply multiplier resistor with high temperature coefficient. - Effect of capacitance of the bridge rectifier: Bridge rectifier has imperfect capacitance thus due to this it byp asses the high frequency currents. Hence there is decrement in the reading. - The sensitivity of Rectifier type instruments is low in case of AC input voltage. Advantages of Rectifier Type Instruments Following are the advantages of the rectifier type of instruments: - The accuracy of rectifier type instrument is about 5 percent under normal operating condition. - The frequency range of operation can be extended to high value. - They have uniform scale on the meter. - They have low operating value of current and voltages. The loading effect of an AC rectifier voltmeter in both the cases (i.e. half wave diode rectifier and full wave diode rectifier) is high as compared to the loading effects of DC voltmeters as the sensitivity of the voltmeter either using in half wave or full wave rectification is less than the sensitivity of DC voltmeters.

The Plemelj Construction of a Triangle: 1 This Demonstration constructs a triangle given the length of its base, the length of the altitude from to and the difference between the angles at and .[more] Step 1: Draw a straight line of length . Draw a line of length perpendicular to . Let be the midpoint of . Step 2: Construct a circle with center such that the chord subtends an angle from points on below the chord. The inscribed angle above the chord is and the corresponding central angle is . Step 3: Construct the point on at a distance from . Step 4: The point is the intersection of the right bisector of and the line through parallel to . Step 5: The triangle meets the stated conditions. Triangle is congruent to . In the isosceles triangle , , so . Therefore the obtuse angle . On the other hand, , so and . This Demonstration shows Plemelj's somewhat complicated construction. Fascinated, his teacher showed him the solution from a textbook unknown to the author. This is shown in The Plemelj Construction of a Triangle: 2. Plemelj then made a construction that is shown in The Plemelj Construction of a Triangle: 3. Plemelj admitted that he found the first construction using trigonometry. Three solutions of the triangle construction problem are in . Here is the trigonometric proof. The altitude from divides into two parts of length and . So , or , which can be rewritten as . Let be the angle of at . Since , the equation can be read as From this equation, we must determine ; it can be transformed to a quadratic equation in the unknown . Introduce the angle as or , where . Then , . The equation for is now . This equation can be thought of as the law of sines of the triangle with sides and and opposite angles and .[less] This problem was posed to Josip Plemelj (1873–1967) in 1891 when he was in secondary school by his mathematics teacher in Ljubljana, then in the Austro-Hungarian Empire, now in Slovenia. Plemelj noted that he had made nine original solutions of the problem and that he knew two textbook solutions (one from his teacher's textbook and the other from ). Plemelj stated that the equations produced six different solutions . The first publication of the construction problem by Plemelj was published in Proteus, the natural science journal for students. It included his first three solutions as well as two solutions by readers of the journal, which the editor wrote were already in Plemelj's collection. Plemelj mentioned that he had a large collection of solutions of the problem, with the last entry on December 31, 1939. The author of this Demonstration visited the Archives of the Republic of Slovenia in May 2017, but found only a page of a calendar for November 1939, with three constructions on the other side (see photograph). The Plemelj construction of Triangle 4 describes the first of these three constructions, shown at the top-left in the photograph. At the bottom-left is a construction based on a problem in Bland and Wiegand's book [5, pp. 147]. On the bottom-right is probably Plemelj's last construction. When Professor Plemelj was in Chernivtsi (1908–1914, then in the Austrian-Hungarian Empire, now Ukraine), he talked with two students about the problem. They brought him a copy of the textbook, but Plemelj forgot for a second time to write down the title, so we still don't know its title. He looked for the book in Ljubljana after the First World War; he could not find it, but he found a similar problem in . Using the construction from , Plemelj found a nice solution, which is given in The Plemelj Construction of a Triangle: 5. In the photograph, it is the second on the left. The last construction on the right in the photograph is shown in The Plemelj Construction of a Triangle: 6. (The circle was drawn on the opposite side of the line segment .) This seems to be Plemelj's last construction and was found in . This is the only construction that begins with the point . Our construction is adapted from [6, pp. 93]. A version of the construction by a reader of Proteus was published in . So far we have mentioned eight original solutions. It seems that the ninth solution is a simple modification of the solution in Plemelj's teacher's textbook. It is published in [6, pp. 93]. It is evident that Plemelj did not mention various constructions on the basis of using geometric methods for solving quadratic equations. Plemelj's most original contribution in mathematics is the elementary solution he provided for the Riemann–Hilbert problem about the existence of a differential equation with given monodromy group [1, 3]. Wikipedia. "Josip Plemelj." (Aug 9, 2017) en.wikipedia.org/wiki/Josip_Plemelj. J. Plemelj, Iz mojega življenja in dela (From My Life and Work), Obzornik za matematiko in fiziko, 39, 1992 pp. 188–192. J. J. O'Connor and E. F. Robertson. "Josip Plemelj." MacTutor. www-history.mcs.st-andrews.ac.uk/Biographies/Plemelj.html. J. Plemelj, Proteus (year 12, 1949–1950), 4–5, p.166; 7, pp. 243–245; 8, pp. 285. M. Bland and A. Wiegand, Geometrische Aufgaben für Hohëre Lehranstalten, Braunschweig: Schwetschke und Sohn, 1865. D. S. Modic, Trikotniki, Konstrukcije, Algebrske Rešitve, Ljubljana: Math d.o.o., 2009. I. Pucelj, "Plemelj's Triangle and Fixed Points of Transformations," (in Slovenian), Obzornik za matematiko in fiziko, 62(1), 2015 pp.12–14. Archives of the Republic of Slovenia, Plemelj Fond (SI AS 2012), PE19, Box 3 (manuscripts).

« السابقةمتابعة » All the difficulty therefore reduces itself to finding a function V which satisfies the partial differential equation, becomes equal to the known value of V at the surface, and is moreover such that none of its differential coefficients shall be infinite when p is within A. In like manner, in order to find V, we shall obtain V, its value at A, by means of the equation (a), since this evidently becomes a=T'-B, i.e. V =7. Moreover it is clear, that none of the differential coefficients of V = j-— can be infinite when p is exterior to the surface A, and when p is at an infinite distance from A, V is equal to zero. These two conditions combined with the partial differential equation in V, are sufficient in conjunction with its known value V7 at the surface A for the complete determination of V, since it will be proved hereafter, that when they are satisfied we shall have the integral, as before, extending over the whole surface A, and (p) being a quantity dependent upon the respective position of p and da. It only remains therefore to find a function V which satisfies the partial differential equation, becomes equal to V when p is upon the surface A, vanishes when p is at an infinite distance from A, and is besides such, that none of its differential coefficients shall be infinite, when the pointy is exterior to A. All those to whom the practice of analysis is familiar, will readily perceive that the problem just mentioned, is far less difficult than the direct resolution of the equation (a), and therefore the solution of the question originally proposed has been rendered much easier by what has preceded. The peculiar consideration relative to the differential coefficients of V and V, by restricting the generality of the integral of the partial differential equation, so that it can in fact contain only one arbitrary function, in the place of two which it ought otherwise to have contained, and, which has thus enabled us to effect the simplification in question, seems worthy of the attention of analysts, and may be of use in other researches where equations of this nature are employed. We will now give a brief account of what is contained in the following Essay. The first seven articles arc employed in demonstrating some very general relations existing between the density of the electricity on surfaces and in solids, and the corresponding potential functions. These serve as a foundation to the more particular applications which follow them. As it would be difficult to give any idea of this part without employing analytical symbols, we shall content ourselves with remarking, that it contains a number of singular equations of great generality and simplicity, which seem capable of being applied to many departments of the electrical theory besides those considered in the following pages. In the eighth article we have determined the general values of the densities of the electricity on the inner and outer surfaces of an insulated electrical jar, when, for greater generality, these surfaces are supposed to be connected with separate conductors charged in any way whatever; and have proved, that for the same jar, they depend solely on the difference existing between the two constant quantities, which express the values of the potential functions within the respective conductors. Afterwards, from these general values the following consequences have been deduced:— When in an insulated electrical jar we consider only the electricity accumulated on the two surfaces of the glass itself, the total quantity on the inner surface is precisely equal to that on the outer surface, and of a contrary sign, notwithstanding the great accumulation of electricity on each of them: so that if a communication were established between the two sides of the jar, the sum of the quantities of electricity which would manifest themselves on the two metallic coatings, after the discharge, is exactly equal to that which, before it had taken place, would have been observed to have existed on the surfaces of the coatings farthest from the glass, the only portions then sensible to the electrometer. If an electrical jar communicates by means of a long slender wire with a spherical conductor, and is charged in the ordinary way, the density of the electricity at any point of the interior surface of the jar, is to the density on the conductor itself, as the radius of the spherical conductor to the thickness of the glass in that point. The total quantity of electricity contained in the interior of any number of equal and similar jars, when one of them communicates with the prime conductor and the others are charged by cascade, is precisely equal to that, which one only would receive, if placed in communication with the same conductor, its exterior surface being connected with the common reservoir. This method of charging batteries, therefore, must not be employed when any great accumulation of electricity is required. It has been shown by M. PoisSON, in his first Memoir on Magnetism (Mem. de i'Acad. de Sciences, 1821 et 1822), that when an electrified body is placed in the interior of a hollow spherical conducting shell of uniform thickness, it will not be acted upon in the slightest degree by any bodies exterior to the shell, however intensely they may be electrified. In the ninth article of the present Essay this is proved to be generally true, whatever may be the form or thickness of the conducting shell. In the tenth article there will be found some simple equations, by means of which the density of the electricity induced on a spherical conducting surface, placed under the influence of any electrical forces whatever, is immediately given; and thence the general value of the potential function for any point either within or without this surface is determined from the arbitrary value at the surface itself, by the aid of a definite integral. The proportion in which the electricity will divide itself between two insulated conducting spheres of different diameters, connected by a very fine wire, is afterwards considered; and it is proved, that when the radius of one of them is small compared with the distance between their surfaces, the product of the mean density of the electricity on either sphere, by the radius of that sphere, and again by the shortest distance of its surface from the centre of the other sphere, will be the same for both. Hence when their distance is very great, the densities are in the inverse ratio of the radii of the spheres. When any hollow conducting shell is charged with eleccricity, the whole of tbc fluid is carried to the exterior surface, without leaving any p rtion on the interior one, as may he immediately shown from the fourth and fifth articles. In the experimental verification of this, it is necessary to leave a small orifice in the shell: it became therefore a problem of some interest to determine the modification which this alteration would produce. We have, on this account, terminated the present article, by investigating the law of the distribution of electricity on a thin spherical conducting shell, having a small circular orifice, and have found that its density is very nearly constant on the exterior surface, except in the immediate vicinity of the orifice; and the density at any point p of the inner surface, is to the constant density on the outer one, as the product of the diameter of a circle into the cube of the radius of the orifice, is to the product of three times the circumference of that circle into the cube of the distance of p from the centre of the orifice; excepting as before those points in its immediate vicinity. Hence, if the diameter of the sphere were twelve inches, and that of the orifice one inch, the density at the point on the inner surface opposite the centre of the orifice, would be less than the hundred and thirty thousandth part of the constant density on the exterior surface. In the eleventh article some of the effects due to atmospherical electricity are considered; the subject is not however insisted upon, as the great variability of the cause which produces them, and the impossibility of measuring it, gives a degree of vagueness to these determinations. The form of a conducting body being given, it is in general a problem of great difficulty, to determine the law of the distribution of the electric fluid on its surface: but it is possible to give different forms, of almost every imaginable variety of shape, to conducting bodies; such, that the values of the density of the electricity on their surfaces may be rigorously assignable by the most simple calculations: the manner of doing this is explained in the twelfth article, and two examples of its use are given. In the last, the resulting form of the conducting body is an oblong spheroid, and the density of the electricity on its surface, here found, agrees with the one long since deduced fronother methods. Thus far perfect conductors only have been considered. In order to give an example of the application of theory to bodies which are not so, we have, in the thirteenth article, supposed the matter of which they are formed to be endowed with a constant coercive force equal to /9, and analogous to friction in its operation, so that when the resultant of the electric forces acting upon any one of their elements is less than /3, the electrical state of this element shall remain unchanged; but, so soon as it begins to exceed y9, a change shall ensue. Then imagining a solid of revolution to turn continually about its axis, and to be subject to a constant electrical force / acting in parallel right lines, we determine the permanent electrical state at which the body will ultimately arrive. The result of the analysis is, that in consequence of the coercive force fi, the solid will receive a new polarity, equal to that which would be induced in it if it were a perfect conductor and acted upon by the constant force /8, directed in lines parallel to one in the body's equator, making the angle 90° + 7, with a plane passing through its axis and parallel to the direction of/: f being supposed resolved into two forces, one in the direction of the body's axis, the other b directed along the intersection of its equator with the plane just mentioned, and 7 being determined by the equation sm7 = ^ . In the latter part of the present article the same problem is considered under a more general point of view, and treated by a different analysis: the body's progress from the initial, towards that permanent state it was the object of the former part to determine is exhibited, and the great rapidity of this progress made evident by an example. The phenomena which present themselves during the rotation of iron bodies, subject to the influence of the earth's magnetism, having lately engaged the attention of experimental philosophers, we have been induced to dwell a little on the solution of the preceding problem, since it may serve in some measure to illustrate what takes place in these cases. Indeed,

Name: Sarah Davis City: Los Angeles "I have always been curious about the relationship between risk ratio and odds ratio in statistical analysis. When I stumbled upon the keyword 'when can the risk ratio be approximated by the odds ratio?', I was thrilled to find relevant information. The search results provided me with a comprehensive understanding of this topic, making it easier for me to interpret research findings accurately. I am truly grateful for the clarity and depth of knowledge I gained from the search. Now, I can confidently apply these concepts to my work as a data analyst. Thanks to this valuable resource, I can confidently say that my statistical prowess has reached new heights!" "As a seasoned researcher, I am always on the lookout for reliable and informative content. When I stumbled upon the query 'when can the risk ratio be approximated by the odds ratio?', I was amazed by the wealth of knowledge that I found. The search results provided me with a clear understanding of the circumstances where the risk ratio can be approximated by the odds ratio. The explanations were concise yet comprehensive, allowing me to delve deeper into statistical analysis with confidence. The How to convert hazard ratio to odds ratio Meta Tag Description: Explore the process of converting hazard ratio to odds ratio in the US region. This expert review provides informative and easy-to-understand insights, guiding you through the steps of this essential statistical conversion. In the field of medical research and epidemiology, hazard ratio (HR) and odds ratio (OR) are widely used statistical measures. While HR estimates the relative risk of an event occurring over time, OR quantifies the association between exposure and outcome in a case-control study. Understanding how to convert HR to OR is crucial for data interpretation, as it allows for a more comprehensive analysis of the data. In this review, we will explore the process of converting hazard ratio to odds ratio specifically for the US region. Converting Hazard Ratio to Odds Ratio: Converting HR to OR involves a mathematical transformation, which is dependent on the baseline hazard rate and the incidence of the outcome in the control group. The formula for converting HR to OR is as follows: OR = (HR * p) / (1 - p + (HR * p)) Here, "p" represents the incidence of the outcome in the control group. Let's consider an example to illustrate the conversion process What does HR mean in statistics? The hazard ratio (HR) is the main, and often the only, effect measure reported in many epidemiologic studies. For dichotomous, non–time-varying exposures, the HR is defined as the hazard in the exposed groups divided by the hazard in the unexposed groups. What is HR and RR in statistics? |Static – does not consider rates. Summarizes an overall study. |Based on rates. Provides information about the way a study progresses over time. Is HR and RR the same thing? What does hazard ratio of 1.5 mean? Can you convert odds ratio to hazard ratio? Frequently Asked Questions How do you calculate the hazard rate? What is the formula for the odds ratio of risk? |Relative risk (risk ratio) |EER / CER |Relative risk reduction |(CER − EER) / CER, or 1 − RR |Preventable fraction among the unexposed |(CER − EER) / CER |(EE / EN) / (CE / CN) What is the alpha level that is interpreted by the nurse researcher as a highly statistically significant result? - When a value is obtained that shows no difference in an experiment? - If the null value (the value that indicates no difference and is usually zero or one) is included in the confidence interval, then the result is not statistically significant. - How do you calculate the hazard ratio? - The HR has also been defined as, the ratio of (risk of outcome in one group)/(risk of outcome in another group), occurring at a given interval of time (21). In the situation where the hazard for an outcome is exactly twice in Group A than in Group B, the value of the hazard ratio can be either 2.0 or 0.5. - What is the difference between odds ratio and incidence rate ratio? - The normally used odds ratio from a classical case-control study measures the association between genotype and being diseased. In comparison, under incidence density sampling, the incidence rate ratio measures the association between genotype and becoming diseased. How to get hazard ratio from odds ratio |What is the formula for hazard? |(7.3) λ ( t ) = f ( t ) S ( t ) , which some authors give as a definition of the hazard function. In words, the rate of occurrence of the event at duration equals the density of events at , divided by the probability of surviving to that duration without experiencing the event. λ ( t ) = − d d t log |How do you calculate odds ratio from hazard ratio? |The odds are equal to the hazard ratio, which is 1.9 in the present case. The probability of healing sooner can be derived from the hazard ratio by the following formula: HR = odds = P/(1 − P); P = HR/(1 + HR). And so, in this example, P = 1.9/2.9 = 0.67. |How do you convert risk ratio to odds? |To convert an odds ratio to a risk ratio, you can use "RR = OR / (1 – p + (p x OR)), where p is the risk in the control group" (source: http://www.r-bloggers.com/how-to-convert-odds-ratios-to-relative-risks/). - What is the formula for the odds ratio? - In a 2-by-2 table with cells a, b, c, and d (see figure), the odds ratio is odds of the event in the exposure group (a/b) divided by the odds of the event in the control or non-exposure group (c/d). Thus the odds ratio is (a/b) / (c/d) which simplifies to ad/bc. - What does a 1.5 hazard ratio mean? - If the ratio is 1 that means that the risks are the same. If it is greater than 1, then the risk is higher, and vice versa. The drug is usually the denominator, so 1.5 means for example, that the risk of dying is higher on the drug by about 50%. - Is odds ratio same as hazard ratio? - In logistic regression, an odds ratio of 2 means that the event is 2 time more probable given a one-unit increase in the predictor. In Cox regression, a hazard ratio of 2 means the event will occur twice as often at each time point given a one-unit increase in the predictor.

The conclusion of my lesson on units: More conversion examples, proper notation for units, scientific notation and prefixes. Be sure to get your free worksheet that goes with this video: Hey there, Doc Bateman here. This is part 2 of my Physics Done Right lesson about units. If you haven’t seen part 1, please go watch it first. In this video, I’ll do some more examples and then talk about scientific notation, the proper way to write units, and how prefixes work and how to write them. Please click like and subscribe below and also leave me a comment. If you think these videos are helpful, please share them with a friend. Let’s get started. Next, let’s do a time conversion. How many seconds are in January? Draw a bracket with January. I need that to cancel, so I put January in the denominator and I know that January equals 31 days. Now let’s work our way down to seconds. We need a day in the bottom to cancel: one day = 24 hours, 1 hour = 60 minutes and 1 minute = 60 seconds. Let’s check the cancels. January cancels, days cancel, hours cancel and minutes cancel. What I have left is this single unit of seconds, which is good because that’s what I want. Now collect numbers: 31 x 24 x 60 x 60 and the unit I had left was seconds. This equals a very large number: 2 678 400 seconds. For our final conversion example, I wanted to show you how to do a volume conversion. In this question, a concrete truck holds 10 yards. “Yards” is a jargon term that means cubic yards. So I have 10 cubic yards of concrete. How much is that in cubic meters? I need one conversion factor and that is 1 m = 1.094 yards. Draw a bracket with 10 yards cubed. So I need yards in the denominator. I use my conversion factor but I’m not done. I have one yard in the denominator, but I’ve got three (units of) yards in the numerator, so I have to do the same conversion three times to cancel all three of those yards. So let’s do it again, and a third time. Now let’s check the cancels: 3 yards cancels 3 yards, so I have left a meter, and another meter, and another meter, so that’s meters cubed, which is good because that’s what I want. Now gather the numbers 10 over I’m going to shorthand this 1.094 cubed and the units I have left are meters cubed. A calculator tells me that this equals 7.64 cubic meters. Let’s talk about notation. When you write quantities with units there’s a convention. Convention means a standard practice. And this convention says: if the unit is based on a person’s name then you capitalize the abbreviation. But if you need to write out the full word for the unit, don’t capitalize it, because that would confuse people whether you’re talking about the unit or the person. Let’s give some examples. The unit of force is called the newton, after Sir Isaac Newton. So if you write the full name you don’t capitalize it, but if you abbreviate it, the abbreviation is a capital N. The same for the joule, the watt, the amp, the volt – there are many units that are named after physicists and other scientists and engineers. So that’s the convention: capitalize the abbreviation but not the complete word. Next let’s talk about scientific notation. Scientific notation is a way to write very large or very small numbers in a compact way and to keep track of where the decimal point is. So here’s the recipe: if you move the decimal point left then you multiply by 10 to the positive (whatever the count is) if you move the decimal point right, then you multiply by 10 to the minus (whatever the count is). For example, here I have 100 000 000. I want to move the decimal point here right after the one so if I move it I’d have to move it 7 places left since I moved left that means I have 107 giving 1.0 x 107. Then I have this very small number (0.00001). I want the decimal point to be right after the one. That means I have to move this decimal point 5 places to the right which means I have 1.0 x 10-5 and just for fun and a really good illustration, there’s a nice video that was made in the late ’70s called Powers of Ten and it nicely illustrates the very large and the very small. It’s on YouTube and I’ll put a link below so you can go watch it. It’s not very long and it’s very illustrative of powers of 10. Next we talk about prefixes. Units can be scaled up or down with prefixes. Prefixes that multiply or scale up are written with capital letters, and prefixes that divide or scale down are written with lowercase letters. But there are exceptions to this rule. The first three “multiply” or “scale up” prefixes – deca, hecto, and kilo – are written with lowercase letters. Really the only one you need to remember is kilo (lowercase k) because deca and hecto are not widely used. The table next lists the standard prefixes. Notice that they mostly step up or down by a factor of 1,000 which is 103. Here’s the table of prefixes. Notice that right here in the middle is 100 or 1. The prefixes that make units bigger go up from there: deca, hecto, and kilo. Notice that they are lower case like I mentioned: kilo is 1,000, mega is a million, giga is a billion, tera is a trillion, and so on. Then the prefixes that make units smaller go down from there: deci, centi, milli. You’ve heard of centimeter that’s a 1/100, milli that’s 1/1000. Here’s a strange one: this is micro and it’s represented by a lowercase Greek letter μ (“moo”) and it means one-one millionth; nano is one-one-billionth, etc. on down. So there you have it. I hope you know a little more about units now, how to do conversions, and what prefixes are. Please click like and subscribe below and leave me a comment. If you think these videos are helpful, please share them with a friend. Thanks for watching.

Veronika E. Hubeny and Mukund Rangamani ††, Department of Physics, Stanford University, Stanford, CA 94305, USA Department of Physics, University of California, Berkeley, CA 94720, USA Theoretical Physics Group, LBNL, Berkeley, CA 94720, USA We argue that pp-wave backgrounds can not admit event horizons. We also comment on pp-wave generalizations which would admit horizons and show that there exists a black string solution which asymptotes to a five dimensional plane wave background. Plane wave†† To avoid any confusion later, let us clarify the terminology from the very outset: pp-waves (or “plane-fronted waves with parallel rays”) are all spacetimes with covariantly constant null Killing field; plane waves are a subset of these which have in addition an extra “planar” symmetry along the wavefronts. These are in fact the spacetimes that much of the recent literature discussing Penrose limits has been calling “pp-waves”. spacetimes have a special importance in theoretical physics. In general relativity, they form simple solutions to Einstein’s equations with many curious properties. They can be thought of as arising from the so-called Penrose limit of any spacetime, which essentially consists of zooming in onto any null geodesic in that spacetime. Being a subset of pp-waves, they admit a covariantly constant null Killing field, which in turn implies that all curvature invariants vanish. Nevertheless, they are distinct from flat spacetime and their structure is much richer. Interestingly, as shown by Penrose in , these spacetimes are not globally hyperbolic, so that there exists no Cauchy hypersurface from which a causal evolution would cover the entire spacetime. This automatically implies that even the causal structure of plane waves is different from that of flat spacetime. pp-wave spacetimes are especially important within the context of string theory. This is because they yield exact classical backgrounds for string theory, since all curvature invariants, and therefore all corrections, vanish [3,4]. Hence the pp-wave spacetimes correspond to exact conformal field theories. Because of this fact, they provide much-needed examples of classical solutions in string theory, which can in turn be used as toy models for studying its structure and properties. Plane waves happen to be even simpler, for the action in light cone gauge is quadratic. While this fact has been appreciated for some time [3,4], only recently have plane waves received significant attention, mainly initiated by the work of Berenstein, Maldacena, and Nastase (BMN) , based on the AdS/CFT correspondence [6,7,8,9]. These authors proposed a very interesting solvable model of string theory in Ramond-Ramond backgrounds by taking the Penrose limit of spacetime [5,10], the holographic dual of , Super-Yang-Mills theory. This limit corresponds to a particular sector of the gauge theory, with large dimensions of operators and large charges, but with a finite difference between the charges and the dimensions. Part of the importance of this result stems from the fact that since the dual background is exactly solvable as a string theory, we can claim to have understood, at least in principle, this particular sector of the gauge theory. The BMN “correspondence” has since been examined and generalized by many authors and Penrose limits of various supergravity solutions have been considered in the recent literature. One interesting avenue for exploration concerns the addition of black holes into the plane wave spacetime. Naively, this might correspond to “thermalizing” the high energy sector of the gauge theory. While this generalization is rather suggestive and follows in close analogy with the corresponding developments in the AdS/CFT correspondence, where adding a large Schwarzschild black hole into AdS corresponds to thermalizing the gauge theory, no concrete solutions or understanding have yet been reached. Partly, this sector of gauge theory does not yet stand on its own as a well-defined theory without invoking the limit; but more importantly, no appropriate black hole solution has yet been found. In the case of the maximally supersymmetric homogeneous plane-wave discussed by BMN, exact quantization of the light-cone string Hamiltonian helps in the analysis of the thermal partition function [11,12,13]. One might naively hope to obtain a black hole by taking an appropriate Penrose limit of a more general spacetime. In this paper, we argue that this is not possible. In particular, we show that no plane wave can admit event horizons. We will in fact make the stronger claim that no pp-waves can admit event horizons. While the latter may not seem as interesting in the context of the recent excitement about Penrose limits, it is nevertheless of interest to string theory, because, as mentioned above, pp-waves are exact classical solutions in string theory. Furthermore, they too can have interesting duals. The most obvious way to prove the absence of black holes is to examine the global causal structure of a general pp-wave. Although it turns out to be a rather formidable task to examine the causal structure in full generality, in the ensuing paper we will discuss specific examples and some of the causal properties we expect the general pp-wave to carry. It is somewhat simpler to concentrate on just the plane waves. Indeed, the causal structure of certain plane waves has recently been studied by Marolf and Ross , who use the approach introduced by Geroch, Kronheimer, and Penrose , which is based on completing the spacetime by “ideal points” reflecting its causal structure. Marolf and Ross demonstrate that for homogeneous plane waves, the conformal boundary consists of a one-dimensional null line plus two points corresponding to future and past infinity. This result agrees with and generalizes that of , who obtain the asymptotic structure of the BMN plane wave by conformally mapping it into the Einstein static universe. In the examples studied thus far it is clear that the entire spacetime manifold is in the causal past of infinity, thereby precluding the presence of event horizons. While the study of pp-wave causal structure has not been completed in full generality, in this paper, we shall content ourselves with examining the more limited (but, from string theory point of view, perhaps the most interesting) aspect of causal structure, which can be addressed in generality. Specifically, we will ask the question can pp-waves admit horizons? As revealed above, we will argue that the answer is no. This, however, does not mean that there cannot exist black hole/string solutions which are asymptotically plane or pp-wave, though they do not respect the plane or pp-wave symmetries everywhere. We offer a particular simple example in section 5, but work is underway to find more physically interesting solutions. The outline of this paper is as follows. In the following short section, we review certain basic aspects of plane wave and pp-wave spacetimes, mainly with the view of setting up notation, and offer a definition of event horizons and asymptopia in spacetimes which are not asymptotically flat. We will then motivate an argument for the absence of a horizon in the plane wave spacetime, by showing that any point in the spacetime can communicate to arbitrarily large distances using a symmetry argument. Section 4 presents the no-horizon argument for pp-waves. This is distinct from the arguments presented in section 3, but it simultaneously provides an alternate proof that plane waves can’t admit event horizons. While, from the point of view of constructing interesting black hole solutions in pp-waves, up to here our results were negative, in section 5 we try to remedy this by discussing generalizations of pp-waves which would admit event horizons. Finally, we end in section 6 with a more general discussion. 2. Terminology and definitions To pave the way for arguing why pp-waves cannot admit event horizons, we first explain what are pp-waves and plane waves by writing the corresponding metrics. We then discuss what it would mean for these metrics to admit black holes, and offer a criterion for absence of black holes. In the subsequent sections, we use this criterion to argue that pp-waves do not admit horizons. Note that we shall be concerned with physical spacetimes i.e., ones which are solutions to the Einstein-Hilbert action with matter content satisfying appropriate energy conditions. To set the notation and re-emphasize terminology, we will write explicitly three classes of spacetimes, in decreasing generality. The pp-wave spacetimes are defined as spacetimes admitting a covariantly constant null Killing field. The most useful ones†† Generically, a background admitting a covariantly constant null Killing field can have non-vanishing components of the metric. Also there is no requirement that the transverse space be flat; for vacuum solutions we could have easily considered Ricci flat transverse metrics. While our arguments are expected to hold for these cases, we will restrict our discussion to metrics of the form presented in Eq.(2.1). can be written as where the vacuum Einstein’s equations dictate that satisfy the transverse Laplace equation for each . , however, can be an arbitrary function of . Plane wave spacetimes are those where this harmonic function is in fact quadratic, in (2.1), so that plane waves can be written as Here, can be any function of , subject to the constraint that for each , is symmetric and, for vacuum solutions, traceless. As suggested by the name, these metrics have an extra “plane” symmetry, which contains the translations along the wave-fronts in the transverse directions. This can be seen explicitly by casting (2.2) into the Rosen form,†† Typically, this metric is not geodesically complete because of coordinate singularities, but the Brinkman form (2.2) does cover the full spacetime. The coordinate transformation from one form into the other is given e.g., in . For metric of the Brinkman form , the coordinate transformation where satisfies , casts this metric into the Rosen form . The homogeneous plane waves further specialize (2.2) by taking out ’s dependence on , The BMN plane wave metric , found earlier by , belongs to this last class, for the special case , and in their notation. In all the aforementioned spacetimes we have a covariantly constant null Killing vector given as . The fact that this is a null Killing vector is obvious from the metric, while its being covariantly constant may be inferred from the vanishing of the Christoffel symbols . 2.2. Event horizons and asymptopia Black holes are defined as regions of spacetime inside event horizons. Hence, to show that a particular class of spacetimes cannot represent black holes, we need to show that these spacetimes do not admit event horizons. However, in order to do so, we first need to specify a suitable definition of an event horizon. In asymptotically flat spacetimes, an event horizon is defined simply as the boundary of the causal past of the future null infinity, , where the past of future null infinity is defined as the union of the pasts of all the points at infinity , i.e., . Physically, this just says that an asymptotic observer can’t see inside black holes. However, as is well-known, when the spacetime in question is not asymptotically flat, this simple definition may not work. First, the notion of asymptopia may be more murky. For instance, as in the case of closed FRW universe, there can be a big crunch, so that there does not exist any asymptotic region at all. Similarly, for some of the presently-studied pp-waves, the “asymptotic” region may be singular if for some in the pp-wave metric (2.1); or the spacetime may terminate at finite if as . We will therefore adopt a more universal definition of a black hole, or rather the absence of black hole, which, instead of requiring that any point in the spacetime is “visible” (i.e., causally connected) to asymptotic observer, merely requires that any point in the spacetime is visible to an observer who is “arbitrarily far.” This last phrase needs a bit more qualification. One might naively try to define “arbitrarily far” by “some spatial coordinate getting arbitrarily large,” but this is too glib since it is a coordinate-dependent statement. First of all, if the coordinates don’t cover the entire spacetime, reaching arbitrarily large values of the coordinates would merely indicate coming closer to the boundary of our coordinate patch, not the spacetime. Also, for all geodesics which don’t terminate at a singularity, the affine parameter gets arbitrarily large; but in the case of pp-waves, this will be one of the coordinates, , which has a spatial component. The first objection can be bypassed in the case of pp-waves: the coordinate patch of the metric (2.1), with all the coordinates ranging form to , does cover the full spacetime. The second objection might be mollified by noting that plays the role of time rather than a spatial coordinate, but that does not suffice. Specifically, there always exists a geodesic, for which , and is the affine parameter, so that from any point in the spacetime, we can causally communicate to .†† This may be restated in a more covariant fashion as follows: For all spacetimes admitting a null Killing field, the integral curves along these Killing vectors actually describe null geodesics. To see this, denote the Killing vector by . Then , where the first equality follows from the definition of Killing vector, the second from product rule, and the last from being null and thus having a constant norm. But is just the geodesic equation, with being the tangent vector to the geodesic. Since this point is part of , the null infinity, one might be tempted to argue straight-off that there can’t be event horizons in spacetimes with a null Killing field. The reason we do not wish to do so is that we don’t want to preclude horizons stretched along the direction, (thus respecting the null symmetry), but separating say, a region from which no causal curve can reach arbitrarily large transverse distance, . (Also, there are counter-examples, such as the black holes studied by ,.) Given the above considerations, we will adopt the following criterion for absence of black holes in pp-wave spacetimes. Def: A pp-wave spacetime does not admit an event horizon iff from any point in the spacetime, say , there exists a causal curve to some point , where is arbitrary, while . The important aspect is that not just , but also at least one of the transverse coordinates, , gets arbitrarily big along a causal curve. We will in fact use a stronger version of this criterion; namely, we will require , for arbitrarily small . This will allow us to use the criterion in greater generality, in particular even in the cases where our spacetime terminates at some finite , i.e., as . One more side comment on terminology is in order: We are using the term event horizon in an unconventional (generalized) way, as defined above, rather than as something fundamentally related to . However, it seems likely that if any part of spacetime is visible to an observer who is arbitrarily far, it will also be visible to an asymptotic observer. Also, in the present discussion, horizon is used as shorthand for “event horizon” as defined above. It is perhaps worth stressing that there of course are Rindler horizons in pp-waves, just as there are Rindler horizons in e.g. the flat spacetime. These, however, are rather trivial, and don’t carry any globally special properties. In particular, they do not bound a black hole. 3. Heuristic motivation for no horizons in plane wave In the present section we will first give a heuristic motivation for no horizons in plane wave, and then try to argue that in plane wave spacetimes, causal communication from a given point in the spacetime to asymptotically large distances is always possible. As discussed above, this automatically precludes the presence of horizons. A heuristic argument for the absence of black holes as plane waves is as follows. As shown originally by Penrose in the context of classical general relativity, and later extended to supergravity by , a plane wave spacetime can be viewed as a Penrose limit of some spacetime. In this limit, one zooms arbitrarily close to a null geodesic and reexpands the transverse directions—a procedure analogous to obtaining a tangent space by zooming in to a point in a manifold—so that the only nontrivial information which survives is the 1-dimensional structure along the geodesic, parameterized by its affine parameter. The “blowing up” of the transverse directions gives rize to the covariantly constant null Killing field mentioned above. This zooming and reexpanding effectively washes out most of the global information contained in the spacetime in all directions excepting that along the null geodesic. In particular, the limit retains local information about the spacetime, albeit in a more general fashion than the tangent space, but at the expense of losing global information such as that pertaining to event horizons. In the following subsection, we will illustrate this point with a specific example, the four dimensional Schwarzschild black hole. We will then give a more rigorous argument, essentially based on the symmetries of the plane wave spacetimes. 3.1. Penrose limits of black hole spacetimes We demonstrate that Penrose limits of black hole spacetimes do not retain information about the event horizon (cf. , for considerations of Penrose limits in AdS-Schwarzschild spacetimes). While this is somewhat obvious from the preceding discussion, it may nevertheless serve as an intuition-building exercise. Consider, for instance, the asymptotically-flat Schwarzschild black hole. The causal structure is as given by the Penrose diagram of Fig.1. Since the Penrose limit requires us to consider the neighbourhood of null geodesics, let us see what sorts of null geodesics are allowed in the spacetime. First of all, it is clear that there are radially infalling geodesics, such as of Fig.1, which describe the trajectory of a photon falling into the black hole. There can also be ones with angular momentum (which in the plane of the Penrose diagram would appear timelike), which likewise fall into the singularity. These geodesics intersect the horizon at a single point and terminate at finite value of affine parameter upon hitting the singularity. The resulting plane wave spacetime will have a singularity reflecting this, and in fact the spacelike singularity of the Schwarzschild black hole will be converted to a “cosmological” null singularity. From this construction it is clear that the resulting spacetime will not have a horizon since we keep only a small region close to the point on the horizon where the geodesic intersects the same. In particular, this geodesic would be completely insensitive to, for instance, a null shell which might fall into the black hole later, thus shifting the position of the event horizon of the original spacetime. A second class of null geodesics are those which are carrying some angular momentum, but staying put at constant values of the radial coordinate, such as the curve labeled by in Fig.1. This physically corresponds to photon orbits in the black hole spacetime. For the four dimensional Schwarzschild black hole this happens to be at . However, the neighbourhood of this region is completely smooth and the resulting Penrose limit is just the flat space. The last interesting geodesic which may be considered is one which is sitting at the horizon i.e., , and some constant angle ; this is labeled by in Fig.1. This would be the geodesic just skimming the horizon and one would be most tempted to consider this as the one which can lead to an interesting spacetime in the Penrose limit. This geodesic also leads to a flat space in the Penrose limit for reasons similar to the previous case. To summarize, Penrose limits of black hole spacetimes are incapable of retaining the global structure of the event horizon. We note in passing that we are here strictly considering Penrose limits of given spacetimes which has a well-defined algorithmic prescription. It is less clear whether there exist other limiting procedures (probably double/mutiple scaling limits), wherein we start with a spacetime with a horizon and end up with a resulting simple spacetime (some analog of plane wave), whilst retaining interesting information about the global causal structure. 3.2. geodesics and symmetries in general plane waves Let us consider the general plane wave metric (2.2). As can be easily checked, the null geodesics are given by and the integral constraint where is an arbitrary integration constant which is fixed by the initial conditions. The null Killing field implies that is a constant of motion, so that we can take to be the affine parameter along the geodesic, and the derivative . Let us now make the following simple observation: Under the constant rescaling , , the metric remains physically the same (only the “units” get rescaled). Therefore any geodesic remains the same under this rescaling. Note that this is exactly what we would expect from the geodesic equation: since (3.1) is linear in , we are free to rescale ; and from (3.2), is rescaled as . This rescaling freedom suggests that if a geodesic can make it to some distance , it can make it to arbitrarily large , so there couldn’t be a horizon at any finite value of . Note that this is already obvious for , since is a Killing field, so no value of can be physically distinguished from any other. While these arguments were mostly motivational, in the next section we shall present more rigorous proof of nonexistence of event horizon in pp-waves, which include plane waves as a subset. 4. No horizons in pp-waves In the above, we have shown that there can be no horizons in a generic plane wave. Let us now ask a more general question, namely, can there be horizons in a pp-wave? We shall be working with the pp-wave metric (2.1), but since we will be interested in curves which reach large transverse directions , it is more convenient to rewrite (2.1) in spherical coordinates , where only can get large. The -dimensional pp-wave metric then becomes As indicated above, in order to demonstrate the absence of horizons, it suffices to show that from any point of the spacetime, there exists a causal curve which reaches arbitrarily large values of and in arbitrarily small . Below, we will first try to construct such curves explicitly, and then offer a more elegant proof. As apparent from (4.1), any causal curve must satisfy where . Then is a causal (and in fact null) curve which reaches arbitrarily large values of . However, it stays at constant . Since all curves with are simply related to , let us now consider curves with . This will be necessary in order for the curve to reach arbitrarily large . We can then rewrite the causal relation (4.2) simply as where now . Let be a curve such that where is some constant, to be chosen later, and the initial conditions are given by . Now, if along the full curve, we can approximate the spacetime region through which such a curve propagates by replacing with . Furthermore, since our curve stays at fixed , we can “freeze” that dependence in as well. Thus, letting , our curve is arbitrarily well approximated by Since is constructed so as to satisfy (4.3), it is clearly a causal curve. Hence we only need to show that exists and can reach arbitrarily large and for some choice of . But (4.5) is a first order system, which we can just integrate forward; the requirement for the existence of the solution is that both and remain non-negative. If is bounded from below, we can pick such that the radial velocity of the curve is always positive, and reaches arbitrarily large . Construction of a suitable curve is going to be more problematic in regions where changes sign. To specify the curve completely we first pick the sign of depending on the sign of near . Now suppose passes through zero for some . At we also flip the sign of and continue with the construction of the causal curve. In other words, in each interval in where doesn’t change sign, we can solve the equation , and then patch the outgoing pieces together. Of course, such a curve will be continuous but will have discontinuities in its second derivatives. While this explicitly constructs a causal curve, the technique used to construct the same is not quite elegant. In particular, it requires us to separate the spacetime into various regions depending on the sign of , construct a causal curve which reaches maximal in each specific region, and then patch the pieces together. To avoid this cumbersomeness, we will now present an alternate proof which is more universal. Pick any point in the spacetime, and any , (which will represent the arbitrarily large distances that we want to reach by a causal curve). To prove the absence of horizons, we want to show that there exists a point , with some , , and arbitrarily small, which lies on a causal curve from . Pick a constant, , such that . This will clearly be possible if is not singular in this region, but we can generalize the proof for singularities as well. We will discuss the existence of below; but for the moment, we will assume that does exist. Now, consider the fiducial metric, We want to claim that any curve which is causal in is also causal in . But this is easily shown:†† We will now revert back to parameterizing our curve, so . Any curve which is causal in must satisfy , but since , this automatically implies that (4.3) is also satisfied. This means that if we can find a curve from to which is causal in , we are done. But that is also easy. To find such that let be given by e.g. with and fixed, and to be chosen so as to satisfy the causality condition of (4.7), namely for . Let Then satisfies all the requirements of (4.7); in particular, it is causal in . As argued above, this also means that it is causal in the pp-wave spacetime . In fact, this is easy to see, since is just the flat spacetime. Explicitly, if , consider the coordinate transformation and ; while if , consider the coordinate transformation and . In both cases, the metric (4.6) becomes , which is clearly the -dimensional Minkowski spacetime. But we know that Minkowski spacetime has no horizons, so that from any point, there exists a causal curve which can attain arbitrarily large values of and . Thus, we have found a causal curve starting from an arbitrary point of the pp-wave spacetime and attaining arbitrarily large values of the coordinates. Hence no point can be inside an event horizon, so that there can’t be black holes. The only step which still needs to be discussed is the existence of , to which we turn next. Let us first concentrate on the class of pp-waves which are solutions to vacuum Einstein’s equations. Since the Einstein tensor is given by , where is the transverse Laplacian, of (4.1) must satisfy the transverse Laplace equation, . This is a very remarkable result, since this implies that, due to the linearity of Laplace equation, we may superpose the solutions. In particular, we can decompose in terms of the generalized -dimensional spherical harmonics , where : Therefore along our curve , this becomes We see that there can be singularities at and . Now, all “singularities” are by definition excluded from our spacetime, in the sense that all points which are part of the physical spacetime must be nonsingular. In particular all starting points must be of that kind. Thus, if there is a singularity at , we must chose . Similarly, we must chose . But then in the region of interest, , is bounded. This shows that must always exist for vacuum pp-waves. What about non-vacuum solutions? This is much more complicated to analyse, since can in principle be anything as long as we have the appropriate matter content. A-priori, it could obstruct the proposed path of by a singularity. For the case of pp-wave solutions that lead to integrable sigma models in light-cone quantization of the world-sheet superstring theory , , it is possible to see that we can construct causal curves reaching the asymptotic regions of the spacetime. 5. Generalizations admitting horizons Above, we have seen that for pp-wave spacetimes, namely those admitting a covariantly constant null Killing field, there can’t be any event horizons. The existence of this null Killing field played an important role in this observation; in fact, the integral curves of this null Killing vector define null geodesics, so that as we have argued at the beginning, from any point in such a spacetime we can “communicate out to infinity” at . However, as we also cautioned, this does not automatically guarantee the absence of horizons: without horizons, we should be able to reach infinity in all directions. What is most remarkable is the fact that one is able to communicate causally also in the transverse directions out to large distances. We therefore want to ask, how many of the properties of pp-waves do we need to relax, in order for the existence of horizons to become possible. We would like to claim that we can find black hole solutions admitting a null Killing field, which however is not covariantly constant. While we have no real evidence for this claim, it is easy to see post facto that the relaxation of the covariant constancy requirement does lead to spacetimes with an event horizon. In fact, such spacetimes already exist in literature. The simplest such example is the case of traveling waves on a five-dimensional black string as discussed in (cf., , , and for additional discussions on related issues). The solution studied in is a solution to the low energy effective action for the heterotic string in five dimensions. The metric and the dilaton for the solution are given by The metric has been written in the string frame and the Einstein frame metric is given as . The functions appearing in the metric are given by There are other fields which need to be turned on for the above metric to solve the equations of motion and we refer the reader to the original source for explicit expressions of the same. By a judicious choice of the charges we can even set the dilaton to be constant; setting will suffice for the same. These spacetimes have an event horizon at , which has a finite area. As we see that the radius of the two-sphere takes the constant value . So we have a finite area and therefore a solution with finite entropy. While the coordinates in which the above metric is written degenerate near the horizon, it is possible to find a set of regular coordinates . The solution is asymptotically flat since the functions appearing in the metric go over to unity for large values of the radial coordinate. One can imagine recovering a spacetime with a covarinatly constant null Killing vector†† We would like to thank Gary Horowitz for bringing this to our attention. from the solution given in (5.1), by setting , which requires the choice . This gives a string frame metric with a regular horizon at and is a covariantly constant null Killing field, seemingly violating the claims we have made hitherto. However, this is illusory. In the physical spacetime i.e., in the Einstein frame metric, we see that is no longer covariantly constant. In addition is a singular point in the spacetime, for the curvature invariants blow up there. It is interesting, however, that while there is no contradiction in the physical spacetime, there apparently is a violation of our claims in the string frame metric. One can use the above charged black string solution and generate a solution which is asymptotically plane-wave. The idea is to use the Garfinkle-Vachaspati construction , to make the spacetime asymptotically plane-wave. As explained in , this is possible for spacetimes which admit a null Killing vector which is hypersurface orthogonal and also show that this procedure leads to spacetimes which have the same set of curvature invariants as the original spacetime. For the particular case of the asymptotically flat black string, this construction implies that we can add a term to the metric appearing in (5.1); the resulting metric is a solution to Einstein’s equations with all other fields unaltered, so long as is a harmonic function in the transverse space. In particular, we can have with arbitrary functions . All spacetimes with for suffer from singular behaviour at , while those with with are singular at . By singular we mean that there are divergent tidal forces on finite-sized observers. Addition of and lead to spacetimes which are diffeomorphic to the original, and the monopole solution leads to a regular spacetime, as was demonstrated by . These cases are the most uninteresting ones as far as constructing a black hole spacetime which is asymptotically plane wave. The interesting case therefore is the case when , reverting back to cartesian coordinates. Now it is clear that is an asymptotically plane wave spacetime, whilst retaining the regular black hole horizon at . These statements of course remain true in the Einstein frame as well. The string frame metric for the solution is then given as (setting ), where the functions , , , are given in (5.2). Once again the Einstein frame metric is . The essential trick in constructing the same is that close to the origin, the plane wave is identical to flat space and so given a spacetime which has a horizon at , we can make it asymptotically plane wave whilst keeping the horizon. The above construction provides an example of a charged black string which is asymptotically of the plane wave form. This naturally begs the question whether there isn’t a neutral black string which is asymptotically a plane wave. We cannot “uncharge” the above solution to get a neutral solution, as then the horizon shrinks to zero size; setting any of the charges to be zero causes the horizon to shrink toward the singularity. However, we should be able to take two such solutions with opposite charges and collide them. Since each such black string has a finite horizon area, the area theorem will tell us that the resulting solution ought to have a horizon of finite area. Colliding two such solutions shouldn’t change the asymptotics and hence we should have a spacetime which is a neutral black string with a finite entropy and asymptoting to a plane-wave spacetime. In the first four sections of this paper, we have established that pp-waves cannot admit event horizons. While this is easily motivated for plane waves, we have provided an alternate argument for the more general pp-waves, which in particular applies to plane waves. For the particular case of plane waves one can argue for the absence of horizons in a more rigorous fashion following the analysis of the causal structure of these spacetimes in . We shall present similar arguments for pp-waves in a future work . The reason for considering pp-waves rather than just plane waves is that for both classes, all the curvature invariants vanish, so that they represent exact classical solutions to string theory. Also, given a plane wave background, one simple deformation of the same is to convert it into a pp-wave background. This follows trivially from the fact that the Einstein’s equations for the metric ansatz in (2.1) are linear, enabling superposition of the solutions. In fact, this is the simplest example of the Garfinkle-Vachaspati construction ,. In a sense, these deformations are similar to exactly marginal deformations in the world-sheet theory, though they are non-normalizable. It is then perhaps somewhat disappointing that these classes of exact classical solutions cannot admit horizons. However, this does not mean that these solutions cannot be modified to include horizons. After all, on physical grounds, one would expect that if one puts some matter into a plane wave which respects the necessary symmetries, this matter may nevertheless be Jeans-unstable to collapsing. Naively, one may then expect to obtain a black hole. This of course does not conflict with our previous conclusions, since such black holes would break the original symmetries. In particular, the geometry would no longer support a covariantly constant null Killing field. It would be very interesting to study this Jeans instability, and to follow the evolution dynamically, but that is beyond the scope of the present paper. Instead, in the previous section, we have explored only a mild relaxation of the pp-wave symmetries, namely, keeping the null Killing field, and dropping only the requirement that it be covariantly-constant. Hence, this is not a pp-wave, and the curvature invariants do not vanish. We have presented an explicit solution of a black string with a horizon which is asymptotically a plane wave, in five-dimensions. It would be very interesting to study solutions which do not carry any charges. One strategy as mentioned earlier would be try to collide two oppositely charged asymptotically plane wave black strings. The collision of plane waves is a problem that has been discussed hitherto in , , . Perhaps it would be possible to extend these discussions to the charged black string discussed above. It is a great pleasure to thank Gary Horowitz, Nemanja Kaloper, and Don Marolf for illuminating discussions. 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? 1. Stephen hawking believes that the earth is unlikely to be the only planet ___ life has developed gradually. （2010福建卷） A. that B . where C. which D. whose 解析：先行词为planet，表示地点，从句中的动词develop为 不及物动词,从句不缺少成分，故排除选项中的关系代词that, which, whose，选择where在从句中作状语。 ? 2. Some pre-school children go to a day-care center, ___ they learn simple games and songs. (2007全国卷?) A. then B. there C. while D. where 解析：先行词为a day-care center,表地点，选择where在从句 中作地点状语。 3. ---Can you believe I had to pay 30 dollars for a haircut ? ---You should try the barber’s ___ I go. it’s only 15. （2010天津卷） A. as B. which C. where D. that 从句中的go是不及物动词，从句不缺少成分，所以排除选项中的 关系代词as, which, that。the barber’s = the barber’s house 表地点，故应选择where在从句中作状语。 1. 先行词虽然是表地点的名词，但引导定语从句的关系 词不一定用where。 New York, ___ last year is a nice old city. （2003北京卷） A. that I visited B. which I visited C. where I visited D. in which I visited 先行词为New York，地点名词，从句中谓语动词visit是及物动 词，用关系代词作从句的宾语成分，故应选B. which I visited。 ? 2. 先行词为地点名词，且在从句中作状语成分，关系词 也可以用in / at /on which替代。in用于某一空间范围， at用于平面上的某一点，有时也可以用on。 (1) The place ___ the bridge is supposed to be built should be ___ the cross-river traffic is the heaviest. （2005江苏卷） A. which; where B. at which; which C. at which; where D. which; in which 第一个空格后的定语从句不缺少成分，先行词又是地点名词 place，在从句中作状语，故可填at which或where。第二个空格 后的句子也不缺少成分，由题意可知，应用where引导表语从 句，故应选C. at which; where。 (2) Is this the house ___ shakespeare was born? （1988全 国卷） A. at where B. which C. in which D. at which 先行词the house，表空间范围的地点名词，从句中不缺少成 分，应用关系副词where或in which作状语，故应选C. The house I grew up ___ has been taken down and replaced by an office building. （2009江西卷） A. in it B. in C. in that D. in which the house后面i grew up这个句子是定语从句，grew up是不及物动词，而the house在从句中作宾语，显然需要加 介词，构成the house (which / that) i grew up in...的结构， 而which / that关系代词可以省略，故选B. in。 ? 二.先行词（表示家具、衣物、工具等名词）+关系副词 （where）+定语从句 1. If a shop has chairs ___ women can park their men, women will spend more time in the shop. （2005，上海卷） A. that B. which C. when D. where 解析：先行词为chairs，是表示具体事物的名词，又知定语从句 中不缺少成分。根据句意可把 chairs看作表地点的名词，在从 句中作状语，故选d. where。 ? 2. There were dirty marks on her trousers ___ she had wiped her hands. （2004四川卷） A. where B. which C. when D. that ? 解析：由题意可知先行词应为trousers，而不是marks。从句 中不缺少成分，故排除选项中的关系代词which, that，而应选 择where在从句中作状语。 三.先行词（抽象名词）+关系副词（where）+定语从句 I can think of many cases ___ students obviously knew a lot of English words and expressions but couldn’t write a good essay. A. why B. which C. as D. where 先行词为case，属抽象名词，且从句中不缺少成分，故排除选项 中的关系代词which和as. 2. It’s helpful to put children in a situation ___ they can see themselves differently. （2009福建卷） A. that B. when C. which D. where 本句先行词为situation，译作“环境，境遇”，用于表示地点的 抽 象名词，且从句中不缺少成分，故应选D. where。 3. we’re just trying to reach a point ___ both sides will sit down together and talk. （2006山东卷） A. where B. that C. when D. which 先行词为point，可译作“目标，目的”，属表地点的抽象名词， 从 句中不缺少成分，故应选A. where。 4. Those successful deaf dancers think that dancing is an activity ___ sight matters more than hearing. （2007天津卷 A. when B. whose C. which D. where 先行词为activity，属表示地点的抽象名词，而从句中的谓词动词 matter为不及物动词，不需要宾语，故应选D. where。 5. Many people who had seen the film were afraid to go to the forest when they remembered the scenes ___ people were eaten by the tiger. （2005广东卷） A. in which B. by which C. which D. that 先行词为scene，属于表地点的抽象名词，从句为被动语态，且不 缺少成分,故选A. in which。 6. I’ll give you my friend’s home address, ___ I can be reached most evenings. （2008北京卷) A. which B. when C. whom D. where 先行词为address，属表地点的抽象名词，且从句中不缺少成分,故 应选D. where。 ? 7. All the neighbors admire this family, ___ the parents are treating their child like a friend. （2008安徽卷） A. why B. where C. which D. that ? 先行词为family，属表地点的抽象名词，且从句中不缺少成分， 故选B. where。 8. I work in a business ___ almost everyone is waiting for a great chance. （2004湖南卷） ? A. how B. which C. where D. that ? 先行词为business，属表地点的抽象名词，且从句中不缺少成 分，故选C. where。 9. — What do you think of teaching, Bob ? — I find it fun and challenging. It is a job ___ you are doing something serious but interesting. （2009北京卷） ? A. where B. which C. when D. that 先行词为job，属表地点的抽象名词，且从句中不缺少成分，故 选A. where。 ? ? ? ? ? ? ? ? ? 例题展示： ①The door look very nice after white. A. painting B. being painted C. to be painted D. painted ②Before to a university, you are supposed to work harder and make preparations. A. be admitted B. being admitted C. admitting D. admit 考点提示：where引导的定语从句。 特殊的先行词（抽象名词）+关系副词（where/ when） +定语从句(从句中不缺充分)。 这些特殊的先行词有 case（情况）/ condition(状况)/ point（阶段）/ position（处境）/ situation（情形）/ occasion(场合) / circumstances（境况）/ scene（情景，场面） ? 例题展示： ? ①After graduating from high school, you will reach a point in your life ________you need to decide what to do. ? A. that B. what C. which D. where ? ②In our next English class the cases will be offered to you ________these phrases can be used together. ? A. that B. which C. who D. where ? ③（2008· 山东） Occasions are quite rare ___________I have the time to spend a day with my kids. ? A. who B. which C. where D. when ? ④（2009· 福建） It’s helpful to put children in a situation ________they can see themselves differently. ? A. that B. when C. which D. where ? ⑤The head of the company is in a slightly awkward position___ he can’t handle the problems he is faced with. ? A. that B. when C. which D. where ? ⑥（2005广东）Many people who had seen the film were afraid to go to the forest when they remembered the scenes ___ people were eaten by the tiger. A. in which B. by which C. which D. that

« PreviousContinue » 11. An obtuse angle is that which is greater than a right angle. 12. An acute angle is that which is less than a right angle. 13. A term or boundary is the extremity of anything. 14. A figure is that which is enclosed by one or more boundaries. 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 19. A segment of a circle is the figure contained by a straight line, and the circumference it cuts off. 20. Rectilineal figures are those which are contained by straight lines. 21. Trilateral figures, or triangles, by three straight lines. 22. Quadrilateral by four straight lines. 23. Multilateral figures, or polygons, by more than four straight lines. 24. Of three-sided figures, an equilateral triangle is that which has three equal sides. 25. An isosceles triangle is that which has two sides equal. 26. A scalene triangle is that which has three unequal sides. 27. A right-angled triangle is that which has a right angle. 28. An obtuse-angled triangle is that which has an obtuse angle. 29. An acute-angled triangle is that which has three acute angles. 30. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. 31. An oblong is that which has all its angles right angles, but has not all its sides equal. 32. A rhombus is that which has all its sides equal, but its angles are not right angles. 33. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. 34. All other four-sided figures besides these are called Trapeziums. 35. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. 1. Let it be granted that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any centre, at any distance from that centre. 1. Things which are equal to the same are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double of the same are equal to one another. 7. Things which are halves of the same are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. 9. The whole is greater than its part. 10. Two straight lines cannot enclose a space. 11. All right angles are equal to one another. 12. If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. PROBLEM. To describe an equilateral triangle upon a given finite straight line. Let AB be the given straight line; it is required to describe an equilateral triangle upon it. From the centre A, at the distance AB, describe (3 Postulate) the circle BCD, and from the centre B, at the distance BA, describe the circle ACE; and from the point C, in which the circles cut one another, draw the straight lines (1 Post.) CA, CB, to the points A, B: ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal (15 Definition) to AB; and because the point B is the centre of the circle ACE, BC is equal to BA: But it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB: But things which are equal to the same are equal to one another (1 Axiom); therefore CA is equal to CB; wherefore CA, AB, BC are equal to one another; and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB. Which was required to be done. PROB. From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line; it is required to draw from the point A a straight line equal to BC. From the point A to B draw (1 Post.) the straight line AB; and upon it describe (1. 1.) the equilateral triangle DAB, and produce (2 Post.) the straight lines DA, DB, to E and F; from the centre B, at the distance BC describe (3 Post.) the circle CGH, and from the centre D, at the distance DG, describe the circle GKL: AL shall be equal to BC. Because the point B is the centre of the circle CGH, BC is equal (15 Def.) to BG; And because D is the centre of the circle GKL, DL is equal to DG, and (Constr.) DA, DB, parts of them, are equal; therefore the remainder AL is equal to the remainder (3 Ax.) BG ; But it has been shown, that BC is equal to BG; wherefore AL and BC are each of them equal to BG; and things that are equal to the same are equal (1 Ax.) to one another; therefore the straight line AL is equal to BC. Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC. Which was to be done. PROB. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, whereof AB is the greater. It is required to cut off from AB, the greater, a part equal to C, the less. From the point A draw (2. 1.) the straight line AD equal to C; and from the centre A, and at the distance AD describe (3 Post.) the circle DEF; And because A is the centre of the circle DEF, AE is equal (15 Def.) to AD; but the straight line C is likewise equal (Constr.) to AD; whence AE and C are each of them equal to AD; wherefore the straight line AE is equal (1 Ax.) to C, and from AB, the greater of two straight lines, a part AE has been cut off equal to C the less. Which was to be done. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each; and have likewise the angles contained by those sides equal to one another; they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. Let ABC, DEF be two triangles, which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz., AB to DE, and AC to DF; and the angle BAC equal to the angle EDF, the base BC shall be equal to the base EF; and the triangle ABC to the triangle DEF; and the other angles to which the equal sides are opposite, shall be equal, each to each, viz., the angle ABC to the angle DEF, and the angle ACB to DFE. For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE, the point B shall coincide with the point E, because AB is equal (Hyp.) to DE; and AB coinciding with DE, AC shall coincide with DF, because the angle BAC is equal (Hyp.) to the angle EDF; wherefore also the point C shall coincide with the point F, because the straight line AC (Hyp.) is equal to DF: But the point B coincides with the point E; wherefore the base BC shall coincide with the base EF, because the point B coinciding with E, and C with F, if the base BC does not coincide with the base EF, two straight lines would enclose a space, which is impossible (10 Ax.). Therefore the base BC shall coincide with the base EF, and be equal to it. Wherefore the whole triangle ABC shall coincide with the whole triangle DEF, and be equal to it (8 Ax.); and the other angles of the one shall coincide with the remaining angles of the other, and be equal to them, viz., the angle ABC to the angle DEF, and the angle ACB to DFE. Therefore, if two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to one another, their bases shall likewise be equal, and the triangles be equal, and their other angles to which the equal sides are opposite shall be equal, each to each. Which was to be demonstrated. THEOR. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles upon the other side of the base shall be equal. Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC be produced to D and E, the angle ABC shall be equal to the angle ACB, and the angle CBD to the angle BCE. In BD take any point F, and from AE the greater, cut off AG equal to (3. 1.) AF, the less, and join FC, GB. Because AF is equal (Constr.) to AG, and AB to (Hyp.) AC, the two sides FA, AC are equal to the two GA, AB, each to each; and they contain the angle FAG common to the two triangles AFC, AGB; therefore the base FC is equal (4. 1.) to the base GB, and the triangle AFC to the triangle AGB; and the remaining angles of the one are equal (4. 1.) to the remaining angles of the other, each to each, to which the equal sides are opposite, viz., the angle ACF to the angle ABG, and the angle AFC to the angle AGB: And because the whole AF is equal to the whole AG (Constr.), of which the parts AB, AC, are equal (Hyp.); the remainder BF shall be equal (3 Ax.) to the remainder CG; and FC was proved to be equal to GB; therefore the two sides BF, FC are equal to the two CG, GB, each to each; and the angle BFC is equal to the angle CGB;

So there is some variation involved. Also notice that there were 7 df on top and df on bottom. This requires that you have all of the sample data available to you, which is usually One way analysis of variance case, but not always. Multiple Comparisons Results In the "Tukey Grouping" table, means with the same letter are not significantly different. The standard error of the difference is an estimate of the variability within each group assumed to be the same. Select o3 as the dependent variable. Finishing the Test Well, we have all these wonderful numbers in a table, but what do we do with them? If the chance of a type I error in one such comparison is 0. Abstract This review introduces one-way analysis of variance, which is a method of testing differences between more than two groups or treatments. The researcher randomly assigns a group of volunteers to either a group that a starts slow and then increases their speed, b starts fast and slows down or c runs at a steady pace throughout. Review the Results This analysis tests whether the independent variable shift is a significant factor in accounting for the variation in ozone levels. She conducted a randomized clinical trial to see the differences between 3 different groups of high school students attending one sexual education class, attending one sexual education plus one skills training class, and attending regular class- no sexual educational class provided. The total variation not variance is comprised the sum of the squares of the differences of each mean with the grand mean. What two number were divided to find the F test statistic? One-way analysis of variance In an independent samples t-test, the test statistic is computed by dividing the difference between the sample means by the standard error of the difference. To determine which specific groups differed from each other, you need to use a post hoc test. The number of pairwise comparisons is given by 4C2 and is equal to 4! If the sample means are close to each other and therefore the Grand Mean this will be small. Which means are different? Within Group Variation Error Is every data value within each group identical? This study design is illustrated schematically in the diagram below: The questionnaires were given to the participants of the study, 4 weeks after attending the classes. Calculations of the means and the variance are performed as part of the hypothesis test. If the variances are not known to be equal, a generalization of 2-sample Welch's t-test can be used. Between Group Variation Treatment Is the sample mean of each group identical to each other? The general conclusion from these studies is that the consequences of such violations are less severe than previously thought. I am writing my hypotheses like this: We need a critical value to compare the test statistic to. Each sample is considered independently, no interaction between samples is involved. The commonly used normal linear models for a completely randomized experiment are: The standard error of the difference is an estimate of the variability within each group assumed to be the same. The weight applied is the sample size. This is the total variation. There are k samples involved with one data value for each sample the sample meanso there are k-1 degrees of freedom. Are you ready for some more really beautiful stuff? There is the between group variation and the within group variation. Tiku found that "the non-normal theory power of F is found to differ from the normal theory power by a correction term which decreases sharply with increasing sample size. However, as either the sample size or the number of cells increases, "the power curves seem to converge to that based on the normal distribution". Remember that error means deviation, not that something was done wrong. Also recall that the F test statistic is the ratio of two sample variances, well, it turns out that's exactly what we have here. The one-way analysis of variance ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent unrelated groups. Notice that the between group is on top and the within group is on bottom, and that's the way we divided. The case of fixed effects, fully randomized experiment, unbalanced data[ edit ] The model[ edit ] The normal linear model describes treatment groups with probability distributions which are identically bell-shaped normal curves with different means.The One-Way ANOVA task enables you to perform an analysis of variance when you have a continuous dependent variable and a single classification variable. For example, consider the data set on air quality (Air), described in the preceding section. Analysis of Variance, or ANOVA for short, is a statistical test that looks for significant differences between means. For example, say you are interested in studying the education level of athletes in a community, so you survey people on various teams. A One-Way Analysis of Variance is a way to test the equality of three or more means at one time by using variances. Assumptions The populations from which the samples were obtained must be normally or approximately normally distributed. One-way Analysis of Variance (ANOVA) Essentially Analysis of Variance (ANOVA) is an extension of the two sample hypothesis testing for comparing means (when variances are unknown) to. Mar 01, · One-way analysis of variance is the simplest form. It is an extension of the independent samples t-test (see statistics review 5 [ 1 ]) and can be used to. The one-way analysis of variance compares the means of two or more groups to determine if at least one group mean is different from the others. The F-ratio is used to determine statistical significance. The tests are non-directional in.Download

✍️ Free Response Questions (FRQs) 👆 Unit 1 - Exploring One-Variable Data 1.4Representing a Categorical Variable with Graphs 1.5Representing a Quantitative Variable with Graphs 1.6Describing the Distribution of a Quantitative Variable 1.7Summary Statistics for a Quantitative Variable 1.8Graphical Representations of Summary Statistics 1.9Comparing Distributions of a Quantitative Variable ✌️ Unit 2 - Exploring Two-Variable Data 2.0 Unit 2 Overview: Exploring Two-Variable Data 2.1Introducing Statistics: Are Variables Related? 2.2Representing Two Categorical Variables 2.3Statistics for Two Categorical Variables 2.4Representing the Relationship Between Two Quantitative Variables 2.8Least Squares Regression 🔎 Unit 3 - Collecting Data 3.5Introduction to Experimental Design 🎲 Unit 4 - Probability, Random Variables, and Probability Distributions 4.1Introducing Statistics: Random and Non-Random Patterns? 4.7Introduction to Random Variables and Probability Distributions 4.8Mean and Standard Deviation of Random Variables 4.9Combining Random Variables 4.11Parameters for a Binomial Distribution 📊 Unit 5 - Sampling Distributions 5.0Unit 5 Overview: Sampling Distributions 5.1Introducing Statistics: Why Is My Sample Not Like Yours? 5.4Biased and Unbiased Point Estimates 5.6Sampling Distributions for Differences in Sample Proportions ⚖️ Unit 6 - Inference for Categorical Data: Proportions 6.0Unit 6 Overview: Inference for Categorical Data: Proportions 6.1Introducing Statistics: Why Be Normal? 6.2Constructing a Confidence Interval for a Population Proportion 6.3Justifying a Claim Based on a Confidence Interval for a Population Proportion 6.4Setting Up a Test for a Population Proportion 6.6Concluding a Test for a Population Proportion 6.7Potential Errors When Performing Tests 6.8Confidence Intervals for the Difference of Two Proportions 6.9Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions 6.10Setting Up a Test for the Difference of Two Population Proportions 😼 Unit 7 - Inference for Qualitative Data: Means 7.1Introducing Statistics: Should I Worry About Error? 7.2Constructing a Confidence Interval for a Population Mean 7.3Justifying a Claim About a Population Mean Based on a Confidence Interval 7.4Setting Up a Test for a Population Mean 7.5Carrying Out a Test for a Population Mean 7.6Confidence Intervals for the Difference of Two Means 7.7Justifying a Claim About the Difference of Two Means Based on a Confidence Interval 7.8Setting Up a Test for the Difference of Two Population Means 7.9Carrying Out a Test for the Difference of Two Population Means ✳️ Unit 8 Inference for Categorical Data: Chi-Square 📈 Unit 9 - Inference for Quantitative Data: Slopes 🧐 Multiple Choice Questions (MCQs) Best Quizlet Decks for AP Statistics ⏱️ 3 min read June 3, 2020 The least squares regression line is the best linear regression line that exists. It’s made by minimizing the sum of the squares of the residuals. Why square the residuals? This is because if we didn’t, negative and positive residuals would cancel out, reducing the impact of the residuals. Like regular regression models, the LSRL has a formula of ŷ=a+bx, with a being y-intercept and b being slope with each having their own formula using one-variable statistics of x and y. The slope is the predicted increase in the response variable with an increase of one unit of the explanatory variable. To find the slope, we have the formula: image courtesy of: codecogs.com This is basically saying that the slope is the average deviation of y over the average deviation of x with the correlating coefficient as a correcting factor. When asked to interpret a slope of a LSRL, follow the template below: There is a predicted increase/decrease of ______ (slope in unit of y variable) for every 1 (unit of x variable). Once we have a slope, we can get the y-intercept and general formula of the LSRL from point-slope form given that we have a point. Fortunately, we have a point that we can use for this. An important thing to note is that the LSRL always passes through the point (x̄,ȳ). Thus, from point-slope form we have ŷ-ȳ=b(x-x̄) and ŷ=bx+(-bx̄+ȳ). The expression in parentheses is the y-intercept, but usually, you can derive it from the point-slope form. We can interpret the y-intercept as the value the response variable would take if the explanatory variable is 0. When asked to interpret a y-intercept of a LSRL, follow the template below: The predicted value of (y in context) is _____ when (x value in context) is 0 (units in context). To determine how well the LSRL fits the data, we can use a statistic called the coefficient of determination, also called r^2 because it is the correlation coefficient squared. This can be a value between 0 and 1, with 0 meaning that the LSRL does not model the data at all, with the equation being ŷ-ȳ, while 1 means that all the points lie on the LSRL. There is also another formula for r^2 as well. This formula is: image courtesy of: codecogs.com This is saying that this is the percent difference between the variance of y and the sum of the residual squared. In other words, this is the reduction in the variation of y due to the LSRL. When interpreting this we say that it is the “percentage of the variation of y that can be explained by a linear model with respect to x.” When asked to interpret a coefficient of determination for a least squares regression model, use the template below: ____% of the variation in (y in context) is due to its linear relationship with (x in context). The last statistic we will talk about is the standard deviation of the residuals, also called s. S is the typical residual by a given data point of the data with respect to the LSRL. The formula for s is given as image courtesy of: apcentral.collegeboard.org which looks similar to the sample standard deviation, except we will divide by n-2 and not n-1. Why? We will learn more about s when we learn inference for regression in Unit 9. On the AP test, it is very likely that you will be expected to read a computer printout of the data. Here is a sample printout with a look at where most of the statistics you will need to use are (the rest you will learn in Unit 9): Courtesy of Starnes, Daren S. and Tabor, Josh. The Practice of Statistics—For the AP Exam, 5th Edition. Cengage Publishing. Always use R-Sq, NEVER R-Sq(adj)! 🎥Watch: AP Stats - Least Squares Regression Lines 2550 north lake drive milwaukee, wi 53211 92% of Fiveable students earned a 3 or higher on their 2020 AP Exams. *ap® and advanced placement® are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product. © fiveable 2020 | all rights reserved.

2 edition of Essays in algebraic simplification found in the catalog. Essays in algebraic simplification Richard J. Fateman |Statement||by Richard J. Fateman.| |LC Classifications||QA154 .F27| |The Physical Object| |Number of Pages||190| |LC Control Number||73172099| free algebraic expressions linear worksheets 9th grade ; sample algebra test glencoe ; convert to square root ; why use 4/3 when calculating the volume of a sphere ; how to do algebra . Exercise 1. Simplify the following, or where they don’t simplify, write “N/A”. 8𝑥+9𝑥−𝑥+𝑦 → 𝟏𝟔𝒙+𝒚4𝑧2+3𝑧 → 𝑵𝑨8𝑥𝑦. quotes from René Girard: 'The anti-Semitic interpretation fails to discern the real intention of the Gospels. It is clearly mimetic contagion that explains the hatred of the masses for . Fractions to Decimals & Percents (easy) – Fractions to Decimals & Percents (difficult) – Decimals to Fractions & Percents – Percents to Fractions and Decimals – Percent of Increase and . Simplifying Algebraic Expressions. By “simplifying” an algebraic expression, we mean writing it in the most compact or efficient manner, without changing the value of the expression. This . Algebra II Free Essay, Term Paper and Book Report This semester I gained knowledge of not only learning math, but also enjoying it. Algebra II has become my favorite subject of this year. . Handbook for effective management of student employees Blair House, the presidents guest house Planning in Northern Ontario (Bulletin / Planning reform in Ontario) Selling skills for the nonsalesperson Tax planning for closely held corporations, 1980 Proposed medical facilities planning annex to the Texas state health plan A short discourse, proving that the Jewish or seventh-day sabbath is abrogated and repealed. Tackling drink driving Fond du Lac County, WI: Digital orthophoto quadrangle data This thesis consists of essays on several aspects of the problem of algebraic simplification by computer. Since simplification is at the core of most algebraic manipulations, efficient and. The Karnaugh Map Boolean Algebraic Simplification. i need help completing this assignment. 5 questions. i have completed the rest of the assignment. Abstract. Some basic techniques for the simplification of terms are surveyed. In two introductory sections the problem of canonical algebraic simplification is formally stated and some. Simplifying Expressions Simplifying Expressions Simplifying expressions is one of the core basics for algebra simplification. The goal is taking the distributive property, which is used to. Simplification of algebraic expressions In order to simplify an algebraic expression, do all possible operations in the correct order. Of course the correct order is based on the order of operations. Like and unlike algebraic terms Like algebraic terms are defined as those terms which are represented by the same algebraic symbol, regardless of the sign or the magnitude. ALGEBRAIC SIMPLIFICATION A GUIDE FOR THE PERPLEXED by Joel Moses Project MAC, MIT Abstract Algebraic simplification is examined first from the point of view of a user needing Cited by: Math solver free, coordinate graph, poems about algebra, algebra 1 student workbook answer key, 9th grade algebra. Common denominator finder, examples of algebraic expression. Rakesh Yadav + Maths Chapterwise PDF. SS Bharti Maths Class Notes PDF Download. Abhinay Sharma Maths PDF Download. SSC English + Chapterwise PDF. More generally, the term algebra encompasses nowadays many other fields of mathematics: geometric algebra, abstract algebra, Boolean algebra,s-algebra, to name a few. Algebra is an. Simplify: to make simpler. One of the big jobs we do in Algebra is simplification. In general, it is simpler when it is easier to use. It is now a little easier to use. "Half" is definitely simpler than. Computer algebra systems began to appear in the s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. S = simplify (expr,Name,Value) S = simplify (expr) performs algebraic simplification of expr. If expr is a symbolic vector or matrix, this function simplifies each element of expr. S = simplify 'default': Use the default (internal) simplification criteria. Simplifying Algebraic Expressions You probably know that if you have an expression like you cannot add those terms to simplify it in any way. That’s because one term in a constant (the 4). A little test on algebraic simplication, includes factorising, index laws and algebraic fractions. Algebraic Academy is founded with the vision to provide an Early Start to the leaders of tomorrow by imparting critical life skills on a Game and Fun platform. To help young children to. Check out a sample textbook solution. Using Composite and Inverse Functions In Exercisesuse the functions f(x)=18x3 and g(x)=x3 to find the i Calculus: Early Transcendental. Excellent resource for visual learners. In this activity, students draw models for algebraic expressions based on real-world situation. They can use the models to write and simplify 4/5(50). Basic Algebraic Properties of Real Numbers 10 October Algebraic Properties of Real Numbers The numbers used to measure real-world quantities such as length, area, volume. An algebraic expression is an expression involving numbers, parentheses, operation signs and pronumerals that becomes a number when numbers are substituted for the pronumerals. For File Size: 1MB. Essays in algebraic simplification, Pdf Download Encyclopedia of Molecular Biology Free Ebook From Moscow to Madrid: European Cities, Postmodern Cinema Pdf Download Eating the IT .Simplifying Algebraic Expressions by Combining Like Terms Objective: Students will identify like terms. Students will simplify algebraic expressions by combining like terms. Term Definition File Size: 1MB.Rewrite each phrase as an algebraic expression. 6. c multiplied by 5 5c 7. 10 larger than y 10 + y 8. 9 less than e e - 9 9. triple r 3r half of p p 2 quadruple f 4f Write your answer to the File Size: KB.

Suppose you receive the following questionnaire in an email: Imagine an urn containing 90 balls of three different colors: red balls, black balls and yellow balls. We know that the number of red balls is 30 and that the sum of the the black balls and the yellow balls is 60. Our questions are about the situation where somebody randomly takes one ball from the urn. - The first question is about a choice between two bets: Bet I and Bet II. Bet I involves winning ‘10 euros when the ball is red’ and ‘zero euros when it is black or yellow’. Bet II involves winning ‘10 euros when the ball is black’ and ‘zero euros when it is red or yellow’. The first question is: Which of the two bets, Bet I or Bet II, would you prefer? - The second question is again about a choice between two different bets, Bet III and Bet IV. Bet III involves winning ‘10 euros when the ball is red or yellow’ and ‘zero euros when the ball is black’. Bet IV involves winning ‘10 euros when the ball is black or yellow’ and ‘zero euros when the ball is red’. The second question is: which of the two bets, Bet III or Bet IV, would you prefer? This are exactly the questions sent out by Diederik Aerts and pals at the Brussels Free University in Belgium. They received replies from 59 people which broke down like this: 34 respondents preferred Bets I and IV, 12 preferred Bets II and III, 7 preferred Bets II and IV and 6 preferred Bets I and III. That most respondents preferred Bets I and IV is no surprise. It’s been verified in countless experiments since the 1960s when the situation was dreamt up by Daniel Ellsberg, a Harvard economist (who more famously leaked the Pentagon Papers later that decade). The situation is interesting because, paradoxically, a branch of science called decision theory, on which modern economics is based, predicts that humans ought to make an entirely different choice. Here’s why. Decision theory assumes that any individual tackling this problem would do it by assigning a fixed probability to the chance of picking a yellow or black ball and then stick with that probability as they chose their bets. This approach leads to the conclusion that if you prefer Bet I, then you must also prefer Bet III. But if you prefer Bet II, then you must also prefer Bet IV. Of course, humans don’t generally think like that, which is why most people prefer Bets I and IV (and why modern economic theory has served us so badly in recent years). At the heart of the Ellsberg paradox are two different kinds of uncertainties. The first is a probability: the chance of picking a red ball versus picking a non-red ball, which we are told is 1/3. The second is an ambiguity: the chance of the non-red ball being black or yellow which is entirely uncertain. Conventional decision theory cannot easily handle both types of uncertainty. But various researchers in recent years have pointed out that quantum theory can cope with with both types and what’s more, can accurately model the patterns of answers that humans come up with. We looked at an example a couple of years ago that showed how quantum probability theory can explain other paradoxical behaviours in humans called the conjunction and disconjunction fallacies. Now Aerts and pals have done the same for the Ellsberg paradox by creating a model of the way humans think about this problem and framing it in terms of quantum probability theory. In fact, these guys go further. The point out that humans can also think in a way that is consistent with decision theory and therefore that this thinking must employ classical logic. So both classical and quantum logic must both be at work at some level in human thought. The big surprise is that quantum theory works at all. Just why quantum probability theory should explain the strange workings of the human mind, nobody is quite sure. Neither is it yet clear how quantum probability theory will help to mould new ideas about economics and broader human behaviour. But that’s why there is so much excitement over this new approach and why you’re likely to hear much more about it in future. Ref: arxiv.org/abs/1104.1459: A Quantum Cognition Analysis of the Ellsberg Paradox You can now follow The Physics arXiv Blog on Twitter DeepMind’s cofounder: Generative AI is just a phase. What’s next is interactive AI. “This is a profound moment in the history of technology,” says Mustafa Suleyman. What to know about this autumn’s covid vaccines New variants will pose a challenge, but early signs suggest the shots will still boost antibody responses. Human-plus-AI solutions mitigate security threats With the right human oversight, emerging technologies like artificial intelligence can help keep business and customer data secure Next slide, please: A brief history of the corporate presentation From million-dollar slide shows to Steve Jobs’s introduction of the iPhone, a bit of show business never hurt plain old business. Get the latest updates from MIT Technology Review Discover special offers, top stories, upcoming events, and more.

posted by Anonymous Please help with this test, I was out of school for awhile due to my dad passing away from heart and kidney failure. I'm trying to catch up on missing work and get good grades to make my mom happy and every time I ask my teachers for help with this test they really don't help at all. The test is lesson 11 unit 3 rational numbers unit test. It's a connections academy test If you choose the 5 most difficult questions to post and tell the tutors what YOU THINK and/or what confuses you, you might get some help. No one, however, will take the whole test for you! Her dad just died. Show some sympathy at least. I mean- how would you feel if your dad just passed away and all you want to do is help your mom out by getting good grades? Also, what grade is it for? I think that I can help you. @Anonymous So so so so so sorry about your dad:( Can someone post the answers to the test I'm so sorry for your loss. Please post the questions that you need the most help with, and I can help you with those questions. Thank You. sorry 4 your loss It better not be a lie if it's not my condenses for your loss When your doing your math classwork try matching up the problems with Khanacademy its free and gives you immediate feedback when you are wrong and how to do the problem. The key is to actually learn this math. Then when you get stuck on classwork you can get the Tutor or teachers help. This guy is a fake his/her dad didn't pass away I saw this excuse like 20 times before. cuz of the meloneeee @dud f off u dont know fo sho I NEED ANSWERS :O Jungkook! from kpop ur kewl. Sorry 4 ur loss, too bad I didn't get to kill em mah self!!! >:D sowwy just kiddin D:> fo give mah and mah insignificance. Nugget not all that bad! :v aww here you go Trust me i just took the test 18/18 i hope i help you xoxo michelle aka shelly My test has 30 questions PLEASE ANSWER THEM SURE Fòx what test are you on? Michelle help me with connections acadamy unit 7 lesson 8 systems of equaitons test plz answers? Michelle which test are those answers for this is for lesson 11 unit 3 rational numbers unit test Michelle can you please help me with connections academy unit 5 lesson 10 question 22? I need the answer for that What about unit 5 lesson 10 functions unit test Anybody have answers for does ok sure anonymous what test is that is it a quiz if so which lesson Lesson 13: Measurement Unit Test Algebra Readiness (Pre-Algebra) B Unit 2: Measurement Can someone please give me the answers i know she/he is lying i have seen this lie before she/he is making you feel bad so u give the answers which you did so good job you caved in... Joesph is my boo Who u talking to disboi 1. Which solid has one base that is a rectangle and four lateral surfaces that are triangles? Answer: C - Rectangular Prism Rectangular prism have one rectangular base. 2. A solid with two parallel and congruent bases cannot be which of the following? Answer: A - Cylinder Since the bases in a cylinder ar circular, they cannot be parallel. 3. Which of the following are considered skey lines? Diagram is a A. AC and DF B. AD and CF C. BE and DF <--- Correct answer. These lines do not intersect and are not parallel. D. BE and BC 4. What is the base plan for the set of stacked cubes? -There are two cubes in the front row ( side by side) - There are two cubes in the middle row ( one on top of the other) - Ther are three cubes in the back row (one on top of the other ) and one cube attached to the right of these three cubes. Correct Answer: D 5. Which of the following is the front few of the model? - There are two cubes in the front row ( side by side) - There are two cubes in the middle row ( one on top of the other) - There are five cubes on the back row ( three stacked on top of each other and two cubes attached (side by side) to those stacked cubes on the left side Correct Answer: B 6. Which solid does the net form? A. hexagonal prism B. hexagonal pyramid <----- ANSWER The bases are hexgons and the lateral faces are rectangles. C. rectangular prism D. rectangular pyramid 7. Which solid does the net form? A. Square Pyramid <----------- ANSWER The base is a square. B. Triangular Prism C. Triangular Pyramid 8. What is the surface area of the given figure? Triangular prism with the following measurements: 20 cm, 32 cm, 16 cm , and 12 cm. Here are the steps: 1) 12 x 16 = 192 2) 192 / 3 = 64 3) 192 / 3 = 64 4) 192 / 3 = 64 5) 12 x 32 = 384 6) 32 x 20 = 640 7) Add Steps 2 - 6 --> 64 + 64 + 64 +384 + 640 = 1,216 cm^2 (ANSWER) 9. Use the net to find the approximate surface area of the cylinder to the nearest square meter. There is a cylinder with a radius of 7m and a height of 11 m. Use the value 3.14 for pi Surface Area = 2 x pi x radius x height = 2 x 3.14 x 7 x 11 = 483.56 m^2 ----> Rounded up ---> 484 m^2 ( ANSWER) 10. What is the volume of the prism to the nearest whole unit. There is a rectangular box with the following measurements 11 in , 9 in, and 3 in Step 1) Add all the measurments together 11 in + 9 in + 3 in = 23 in^3 (ANSWER) 11.) What is the volume of the triangular prism to the nearest whole unit ? There is triangular prism, think triangular box. It has the following dimensions 2 ft , 12 ft , and 18 ft Step 1) Multiply 2 x 12 x 18 = 432 Step 2) Double 432 --- > 432 + 432 = 864 ft^3 (ANSWER) 12.) What is the volume of the cone to the nearest whole unit? There is a cone with with a circular radius of 6 inches and a cone height of 11 inches. Use 3.14 for pi Step 01) 3 x pi x radius x 2 x height Step 02) 3 x 3.14 x 6 x 2 x 11 Step 03) 1,243.44 ----> 1, 244 in^3 ( ANSWER) 13.) What is the volume of the pyramid to the nearest whole unit? There is a pyramid with the measurements 7yds, 9yds, and 7yds (height) Step 01) Multiply 7 x 9 = 63 Step 02) Divide 63 / 2 = 31.5 Step 03) Multiply 31.5 by the height --> 31.5 x 7 = 220.5 Round up to 221 yd^3 (ANSWER) 14.) What is the slant height for the given pyramid to the nearest whole unit? Pyramid base = 6 cm Height = 4 cm 6^2 + 4^2 = C^2 6 x 6 + 4 x 4 = C^2 36 + 16 = C^2 52 = C^2 SquareRoot 52 = SquareRoot C^2 7.2 = C Round 7.2 to the nearest whole ---> 7 cm (ANSWER) 15) What is the length of the diagonal for the given rectangular prism to the nearest whole unit? Length = 12 cm Width = 4 cm Height = 9 cm Step 01) 12 ^ 2 + 4 ^ 2 = C^2 144 + 16 = C^2 160 = C^2 SquareRoot 160 = SquareRoot C^2 12. 6 = C (Diagonal) 12.6 Rounded to the nearest whole ----> 13 cm (ANSWER) 16) The cones below are similar, although not drawn to scale. Small Cone: r = 5ft and 15ft Big Cone = x and 18ft Step 1) 15 / 5(r) = 3 Step 2) 18 / x (r) = 3. What number when you divide 18 by it is 3? .......It is 6 duuuhhhhh B.) 6 (ANSWER) 17. ) A cone has a radius of 40 cm and a volume of 1,875 cm^3. What is the volume of a similar cone with a radius of 16 cm? Step 1) 40/16 = 2.5 Step 2) Divide 1, 875 by 2.5 = 750 cm^3 (ANSWER) 18) What is the area of a sphere with a radius of 5 meters rounded to the nearest square meter? Step 01) A = 4 x pi x r^3 ---> A = 4 x 3.14 x 5^3 ----> A = 4 x 3.14 x 5 x 5 x 5 ---> A = 1,570 Step 02) Divide the answer above , 1, 570 by 3 ---> 1,570 / 3 = 523.33 <---Round down is 523m^2 D) 523 m ^2 (ANSWER) 19.) What is the volume of a sphere with a radius of 4 meters rounded to the nearest square meter? Step 01) Use formula = 4 x pi x r^2 --> 4 x 3.14 x 4^2 --> 4 x 3.14 x 4 x 4 ---> 200.96 m^3 Round 200.96 to the nearest whole is 201 m^3. B) 201 m^3 (ANSWER) 20.) Find the lateral area of the square pyramid. There is a square pyramid with the measurements 8 m , 8m and a height of 22 m Step 01) Area ofa the Square Base 8 x 8 = 64 Step 02) Area of each Triangle 8 x 22 = 176 Step 03) There are four triangles multiply 176 x 4 = 704 Step 05) Divide 704/ 2 = 352 Step 06) Add 64 + 352 = 416 m^2 (ANSWER) 21) Find the surfacea area of the cone. Use 3.14 for pi. There is cone with the diameter of 8 in and 7 in for slant height. Step 01) pi x 8^2 --> 3.14 x 8 x 8 = 200.96 Step 02) 200.96 x 7 = 1 , 406.72 Step 03) 1, 402.77 divide by 3 = 468. 9 in^2 (ANSWER) 22) Find the volume of the cylinder. Use 3.14 for pi There is a cylinder with a radius of 34 m and a height of 27 Step 01) pi x r^2 Step 02) 3.14 x 34^2 = 3.14 x 34 x 34 = 3, 629. 84 Step 03) 3,629 x 27 = 98, 005. 68 m^3 (ANSWER) Can someone help me with the questions on integers lesson 10 unit 4 test i can tell you the ones i need help on in connexus, i need help with most my mom is in work at india so she cant help me and my dad isnt home can you guys help plz your a lying of course her dad didn't die is everyone so gullible I am not taking any side cuz i don't want to be in this i just want help I mean if her dad died he died that is sad if not then don't lie thats that i don't care enough to argue about it Please do not cheat, even if your dad died, or someone died in your family, then that is still not an excuse to cheat. @Anonymous your mother would be even more proud to know that you did your best. Grades can always be fixed, your families trust and your life cant. @Anonymous, even if your Dad isn't dead, why did someone lie for you, and why don't you stand up for yourself? Teachers know what they are doing. Cheating isn't right. And you should know that. Didn't your parents even teach you that? Cheating, if used multiple times, will lead to banning from Connexus. Cheating becomes a habit you forever remember. Don't do it, and you wont be in trouble about it. Thank you for your understanding, and keep rocking it Teachers! would some one beable to help me with a unit test it is unit 3 lesson 13 so sorry for your loss that is aweful:( michelle is wrong i got 1/18 ;( now i am failing math, thanks!! here are the really answers Lesson 7:probibility unit test, unit 4. 19. You are on your own 20. You are on your own. Yes there is a lot of A's but you have to trust be, for your grades! Hey Rosie, would you mind telling me the answers for this please: Lesson 11: Graphing, Equations, and Inequalities Unit Test CE 2015 1205010 M/J Grade 6 Mathematics - T2 Unit 5: Graphing, Equations, and Inequalities ColCA Student 10 (4.26.2018) Does anyone actually have the answers to the: Lesson 9: Introduction to Functions Unit Test CE 2015 Algebra 1 A Unit 5: Introduction to Functions Thank You, ColCA Student 10 i bet your lying about that and even if your not I DONT GIVE A F*CK Is rosie right? Is rosie right? because, I'm pretty sure that questions 16 to 18 are WorkPads. HEY GUYS WHAT ABOUT THE LAST UNIT TEST IN MATH WE DEFINITLY NEED THE ANSWERS TO THAT!!! ANYONE KNOW LESSON 7 UNIT 6 ANSWERS FOR CONEXUS Ur Welcome :> lesson 9 unit 2 algebra 1B correct for connection academy 17 yes A

Primordial black holes as a novel probe of primordial gravitational waves. II: Detailed analysis Department of Physics and Astronomy, Johns Hopkins University 3400 N. Charles Street, Baltimore, Maryland 21218, USA Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan April 9, 2020April 9, 2020 April 9, 2020April 9, 2020 Recently we have proposed a novel method to probe primordial gravitational waves from upper bounds on the abundance of primordial black holes (PBHs). When the amplitude of primordial tensor perturbations generated in the early Universeisfairly large, they inducesubstantial scalar perturbations due to their second-order effects. If these induced scalar perturbations are too large when they reenter the horizon, then PBHs are overproduced, their abundance exceeding observational upper limits.That is,primordial tensor perturbations on superhorizon scalescan be constrained from the absence of PBHs. In our recent paper we have only shown simple estimations of these new constraints, and hence in this paper, we present detailed derivations, solving the Einstein equations for scalar perturbations induced at second order in tensor perturbations. We also derive an approximate formula for the probability density function of induced density perturbations, necessary to relate the abundance of PBHs to the primordial tensor power spectrum, assuming primordial tensor perturbations follow Gaussian distributions.Our newupper bounds from PBHsare compared with other existing bounds obtained from big bang nucleosynthesis, cosmic microwave background, LIGO/Virgo and pulsar timing arrays. Astochastic background of primordial gravitational waves (PGWs) with a huge range of wavelengths may have been generated in the early Universe. Their power spectrum reflects physical conditions in the early Universe, and hence its constraints provide valuable information for cosmology. PGWs of largest observable wavelengthshave been constrained by Planck Ade:2013zuv and BICEP2 Ade:2014xna, while those of shorter wavelengthshave been constrained by limits onNeff, the effective number of degrees of freedom of relativistic fermions, at big bang nucleosynthesis (BBN) through the current abundance of the light elements Allen:1996vm, or at photon decoupling through the anisotropy of cosmic microwave background (CMB) Smith:2006nka; Kikuta:2014eja. RecentlyPGWs on smaller scales have been constrained by upper limits on the deviation of the CMB photons’ energy spectrum from the Planck distribution Ota:2014hha; Chluba:2014qia. Though BBN and CMB constrain PGWs of a wide range of wavelengths,these upper bounds, obtained through Neff, entail an assumption about the number of relativistic species in the early Universe, as is discussed later. Furthermore, to obtain BBN or CMB bounds we implicitly assume that any physical mechanisms, both known and unknown, increaseNeff, from the standard value Neff=3.046Mangano:2005cc. However, Neff can decrease e.g. in brane world scenarios Ichiki:2002eh; Apostolopoulos:2005at; Maartens:2010ar. Recently weproposed a new method to constrain PGWs in our recent work Nakama:2015nea, which is also applicable on a wide range of wavelengths and in addition does not depend on the aforementioned assumptions much.In this paper, we present detailed derivations of the results presented there. As we have briefly discussed in our recent work Nakama:2015nea, PBHs can also be used to constrain tensor perturbations generated in the early Universe, exiting the horizon once and reentering the horizon later.This is because largetensor perturbations induce large scalar perturbations (induced scalar perturbations) at second order in tensor perturbations. If primordial tensor perturbations are too large, induced scalar perturbations become also too large, and then they collapse to overproduce PBHs shortly after their horizon reenty, exceeding existing upper limits.That is, primordial tensor perturbations can be constrainedfrom upper limits on PBHs22†2Second-order effects of scalar perturbations to induce tensor perturbations (termed induced gravitational waves) have been discussed in the literatureMatarrese:1993zf; Matarrese:1997ay; Carbone:2004iv; Ananda:2006af; Baumann:2007zm; Alabidi:2012ex; Bugaev:2010bb; we can place upper bounds on scalar perturbations(, which can be translated into upper bounds on the abundance of PBHs Saito:2008jc; Saito:2009jt; Bugaev:2009zh,) from the non-detection of GWs. Note that our present paper discusses an effect opposite to this generation of induced gravitational waves. The direct gravitational collapse of nonlinear localized gravitational waves has been discussed in the literature Brill:1959zz; Eppley:1977dk; Miyama:1981mh; Shibata:1993fx; Shibata:1995we; Anninos:1996fg; Shibata:1997ix; Alcubierre:2000xu; Pfeiffer:2004qz and so tensor perturbations may also be constrained using this phenomenon. Still, the initial conditions and dynamics of cosmological nonlinear gravitational waves during the radiation-dominated era have not been well understood. Since the dynamics of nonlinear radiation density perturbations is better understood, we consider only scalar perturbations induced by the tensor perturbation.. Whereas we have presented only simple estimations to obtain these new constraints in Nakama:2015nea, in the present paper we show detailed derivations for them. Due to our ignorance of the physics in the early Universe, new upper limits on tensor perturbations on small scales in themselves would be worthwhile. there are models of the early Universe Boyle:2003km; Baldi:2005gk; Copeland:2008kz; Kobayashi:2011nu; Biswas:2014kva; Ashoorioon:2014nta; Cannone:2014uqa; Graef:2015ova can predict large tensor perturbations on small scales, which makes our new upper limits even more valuable (see the next section). Note that, if a model predicts large tensor perturbations on small scales but also large scalar perturbations at the same time, then such a model would be more severely constrained from the absence of PBHs generated from the first-order scalar perturbations. In this paper we consider PBH formation only from induced scalar perturbations, second order in tensor perturbations, and thus our bounds on tensor perturbations are conservative or model-independent, in the sense that these bounds do not depend onfirst-order scalar perturbations on small scales. Importantly, there are models of the early Universe which predict not only large tensor perturbations, but also large tensor-to-scalar ratio on small scales, and our PBH bounds are particularly useful to constrain these types of models, some of which are reviewed in the next section. Ii Early Universe models predicting large tensor-to-scalar ratio on small scales In Boyle:2003km, tensor power spectra were shown to be blue (i.e. larger power on smaller scales) in cyclic/ekpyrotic models, with the spectrum of scalar perturbations kept slightly red (smaller power on smaller scales) to match observations on large scales. The cyclic Universe entails the periodic collisions of orbifold planes moving in an extra spatial dimension, which is equivalently described by a scalar field rolling back and forth in an effective potential. Each cycle consists of an accelerated expansion phase, a slow contraction phase (the ekpyrotic phase), during which the Universe is dominated by the kinetic energy as well as the negative potential energy of the scalar field and primordial fluctuations are generated, a rapid contraction phase followed by a bounce at which matter and radiation are generated, a phase dominated by the kinetic energy of the scalar field, a radiation-dominated, expanding phase, and finally a phase dominated by matter and dark energy. The spectrum of scalar perturbations can be adjusted to be slightly red by tuning the scalar field potential during the ekpyrotic phase, and the tensor spectrum turns out to be blue up to the scale corresponding to the end of the ekpyrotic phase. For early Universe scenarios where the spectrum of tensor perturbations is strongly blue, probing them on CMB scales may be challenging, while constraints on small-scale components, such as those discussed in this paper, may provide useful information. Indeed, they noted that the strongest constraint on their model parameters is obtained from BBN constraints on high-frequency PGWs. If the inflaton violates the null energy condition (NEC, ρ+p≥0), the Hubble parameter increases during inflation (super inflation) and the spectral tilt nT becomes positive, since nT=−2ϵ≡2˙H/H2∝−(ρ+p). In Baldi:2005gk it was shown that NEC can be violated without the instability of fluctuations of the inflaton. There a toy model was introduced, with the energy density of the NEC-violating inflaton ρ=−˙ϕ2/2+V0e−λϕ/Mpl, which leads to a stage of pole-like inflation, when a(t)∼(−t)p,t<0,p=−2/λ2<0. The background and fluctuations are shown to be stable at the classical level. It was noted that in this model, some mechanism, quantum effects or another field, is necessary to avoid singularity at t→0 and to drive the Universe into a radiation-dominated epoch. The spectrum of tensor perturbations generated during a super inflation in the framework of loop quantum cosmology (LQC) is calculated in Copeland:2008kz. There a strong blue tile with nT≃2 was obtained, while the form of the inflaton potential to realize a scale-invariant power spectrum of scalar perturbations was also discussed in their previous works. In their scenario, the nondimensional power spectrum of tensor perturbations on smallest scales is roughly given by the square of the Hubble parameter He at the end of inflation in units of the Planck scale, and this implies that He can be constrained e.g. by our PBH constraints. They note that He is, in principle, also related to the amplitude of scale-invariant curvature perturbations as well, but such a relation has not been obtained yet in the scenarios they consider. Large tensor perturbations on small scales may also be realized in the framework of the so-called generalized G-inflation (G2-inflation) Kobayashi:2011nu. The action of G2-inflation four generic functions K,G3,G4,G5 of ϕ and X=−∂μϕ∂μϕ/2. The quadratic action for the tensor perturbations is The squared sound speed is c2T=FT/GT, which is not necessarily unity in general cases. The parameters ϵ≡−˙H/H2, fT≡˙FT/HFT and gT≡˙GT/HGT are introduced and they are assumed to be nearly constant. The nondimensional power spectrum of the tensor perturbations was obtained as The tensor spectral tilt is given by nT=3−2νT, and the tensor spectrum is blue (0<nT) if 4ϵ+3fT−gT<0. Also, if the sound speed becomes temporarily small, tensor perturbations are enhanced on the corresponding scales. A slightly red spectrum of the curvature perturbation, while keeping the tensor spectrum strongly blue-tilted, was also shown to be realized during a stringy thermal contracting phase at temperatures beyond the so-called Hagedorn temperature (the Hagedorn phase) in Biswas:2014kva, assuming a In that scenario, primordial curvature perturbations originate from statistical thermal fluctuations, not by scalar field quantum fluctuations. Scalar and tensor perturbations in large field chaotic models with non-Bunch-Davies (non-BD) initial states were analyzed in Ashoorioon:2014nta, and it was shown that in that model also gravitational waves can be blue while maintaining slightly red scalar perturbations. Normally, initial states for perturbations are chosen to be Bunch-Davies (BD) vacuum states, namely, perturbation modes on sub-Hubble scales effectively propagate in vacuum states associated with flat space. Non-BD initial states were characterized by the Bogoliubov coefficients for each k mode and for both scalar and tensor perturbations, which were denoted by αSk,βSk,αTk,βTk, corresponding to the standard BD initial states. These parameters are determined by unknown high energy physics, and depending on the choice of the above parameters, blue gravitational waves were obtained while maintaining the scalar perturbations slightly red. Blue gravitational waves with slightly red scalar perturbations were also obtained without violating NEC by breaking the spatial diffeomorphism, usually imposed on the dynamics of perturbations, in the context of effective theory of inflation Cannone:2014uqa; Graef:2015ova. There, breaking of spatial diffeomorphism was considered by effective quadratic mass terms or derivative operators for metric fluctuations in the Lagrangian during inflation without the necessity for specifying the UV completion, while noting that it may be a version of massive gravity coupled to an inflaton, some model of inflation using vectors, or sets of scalars obeying some symmetries. Before closing this section, let us emphasize one important assumption made throughout this paper. We calculate evolution of primordial fluctuations assuming they obey general relativity below some energy scale. That energy scale and comoving wave number k of primordial fluctuations are related as follows. The wave number k is said to reenter the horizon when k=aH, where a and H are the scale factor and the Hubble parameter. The scale factor can be eliminated by the relation H2=H20Ωra−4, where Ωr is the radiation density parameter and H0 is the current Hubble parameter and here they are taken as Ωr=5×10−5 and H0=67km/s/Mpc. The Hubble parameter H and the temperature of the Universe T are related by (in natural units) H2=4π3g∗T4/45, where g∗ is the degrees of freedom of relativistic species here taken as g∗=106.75. From these relations the temperature and comoving wave number are related by For instance, if the theory is reduced to the standard cosmology described by general relativity at T=5×1010GeV, then our upper limits summarized in Fig. 4 are applicable for k<1018Mpc−1. Iii Radiation density perturbations generated from We work in the comoving gauge, in which the metric is written as44†4 Perturbations to the metric and energy momentum tensor are written as (see Weinberg for more details) where the spatial components of the velocity perturbation δuμ are written as δui=δu,i. Let us consider a coordinate transformation of the form xμ→xμ+ϵμ(xμ), with Then E and δu transform as Here we choose ϵ so that E=0, and then choose ϵ0 so that δu=0. Both choices are unique, so that there is no freedom to make further gauge transformations. This choice is sometimes called the comoving gauge (e.g. Baumann:2009ds). where hij is the tensor perturbation satisfying hij,i=hii=0. Throughout this paper it is assumed that the amplitude of initial tensor perturbations is much larger than that of scalar perturbations (schematically, (scalar)≪(tensor)), and so the scalar quantities in the metric above should be regarded as second order in hij. Hence, for scalar perturbations we write down the Einstein equations keeping second-order terms only in hij. As is also mentioned in the Introduction, our upper bounds from PBHs on tensor perturbations thus obtained are applicable even if this initial hierarchy between tensor and scalar perturbations does not hold. This is because if the amplitude of scalar perturbations is as larger as, or larger than that of tensor perturbations, then the abundance of PBHs increases when the amplitude of tensor modes is fixed. Namely, assuming (scalar)≪(tensor) initially is most conservative in placing upper bounds on tensor modes, and hence our bounds are applicable even if that assumption does not hold. Let us write down the fundamental equations in the following. We denote the energy density and pressure of the dominating radiation by ρ and p, respectively, and write p=c2sρ, where cs is the speed of sound. In this paper we restrict our attention to the formation of PBHs due to collapse of radiation density perturbations during the radiation-dominated era, and so we set cs=1/√3 in calculations, though we leave cs unspecified in equations below for generality. We decompose ρ and p as ρ(η,\boldmathx)=ρ0(η)+δρ(η,\boldmathx) and p(η,\boldmathx)=p0(η)+δp(η,\boldmathx). The zeroth-order Einstein equations yield where H≡a′/a with the prime denoting differentiation with respect to the conformal time η. These two equations are combined to give The Einstein equations at first order in hij give the standard evolution equation for tensor modes as follows: The Einstein equations at second order in hij, derived in Appendix A, are as follows: The conservation of the energy-momentum tensor yields From these equations one can derive the evolution equation of Ψ as follows. First, Eqs. (26) and (27) lead to (hereafter we work in Fourier space) The term −k2B of the above can be eliminated by the following relation, obtained from Eqs. (13) and (14): Using these and (16) as well as (11), (15) can be rewritten as is the source term representing generation of scalar perturbations due to the tensor perturbations. From (14) and (27), the energy density perturbation is given by Eq. (30) can be formally solved as55†5 We choose η=0 at the beginning of the radiation-dominated era, and we assume the initial condition is Ψ(0,\boldmathk)=0. Strictly speaking, however, Ψ is also generated before the radiation-dominated era at second order in tensor perturbations, even without intrinsic first-order scalar perturbations. That generation is highly model-dependent, and hence we restrict attention to the generation of Ψ only during the radiation-dominated era to adopt the above initial condition. This neglect of the generation of Ψ before the radiation-dominated era would probably lead to conservative upper bounds on tensor perturbations, since in general Ψ would be larger if the generation before η=0 is additionally taken into account. An analogous assumption is also made in the literature discussing induced gravitational waves (see footnote†2). while for ^k≡\boldmathk/|\boldmathk| in any other direction, erij(^k)(r=+,×) is defined by applying on each of the indices i and j a standard rotation, that takes the z-direction into the direction of ^k (see e.g. Weinberg). Then one can check the following: Let us further decompose the Fourier components as hr(η,\boldmathk)=D(η,k)hr(\boldmathk), where hr(\boldmathk) is the initial amplitude and D(η,k) is the growth factor, which can be obtained by solving the linear evolution equation (12) for hij (dropping the decaying mode): It turns out that the Fourier components of the source S can be written as follows (see Appendix B): and their nonzero components are written as66†6 These expressions are obtained by first setting ^k=^z, which is possible due to isotropy, and by assuming ^k′ is on the z−y plane, which is justified by the rotational invariance of Ers1 and Ers2. The time evolutions of this power spectrum for a few modes are shown in Fig. 1, where A is set to unity. The power spectrum takes the maximum value shortly after the horizon crossing of each k mode (kη=1). After reaching the maximum, it starts oscillations with the amplitude almost constant, similarly to the behavior in the standard linear cosmological perturbation theory. This is because the tensor perturbations decay after the horizon crossing, and so do the source terms, and then our fundamental equations for scalar perturbations are reduced to the standard ones in the linear theory. Iv Upper bounds on PGWS from PBHs In order to place upper bounds on tensor modes from PBHs, the abundance of PBHs needs to be related to the primordial tensor power spectrum, which can be accomplished by integrating the probability density function (PDF) of the induced density perturbation averaged over the horizon. In the following we first estimate the moment when the PBH formation is most efficient for each kp by calculating the dispersion of the induced density perturbation, and then derive the PDF at this moment. Let us begin by noting that the average ⟨δr(η,\boldmathx)⟩ is nonzero, since the density perturbation is generated by the tensor perturbations. To evaluate this average we introduce f3 and f4 by rewriting Frs as Since only the zero-mode δr(η,\boldmathk=\boldmath0) contributes to ⟨δr(η,\boldmathx)⟩, we need f3 and f4 only in the limit of \boldmathk→\boldmath0, which are, under the assumption of the delta-function-type power spectrum (52), When kpη≫1, the time average of this quantity asymptotes to while ⟨δr⟩→0 for kpη→077†7Strictly speaking this effect may be taken into account in the background Friedmann equations (9) and (10), but A2 is mostly less than 0.1 from Fig. 4, so the correction to the upper bounds would be ∼0.1 at most, while a rigorous treatment of this effect would greatly complicate analysis. Hence we neglect this effect. We denote the density perturbation averaged over a sphere with comoving radius R by δr(η,\boldmathx,R), the dispersion of which is related to the power spectrum as follows: where W is the Fourier transform of the top-hat window function: W(x)=3(sinx−xcosx)/x3. Figure 2 shows that the dispersion of the density perturbation at the horizon crossing of some mode k1 smoothed over the horizon scale at that moment (namely, η=k−11), σ(η=k−11,R=k−11), is maximum and is ∼A2 at around

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The total potential return on a stake can be calculated as:. For instance, one of the renowned betting websites priced the candidates to win the U. Presidential Election. Donald Trump: 4. The higher the total payout i. Question: I am playing a game with 5 possible outcomes. It is assumed that the outcomes are random. For sake of his argument let us call the outcomes 1, 2, 3, 4 and 5. I have played the game 67 times. My outcomes have been: 1 18 times, 2 9 times, 3 zero times, 4 12 times and 5 28 times. I am very frustrated in not getting a 3. What are the odds of not getting a 3 in 67 tries? In a greater number of trials there may be an outcome of a 3 so the odds of not getting a 3 would be less than 1. Question: What if someone challenged you to never roll a 3? If you were to roll the dice 18 times, what would be the empirical probability of never getting a three? Question: I have a 12 digit keysafe and would like to know what is the best length to set to open 4,5,6 or 7? 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Then if the question was "what is the expectation of getting a 6 in each trial", then you would multiply the probabilities because it's an "and" situation. Thank you so much for this article. It was most helpful. It answered questions that bothered me since the days in college! Thanks LM, I learned this stuff in school over 30 years ago, but it was refreshing to revisit it! Thanks for sharing and reiterating the basic mathematics we learn in our early years of schooling! Actually, this topic is very useful in real life even if you engange in a field which does not deal much on numbers such as mine. The mathematical concept of odds is related to, yet distinct from the concept of probability. In simplest terms, odds are a way of expressing the relationship between the number of favorable outcomes in a given situation versus the number of unfavorable outcomes. Calculating odds is central to the strategy of many games of chance, like roulette, horse racing and poker. 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Probability Cheat Sheets. Tips and Warnings. Related Articles. Article Summary. Author Info Last Updated: September 11, References. Part 1 of Determine the number of favorable outcomes in a situation. In this case, we'll just wager bets on what number the die will show after we roll it. Let's say we bet that we'll roll either a one or a two. In this case, there's two possibilities where we win - if the dice shows a two, we win, and if the dice shows a one, we also win. Thus, there are two favorable outcomes. Determine the number of unfavorable outcomes. In a game of chance, there's always a chance that you won't win. If we bet that we'll roll either a one or a two, that means we'll lose if we roll a three, four, five, or six. Since there are four ways that we can lose, that means that there are four unfavorable outcomes. Another way to think of this is as the Number of total outcomes minus the number of favorable outcomes. This most likely means " to 1 Odds are against winning" which is exactly the same as "1 to Odds are for winning. This calculator will convert "odds of winning" for an event into a probability percentage chance of success. For example, you win a game if you pull an ace out of a full deck of 52 cards. Pulling any other card you lose. An each-way bet is a combination of win and place bets of equal size. Very popular bet in sports with longer odds, outright markets for things like Premier League top scorer or the Grand National in horse racing. The place part of the bet is determined by the bookmaker you bet with. Place terms can differ from event to event, bookie to bookie.Smarkets Review. Terms apply. As the name suggests, this type of bet amounts to a total of 15 individual bets all wrapped up Online Ausmalbilder one. App-Support Datenschutzrichtlinie.

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Lütfen daha sonra yeniden deneyin. 15 Eyl 2012 tarihinde yayınlandıVisit www.doucehouse.com for more videos like this. TheDouceHouse 761 görüntüleme 5:56 How to use SohcahToa with a right angle triangle to find a missing length - Süre: 7:16. Is equal to 12 squared. The entire figure, the rotated square and the four right triangles, is a large square, too. "Pythagorean Squares" The following explanation requires knowledge of area. So once you have identified the hypotenuse-- and let's say that that has length C. Yükleniyor... Çalışıyor... Answer Questions What is the inverse for 4y^2 ? And let's say that they tell us that this is the right angle. http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U07_L1_T4_text_final.html And it's good to know, because we'll keep referring to it. ideo: How to Use the Pythagorean Theorem: Hypotenuse uizmaster: Find the Length of a Hypotenuse Given Two Sides Finding a Leg This section will explain how to use And that is going to be equal to C squared. But we're dealing with distances, so we only care about the positive roots. Knowing how to identify these triangles is an important part of solving many problems involving these triangles. The example below outlines the process by which we can calculate the hypotenuse. Click "Show third side" to see how the length of the third side is calculated given the lengths of the other two. Pythagorean Theorem Formula To Find B Let's say A is equal to 6. Question on exponential function? The derivation below begins with these areas in step one. So you could say 12 is equal to C. And we know that because this side over here, it is the side opposite the right angle. Skip to main contentSubjectsMath by subjectEarly mathArithmeticPre-algebraAlgebraGeometryTrigonometryPrecalculusStatistics & probabilityCalculusDifferential equationsLinear algebraMath for fun and gloryMath by gradeKindergarten1st2nd3rd4th5th6th7th8thHigh schoolScience & engineeringPhysicsChemistryOrganic ChemistryBiologyHealth & medicineElectrical engineeringCosmology & astronomyComputingComputer programmingComputer scienceHour of CodeComputer animationArts Hypotenuse Leg Theorem And I think you know how to do this already. And we could take the positive square root of both sides. The sum of the area of the small parts is equal to the area of the entire figure. Find The Missing Side Of A Triangle Calculator There is a white rotated square at the center. Beth Johnson 1.018 görüntüleme 4:13 Daha fazla öneri yükleniyor... Hypotenuse Theorem Oturum aç Paylaş Daha fazla Bildir Videoyu bildirmeniz mi gerekiyor? Pythagorean Theorem Calc: Find A, B=n/a, C=n/a Yükleniyor... Algebraically speaking, the relationship looks like... Mr. It's useful in geometry, it's kind of the backbone of trigonometry. Bu özellik şu anda kullanılamıyor. How To Find The Missing Side Of A Right Triangle And we want to figure out this length right over there. Dilinizi seçin. That is, in the above triangle: a2 + b2 = c2 Click here to see a proof. Specifically, the areas of rectangles and triangles must be known to understand the explanation. Oturum aç Çeviri Yazısı 13.430 görüntüleme 38 Bu videoyu beğendiniz mi? How To Find The Missing Side Of A Triangle So if we think about the Pythagorean theorem-- that A squared plus B squared is equal to C squared-- 12 you could view as C. You go opposite the right angle. Let me do one more, just so that we're good at recognizing the hypotenuse. The sum of 9 and 49 is 58, or 9 + 49 = 58. So let's say that that is my triangle, and this is the 90 degree angle right there. Learn more You're viewing YouTube in Turkish. Hypotenuse Formula And this is the same thing. Powers and Exponents How Do You Square a Number? If we are given a triangle's leg and hypotenuse, then we would use the equation to calculate the length of the missing leg. The next section will explain how the equation can be derived. Konuşma metni Etkileşimli konuşma metni yüklenemedi. The right side results when 4(0.5) is simplified to 2. So let's say that C is equal to the length of the hypotenuse. The right side represents the area of the smallest square [c2] and the area of four right triangles [4(0.5ab)]. Yükleniyor... Video kiralandığında oy verilebilir. So let's say that I have a triangle that looks like this. We would still arrive at the same solution, which is left for independant investigation.] The values are squared. 6 x 6 = 36 and 11 x 11 = 121. TheMathsTutorAU 711 görüntüleme 5:52 Pythagoras Theorem - Find Hypotenuse - VividMaths.com - Süre: 2:55. Video kiralandığında oy verilebilir. MyTutorZone 11.938 görüntüleme 3:50 Using the Pythagorean Theorem to solve for "a" & "b" - Süre: 4:13. Uygunsuz içeriği bildirmek için oturum açın. So this is the square root of 36 times the square root of 3. Kapat Evet, kalsın. If the missing side is a, then by pythagoras theorem 102 = a2 + 62 Rearrange to solve for a a2 = 102 − 62 = 100 − 36 = 64 And 3 squared is the same thing as 3 times 3. Yükleniyor... Çalışıyor... Yükleniyor... The right angle (90º) is indicated by a square. So this simplifies to 6 square roots of 3. And let's call this side over here B. Home > College of Sciences > Institute of Fundamental Sciences > Maths First > Online Maths Help > Trigonometry > Pythagoras Theorem and Applications > The Pythagoras Theorem SEARCH MASSEY This is 12, this is 6.

The use of FINDINGS AND ANALYSIS in Business As mentioned in the earlier chapter, the objective of this study is to identify performance for KLCI financial sector price index. The factors used to test the relationship were between the interest rate, inflation rate and the exchange rate. Therefore, a multiple linear regression is used as a method in order to see the relationship between the dependent variable (financial sector price index) and the independent variables (interest rate, inflation rate, and exchange rate). A multiple regression model use monthly data from the year 2005 to 2009. The relationship between the price index and the sectors which represent the dependent variable, and several macroeconomic factors that represent the independent variables will be shown by the regression model. Besides that, it also shows the result from the regression by analyzed the data based on the independent variables used. (Interest rate, inflation rate, and exchange rate). The data gather from 2005 to 2009 shows a pattern of pricing in every index included. This is shown drafty on table 4.1.1. Every information gathers are to analysis (analyze) using SPSS software system to get a linear regression result. We can help you to write your essay! Table 4.1.1: Multiple data from 2005 to 2009 (where do you get this sources) please state sources below the table.. CPI (inflation rate) Table 4.1.2: Multiple Linear Regression Analysis for KLCI Financial Sector (2005-2009) Adjusted R Square Std. Error of the Estimate a. Predictors: (Constant), x3, x1, x2 a. Dependent Variable: y From the Table 4.1.2 above, it shows the result of correlation between the dependent variable and the independent variables. The information represents monthly data from the year 2005 to 2009 in term of finance index, Base Lending Rate, Consumer Price Index and Exchange rate. (actually what is your IV? State clearly.. It must be same with IV in previous chapters) Alpha (a) in the Table 4.1.2 above represents a constant number of equation and it referred as Y intercept in the equation. This essay is an example of a student's work The alpha (a) value stand at 7062.354 and it means that when all independent variables equal to zero, the Y is at. 7062.354. The F-Statistic value is significant at 13.223 at the 0.00 level The R square in the Table 4.1.2 represent as a proportion of variance in the dependent variable, which can be predicted from independent variables. Based on the result, the R square value was stand at 0.826 and this value indicates that 83 percent of the variance in finance sector index can be explained from the independent variables used which is Interest Rate, Inflation Rate and Exchange Rate (USD/RM) 4.2.0 Analysis Statement Thus, following findings and analysis will interpret the result for the independent variables and test on the hypotheses either reject or accept the hypotheses. (have you state the hypothesis statement in this chapter or in previous chapter?) From table 4.1.2, this study concludes that the result can be explained by the following equation. The equation was stated as below: Y = 7062.354 - 86.855x1 -13.425x2 -1116.498x3 ( so it means negative relationship for all variables, right? ) state clearly…. Why it happens ? 4.2.1 Interest Rate Based on the analysis between financial price index and interest rate from the table 4.1.2 above, is shows that the Beta value for the Interest Rate was stand at 86.855. It means that for every one percent increase in Interest Rate, the performance of the sector will decrease by 86.855. The result for t-significant value was stands at 0.091, which is not significant since the value was below the level of significant. ( state what is the level of significant value? ) From the equation above it is been analyze that interest rate(x1) has a negative correlation with financial price(y) in normal economy time. With every changes of one percent in interest rate will affect (positively or negatively affected?) financial price by 86.855. The higher the changes will bring much more effect to financial price performance in term of low price. Therefore, the findings hereby accept the hypothesis statement for Interest Rate, which explained that interest rate, had influence the performance of KLCI financial sector. And it also rejects the null hypothesis statement which explained interest rate does not influence the performance of KLCI financial sector price index. 4.2.2 Inflation Rate For the result on the Inflation rate, it shows that the beta value for the inflation rate was stand at -13.425. It means that for every one percent increase in inflation rate will influence (decrease or increase) the performance of the sector by 13.425. Since it was negative relationship, for every one percent increase in inflation rate will result the decrease in financial performance by 13.425. The result for t-significant value was stands at 0.04, and it means that it was statistically significant. From the equation above it is been analyze that inflation rate(x2) has a negative correlation with financial price(y) in normal economy time. With every changes of one percent in interest rate will affect financial price by13.425. The higher the changes will bring much more effect to financial price performance in term of low price. Therefore, the finding hereby rejects the null hypothesis statement for inflation rate, which explained that inflation rate, is not influences the performance of finance sector. Since it was statistically significant, therefore it also accepts the alternatives hypothesis statement which explained that relationship between interest rate and the performance of finance sector. 4.2.3 Exchange Rate For the result on the exchange rate factor which used US Dollar (USD) against Ringgit Malaysia (RM), it shows that the beta value for the exchange rate was stand at -1116.498. It means that for every one percent increase in exchange rate will influence the performance of the sector by 1116.498. Since it was negative relationship, therefore for every one percent increase in exchange rate will result the decreasing in sector performance by 1116.498. Based on the result for t-significant, the value was stands at 0.00 and it means that it was statistically significant. From the equation above it is been analyze that exchange rate(x3) has a negative correlation with financial price(y) in normal economy time. With every changes of one percent in exchange rate will affect financial price by 13.425. The higher the changes will bring much more effect to financial price performance in term of low price. Earn money as a Freelance Writer! We’re looking for qualified experts As we are always expanding we are looking to grow our team of freelance writers. To find out more about writing with us then please check our freelance writing jobs page. Therefore, the finding hereby rejects the hypothesis statement for exchange rate, which explained that exchange rate, is not influence the performance in finance sector. Since it was statistically significant, therefore it also accepts the alternatives hypothesis statement which explained that there is a relationship between the exchange rate and the performance of finance sector. If you are the original writer of this essay and no longer wish to have the essay published on the UK Essays website then please click on the link below to request removal:

How to Solve Equations with the Distributive Property like a Pro Get the free Equations with the Distributive Property worksheet and other resources for teaching & understanding solving Equations with the Distributive Property How to Solve Equations with the Distributive Property Solving Equations with the Distributive Property happens when a linear equation has a term being distributed to multiple terms inside of a set of parenthesis. The first thing you must do is simplify by using the Distributive Property. You simplify using the Distributive Property by distributing the term in front of the parenthesis by multiplying it by everything on the inside of the parenthesis. After you use the Distributive Property, you solve the equations just like any other Two Step Equation. The first step in solving Two Step Equations is to get all of the constants (numbers) on one side of the equal sign, and the coefficient with the variable on the other side. In order to do this you must use the addition and subtraction property of equality to get the constants on the opposite side as the variable. Once the constants are separated from the variable, you must use the multiplication or division property of equality to cancel out the coefficient on the variable. You can always check your answer by substituting your solution back in to the equation for the variable. Common Core Standard: 8.EE.C.7 Basic Topics: Combining Like Terms, Distributive Property, Two Step Equations, One Step Inequalities, Two Step Inequalities, Multi Step Inequalities Related Topics: Two Step Equations, Multi Step Equation, Equations with Variables on Both Sides More Examples of Multi Step Equations with Distributive Property So what is the distributive property anyway? Solving Equations with Distributive Property happens when a linear equation contains the distributive property. You can tell if the linear equation contains the distributive property if there is a term that is being distributed to multiple terms inside a set of parenthesis. You must first distribute according to the distributive property. Then you can solve the equation for the variable just like you would any two step equation. 4 Easy Steps for Solving Distributive Property Equations Steps for solving the Equation with the Distributive Property above: - Distribute the two to the x and one inside the parenthesis. - Multiply the two times x and the two times one. - Add two to both sides. - Divide both sides by two to get the solution of x equals three. Equations with Distributive Property Practice Problems Quiz Video on the Free Distributive Property Equations Worksheet Watch our free video on how to solve 2 step equations with Distributive Property. This video shows how to solve problems that are on our free solving Equations with Distributive Property worksheet that you can get by submitting your email above. Watch the free Equations with the Distributive Property video on YouTube here: Equations with the Distributive Property This video we’re going to show you some problems from our equations with the two step equations with distributive property worksheet. This video is about learning how to solve two step equations with distributive property. Our first problem on our equations with the distributive property worksheet is 2 times the quantity X minus 1 equals 4. The first step or the first thing you have to do when you have the distributive property is you have to distribute whatever is on the outside of the parenthesis to everything on the inside of the parenthesis. In the case of this problem we have to take 2 times it by X. 2 times X minus and then 2 times 1 and then you bring down your equals 4. We’ve taken what’s on the outside of the parenthesis which in this case is 2 and multiplied it or distribute it it to everything on the inside of the parenthesis. After you do that you have 2 times X minus 2 times 1 equals 4. Of course 2 times X is 2x and then 2 times 1 is 2. Now our equation is 2x minus 2 equals 4. Now we have to get the variable on one side by itself and constants on the other in order to do that we’re going to add 2 to both sides. That the twos will cancel and then you have 2x on this side of the equation and then 4 plus 2 which is 6. On this side of the equation then the last step is to divide both sides by 2 because the coefficient on 2x is 2 and this is like saying 2 times X the opposite of 2 times X is 2 divided by 2. Now we have X on this side and then 6 divided by 2 on this side which is 3 and that’s our solution. This is a short explanation what is distributive property example. The second problem on our equations with distributive property worksheet gives us five times the quantity 5x minus 5 equals 50. Once again the first step is to distribute everything on the outside to everything on the inside of the parenthesis. We will do 5 times 5x minus you keep the sign in the middle the same five times five, then you bring down your equals and then you bring down your constant on this side which is 50. Then to simplify you do five times 5x which is 25 X minus five times 5 which is 25 equals 50. Now we have to solve for X. In order to do that we’re going to add 25 here because we have to get rid of all the constants on the same side as . These 25s cancel you bring down your 25 X and then you do 50 plus 25 over here which is 75. Then the last step is to divide by 25 because we have to cancel the coefficient on the X. These guys cancel you’re left with just X on this side and then 75 divided by 25 is 3. The solution to number 2 is x equals 3. The last problem we’re going to go over on our worksheet for the equations with a distributive property is number 3. Number 3 gives us 20 equals negative 10 times the quantity X plus 8. Once again the first step is to distribute the negative 10 or whatever’s on the outside to everything on the inside and this time you need to be careful because this is a negative 10. You have to distribute a negative 10. You have to include the negative when you distribute. This is like negative 10 times X plus and then negative 10, will write it in parenthesis because it’s negative times 8. Now we have 20 equals negative 10 times X plus negative 10 times 8. When we simplify this negative 10 times X is negative 10 X and then negative 10 times 80 is negative 80. You bring down your equal sign and your constant on the other side. Now we have to get rid of this negative 80 and in order to do that we’re gonna go ahead and add 80 to both sides. This negative 80 and this positive 80 will cancel and then you bring down your negative 10 X and then 20 plus 80 is 100. The final step is to get X by itself. In order to do that we divide both sides by negative 10. These guys will cancel and your have X on this side and then 100 divided by negative 10 is negative 10. That will do it for our 2 step equations with distributive property worksheet. Equations with the Distributive Property Enter your email to download the free Equations with the Distributive Property worksheet Practice makes Perfect. We have hundreds of math worksheets for you to master. Share This Page Get the best educational and learning resources delivered. Join thousands of other educational experts and get the latest education tips and tactics right in your inbox.

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The uncertainty is respect of behaviour pattern of a firm under oligopoly arising our of their unpredictable action and reaction makes a systematic analysis of oligopoly difficult. However, classical and modern economists have developed a variety of models based on different behavior assumptions. These models can broadly be classified into two categories (I) classical duopoly models and modern oligopoly Duopoly Models, when there are only two sellers a product, there, exists duopoly. Duopoly is a special case of oligopoly. Duopoly is a special case in the sense that it is limiting case of oligopoly as there must be at least two sellers to make the market oligopolistic in nature. 1. The Cournot’s Duopoly Model 2. The Chamberlin Duopoly Model 3. The Bertrand’s Duopoly Model 4. The Edgeworth Duopoly Model 1. Cournot’s Duopoly Model: Augustin Cournot, a French economist, was the first to develop a formal duopoly model in 1838. To illustrate his model, Cournot assumed: (a) Tow firms, each owing an artesian mineral water well; (b) Both operate their wells at zero marginal cost2; (c) Both face a demand curve with constant negative slope; (d) Each seller acts on the assumption that his competitor will not react to his decision to change his and price. This is Cournot’s behavioural assumption. On the basis of this model, Cournot has concluded that each seller ultimately supplies one-third of the market and charges the same price. While one-third of the market remains unsupplied. Cournot’s duopoly model is presented in Fig. 1. To begin the analysis, suppose that there are only two firms. A and B, and that, initially. A is the only seller of mineral water in the market. In order to maximize his profits (or revenue), he sells quantity OQ where his MC = O MR, at price OP2 His total profit is OP2PQ. Now let B enters the market. The market open to him is QM which is half of the total market. He can sell his product in the remaining half of the market. He assumes that A will not change his price and output as he is making the maximum profit i.e., A will continue to sell OQ at price OP2 Thus, the market available to B is QM and the demand curve is PM. When to get maximize revenue, B sells ON at price OP1, His total revenue is maximum at QRP’N. Note that B supplies only QN = 1/4 = (l/2)/2 of the market.) With the entry of B, price falls to OP1 Therefore, A’s expected profit falls to OP1 PQ Faced with this situation, A attempts to adjust his price and output to the changed conditions. He assumes that B will not change his output QN and price OP1 as he is making maximum profit. Accordingly, A assumes that B will continue to supply 1/4 of market and he has 3/4 (= 1 – 14) of the market available to him. To maximise his profit. Supplies 1/2 of (3/4), i.e., 3/8 of the market. Note that A’s market share has fallen from 1/2 to 3/8. Now it is B’s turn to react. Considering Cournot’s assumption, B assumes that A will continue to supply only 3/8 of the market and market open to him equals 1 – 3/8 = 5/8. In order to maximise his profit under the new conditions B supplies 1/2 x 5/8 = 5/16 of the market. It is now for A to reappraise the situation and adjust his price and output accordingly. This process of action and reaction continues in successive periods. In the process, A continues to lose his market share and B continues to gain. Finally situation is reached when their market shares equal at 1/3 each. Any further attempt to adjust output produces the same result. The firms, therefore, reach their equilibrium position where each one supplies one-third of the market. The equilibrium of firms, according to Cournot’s model, has been presented in table below: Cournot’s equilibrium solution is stable. For given the action and reaction, it is not possible for any of the two sellers to increase their market share. It can be shown as follows: A’s share= 1/2(1 – 1/3) = 1/3. Similarly B’s share = 1/2 (1 – 1/3) = 1/3. Cournot’s model of duopoly can be extended to the general oligopoly. For example, if there are three sellers, the industry, and firms will be in equilibrium when each firm supplies 1/3 of the market. Thus, the three sellers together supply 3/4 of the market, 1/4 of the market remaining unsupplied. The formula for determining the share of each seller in an oligopolistic market is: Q -f- (n + 1), where Q = market size, and n = number of sellers. Criticism of the Model: Although ournot’s model yields a stable equilibrium, it has been criticised on the following grounds: (1) Curnot’s behavioural assumption [assumption (d) above] is naive to the extent that it implies that firms continue to make wrong calculations about the competitor’s behaviour. Each seller continues to assume that his rival will not change his output even though he reportedly observes that his revel firm does change its output. (2) The assumption of zero cost of production is totally unrealistic. If this assumption is dropped, it does not alter his position. 2. Chamberlin’s Duopoly Model- A Small Group Model: Chamberlin’s model of duopoly recognizes interdependence if firms in such a market. Chamberlin argues that in the real world of oligopoly firms are not so native that they will not learn from the past experience. However, he makes the same assumptions as the exponents of old classical models have done. In other words, his model is also based on the assumption of homogeneous products, firms of equal size with identical costs, no entry by new firms and full knowledge of demand. Recognition of interdependence of firms in an oligopolistic market given us a result quite different from that of Cournot. Chambrilin argues that firms are aware of the fact that their output or price decision will definitely invite reactions of other firms. Therefore, he goes not visualize any price war in oligopolistic markets. He also rules out the possibility of firms adjusting their outputs over a period of time and thus reaching the equilibrium at an output level lower than that would be reached under monopoly. According to Chamberlin, recognition of possible sharp reactions to an oligopolistic firm’s price or output manipulations would avert harmful competition amongst the firms in such a market and would result in a stable industry equilibrium with the monopoly price and monopoly output. He further stated that no collusion is required for obtained this solution. In case farms in an oligopolistic market are aware of their mutual dependence, and willing to learn from their past experience, then in order to maximize their individual and joint profits they will charge the monopoly price. Chamberlin’s model can be explained in the frame work of a dupoly market. Chamberlin, like Cournot, assumes linear demand for the product. For simplicity we assume that even in this case the cost of producing the good is zero. Chamberlin model has been illustrated in Figure 2. In this figure DQ is the market demand curve. If firm A is first to enter the market, it will produce output OQ1 because at this level of output its marginal revenue is equal to marginal cost (MR = MC = 0). The firm can charge price OP1 which is the monopoly price. This will maximise its profits. At price OP) elasticity of demand is unity. Firm B entering market at this stage considers that its demand curve is CQ and will thus produce Q1Q2 so as to maximise its profit. It will charge price OP2. It now realizes that it cannot sell QQ1 quantity at the monopoly price and thus decides to reduce the output to QQ3, which is one-half of the monopoly output QQ1. Firm B can continue to produce quantity Q1Q2 which is same as Q3Q1. The industry output thus is OQ1 and the price rises to the level OP1. This is an ideal situation from the point of view of both firms A and B. In this case, the joint output of the two firms is monopoly output and they charge monopoly price. Thus, considering the assumption of equal costs (costs = 0) the market will be shared equally between firms A and B. Appraisal of the Model: Chamberlin’s model is certainly more realistic than earlier models. It assumes that firms recognize interdependence and then act in a manner that monopoly solution is reached. In the real world of oligopoly there are certain difficulties in reaching this solution. In the absence of collusion, firms must have a good knowledge of market demand curve which is almost impossible to obtain. In case this information is lacking, firms will not know how to reach monopoly solution. Further, Chamberlin ignores entry. In real practice, oligoplistic markets are rarely closed. So if we recognize the fact of entry, it would not be certain that the stable monopoly solution will ever be reached. Differences in costs and market opportunities are also hindrance for attaining a monopoly-type outcome by the independent actions of firms in oligopolies. 3. Bertrand’s Duopoly Model: Bertrand, a French Mathematician developed his own model of duopoly in 1883. Bertrand’s model differs from Cournot’s model in respect of its behavioural assumption. While under Cournot’s model, each seller assumes his rival’s output to remain constant, under Bertrand’s model each seller determines his price on the assumption that his rival’s price, rather than his output, remains constant. Bertrand’s model focuses on price competition. His analytical tools are reaction function of the duopolists. Reaction functions are derived on the basis of iso-profit curves. An iso-profit curve, for a give level of profit, is drawn on the basis of various combinations of prices charged by the rival firms. He assumed only two firms, A and B and their prices are measured along the horizontal and vertical axes, respectively. Their iso-profit curves are drawn on the basis of the prices of the two firms. Iso-profit curves of the two firms are concave to their respective prices axis, as shown in Fig. 3 and 4. Iso- profit curves of firm A are convex to its price axis PA (Fig. 3) and those of firm B are convex to PB (Fig. 4). In Figure 4, we have curve A, which shows that A can earn a given profit from the various combinations of its own and its rival’s price. For example, price combinations at points, a, b and c yield the same level of profit indicated by the iso-profit curve A1. If firms B fixes its prices Pb1– firm A has two alternative prices, Pa1 and Pa2, to make the same level of profits. When B reduces its price, A may either raise its price or reduce it. A will reduce its price when he is at point c and raise its price when he is at point a. But there is a limit to which this price adjustment is possible. This point is shown by point b. So there is a unique price for A to maximize its profits. This unique price lies at the lowest point of iso-profit curve. The same analysis applies to all other iso-profit curves, A1 A2 and A3 we get A’s reaction curve. Note that A’s reaction curve has a rightward slant. This is so because, iso-profit curve tends to shift rightward when A gains market from his rival B. Following the same process, B’s reaction curve may be drawn as shown in Fig. 4. The equilibrium of duopolists suggested by Bertrand’s model may be obtained by putting together the reaction curves of the firms A and B as shown in Fig. 5. The reaction curves of A and B intersect at point E where their expectations materialize, point E is therefore equilibrium point. This equilibrium is stable. Fo, if any one of the firms disagrees to this point, it will create a series of actions and reactions between the firms which will lead them back to point E. Criticism of the Model: Bertrand’s model has been criticised on the same grounds as Cournot’s model. Bert- rand’s implicit behavioural assumption that firms never learn from their past experience seems to be unrealistic. If cost is assumed to be zero, price will fluctuate between zero and the upper limit of the price, instead of stabilizing at a point. 4. Edgeworth’s Duopoly Model: Edgeworth developed his model of duopoly in 1897. Edgeworth’s model follows Bertrand’s assumption that each seller assumes his rival’s price, instead of his output, to remain constant. His model is illustrated in Fig. 6. In this figure we have supposed that there are two sellers, A and B, in the market who face identical demand curves. A has his demand curve DDB and as DDB Let us also assume that seller A has a maximum capacity of output OM and B has a maximum output capacity of OM’. The ordinate ODA measures the price. To explain Edgeworth’s model, let us assume, to begin with, that A is the only seller in the market. Following the profit maximising rule of a monopoly seller, he sells OQ and charges a price, OP2. His monopoly profit under zero cost, equals OP2EQ Now, let B enter the market. B assumes that A will not change his price since he is making maximum profit. He sets his price slightly below A’s price (OP2) and is able to sell his total output. At this price, he captures a substantial part of A’s market. Seller A, on the other hand, that his sales have gone down. In order to regain his market, A sets his price slightly below B’s price. This leads to price-war between the sellers. The price- war takes the form of price-cutting which continues until price reaches OP1 At this price both A and B are able to sell their entire output- A sells OQ and B sells OQ The price OP1 could therefore be expected to be stable. But, according to Edgeworth, price OP1 should not be stable. Simple reason is that, once price OP is set in the market, the sellers observe an interesting fact. This is, each seller realise that his rival is selling his entire output and he will therefore not change his price, and each seller thinks that he can raise his price to OP2 and can make pure profit. This realisation forms the basis of their action and reaction. For examples, let seller A take the initiative and raise his price to OP2. Assuming A to retain his price OP2.B finds that if he raises his price at a level slightly below OP2 he can sell his entire output at a higher price and make greater profit. Therefore, B raises his price according to his plan. Now it is A’s turn to know the situation and react. A finds that his price is higher than B’s price and his total sale has fallen. Therefore assuming B to retain his price, A reduces his price slightly below B’s price. Thus, the price-war between A and B begins once again. This process continues indefinitely and price keeps moving up and down between OP1 and OP2 Obviously, according to Edgeworth’s model of duopoly, equilibrium is unstable and indeterminate since price and output are never determined. In the words form Edgeworth, “there will be an indeterminate tract through which the index of value will oscillate, or, rather will vibrate irregularly for an indefinite length of time. In a net shell Edgeworth’s model, like Cournot’s is based on a native assumption, i.e. each seller continues to assume that his rival will never change his price even though they are proved repeatedly wrong. But according to Hotelling Edgeworth’s model is definitely an improvement upon Cournot’s model in that it assumes price, rather than output, to be the relevant decision variable for the sellers.

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Unformatted text preview: (c) Are the answers to parts (a) and (b) the same? Explain. ins5-2h22-28 2 5 2 In Problems ??, find the area of the regions between the curve Math for Econ II, Written Assignment 8 (28 points) and the horizontal axis 5-2h29 Due Friday, November 7 x 22. Under y = 6x3 2 for 5 x 10. Jankowski, Fall 2014 23. Under the curve y = cos t for 0 t Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, 24. you will = ln be given x 4. Under y not x for 1 25. Under y = 2 cos(t/10) for 1 t 2. Figure 5.39: Graph consists of a semicircle and • Write your name x for 0 x 2. 26. Under the curve y = cos and recitation section number. 27. Under the curve y =homework above the x-axis. 7 x2 and if you have multiple pages! • Staple your 28. Above the curve y = x4 8 and below the x-axis. 1. (2 pts to find the each) 29. Use Figure ??total; 1 ptvalues ofUse the figure below to find the values of 5-2h33 33. 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(4 pts) Use the Evaluation Theorem to show 6000 and the supply curve is given by P = Q + 10. Find Q + 50 the equilibrium price and quantity, and compute the consumer and producer surplus. 8. (5 pts) Suppose the demand curve is given by P = Some extra practice (not to be handed in) f (x) dx using R5 and L5 . −2 −1 1 2 3 −5 −4 −3 −2 −3 −1 1 2 3 4 5 2 2. Suppose h is a function such that h(1) = 2, h0 (1) = 3, h00 (1) = 4, h(2) = 6, h0 (2) = 5, h00 (2) = 13, and h00 is continuous everywhere. Find 1 h00 (u) du. ... View Full Document

David Houle Department of Biological Science, Florida State University, Tallahassee, FL 32306–1100, USA. Tel.: +1 850 645 0388; fax: +1 850 644 9829; e-mail: email@example.com The relationship between developmental stability and morphological asymmetry is derived under the standard view that structures on each side of an individual develop independently and are normally distributed. I use developmental variance of sizes of parts, VD, as the converse of developmental stability, and assume that VD follows a gamma distribution. Repeatability of asymmetry, a measure of how informative asymmetry is about VD, is quite insensitive to the variance in VD, for example only reaching 20% when the coefficient of variation of VD is 100%. The coefficient of variation of asymmetry, CVFA, also increases very slowly with increasing population variation in VD. CVFA values from empirical data are sometimes over 100%, implying that developmental stability is sometimes more variable than any previously studied type of trait. This result suggests that alternatives to this model may be needed. There has been considerable speculation about the degree of variation among individuals in their developmental stability, their ability predictably to complete development to an optimum state ( Palmer, 1996; Møller & Swaddle, 1997). The principal hurdle to empirical studies of developmental stability is that, in most cases, we do not know what the optimum state of a trait is, so we cannot say how much of the variation is due to variation in the optimum and how much to a failure to develop to that optimum. When the same structure develops on either side of a symmetrical body however, we can assume that the optimum state is often one of perfect symmetry. Where this symmetry of the optimum can be assumed, asymmetry is referred to as fluctuating asymmetry (FA) ( Palmer & Strobeck, 1986; Palmer, 1994). A large body of work now assumes that FA is a good indicator of developmental stability ( Møller & Swaddle, 1997). Surprisingly, given these attractions of FA as an indicator of developmental stability, the precise relationship between the two has never been addressed analytically. Previous studies have, with few exceptions, relied on the negative relationship between FA and developmental stability that exists by definition. There are several barriers to studying this relationship. First, developmental stability is currently a hypothetical entity that has no unique relationship with observable properties of organisms. For example, FA is known to be affected by the environment an individual develops in ( Parsons, 1990) as well as by variation in the developmental stability of individuals. It is also not clear that there is any single property of an individual that can be labelled developmental stability, as each trait of an individual may to some degree have different stability properties from every other trait ( Møller & Swaddle, 1997, pp. 53–55; Leung & Forbes, 1997). For example, traits that develop at different times, or in different parts of the body, may differ in their developmental stability. Taken together, these complications have so far precluded empirical characterization of the relationship between developmental stability and FA, despite a number of demonstrations that variation in developmental stability of some characters does exist ( Parsons, 1990; Whitlock & Fowler, 1997; Gangestad & Thornhill, 1999). In the absence of detailed empirical data, we must depend on models to develop our intuition. The majority of workers have adopted the following standard model of fluctuating asymmetry ( Palmer & Strobeck, 1986; Palmer, 1996). Each individual is characterized by its developmental variance, the converse of developmental stability. Paired structures are assumed to develop independently towards the same expected size but to show some normally distributed variation around that expectation. The amount of variation around the expected size is determined by the developmental variance. In this model, the difference between paired structures is therefore also normally distributed, with variance twice that of the variance of each paired structure. Several recent explorations of this standard model suggest that the expected relationship between FA and developmental stability is very weak. FA essentially measures developmental variance, the variance in size of body parts when they develop in the same environment, which is the converse of developmental stability. Variances are more difficult to estimate well than means, so we should expect that more sampling effort would be needed to study FA than a typical trait. Unfortunately, FA must usually be estimated from a single pair of measurements, yielding a poor estimate of the variance and therefore of the proportion of variation in FA that could be due to real variation in developmental stability ( Whitlock, 1996; Houle, 1997). The likelihood that only a small amount of information about individual developmental stability is gained from a single measure of FA has led to two efforts to quantify how much of the variation in FA could be due to variation in developmental stability, both assuming the simple model of the development of paired structures outlined above. One such effort used the relationship between the mean and variance of FA ( Whitlock, 1996), whereas the other exploited the kurtosis in FA expected to arise from variation in developmental stability ( Gangestad & Thornhill, 1999). Whitlock (1996) observed that there is a simple relationship between the mean FA and the variance in FA for individuals with the same developmental stability. Therefore, one can calculate by subtraction the proportion of the total variance in FA that can be due to differences in developmental stability. This quantity is familiar from quantitative genetics as the repeatability. Although this insight is correct, the formulas given by Whitlock (1996) were incorrect; corrected formulas have now been published ( Van Dongen, 1998b; Whitlock, 1998). The repeatability provides an intuitive measure of the reliability of individual measurements. More importantly, the repeatability sets an upper limit to the heritability, the proportion of the variance that can be due to genetic causes. It also sets an upper limit on the phenotypic correlation between the asymmetries of different pairs of traits on the same individual. Whitlock showed that the maximum repeatability of FA is 0.64 and that the coefficient of variation of FA is sometimes so large that the repeatability would be expected to approach this maximum value ( Whitlock, 1996). Paradoxically, because FA is such a poor measure of variance, even traits with low repeatability and small correlations of FA among traits may reflect a great deal of variation in developmental stability. Using simulations, Gangestad & Thornhill (1999) derived an empirical relationship between repeatability and kurtosis in signed FA, the difference in size between paired structures. They come to conclusions similar to those of Whitlock, arguing that, despite the low repeatability of many estimates of FA, the heritability of developmental stability itself may be high. In this paper, I extend Whitlock’s (1996, 1998) work to consider the relationship between the distribution of developmental stabilities and the variance of FA. Whitlock’s approach leads to an estimate of the proportion of the variation in FA that can be due to variation in developmental stabilities, but it does not consider variation in developmental stability explicitly. The results presented here go the next step and allow inferences about the amount of variation in developmental stability from the repeatability of FA, based on the standard model. Previous work that has explicitly included variation in developmental stability has considered mixtures of individuals with two or three different stabilities ( Houle, 1997; Van Dongen, 1998b), rather than more realistic continuous distributions. Other work has relied on simulations ( Leung & Forbes, 1997; Van Dongen, 1998b; Gangestad & Thornhill, 1999), which are difficult to generalize. The principal result of this model is that, in order for measures of FA to have the substantial repeatabilities implied by some data, mean-standardized variation in developmental stability would have to be higher than for most previously studied traits. In the next section I present an intuitive introduction to the model. The Mathematical results section then derives the relationship between developmental stability and measures of asymmetry based on this model. From these relationships, I then obtain Numerical results. The reader who wishes to obtain the main results without mathematical details may skip the Mathematical results section. I assume a population of organisms that are unable to regulate development perfectly. This imperfect development is studied by measuring a pair of structures on either side of an axis of symmetry, such as right and left limb lengths. I start with the commonly accepted model for the development of bilaterally paired traits, which imagines that each side of the organism develops independently of the other and that the variation in each side is normally distributed ( Palmer & Strobeck, 1986; Palmer, 1996). This model of asymmetry has a pragmatic basis and is not directed at attempting to discern the causes of variation in development. It merely assumes that developmental variance exists and that developmental variance captures something about what we intuitively mean when we discuss developmental stability. If we understood the details of development of the morphological structures, we could make the relationship between asymmetry and development explicit. Clearly, the present state of knowledge does not allow this step, although a number of speculative efforts in this direction have been undertaken ( Graham et al., 1993 ; Klingenberg & Nijhout, 1999). Variation in developmental stability has often been shown to be caused by environmental variation ( Parsons, 1990) and in some cases to have a genetic basis as well ( Parsons, 1990; Whitlock & Fowler, 1997; Gangestad & Thornhill, 1999). Developmental stability can in principle be decomposed into developmental noise, factors that cause variation in development, and developmental homoeostasis, processes that damp out the effects of developmental noise ( Palmer, 1996; Leung & Forbes, 1997). In practice these are usually indistinguishable, so they are considered together here. One must model at least four kinds of variances to investigate the relationship between variation in developmental stability and variation in asymmetry. The most familiar of these is the variance of asymmetry itself, which I symbolize σ2. The variance of asymmetry depends on the variance in the traits from which asymmetries are calculated, that is, on the developmental variance, VD. VD is the converse of developmental stability. In addition, the variance of sides may contain measurement error, Ve. The fourth sort of variance is variation in the developmental variance, which has not been explicitly included in previous analyses. I assume that each individual offers two or more examples or realizations of the same trait, S, which I will refer to as ‘sides’, although their spatial arrangement is not important. On the ith individual, the S-values are drawn independently from the same normal distribution with variance VD. The mean of this distribution must be much greater than √(VD) in order to preserve approximate normality but is otherwise free to vary. For simplicity, I assume that the mean of the distribution of sides is uncorrelated with VD. The developmental variance of the ith individual will be represented as VDi. It may consist of variation caused by both genotype and environment. I consider two statistics to measure asymmetry. First is the absolute value of the difference between sides FAi=|Si1 – Si2|. Second is the variance of sides where n is the number of ‘sides’ measured. The variance of sides has statistical properties superior to those of FA, even when there are only two sides ( Palmer & Strobeck, 1986). My goal is to model the variance in asymmetry as a function of population variation in developmental variance, VVD, so we need to consider what the distribution of VD would look like. The family of distributions I have chosen to represent this situation is shown in Fig. 1. Before I give the mathematical basis for these distributions, I give the following intuitive justification. The choice of a distribution for developmental variances, VD, must take into account the fact that a variance cannot be negative. Consider a series of populations with the same mean developmental variance, V¯D, but that differ in the population variance of developmental variance, VVD. When VVD is small (as shown in the curve labelled α=400 in Fig. 1) the distribution can be nearly symmetrical, as it is very unlikely that a value will fall near VD=0. However, as the population variance of VD increases, the fact that VD cannot be negative has a larger and larger effect on the distribution. If the mean is to be held constant, the lack of negative values means that the likelihood that a value falls between zero and the mean must increase to compensate for the unconstrained tail of large values to the right of the mean. The result is that the distribution must become increasingly skewed as the variance goes up (represented by decreasing α values in Fig. 1), and the mode of the distribution must shift to the left. When VVD is very large relative to the mean, the mode is very close to 0 but balanced by an increasingly long tail of large VD values. A distribution that has these properties is the gamma distribution, which is only defined for values of VD > 0. For the gamma, the probability that individual i has developmental variance VD is where Γ() denotes the gamma function. The gamma distribution has two parameters, α and β. A principal attraction of the gamma distribution is the variety of shapes it can assume, depending on the value of the ‘shape’ parameter α. When α is large, the gamma approaches a normal distribution. Both the exponential and the χ2 distributions are special cases of the gamma distribution. β is the ‘scale’ parameter. The mean of the gamma distribution is αβ, and the variance is αβ2=VVD. Note that the coefficient of variation of a gamma-distributed variable such as VD is In addition to the developmental variance, VD, I assume that the observed variance of S may also include measurement error, Ve. Measurement error is assumed to be constant over all individuals. To incorporate measurement error, the distribution Vi=VD,i + Ve can be modelled as a gamma distribution with a minimum value at Ve, rather than 0. Most elements of this model are shown in Fig. 2, which shows the distribution of the developmental variance, VD, sides S, and FA for two different distributions of VD. The first row of the figure shows the distributions of VD. On the left, α=0.5 resulting in a highly skewed distribution with a high CV; the column on the right shows α=100, a fairly symmetrical distribution with a low CV. Measurement error is assumed to be absent. The second row of the plots shows distributions of sides for representative values of VD. The mean of the sides is always equal to 10, and the distributions are always normal. In each case, the upper right panel of each triplet gives the distribution for a value of VD at the 95th percentile of the distribution, the lower left panel gives the distribution for a value of VD at the 5th percentile of the distribution, and the middle panel shows the value at the median value of VD. For the small α value, the difference in the distributions of S is immediately apparent; the distribution at the 5th percentile has such a small variance that the peak is off the scale chosen. Note that the 95th percentile for this distribution is at VD=6.05, emphasizing the presence of a long tail of large VD values that is not otherwise apparent in the figure. For the large-α case, however, the distributions of S are so similar that no difference is apparent to the eye. These distributions of S will not be observed directly, as each individual has only two sides. Instead, we directly observe the distribution of sides in a population of individuals, where each individual has a different developmental variance drawn from the distributions at the top of the figure. This distribution is shown in the third row of the figure. Note that the distribution with small α results in a kurtotic distribution of sides, whereas the large-α case has a nearly normal distribution of sides. This kurtosis is expected because the distribution is a combination of normal distributions with very different variances ( Wright, 1968; Houle, 1997; Leung & Forbes, 1997; Gangestad & Thornhill, 1999). Finally, we calculate FA by taking the absolute value of the difference in two sides of the same individual, resulting in the peaky, long-tailed distribution of FA in the small-α case, and a distribution close to the half normal in the large-α case. Given this model, we are interested in how informative these measures of asymmetry are concerning the developmental variance of an individual, when individuals vary in VD. As pointed out by Whitlock (1996, 1998) a good measure for this purpose is the repeatability, symbolized ℜ, the proportion of the variance in asymmetry that is due to real differences in developmental variance. The repeatability sets an upper limit both to the heritability of asymmetry and to the correlation of asymmetries of different structures on an individual. To calculate the repeatability of a measure of asymmetry, we need to know the total observed variance in asymmetry in the whole population, σT2, and the realization variance, σR2, the variance in observed asymmetry among individuals with the same VD values. This σR2 term includes any measurement error. The remaining variance, σI2, is the true variance among individuals remaining after the realization variation is removed. By definition, σT2=σR2+ σI2. Note that both Whitlock (1998) and Van Dongen (1998b) treat the parameter σI2 as the variance in developmental stability. (Their notation differs from mine: σI2 is VDS in Whitlock; Vind in Van Dongen.) I reserve the term developmental stability for the inverse of the developmental variance of sides, VD, and the term variance in developmental stability for the variance of developmental stability, VVD=αβ2. Although in some cases (see below) VVD=σI2, defining some aspect of the variance in symmetry as developmental stability risks losing track of the important distinction between developmental stability and the effects it has on a particular phenotype. The expected value of FAi is and the variance is where the expectations are over hypothetical replicate individuals with the same VD,i values. The realization variance is The other variance components cannot be obtained in closed form when there is measurement error, so I first consider the case of no measurement error, Ve=0. Then the mean FA is The total variance in FA is Finally, the true variance in FA among individuals can be obtained as the difference between eqns 9 and 6, The repeatability of FA when there is no measurement error is then The results in eqns 6–11 were checked by simulations in SAS (results not shown; SAS Institute, 1990). Although I was not able to obtain general analytical results for the gamma distribution with error variance, numerical results were obtained by numerical integration of eqns 6 and 9 in Maple V ( Waterloo Maple, 1997), with Ve + VD.i substituted for VD.i. It is also useful to consider the coefficient of variation of FA itself, which is readily measured (e.g. Whitlock, 1996). For the case of no error variance, Just as Whitlock showed that the maximum repeatability of FA measures is a function of CVFA, eqn 12 can be solved iteratively to yield an estimate of α, and CVVD from CVFA. Because the alternative measure of asymmetry, s2, is a variance, it follows a χ2 distribution with n – 1 degrees of freedom, and therefore has expected value VD·i + Ve, and variance 2(VD·i + Ve)2/(n – 1). The mean is therefore just V¯D + Ve, or αβ + Ve. The realization variance is and the total variance is Taking the difference between eqns 13 and 14 gives the individual variance σ2I·s2=αβ2. Thus, the true variance of sides among individuals is the variance in developmental variance. The repeatability is then readily calculated as ℜs2=σ2I·s2/σ2T·s2. These results were checked by simulations in SAS (results not shown; SAS Institute, 1990). Figure 3 shows ℜFA, the repeatability of FA as a function of CVVD, the coefficient of variation of developmental variance. In all cases, the mean developmental variance was held constant at 1. The small graphs along the top show the very wide range of distributions considered, from symmetrical distributions with small CVs on the left to highly skewed distributions with a very strong mode at 0 on the right. The chief result is that ℜFAincreases rather slowly with CVVD. Only when CVVD is very large does the repeatability reach substantial values. The effect of measurement error, Ve, on repeatability is shown in the lower curves in Fig. 3. The range of measurement error considered is quite large; when Ve=1 the variance due to measurement error is as large as the mean variance of sides. Increasing measurement error lowers the repeatability. Figure 4 shows the coefficient of variation of fluctuating asymmetry, CVFA, values calculated for the same parameter set used in Fig. 3. When α is large and the coefficient of variation of developmental variance, CVVD, is small, CVFA is very close to the theoretical minimum of √[(π – 2)/2] ≈ 76%, which follows from eqns 4 and 5. Only when CVVD is very large does CVFA increase substantially. Increasing measurement error lowers CVFA, as it increases the mean fluctuating asymmetry, as well as the variance. Figure 5 shows ℜs2, the repeatability of s2, the alternative estimator of asymmetry, for the same parameter set used in Figs 3 and 4. The overall shape of these curves is similar to that for ℜFA, in that the repeatabilities only become substantial when the coefficient of variation of developmental variance is quite large. ℜs2 is always lower than ℜFA, although this difference becomes less marked when Ve is large. One advantage of s2 as a measure of asymmetry is that it can incorporate information on more than two sides. Figure 6 shows how repeatability increases with n, the number of sides measured. This relationship emphasizes the usefulness of organisms with multiple realizations of traits per individual for the investigation of developmental stability ( Leung et al., 2000 ). In this paper I have extended a standard model of fluctuating asymmetry to explicitly include variance in the developmental stability parameter (here represented by its converse developmental variance) assumed to underlie variation in asymmetry. The main new result of the model is the very weak relationship between variance in developmental stability and fluctuating asymmetry. Only when the underlying variance in developmental stability is enormous does that variation become apparent in measures of asymmetry. To see this pattern, note (in Figs 3 and 5) that, in the best case when there is no error variance, the repeatability of each asymmetry measure reaches the very modest value of 20% only when the coefficient of variation of developmental variance is about 100%. A coefficient of variation of 100% means that the standard deviation is equal to the mean. Since variances must be positive, this 100% coefficient of variation is accompanied by a highly skewed distribution of developmental variances. Perhaps even more striking is that the coefficient of variation of FA in the population is only about 30% higher than its minimum expected value when the coefficient of variation of developmental variance is 100%, as shown in Fig. 4. One reason for presenting results in terms of repeatability is that it sets an upper limit to the heritability of asymmetry. If the heritability of developmental variance is 1, the heritability of asymmetry is its repeatability. In reality, there is ample evidence that individuals differ in developmental stability because of environmental factors as well as genetic ones ( Parsons, 1990), and the heritability of asymmetry would certainly be lowered as a result, perhaps quite substantially. When the heritability of developmental variance is less than 1, we can be sure that the heritability of asymmetry will be less than its repeatability. One useful and simple result is that adding a given environmental variance to the developmental variance lowers the repeatability of asymmetry in exactly the same manner as does measurement error. For example, the line corresponding to Ve=1 in Fig. 3 is equivalent to a heritability of asymmetry of 0.5. It is therefore not surprising that asymmetry seems to have very low heritability, usually less than 5% ( Whitlock & Fowler, 1997; Gangestad & Thornhill, 1999). Another very useful interpretation of the repeatability is as the correlation of asymmetry values for sets of traits with the same developmental variances. If the developmental variances of different traits are proportional – that is the variance of one trait is a multiple of the variance in some other trait – the correlation of asymmetries of these traits on the same individual will equal the repeatability. However, different traits on an individual may differ in their developmental stabilities for a variety of reasons ( Møller & Swaddle, 1997, pp. 53–55; Leung & Forbes, 1997), so correlations may be substantially less than the repeatability. Gangestad & Thornhill (1999) reviewed a number of large data sets, however, and argued that correlations in FA among traits are close to those expected on the basis of kurtosis and average trait repeatabilities under their model. Future work comparing repeatabilities estimated from CVs and from kurtosis, trait correlations, and the heritability of FA in the same population could provide a means for testing models of FA, such as the one presented in this paper. Given the general result that asymmetry rises slowly with the variation in developmental stability, it is interesting to observe that the coefficient of variation of fluctuating asymmetry is sometimes greater than 100% (see, e.g., Whitlock, 1996; Van Dongen, 1998a). Above CVFA=100%, the CV of developmental stability rises extremely rapidly, as shown in Fig. 4. For example, CVFA=170% for a sample of 188 tarsus lengths in the olive sunbird, Nectarina olivacea ( Van Dongen, 1998a). This result implies a CV of developmental stability of 220%. On the other hand, the available data sets with the largest sample sizes imply modest CVs for developmental stability ( Gangestad & Thornhill, 1999). To compare these inferred coefficients of variation for developmental stability with those for other traits, CVFA should be divided by two, because they have units of the trait squared ( Lande, 1977; Houle, 1992). The phenotypic coefficients of variation of morphological traits are generally between 2 and 20%, whereas fitness components have values generally between 10 and 100% ( Houle, 1992). Thus, in cases such as the olive sunbird ( Van Dongen, 1998a), where the predicted variation in developmental stability is extremely high, either the model is false, or developmental stability is sometimes more variable than for any previously studied traits. Asymmetry has attracted attention because it potentially captures information about developmental stability, which may be of fundamental importance to fitness. If developmental stability has higher variance than typical for fitness, this raises an interesting paradox. The standard model makes a number of questionable assumptions to the standard model that could increase the CVFA when violated. First, the model assumes that the distribution of sides is normal for a given level of developmental stability. The distribution of sides could instead be a mixture of different distributions, perhaps reflecting discrete events that deflect development into alternative pathways. For example, asymmetrical use, injury, starvation, or other traumas might have large effects on development, although not necessarily so large as to lead to the rejection of an individual as a statistical or biological outlier. Extreme individuals have a disproportionate impact on measures of variation. For example, Whitlock (1996) notes that exclusion of a single highly asymmetrical individual lowers CVFA in his sample of wolf jaws from 145% to about half that, close to the minimum value expected if there were no variation in developmental stability. In correlational studies, extreme individuals can account for much of the apparent power of a model to explain the data ( Leung & Forbes, 1997). The developmental model of Klingenberg & Nijhout (1999) provides a very different explanation for how normality could be violated. These authors point out that developmental stability is likely to be an epiphenomenon of the parameters of the developmental system, rather than a single property of that system. They expect that variation in the fundamental developmental properties will have nonlinear effects on morphology, which could easily lead to non-normal distributions of sizes. Similarly, nonlinear interactions during development could result in distributions of developmental variances that are very different from the gamma distribution I assumed. Another possibly incorrect assumption is that each side develops independently. Because sides usually develop simultaneously on the same organism, there are opportunities for interactions during development ( Graham et al., 1993 ; Klingenberg & Nijhout, 1998). The most plausible kind of interaction would be competition for resources during development ( Klingenberg & Nijhout, 1998). This would tend to cause antisymmetry and lower CVFA ( Van Dongen, 1998a) and so cannot help to explain the high values that raise questions under the present model. The finding that some distributions of FA are consistent with condition-dependent antisymmetry ( Rowe et al., 1997 ) may indicate that models of this type should be taken seriously. In addition to these assumptions common to the standard models of asymmetry, I also had to make an assumption about the actual distribution of developmental stabilities. I chose the gamma distribution, but other distributions, such as the log-normal, may be worth considering. My principal conclusion, that the usefulness of FA as an indicator of individual developmental stability is poor unless the variance of developmental stability is extremely large, does not depend on the choice of distribution. Previous numerical or simulation studies using mixtures of two developmental stabilities ( Houle, 1997), three developmental stabilities ( van Dongen, 1998b), and normal, half-normal and uniform distributions of developmental stabilities ( Gangestad & Thornhill, 1999) yield similar conclusions. In summary, this elaboration of a basic model in asymmetry studies is consistent with many of the largest experimental studies in suggesting that the proportion of the within-population variation in fluctuating asymmetry that can be explained by variation in developmental stability is small ( Gangestad & Thornhill, 1999). If the variation in developmental stability is typical of that found for other sorts of traits, then the value of FA as an indicator of developmental stability is low. On the other hand, a number of smaller studies report distributions of FA that imply enormous variation in developmental stabilities ( Whitlock, 1996; Van Dongen, 1998a; Lens & van Dongen, 1999; Lens et al., 1999 ). In these cases, we need either to explain how the variance in developmental stability can be so high or to modify this standard model of the relationship between developmental stability and fluctuating asymmetry. I thank L. Rowe, R. Palmer, M. Whitlock, G. Bell, S. W. Gangestad, and anonymous reviewers for comments on the manuscript, and for helpful discussions. A. Thistle helped to edit the manuscript. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

Customer: SACO CONTROLS Prep by: Sam Larizza Project #: P7-03-01 BILL OF MATERIALS Date: July 11, 2003 P.O. #: 2003/10781 Project: Page: 1 of 4 Report CopyRight/DMCA Form For : Feedback Control Theory System Control Group At Preface iii,1 Introduction 1,1 1 Issues in Control System Design 1. 1 2 What Is in This Book 7,2 Norms for Signals and Systems 13. 2 1 Norms for Signals 13,2 2 Norms for Systems 15,2 3 Input Output Relationships 18. 2 4 Power Analysis Optional 19,2 5 Proofs for Tables 2 1 and 2 2 Optional 21. 2 6 Computing by State Space Methods Optional 24,3 Basic Concepts 31. 3 1 Basic Feedback Loop 31,3 2 Internal Stability 34. 3 3 Asymptotic Tracking 38,3 4 Performance 40,4 Uncertainty and Robustness 45. 4 1 Plant Uncertainty 45,4 2 Robust Stability 50,4 3 Robust Performance 53. 4 4 Robust Performance More Generally 58,4 5 Conclusion 59. 5 Stabilization 63,5 1 Controller Parametrization Stable Plant 63. 5 2 Coprime Factorization 65, 5 3 Coprime Factorization by State Space Methods Optional 69. 5 4 Controller Parametrization General Plant 71,5 5 Asymptotic Properties 73. 5 6 Strong and Simultaneous Stabilization 75,5 7 Cart Pendulum Example 81. 6 Design Constraints 87,6 1 Algebraic Constraints 87. 6 2 Analytic Constraints 88,7 Loopshaping 101,7 1 The Basic Technique of Loopshaping 101. 7 2 The Phase Formula Optional 105,7 3 Examples 108. 8 Advanced Loopshaping 117,8 1 Optimal Controllers 117. 8 2 Loopshaping with C 118,8 3 Plants with RHP Poles and Zeros 126. 8 4 Shaping S T or Q 135,8 5 Further Notions of Optimality 138. 9 Model Matching 149,9 1 The Model Matching Problem 149. 9 2 The Nevanlinna Pick Problem 150,9 3 Nevanlinna s Algorithm 154. 9 4 Solution of the Model Matching Problem 158,9 5 State Space Solution Optional 160. 10 Design for Performance 163,10 1 P 1 Stable 163,10 2 P 1 Unstable 168. 10 3 Design Example Flexible Beam 170,10 4 2 Norm Minimization 175. 11 Stability Margin Optimization 181,11 1 Optimal Robust Stability 181. 11 2 Conformal Mapping 185,11 3 Gain Margin Optimization 187. 11 4 Phase Margin Optimization 192,12 Design for Robust Performance 195. 12 1 The Modified Problem 195,12 2 Spectral Factorization 196. 12 3 Solution of the Modified Problem 198,12 4 Design Example Flexible Beam Continued 204. References 209, Striking developments have taken place since 1980 in feedback control theory The subject has be. come both more rigorous and more applicable The rigor is not for its own sake but rather that even. in an engineering discipline rigor can lead to clarity and to methodical solutions to problems The. applicability is a consequence both of new problem formulations and new mathematical solutions. to these problems Moreover computers and software have changed the way engineering design is. done These developments suggest a fresh presentation of the subject one that exploits these new. developments while emphasizing their connection with classical control. Control systems are designed so that certain designated signals such as tracking errors and. actuator inputs do not exceed pre specified levels Hindering the achievement of this goal are. uncertainty about the plant to be controlled the mathematical models that we use in representing. real physical systems are idealizations and errors in measuring signals sensors can measure signals. only to a certain accuracy Despite the seemingly obvious requirement of bringing plant uncertainty. explicitly into control problems it was only in the early 1980s that control researchers re established. the link to the classical work of Bode and others by formulating a tractable mathematical notion. of uncertainty in an input output framework and developing rigorous mathematical techniques to. cope with it This book formulates a precise problem called the robust performance problem with. the goal of achieving specified signal levels in the face of plant uncertainty. The book is addressed to students in engineering who have had an undergraduate course in. signals and systems including an introduction to frequency domain methods of analyzing feedback. control systems namely Bode plots and the Nyquist criterion A prior course on state space theory. would be advantageous for some optional sections but is not necessary To keep the development. elementary the systems are single input single output and linear operating in continuous time. Chapters 1 to 7 are intended as the core for a one semester senior course they would need. supplementing with additional examples These chapters constitute a basic treatment of feedback. design containing a detailed formulation of the control design problem the fundamental issue. of performance stability robustness tradeoff and the graphical design technique of loopshaping. suitable for benign plants stable minimum phase Chapters 8 to 12 are more advanced and. are intended for a first graduate course Chapter 8 is a bridge to the latter half of the book. extending the loopshaping technique and connecting it with notions of optimality Chapters 9 to. 12 treat controller design via optimization The approach in these latter chapters is mathematical. rather than graphical using elementary tools involving interpolation by analytic functions This. mathematical approach is most useful for multivariable systems where graphical techniques usually. break down Nevertheless we believe the setting of single input single output systems is where this. new approach should be learned, There are many people to whom we are grateful for their help in this book Dale Enns for. sharing his expertise in loopshaping Raymond Kwong and Boyd Pearson for class testing the book. and Munther Dahleh Ciprian Foias and Karen Rudie for reading earlier drafts Numerous Caltech. students also struggled with various versions of this material Gary Balas Carolyn Beck Bobby. Bodenheimer and Roy Smith had particularly helpful suggestions Finally we would like to thank. the AFOSR ARO NSERC NSF and ONR for partial financial support during the writing of this. Introduction, Without control systems there could be no manufacturing no vehicles no computers no regulated. environment in short no technology Control systems are what make machines in the broadest. sense of the term function as intended Control systems are most often based on the principle. of feedback whereby the signal to be controlled is compared to a desired reference signal and the. discrepancy used to compute corrective control action The goal of this book is to present a theory. of feedback control system design that captures the essential issues can be applied to a wide range. of practical problems and is as simple as possible. 1 1 Issues in Control System Design, The process of designing a control system generally involves many steps A typical scenario is as. 1 Study the system to be controlled and decide what types of sensors and actuators will be used. and where they will be placed,2 Model the resulting system to be controlled. 3 Simplify the model if necessary so that it is tractable. 4 Analyze the resulting model determine its properties. 5 Decide on performance specifications,6 Decide on the type of controller to be used. 7 Design a controller to meet the specs if possible if not modify the specs or generalize the. type of controller sought, 8 Simulate the resulting controlled system either on a computer or in a pilot plant. 9 Repeat from step 1 if necessary, 10 Choose hardware and software and implement the controller. 11 Tune the controller on line if necessary,2 CHAPTER 1 INTRODUCTION. It must be kept in mind that a control engineer s role is not merely one of designing control. systems for fixed plants of simply wrapping a little feedback around an already fixed physical. system It also involves assisting in the choice and configuration of hardware by taking a system. wide view of performance For this reason it is important that a theory of feedback not only lead. to good designs when these are possible but also indicate directly and unambiguously when the. performance objectives cannot be met, It is also important to realize at the outset that practical problems have uncertain non. minimum phase plants non minimum phase means the existence of right half plane zeros so the. inverse is unstable that there are inevitably unmodeled dynamics that produce substantial un. certainty usually at high frequency and that sensor noise and input signal level constraints limit. the achievable benefits of feedback A theory that excludes some of these practical issues can. still be useful in limited application domains For example many process control problems are so. dominated by plant uncertainty and right half plane zeros that sensor noise and input signal level. constraints can be neglected Some spacecraft problems on the other hand are so dominated by. tradeoffs between sensor noise disturbance rejection and input signal level e g fuel consumption. that plant uncertainty and non minimum phase effects are negligible Nevertheless any general. theory should be able to treat all these issues explicitly and give quantitative and qualitative results. about their impact on system performance, In the present section we look at two issues involved in the design process deciding on perfor. mance specifications and modeling We begin with an example to illustrate these two issues. Example A very interesting engineering system is the Keck astronomical telescope currently. under construction on Mauna Kea in Hawaii When completed it will be the world s largest The. basic objective of the telescope is to collect and focus starlight using a large concave mirror The. shape of the mirror determines the quality of the observed image The larger the mirror the more. light that can be collected and hence the dimmer the star that can be observed The diameter of. the mirror on the Keck telescope will be 10 m To make such a large high precision mirror out of. a single piece of glass would be very difficult and costly Instead the mirror on the Keck telescope. will be a mosaic of 36 hexagonal small mirrors These 36 segments must then be aligned so that. the composite mirror has the desired shape, The control system to do this is illustrated in Figure 1 1 As shown the mirror segments. are subject to two types of forces disturbance forces described below and forces from actuators. Behind each segment are three piston type actuators applying forces at three points on the segment. to effect its orientation In controlling the mirror s shape it suffices to control the misalignment. between adjacent mirror segments In the gap between every two adjacent segments are capacitor. type sensors measuring local displacements between the two segments These local displacements. are stacked into the vector labeled y this is what is to be controlled For the mirror to have the. ideal shape these displacements should have certain ideal values that can be pre computed these. are the components of the vector r The controller must be designed so that in the closed loop. system y is held close to r despite the disturbance forces Notice that the signals are vector valued. Such a system is multivariable, Our uncertainty about the plant arises from disturbance sources. As the telescope turns to track a star the direction of the force of gravity on the mirror. During the night when astronomical observations are made the ambient temperature changes. 1 1 ISSUES IN CONTROL SYSTEM DESIGN 3,disturbance forces. r u mirror y,controller actuators, Figure 1 1 Block diagram of Keck telescope control system. The telescope is susceptible to wind gusts,and from uncertain plant dynamics. The dynamic behavior of the components mirror segments actuators sensors cannot be. modeled with infinite precision, Now we continue with a discussion of the issues in general. Control Objectives, Generally speaking the objective in a control system is to make some output say y behave in a. desired way by manipulating some input say u The simplest objective might be to keep y small. or close to some equilibrium point a regulator problem or to keep y r small for r a reference. or command signal in some set a servomechanism or servo problem Examples. On a commercial airplane the vertical acceleration should be less than a certain value for. passenger comfort, In an audio amplifier the power of noise signals at the output must be sufficiently small for. high fidelity, In papermaking the moisture content must be kept between prescribed values. There might be the side constraint of keeping u itself small as well because it might be constrained. e g the flow rate from a valve has a maximum value determined when the valve is fully open. or it might be too expensive to use a large input But what is small for a signal It is natural to. introduce norms for signals then y small means kyk small Which norm is appropriate depends. on the particular application, In summary performance objectives of a control system naturally lead to the introduction of. norms then the specs are given as norm bounds on certain key signals of interest. 4 CHAPTER 1 INTRODUCTION, Before discussing the issue of modeling a physical system it is important to distinguish among four. different objects,1 Real physical system the one out there. 2 Ideal physical model obtained by schematically decomposing the real physical system into. ideal building blocks composed of resistors masses beams kilns isotropic media Newtonian. fluids electrons and so on, 3 Ideal mathematical model obtained by applying natural laws to the ideal physical model. composed of nonlinear partial differential equations and so on. 4 Reduced mathematical model obtained from the ideal mathematical model by linearization. lumping and so on usually a rational transfer function. Sometimes language makes a fuzzy distinction between the real physical system and the ideal. physical model For example the word resistor applies to both the actual piece of ceramic and. metal and the ideal object satisfying Ohm s law Of course the adjectives real and ideal could be. used to disambiguate, No mathematical system can precisely model a real physical system there is always uncertainty. Uncertainty means that we cannot predict exactly what the output of a real physical system will. be even if we know the input so we are uncertain about the system Uncertainty arises from two. sources unknown or unpredictable inputs disturbance noise etc and unpredictable dynamics. What should a model provide It should predict the input output response in such a way that. we can use it to design a control system and then be confident that the resulting design will work. on the real physical system Of course this is not possible A leap of faith will always be required. on the part of the engineer This cannot be eliminated but it can be made more manageable with. the use of effective modeling analysis and design techniques. Mathematical Models in This Book, The models in this book are finite dimensional linear and time invariant The main reason for this. is that they are the simplest models for treating the fundamental issues in control system design. The resulting design techniques work remarkably well for a large class of engineering problems. partly because most systems are built to be as close to linear time invariant as possible so that they. are more easily controlled Also a good controller will keep the system in its linear regime The. uncertainty description is as simple as possible as well. The basic form of the plant model in this book is, Here y is the output u the input and P the nominal plant transfer function The model uncertainty. comes in two forms,n unknown noise or disturbance,unknown plant perturbation. 1 1 ISSUES IN CONTROL SYSTEM DESIGN 5, Both n and will be assumed to belong to sets that is some a priori information is assumed. about n and Then every input u is capable of producing a set of outputs namely the set of. all outputs P u n as n and range over their sets Models capable of producing sets of. outputs for a single input are said to be nondeterministic There are two main ways of obtaining. models as described next,Models from Science, The usual way of getting a model is by applying the laws of physics chemistry and so on Consider. the Keck telescope example One can write down differential equations based on physical principles. e g Newton s laws and making idealizing assumptions e g the mirror segments are rigid The. coefficients in the differential equations will depend on physical constants such as masses and. physical dimensions These can be measured This method of applying physical laws and taking. measurements is most successful in electromechanical systems such as aerospace vehicles and robots. Some systems are difficult to model in this way either because they are too complex or because. their governing laws are unknown,Models from Experimental Data. The second way of getting a model is by doing experiments on the physical system Let s start. with a simple thought experiment one that captures many essential aspects of the relationships. between physical systems and their models and the issues in obtaining models from experimental. data Consider a real physical system the plant to be controlled with one input u and one. output y To design a control system for this plant we must understand how u affects y. The experiment runs like this Suppose that the real physical system is in a rest state before. an input u is applied i e u y 0 Now apply some input signal u resulting in some output. signal y Observe the pair u y Repeat this experiment several times Pretend that these data. pairs are all we know about the real physical system This is the black box scenario Usually we. know something about the internal workings of the system. After doing this experiment we will notice several things First the same input signal at different. times produces different output signals Second if we hold u 0 y will fluctuate in an unpredictable. manner Thus the real physical system produces just one output for any given input so it itself. is deterministic However we observers are uncertain because we cannot predict what that output. Ideally the model should cover the data in the sense that it should be capable of producing. every experimentally observed input output pair Of course it would be better to cover not just. the data observed in a finite number of experiments but anything that can be produced by the real. physical system Obviously this is impossible If nondeterminism that reasonably covers the range. of expected data is not built into the model we will not trust that designs based on such models. will work on the real system, In summary for a useful theory of control design plant models must be nondeterministic having. uncertainty built in explicitly,Synthesis Problem, A synthesis problem is a theoretical problem precise and unambiguous Its purpose is primarily. pedagogical It gives us something clear to focus on for the purpose of study The hope is that. 6 CHAPTER 1 INTRODUCTION, the principles learned from studying a formal synthesis problem will be useful when it comes to. designing a real control system, The most general block diagram of a control system is shown in Figure 1 2 The generalized plant. generalized,controller,Figure 1 2 Most general control system. consists of everything that is fixed at the start of the control design exercise the plant actuators. that generate inputs to the plant sensors measuring certain signals analog to digital and digital. to analog converters and so on The controller consists of the designable part it may be an electric. circuit a programmable logic controller a general purpose computer or some other such device. The signals w z y and u are in general vector valued functions of time The components of w. are all the exogenous inputs references disturbances sensor noises and so on The components of. z are all the signals we wish to control tracking errors between reference signals and plant outputs. actuator signals whose values must be kept between certain limits and so on The vector y contains. the outputs of all sensors Finally u contains all controlled inputs to the generalized plant Even. open loop control fits in the generalized plant would be so defined that y is always constant. Very rarely is the exogenous input w a fixed known signal One of these rare instances is where. a robot manipulator is required to trace out a definite path as in welding Usually w is not fixed. but belongs to a set that can be characterized to some degree Some examples. In a thermostat controlled temperature regulator for a house the reference signal is always. piecewise constant at certain times during the day the thermostat is set to a new value The. temperature of the outside air is not piecewise constant but varies slowly within bounds. In a vehicle such as an airplane or ship the pilot s commands on the steering wheel throttle. pedals and so on come from a predictable set and the gusts and wave motions have amplitudes. and frequencies that can be bounded with some degree of confidence. The load power drawn on an electric power system has predictable characteristics. Sometimes the designer does not attempt to model the exogenous inputs Instead she or he. designs for a suitable response to a test input such as a step a sinusoid or white noise The. designer may know from past experience how this correlates with actual performance in the field. Desired properties of z generally relate to how large it is according to various measures as discussed. 1 2 WHAT IS IN THIS BOOK 7, Finally the output of the design exercise is a mathematical model of a controller This must. be implementable in hardware If the controller you design is governed by a nonlinear partial. differential equation how are you going to implement it A linear ordinary differential equation. with constant coefficients representing a finite dimensional time invariant linear system can be. simulated via an analog circuit or approximated by a digital computer so this is the most common. type of control law, The synthesis problem can now be stated as follows Given a set of generalized plants a set. of exogenous inputs and an upper bound on the size of z design an implementable controller to. achieve this bound How the size of z is to be measured e g power or maximum amplitude. depends on the context This book focuses on an elementary version of this problem. 1 2 What Is in This Book, Since this book is for a first course on this subject attention is restricted to systems whose models. are single input single output finite dimensional linear and time invariant Thus they have trans. fer functions that are rational in the Laplace variable s The general layout of the book is that. Chapters 2 to 4 and 6 are devoted to analysis of control systems that is the controller is already. specified and Chapters 5 and 7 to 12 to design, Performance of a control system is specified in terms of the size of certain signals of interest For. example the performance of a tracking system could be measured by the size of the error signal. Chapter 2 Norms for Signals and Systems looks at several ways of defining norms for a signal u t. in particular the 2 norm associated with energy,the norm maximum absolute value. and the square root of the average power actually not quite a norm. lim u t dt, Also introduced are two norms for a system s transfer function G s the 2 norm. kGk2 G j d,and the norm,kGk max G j, Notice that kGk equals the peak amplitude on the Bode magnitude plot of G Then two very. useful tables are presented summarizing input output norm relationships For example one table. gives a bound on the 2 norm of the output knowing the 2 norm of the input and the norm of the. 8 CHAPTER 1 INTRODUCTION,Figure 1 3 Single loop feedback system. transfer function Such results are very useful in predicting for example the effect a disturbance. will have on the output of a feedback system, Chapters 3 and 4 are the most fundamental in the book The system under consideration is. shown in Figure 1 3 where P and C are the plant and controller transfer functions The signals are. as follows,r reference or command input,e tracking error. u control signal controller output,d plant disturbance. y plant output,n sensor noise, In Chapter 3 Basic Concepts internal stability is defined and characterized Then the system is. analyzed for its ability to track a single reference signal r a step or a ramp asymptotically as. time increases Finally we look at tracking a set of reference signals The transfer function from. reference input r to tracking error e is denoted S the sensitivity function It is argued that a useful. tracking performance criterion is kW1 Sk 1 where W1 is a transfer function which can be tuned. by the control system designer, Since no mathematical system can exactly model a physical system we must be aware of how. modeling errors might adversely affect the performance of a control system Chapter 4 Uncertainty. and Robustness begins with a treatment of various models of plant uncertainty The basic technique. is to model the plant as belonging to a set P Such a set can be either structured for example. there are a finite number of uncertain parameters or unstructured the frequency response lies in. a set in the complex plane for every frequency For us unstructured is more important because it. leads to a simple and useful design theory In particular multiplicative perturbation is chosen for. detailed study it being typical In this uncertainty model there is a nominal plant P and the family. P consists of all perturbed plants P such that at each frequency the ratio P j P j lies in a. disk in the complex plane with center 1 This notion of disk like uncertainty is key because of it. the mathematical problems are tractable, Generally speaking the notion of robustness means that some characteristic of the feedback. system holds for every plant in the set P A controller C provides robust stability if it provides. internal stability for every plant in P Chapter 4 develops a test for robust stability for the multi. plicative perturbation model a test involving C and P The test is kW2 T k 1 Here T is the. 1 2 WHAT IS IN THIS BOOK 9, complementary sensitivity function equal to 1 S or the transfer function from r to y and W2. is a transfer function whose magnitude at frequency equals the radius of the uncertainty disk at. that frequency, The final topic in Chapter 4 is robust performance guaranteed tracking in the face of plant. uncertainty The main result is that the tracking performance spec kW1 Sk 1 is satisfied for all. plants in the multiplicative perturbation set if and only if the magnitude of W1 S W2 T is less. than 1 for all frequencies that is,k W1 S W2 T k 1 1 1. This is an analysis result It tells exactly when some candidate controller provides robust perfor. Chapter 5 Stabilization is the first on design Most synthesis problems can be formulated like. this Given P design C so that the feedback system 1 is internally stable and 2 acquires some. additional desired property or properties for example the output y asymptotically tracks a step. input r The method of solution presented here is to parametrize all Cs for which 1 is true and. then to find a parameter for which 2 holds In this chapter such a parametrization is derived it. has the form, where N M X and Y are fixed stable proper transfer functions and Q is the parameter an. arbitrary stable proper transfer function The usefulness of this parametrization derives from the. fact that all closed loop transfer functions are very simple functions of Q for instance the sensitivity. function S while a nonlinear function of C equals simply M Y M N Q This parametrization. is then applied to three problems achieving asymptotic performance specs such as tracking a. step internal stabilization by a stable controller and simultaneous stabilization of two plants by a. common controller, Before we see how to design control systems for the robust performance specification it is. important to understand the basic limitations on achievable performance Why can t we achieve. both arbitrarily good performance and stability robustness at the same time In Chapter 6 Design. Constraints we study design constraints arising from two sources from algebraic relationships that. must hold among various transfer functions and from the fact that closed loop transfer functions. must be stable that is analytic in the right half plane The main conclusion is that feedback control. design always involves a tradeoff between performance and stability robustness. Chapter 7 Loopshaping presents a graphical technique for designing a controller to achieve. robust performance This method is the most common in engineering practice It is especially. suitable for today s CAD packages in view of their graphics capabilities The loop transfer function. is L P C The idea is to shape the Bode magnitude plot of L so that 1 1 is achieved at. least approximately and then to back solve for C via C L P When P or P 1 is not stable L. must contain P s unstable poles and zeros for internal stability of the feedback loop an awkward. constraint For this reason it is assumed in Chapter 7 that P and P 1 are both stable. Thus Chapters 2 to 7 constitute a basic treatment of feedback design containing a detailed. formulation of the control design problem the fundamental issue of performance stability robustness. tradeoff and a graphical design technique suitable for benign plants stable minimum phase. Chapters 8 to 12 are more advanced,10 CHAPTER 1 INTRODUCTION. Chapter 8 Advanced Loopshaping is a bridge between the two halves of the book it extends the. loopshaping technique and connects it with the notion of optimal designs Loopshaping in Chapter 7. focuses on L but other quantities such as C S T or the Q parameter in the stabilization results. of Chapter 5 may also be shaped to achieve the same end For many problems these alternatives. are more convenient Chapter 8 also offers some suggestions on how to extend loopshaping to handle. right half plane poles and zeros, Optimal controllers are introduced in a formal way in Chapter 8 Several different notions of. optimality are considered with an aim toward understanding in what way loopshaping controllers. can be said to be optimal It is shown that loopshaping controllers satisfy a very strong type. of optimality called self optimality The implication of this result is that when loopshaping is. successful at finding an adequate controller it cannot be improved upon uniformly. Chapters 9 to 12 present a recently developed approach to the robust performance design prob. lem The approach is mathematical rather than graphical using elementary tools involving interpo. lation by analytic functions This mathematical approach is most useful for multivariable systems. where graphical techniques usually break down Nevertheless the setting of single input single. output systems is where this new approach should be learned Besides present day software for. control design e g MATLAB and Program CC incorporate this approach. Chapter 9 Model Matching studies a hypothetical control problem called the model matching. problem Given stable proper transfer functions T1 and T2 find a stable transfer function Q to. minimize kT1 T2 Qk The interpretation is this T1 is a model T2 is a plant and Q is a cascade. controller to be designed so that T2 Q approximates T1 Thus T1 T2 Q is the error transfer function. This problem is turned into a special interpolation problem Given points ai in the right half. plane and values bi also complex numbers find a stable transfer function G so that kGk 1. and G ai bi that is G interpolates the value bi at the point ai When such a G exists and how. to find one utilizes some beautiful mathematics due to Nevanlinna and Pick. Chapter 10 Design for Performance treats the problem of designing a controller to achieve the. performance criterion kW1 Sk 1 alone that is with no plant uncertainty When does such a. controller exist and how can it be computed These questions are easy when the inverse of the. plant transfer function is stable When the inverse is unstable i e P is non minimum phase the. questions are more interesting The solutions presented in this chapter use model matching theory. The procedure is applied to designing a controller for a flexible beam The desired performance is. given in terms of step response specs overshoot and settling time It is shown how to choose the. weight W1 to accommodate these time domain specs Also treated in Chapter 10 is minimization. of the 2 norm of some closed loop transfer function e g kW1 Sk2. Next in Chapter 11 Stability Margin Optimization is considered the problem of designing a. controller whose sole purpose is to maximize the stability margin that is performance is ignored. The maximum obtainable stability margin is a measure of how difficult the plant is to control. Three measures of stability margin are treated the norm of a multiplicative perturbation gain. margin and phase margin It is shown that the problem of optimizing these stability margins can. also be reduced to a model matching problem, Chapter 12 Design for Robust Performance returns to the robust performance problem of. designing a controller to achieve 1 1 Chapter 7 proposed loopshaping as a graphical method. when P and P 1 are stable Without these assumptions loopshaping can be awkward and the. methodical procedure in this chapter can be used Actually 1 1 is too hard for mathematical. 06-Apr-2020 1 Views 13 Pages Customer: SACO CONTROLS Prep by: Sam Larizza Project #: P7-03-01 BILL OF MATERIALS Date: July 11, 2003 P.O. #: 2003/10781 Project: Page: 1 of 4 06-Apr-2020 0 Views 15 Pages properties can be improved with the amendment using an organic source, usually sewage sludge. 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A zener diode is always operated in reverse bias, so the characteristic curve only shows the reverse characteristic. 06-Apr-2020 0 Views 18 Pages Ophthalmic Antibiotics Therapeutic Class Review (TCR) April 19, 2017 No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, digital scanning, or via any information storage or 06-Apr-2020 0 Views 238 Pages The Hillfolk. 69 Rivals 70 Outlanders. 70 Shell-Grinders 70 Domers. 72 Iron-Makers 72 Rockheads. 72 Saltmen 75 Threshers. 75 Tridents 75 Sample Names. 75 Why a Fictionalized 10th. century BCE Levant? 77 Additional Settings Under Hollow Hills,78 Hollywoodland, Jason Morningstar 81 Mad Scientists Anonymous, Michelle Nephew 85 Moscow Station, Kenneth Hite 91 World War 2.1, Matt Forbeck 95 Malice ... 06-Apr-2020 0 Views 5 Pages This is where the Free Hillfolk live. 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Presentation on theme: "Agenda Blog signup… First week impressions – High School vs. University.. Career Night – Sep 12 th (Wed) CS outcomes Complexity."— Presentation transcript: Agenda Blog signup… First week impressions – High School vs. University.. Career Night – Sep 12 th (Wed) CS outcomes Complexity First Week Impressions? How was your first week at ASU? Good things? Frustrating things? Differences from High School? Mark your calendars for a special evening event (approx. 4:00-8:00PM) in the Memorial Union (Second Floor) At the event, you will have the opportunity to talk to working engineers from a variety of industries and companies, to find out more about engineering careers. You will have the chance to ask them anything you want to know about including what it’s like to be an engineer, or what you need to do to be successful. Attendance will be taken; – there will be no class that week (no class on 9 th September) More information to come later Career Exploration Night Wednesday September 12th http://more.engineering.asu.edu/career/event s/freshman-engineering-career-exploration- night/ Sorting… When is a sequence in sorted order? – How many pairwise comparisons do I need to do to check if a sequence of n-numbers is sorted? – If I have a procedure for checking whether a sequence is sorted, is it reasonable to sort a sequence of numbers by generating permutations and testing if any of them are sorted? Intelligence is putting the “test” part of Generate&Test into generate part… Career Fair Activity Career Exploration Module Part 1 Homework: Prepare for Career Exploration Night To prepare for Career Exploration Night you should think about what you want to know or learn from engineering professionals. This may include information about the following: what they do in their jobs (day to day tasks; responsibilities; etc..) what career path did they take from college to where they are now? how they chose their careers advice they have about skills and experience you need to be successful etc, etc, etc… In preparation for the event, create a list of at least 5 questions that you would like to ask the professionals at the event. When you have completed your list, submit it to the Blog You should, of course, ask more than 5 questions at the event, but this list will at least help you to start conversations to get the information you want to know. There is no required set of questions; you should ask the questions you are most interested in. If you are having a hard time thinking of questions, look at the “Student Information for Engineering Career Exploration Night” document (posted on Blackboard) for some ideas to help stimulate your thinking. Can you think of additional questions you want answered? Event on Wed 9/12 4-8pm NO separate class next Friday (the event attendance is the class attendance) Career Exploration Night Experience Topics for Group Discussion: – Share the majors, job titles, and companies of the individuals you met. – What was the most important thing you learned about engineering? Most surprising? Most interesting? – What kind of projects/work activities were mentioned? – What differences did you find between “real people” and what you had read beforehand? Similarities? – What was the best advice that you received? Discussion Highlights (Group reporting) What was the most important thing you learned? – Most surprising? – Most interesting? Major differences/similarities between information from ‘real people’ and information you read? Types of projects/work activities? Did you see different job functions within a single discipline (or similar job functions across multiple disciplines)? What was the best advice that you received? Company Requires Or DesiresMy QualificationDifference Between The Two What I Am Going To Do To Make Up The Difference or Gain a Measurable Accomplishment Job Title: ___________________________ Page # ______ Summary of the points from the three groups (9/21) Exactly when do we say an algorithm is “slow”? We kind of felt that O(N! * N) is a bit much complexity How about O(N 2 )? O(N 10 )? Where do we draw the line? – Meet the Computer Scientist Nightmare – So “Polynomial” ~ “easy” & “exponential” ~ “hard” – 2 n eventually overtakes any n k however large k is.. How do we know if a problem is “really” hard to solve or it is just that I am dumb and didn’t know how to do better? – If checking the correctness of the solution itself takes more than polynomial time, then we know the problem must be hard.. – But there are many problems, for which checking the correctness is polynomial, and yet we don’t know any efficient algorithm 2n2n Classes P and NP Class P If a problem can be solved in time polynomial in the size of the input it is considered an “easy” problem – Note that your failure to solve a problem in polynomial time doesn’t mean it is not polynomial (you could come up with O(N* N!) algorithm for sorting, after all Class NP Technically “if a problem can be solved in polynomial time by a non-deterministic turing machine, then it is in class NP” Informally, if you can check the correctness of a solution in polynomial time, then it is in class NP – Are there problems where even checking the solution is hard? Tower of Hanoi (or Brahma) Shift the disks from the left peg to the right peg – You can lift one disk at a time – You can use the middle peg to “park” disks – You can never ever have a larger disk on top of a smaller disk (or KABOOM) How many moves to solve a 2-disk version? A 3-disk one? An n-disk one? – How long does it take (in terms of input size), to check if you have a correct solution? How to explain to your boss as to why your program is so slow… I can't find an efficient algorithm, I guess I'm just too dumb. I can't find an efficient algorithm, because no such algorithm is possible. I can't find an efficient algorithm, but neither can all these famous people. The P=NP question Clearly, all polynomial problems are also NP problems Do we know for sure that there are NP problems that are not polynomial? If we assume this, then we are assuming P != NP If P = NP, then some smarter person can still solve a problem that we thought can’t be solved in polytime – Can imply more than a loss of face… For example, factorization is known to be an NP-Complete problem; and forms the basis for all of cryptography.. If P=NP, then all the cryptography standards can be broken! NP-Complete: “hardest” problems in class NP [the giants of NP-world] EVERY problem in class NP can be reduced to an NP-Complete problem in polynomial time --So you can solve that problem by using an algorithm that solves the NP-complete problem Reducing Problems… Mathematician reduces “mattress on fire” problem Make Rao Happy Make Everyone in ASU Happy Make Little Tommy Happy Make his entire family happy General NP- problem Boolean Satisfiability Problem 3-SAT Thus, SAT is NP-Complete Thus, 3SAT is also NP-Complete..but of course! Tommy is a fussy dude! Academic Integrity What it means Typical ASU policy – Homeworks – Exams Take-Home Exams – Term papers Scholarship Opportunities General Scholarships The FURI program NSF REU program Is exponential complexity the worst? After all, we can have 2 2 More fundamental question: Can every computational problem be solved in finite time? “Decidability” --Unfortunately not guaranteed [and Hilbert turns in his grave] 2n2n Some Decidability Challenges In First Order Logic, inference (proving theorems) is semi-decidable – If the theorem is true, you can show that in finite time; if it is false you may never be able to show it In First Order Logic + Peano Arithmatic, inference is undecidable – There may be theorems that are true but you can’t prove them [Godel] Practical Implications of Intractability A class of problems is said to be NP-hard as long as the class contains at least one instance that will take exponential time.. What if 99% of the instances are actually easy to solved? --Where then are the wild things? Satisfiability problem Given a set of propositions P 1 P 2 … P n..and a set of “clauses” that the propositions must satisfy – Clauses are of the form P 1 V P 7 VP 9 V P 12 V ~P 35 – Size of a clause is the number of propositions it mentions; size can be anywhere from 1 to n Find a T/F assignment to the propositions that respects all clauses Is it in class NP? How many “potential” solutions? Canonical NP-Complete Problem. 3-SAT is where all clauses are of length 3 Example of a SAT problem P,Q,R are propositions Clauses – P V ~Q V R – Q V ~R V ~P Is P=False, Q=True, R=False as solution? Is Boolean SAT in NP? Hardness of 3-sat as a function of #clauses/#variables #clauses/#variables Probability that there is a satisfying assignment Cost of solving (either by finding a solution or showing there ain’t one) p=0.5 You would expect this This is what happens! ~4.3 Phase Transition! Phase Transition in SAT Theoretically we only know that phase transition ratio occurs between 3.26 and 4.596. Experimentally, it seems to be close to 4.3 (We also have a proof that 3-SAT has sharp threshold)

Mathematics for Economists. Made Simple mathematics and statistics paperback, 366 pp., 1. edition published: august 2010 recommended price: 275 czk As the field of economics becomes ever more specialized and complicated, so does the mathematics required of economists. With Mathematics for Economists, expert mathematician Viatcheslav V. Vinogradov offers a straightforward, practical textbook for students in economics?for whom mathematics is not a scientific or philosophical subject but a practical necessity. Focusing on the most important fields of economics, the book teaches apprentice economists to apply mathematics algorithms and methods to economic analysis, while abundant exercises and problem sets allow them to test what they?ve learned. ?For non-mathematicians who just use math in their professional activity I believe this is a very helpful source of knowledge, and also a very efficient reference.??Elena Kustova, Saint Petersburg University Viatcheslav V. Vinogradov is a researcher at the Economics Institute of the Academy of Sciences of the Czech Republic and a consultant to the World Bank. table of contents 0.1 Basic Mathematical Notation 0.2 Methods of Mathematical Proof. 0.3 Powers, Exponents, Logs and Complex Numbers 1 Linear Algebra 1.1 Matrix Algebra. 1.2 Systems of Linear Equations 1.3 Quadratic Forms 1.4 Eigenvalues and Eigenvectors 1.5 Diagonalization and Spectral Theorems 1.6 Appendix: Vector Spaces 2.1 The Concept of Limit. 2.2 Differentiation - the Case of One Variable. 2.3 Rules of Differentiation 2.4 Maxima and Minima of a Function of One Variable 2.6 Functions of More than One Variable 2.7 Multivariate Unconstrained Optimization 2.8 The Implicit Function Theorem. 2.9 (Quasi)Concavity and (Quasi)Convexity 2.10 Appendix: Matrix Derivatives 2.11 Appendix: Topological Structure and Its Implications 2.12 Appendix: Correspondences and Fixed-Point Theorems 3 Constrained Optimization 3.1 Optimization with Equality Constraints 3.2 The Case of Inequality Constraints 3.2.1 Non-Linear Programming 3.2.2 Kuhn-Tucker Conditions. 3.3 Appendix: Linear Programming. 4.1 Differential Equations. 4.1.1 Differential Equations of the First Order 4.1.2 Qualitative Theory of First-Order Differential Equations 4.1.3 Linear Differential Equations of a Higher Order with Constant Coefficients 4.1.4 Systems of First-Order Linear Differential Equations 4.1.5 Simultaneous Differential Equations and Types of Equilibria. 4.2 Difference Equations. 4.2.1 First-Order Linear Difference Equations 4.2.2 Second-Order Linear Difference Equations 4.2.3 The General Case of Order n. 4.2.4 Systems of Simultaneous First-Order Difference Equations with Constant Coefficients 4.3 Introduction to Dynamic Optimization 4.3.1 First-Order Conditions 4.3.2 Present-Value and Current-Value Hamiltonians 4.3.3 Dynamic Problems with Inequality Constraints 5.1 Practice Problems 5.2 Solved Problems 5.2.1 Linear Algebra. 5.2.3 Constrained Optimization 5.3 Economics Applications 5.4 Written Assignments 5.5 Sample Problem Sets 5.6 Unsolved Problems 5.6.1 More Problems 5.6.2 Sample Tests To my knowledge, in modern economics most scholars use mathematics, but for the majority of them math is rather a kind of toolbox than a rigorous science they would like to advance (I am not talking here about prominent cases such as, for instance, Arrow, Debrey, Intrilligator, Kamien, or Samuelson, though). Therefore, from my point of view the book exactly serves the purpose. It gives the reader fairly precise understanding of how to apply mathematical algorithms, tools and methods in economic analysis without getting into deep details of math background. If I were asked whether this textbook would be helpful for Ph.D. students majoring in malhematics, I would probably be more skeptieal. but for non-mathematicians who just use math in their professional activity I believe this is very helpful source ofknowledge, and also a very efficient reference point. Furthermore, it was indeed a great idea to include a chapter solely dedicated to problems and solutions. Another value added is that the structure of this book clearly makes it a good text for distance learning students. Elena Kustova, Professor at the Department of Mathematics and Mechanic, Saint Peterburg University The book under review is designed as a textbook for the first-year doctoral students in economics, but at least some parts of it are equally suitable for mathematically oriented students in the MA program in economics and in other social sciences. It consists of four chapters covering virtually all areas of mathematics relevant to modern economic analysis. In addition, there is an introductory section reviewing basic mathematical prerequisites needed for entering the doctoral program, and a lengthy final section of exercises. The introductory section,-labelled as Chapter 0-starts with outlining basic mathematical notation, continuing with explaining methods of mathematical proofs, and finishing with describing powers, logs and complex numbers. The knowledge of the contents ofthis section is indispensable for anybody claiming to be mathematically literate. A shiningjewel of it is the part on mathematical proofs which is exhaustive, remarkably clear and beautifully logical. The first chapter covers linear algebra which is absolutely essential for most econometric work. The formulation of systems of linear equations in terms of matrix algebra is well explained and clearly shows the efficiency gained. Particularly commendable is the explanation of eigenvalues and eigenvectors whose abstract nature is frequently perplexing for students. The second chapter deals with basic calculus, starting with the concept of limit and ending with the notions of concavity and convexity. This is a crucial part of mathematics for economists as recognized by the leading US universities where the knowledge of calculus is a prerequisite for all undergraduates majoring in economics. The author does a very good job of covering this topic. The subject ofthe third chapter ofthe book is constrained optimization, the basic and universal problem of economics. The recognition of this forces the students to approach each economic problem by identifying the economic agents, their objective functions and their constraints. The chapter deals well with both equality and inequality constraints and gives a good exposition of linear programming in the appendix. The fourth chapter extends the realm of economic problems to dynamic situations. It starts with differential equations, goes on to difference equations, leading to dynamic optimisation which is quite a fashionable topic in modern economic research. The final section on dynamic programming is a special bonus for students and researchers interested in dynamic problems. An appendix on optimal control theory helps students comprehend its popularity initiated by Gregory Chow in the 1970s. The last section ofthe book-labelled as Chapter 5-contains several sets ofexercises. These include practice problems (with solutions), solved problems for each of the preceding four chapters, problems involving economic applications, written assignments, sample problem sets, and unsolved problems. The whole section takes up almost one half of the whole book and is not only unique for a text of this kind but also invaluable. It is extremely well done and provides a wonderful resource for students in mastering mathematics needed for serious study of economics. In my opinion the book is definitely commendable for the purpose for which it was written. The author has wisely decided to put emphasis on understanding over abstract proofing, which for economists would be more of an intellectual luxury than of practical use. This general orientation of the book makes it also a good reference text on the bookshelves of economic researchers. Jan Kmenta, Professor Emeritus of Economics, University of Michigan and Visiting Professor CERGE-EI, Prague

In this lesson, we will learn about the Pythagorean Theorem. or the Pythagorean theorem as it is sometimes called. And you might be wondering, “What is the theorem?” and “Who in this world is Pythagoras?” Now, in mathematics, theorems are only statements that have been proven to be true. from other things that are known or accepted to be true. And Pythagoras, well, he’s a very smart man. who lived a long time in ancient Greece. and he proved the theorem. Well historians are not entirely sure it was actually Pythagoras theorem who proved it. It could be one of the students or followers. but he usually gets praise for it. However, the main thing you need to know is that. Pythagoras’s theorem illustrates important geometric relationships. between the three sides of a right triangle. We will learn what the relationship is in just one minute. but first there are a few things you need to know. before you can really understand the Pythagorean Theorem or use it to solve problems. First of all, to understand the Pythagorean Theorem you need to know about angles and triangles. and you also need to know a little about exponents and square roots. So, if the topic is new to you, be sure to watch our videos about them first. And secondly, although the Pythagorean Theorem is about geometry. You need to know some basic algebra to really use it. Specifically, you have to know about variables. and how to solve basic algebraic equations involving exponents. We discussed many of these topics in the first five videos of our Basic Algebra series. Read also : Algebra Basic Okay, now you have all the background information discussed. let’s see what the Pythagorean Theorem says. The theorem can be expressed in several different ways. but what we like the most is like this: For right triangles with feet ‘a’ and ‘b’ and sloping sides ‘c’. ‘a squared’ plus ‘b squared’ is equal to ‘c squared’. As you can see from this definition, the Pythagorean Theorem does not apply to ALL triangles. ONLY applies to the RIGHT triangle. As you know, a right triangle always includes a right angle. which is usually marked with a ‘right-angle symbol’ of a square to help you identify it. And you need to know which angle is the right angle because. this helps you identify the important side of the triangle called the hypotenuse. The hypotenuse is the longest side of a right triangle. and always the ‘opposite’ side from the right angle. In other words, the side that doesn’t touch (or helps shape) the right angle itself. To use the Pythagorean Theorem, you must be able to identify the hypotenuse. because that’s what is the variable ‘c’ for the theorem. ‘c’ is the length of the hypotenuse. The other two sides of the triangle. (which touches or forms a right angle) is called “his feet”. Our Pythagorean Theorem definition uses the variable names ‘a’ and ‘b’ to represent their length. Oh and which foot is called ‘a’ and which foot is called ‘b’. as long as you keep track of which after you make your initial choice. Okay, now we know the various parts of the Pythagorean Theorem. let’s think about what relations or equations (‘a squared’ plus ‘b squared’ equal to ‘c squared’) really tell us. This tells us that if we take the lengths of both legs (sides ‘a’ and ‘b’) and their ‘square’. which means multiplying it yourself. (‘a squared’ is ‘a’ times ‘a’ and ‘b squared’ is ‘b’ times ‘b’). and then if we add the two ‘sums of squares’ together. they will EQUAL the amount that you will get if you ‘square’ the hypotenuse. (which will be ‘c squared’ or c times c). That might sound a little confusing at first. So let’s look at a specific example of a right triangle. it will help the Pythagorean theorem make more sense. This right triangle is called “3, 4, 5 triangles”. because the sides have relative lengths 3, 4, and 5. And with “relative length”, I mean that the unit of length doesn’t really matter. Sides can be expressed in ANY unit (inches, meters, miles, whatever). So a triangle can be of various sizes along its length. will have proportions of 3, 4, and 5 relative to each other. Starting with the side that is 3 units long, which we call “side a”. what do we get if we square that side ? Now in arithmetic, quadratic 3 means multiplying 3 times 3 which is equal to 9. And the geometric equation of squaring something actually produces a square shape. As you can see, this box contains 9 unit boxes. So this red area represents the value of ‘a’ which is squared in the Pythagorean Theorem. Next let’s look at the side which is 4 units in length, which we call “side b”. Squaring 4 means multiplying 4 times 4, which is 16. Again, the geometric equivalent is a literal square. i.e. 4 units on each side and cover a total area of 16 units. So this blue area represents ‘b squared’ in the Pythagorean Theorem. And finally, let’s deal with the hypotenuse, or “c side”, which is the longest side. 5 units in length. Squaring 5 means multiplying 5 times 5, which is 25. And the geometric equation is 5 x 5 square which has an area of 25 units. So this green area represents ‘c squared’ in the Pythagorean Theorem. Now you can see how the arithmetic part of the Pythagorean Theorem. related to the geometric parts of this right triangle. let’s check to see if the Pythagorean Theorem is really true (at least in this particular case). On the arithmetic side, if you add up the sum of ‘squared’ and ‘b squared’. they are really the same as ‘c squared’ because 9 + 16 = 25. And by slightly rearranging our unit box. You can see that the area of the box is formed by two legs. really the same as the area of the square formed by the hypotenuse. Wow, those ancient Greek men are really smart! Okay, but I know what you think. “It’s okay, but why should I care about the Pythagorean Theorem? What are the benefits? “. As always, that’s a good question. And the answer is, like many things in mathematics. The Pythagorean theorem is a useful tool. that can help you use what you do NOT know to find out what you do NOT know. Specifically, if you have a right triangle but you only know how long the two sides are. The Pythagorean theorem tells you how to find out the length of an unknown third side. For example, imagine that you have a right triangle. 2 cm long on this side and 3 cm long on this side. but we don’t know how long the tilt is. No problem!. The Pythagorean theorem tells us the relationship between the three sides of ANY right triangle. so we can find out. We know that ‘a squared’ plus ‘b squared’ is equal to ‘c squared’. so let’s plug what we know into the equation. and then solve it for what we don’t know. Again, it doesn’t matter which leg is called ‘a’ or ‘b’. so let’s label it like this. and then substitute ‘2’ for ‘a’ and ‘3’ for ‘b’ in the Pythagorean Theorem equation. That gives us an algebraic equation that has only one unknown, ‘c’. If we solve this equation for ‘c’ in other words. if we rearrange the equation so that ‘c’ by itself on one side of the sign is equal to. then we will know exactly what ‘c’ is. We will know the length of the side of the triangle. First, we need to simplify the left side of the equation because it contains known numbers. And according to the order of operations, we need to simplify the exponent first. ‘2 squared’ is 4 and ‘3 squared’ is 9. Then, we add the result (4 + 9 = 13). and we have equation 13 = ‘c squared’. which is equal to ‘c squared’ = 13. Then, to get ‘c’ by itself. we need to do the opposite of what was done to him. Because it is squared, the reverse operation is square root. so we need to take the square root from both sides. Taking the square root of ‘c squared’ only gives us the ‘c’ we want on the side of this equation. but that gives us a little problem on the other side. because it’s not easy to know what the square root of 13 is. This is not a perfect square so it will be decimal. and maybe irrational numbers. But that’s okay because it leaves our answer as ‘square root of 13’. Of course, you can use a calculator to get decimal values if you really need them. but in mathematics it is very common to leave square roots alone unless they are easy to simplify. So the sides of this right triangle are, 2 cm, 3 cm and the ‘square root of 13’ cm. Let’s try another example. For this right triangle, we know the length of the hypotenuse (6 m). and one foot (4 m). but the length of the other leg is unknown. So let’s use the Pythagorean Theorem to find the unknown length. As usual, we call the hypotenuse “c side”. And let’s call our feet know “side a” and our feet don’t know “side b”. Then we can replace the known values into the Pythagorean Theorem. and solve for unknown values. Replacing ‘c’ with 6 and ‘a’ with 4 gives us the equation. ‘4 squared’ plus ‘b squared’ is equal to ‘6 squared’. we need to simplify and solve for ‘b’. First let’s simplify the exponent. ‘4 squared’ is 16, and ‘6 squared’ is 36. Now we need to isolate ‘b squared’. and we can do that by subtracting 16 from both sides of the equation. On this side, (+16) and (-16) leave us only with ‘b squared’. And on the other hand we have 36 minus 16 which is 20. We can now solve this simplified equation for ‘b’. by taking the square root from both sides which gives us. ‘b’ is the same as ‘square root of 20’. Again, it’s okay to leave your answer as a square root like this. And some of you might know that ‘square root of 20’ can be simplified. to ‘2 times the square root of 5’. We will not worry about simplifying the root in this video. but if you know how to do it, great! If you don’t know, just leave the answer as ‘square root of 20’ meters. Here’s another interesting one. What if you have a ‘unit square’ cut in half along the diagonal. Each side of the square is 1 unit. but how far is the distance from one corner of the square to another along the diagonal ? Well, because the diagonal divides the square into two right triangles. we can use the Pythagorean Theorem to tell us the unknown distance. We label the right triangle feet ‘a’ and ‘b’ and the hypotenuse of ‘c’. And because we know that ‘a’ and ‘b’ are equal to 1, we can attach those values. into the equation of the Pythagorean Theorem that gives us us. ‘1 squared’ plus ‘1 squared’ is equal to ‘c squared’. Now we are done with ‘c’. ‘1 squared’ is only 1. so the left side of this equation simplifies to 1 plus 1 which is only 2. That means ‘c squared’ is equal to 2. and if we take the square root from both sides. we get ‘c’ equal to ‘square root of 2’. So that’s how far it crosses the diagonal of the unit square. Okay, so that’s how you use the Pythagorean Theorem. to find the unknown side length of a right triangle, which is the most common use. But there are other ways you can use the Pythagorean Theorem that I want to mention. You can also use the Pythagorean Theorem to TEST triangle. to see if it’s really the RIGHT triangle. Yes, know if you are not sure. For example, what if someone shows you this triangle and asks. “Is this a right triangle?” Yeah, that looks a lot like a right triangle. but does not have a right-angle symbol. and it will be difficult to know whether this angle is exactly 90 degrees just by looking at it. Maybe close to 90, like 89.5 degrees. Don’t worry, the Pythagorean Theorem can tell us for sure. if we know the length of all three sides of the triangle. If we know the length (a, b and c). then we can include them in the Pythagorean Theorem equation to see if that is true. In this special case. because the two shorter sides are 3 cm each and the longest side is 4 cm. we can attach the values to ‘a’, ‘b’ and ‘c’ and simplify to see what we get. ‘3 squared’ is 9, so on this side of the equation we get 9 plus 9 which is 18. And on the other side we have ‘4 squared’, which is 16. Uh oh that’s not right! Our equation is simplified to 18 equals 16. What is clear is NOT a true statement. That means that all three sides of this triangle do not function in the Pythagorean Theorem. they do not match the relation ‘a squared’ plus ‘b squared’ equal to ‘c squared’. And because the Pythagorean Theorem tells us that ALL right triangles fit that relationship. this triangle can’t be the RIGHT triangle. Good. so now you know what the Pythagorean Theorem is and you know how to use it. You can use it to find the missing side of the right triangle. and you can also use it to test a triangle to see if it qualifies as a right triangle. But as you see. many other mathematical skills are needed to be able to use the Pythagorean Theorem effectively. so you might need to brush up on some of those skills before you are ready to try using them yourself. And remember, you can’t master math just by watching videos about it. You really need to practice solving real math problems. Pythagoras Conference Global Evidence Pythagoras signs is frequently called the definition of Pythagoras theorems (Pitagoras). Because as the Elementary school when learning math, we don’t overlook to examine phytagoras this sentence, Pythagoras has to have already been no stranger inside our ears. The phytagoras formula is just actually a formula utilized by way of a scientist. The significance of Pythagoras or even pythagoras theorem reads: The left-hand or the side at the elbow’s triangle is corresponding to the flip hand squared. The triangle above would be just an angled triangle, which includes one side vertical (BC), 1 side (AB), and also yet one side tilts (AC). Evidence pythagoras conference global and also the Pythagoras formula functions to locate 1 side with both sides that are known. Pythagoras theorem calculator - Pythagoras formulation in origin shape - In the event the mirror C - The vertical and forthcoming sides are b and A The consequent Pitagoras formulation: b² = a² + c² Then to calculate the upright side and the upcoming side apply formula: a² = b² – c² c² = b² – a² Pythagoras formula in root form - If the mirror side C - The upright and upcoming sides are A and b Significant NOTE: The Pythagoras formula, just valid on the elbows only (Pythagoras formula in square roots). From the signs of this Pythagorean theorem, There’s a blueprint of amounts that Will Need to be recalled to solve the Challenge of Pythagoras is likely to likely probably be simpler and faster to operate on this, the routine is: - 3 – 4 – 5 - 5 – 12 – 13 - 6 – 8 – 10 - 7 – 24 – 25 - 8 – 15 – 17 - 9 – 12 – 15 - 10 – 24 – 26 - 12 – 16 – 20 - 14 – 48 – 50 - 15 – 20 – 25 - 15 – 36 – 39 - 16 – 30 – 34 To know more information about the signs of Phytagoras, subsequently think about the next instance Cases of pythagoras theorem and conversation A concentric elbow comes with a vertical side (a b ), which is 1-5 cm long, and also alongside, it is (BC ) 8 cm, what’s cm kah side of this Mirror (AC)? A B = 1-5 B C = 8 Said: AC span…??? ⇒ First Way AC² = AB² + BC² AC² = 15² + 8² AC² = 225 + 64 AC² = 289 AC = √289 AC = 17 ⇒ Second way AC = √ AB² + BC² AC = √ 15² + 8² AC = √ 255 + 64 AC = √ 289 AC = 17 So, AC length is 17 cm Just how long is the amount of the borders of an elbow-elbow if it’s called this period of the mirror 1 3 cm and also the data side 5 cm? As an instance: c Tilt side-by-side Flat side, an upright side Not Known: c = 1 3 cm, B = 5 cm Said: a …???? ⇒ First Way a² = c² – b² a² = 13² – 5² a² = 169 – 25 a² = 144 a = √ 144 a = 12 ⇒ Second Way a = √ c² – b² a = √ 13² – 5² a = √ 169 – 25 a = √ 144 a = 12 So, the length of the upright triangle is 12 cm There’s a triangle of ABC, elbows in B. In case the span is a b = 16 cm and BC = 30, then what’s the amount of the side of this triangle (AC)? A B = 16 B C = 30 Said: AC =…? AC = √ AB² + BC² AC = √ 16² + 30² AC = √ 256 + 900 AC = √ 1156 AC = 34 So, AC length = 34 cm Notice the picture below, learn the ABC elbow has a vertical side significance of 6 cm and a bottom side of 8 cm, then calculate just how long would be your medial side of this mirror? A B = 8 cm B C = 6 cm Length: Period of AC (angled side of this elbow-elbow above)…? AC² = AB² + BC² AC² = 8² + 6² AC² = 64 + 36 AC² = 100 AC = √100 AC = 10 These are a few situations of pythagoras calculator and also their talk and responses. To understand you, please perform a little exercise about learning Phytagoras below. 1. There’s just actually a triangle PQR X Y Z called the sides, which can be Y, X, and Z. In the subsequent announcement, the stark reality is…? - A. if y² = x² + z² , < X = 90º - B. if z² = y² – x² , < Z = 90º - C. if z² = x² – y² , < Y = 90º - D. if x² = y² + z² , < X = 90º 2. Not known triangle PQR has elbow at Q, where P Q = 8 cm, PR = 17 cm. Therefore, along this QR will be…? - A. 9 cm - B. 15 cm - C. 25 cm - D. 68 cm 3. There’s a squat triangle, its hypotensive 4 √3 cm along with a single facet of this elbows is 2 √2 cm. The length of time can be just the rear side of the elbow… Cm A. 2 √10 B. 3 √5 C. 8 √2 D. 3 √3 4. The period of this Hepotenusa Tri-angle in which the leg is the span that is elbow and also 16 cm is x 5. Figure out the price of X …. Cm A. 4 √2 C. 8 √2 D. 8 √3 Therefore the reason for this sign pythagoras conference global, ideally helpful and certainly will assist in learning math, which frequently makes us all around. Once the initial when we study, then all of the problematic things will probably be more comfortable. The gist of this Pythagoras formula could be that your angled side adds up to the vertical side on both sides and a horizontal border (however, remember to get acquainted).

Try our Free Online Math Solver! Review of Intermediate Algebra Text/workbook for low level and English as a second language students. Uses applications of algebra to the real world throughout. Annotation copyright Book News, Inc. Portland, Or. --This text refers to an out of print or unavailable edition of this title. Review of Imtermediate Algebra Reviewed by a reader, from Milpitas, This book is recommended for any guy looking to be on top of the world of math. The book clearly and completely shows how to go about solving problems. Every problem in the book is solved in the back, step-by-step, not just giving the answers.also, it also covers a wide array of topics. Review of Interactive Math for Introductory Algebra Key Benefit: Interactive Mathematics is an innovative new learning system that covers the full series of developmental mathematics in an interactive, multimedia environment. Interactive Math uses animation, video, audio, graphics, and math tools to support multiple learning styles. It is a program complete with instruction, practice, applications, and assessment. Key Topics: Introduction — A brief teaching statement directs the reader's attention toward mastering a particular skill. A visual representation of the skill is also presented. Read — Book with accompanying audio (which may be disabled) addresses the needs of those who feel more comfortable reading about the concepts and skills before exploring or working problems. Watch — Visual learners can watch and listen as example problems are worked out clearly and completely by the author in a brief on-screen video. Explore — This feature offers the most interactive learning experience because one can master the skills and learn the concept through discovery. Reviewed by nancyeve, from Orange County, CA United States First, just to clarify, access to the on-line portion of this workbook/CD is through the systems at Prentice Hall, via the educational institution.The only reason I did not give this 5 stars is because they are still working out some of the software bugs. A number of my classmates had problems running this program on their computers at home, though I had virtually none. The few problems I did have were just the sometimes slow response times moving from the problem sets to the syllabus.Basically, I loved using this program! The many options for learning on the computer alone helped me to fly though this class. (As of 11/01 I am still in class but well on my way to receiving an "A"). If I get stuck on a concept, there are many options for working through it from being shown a brief video, reading about it, or just being taken step by step through a problem on which I am working. One finds out immediately if their answer to a problem is right or wrong. If wrong, there are several different ways of finding out why, and actually being taken through THAT PROBLEM step by step in order to assist in understanding the concept so that the next problem is done with greater understandng and ease. All in all, I found the learning process enhanced using this program. It was and is an extremely valuable educational tool for me. We will, I am told, be using a different program next year (Algebra II), I think I will miss this one! Reviewed by Joi Cardinal, from northern I've never done well at math in high school or college, but now that I'm re-entering college in my 40s, I really have to learn it in order to succeed in biology and chemistry courses. Thank heavens my local community college has adopted K. Elayn Martin-Gay's phenomenal Interactive Math program as an online course. The package consists of easy to install and very easy to use software on a CD-ROM which communicates with Prentice Hall's computer, a paperback textbook which contains all the teaching material that is in the software as well as additional problem sets and chapter tests for away-from-computer use, and a detailed student handbook explaining the program and its use as well as offering many valuable study skill tips. The software is broken down by the chapter subheadings with each discrete topic covered having its own section. Each section consists of an introduction screen, then three different options for learning that suit different learning styles. If one is most comfortable with reading a traditional textbook, then the concept can be learned that way. People who learn best by watching and listening to an instructor can watch short video clips of Martin-Gay explaining each concept. Those who are most comfortable with experimenting with numbers will enjoy the explore option which provides tools for playing with numbers and deducing the concepts through the patterns of responses that emerge. Finally, each section ends with a practice set of problems to check understanding, then an assessment set of problems, the results of which are then posted to the grade book which is accessed by the professor for the class. Each chapter ends with a more lengthy test which is also posted to the gradebook. I never thought I'd enjoy math so much that I'd look forward to studying it, but this program has made all the difference. Instead feeling totally lost and only able to solve problems that are like the examples given in the textbook, I'm now perceiving math to be fun puzzles that are satisfying to work out. I sincerely hope the author goes on to create additional programs for more advanced math work in time for me to use them! Review of Algebra: Tools for a Changing World Reviewed by titasusan, from Fort Washington, Maryland United States This book is great for practice but practice only. The methods on how to solve the problems are not explained.This book is for practice and not to be confused as the textbook. Review of Introductory Algebra The second volume of a three-book series designed for students with introductory algebra as a prerequisite of their course. This edition emphasizes real-world applications of algebra, and includes geometric applications and exercise and problem sets. Review of Experiencing Intermediate Algebra A text for a one-semester course in intermediate algebra, designed to help students model real-world situations, reason mathematically, choose appropriate problem-solving methods, connect algebra to other disciplines, and communicate mathematics. Concepts are developed using numeric, graphic, algebraic, and verbal approaches. Numeric presentation emphasizes tables of values, either constructed manually or by using a calculator. Assumes students have a TI-83 graphic calculator available. The authors are affiliated with Pellissippi State Technical Community College. -- Copyright © 2000 Book News, Inc., Portland, OR All rights reserved Review of Intermediate Algebra for College Students (5th Edition) Angel's text is one that students can read, understand, and enjoy. With short sentences, clear explanations, many detailed worked examples, and outstanding pedagogy. Practical applications of algebra throughout make the subject more appealing and relevant for students. Key pedagogical features include: Preview and Perspective at the beginning of each chapter; Helpful Hints; Group Activities/Challenge Problems, Writing, exercises, Real-Life Application Problems; and Calculator and Graphing Calculator Corners. --This text refers to the Hardcover edition. Reviewed by a reader, from NC United I used this book to challenge my Intermediate Algebra course, and I passed the exams. This book explain clearly the steps to the questions. Review of Intermediate Algebra Concepts and Applications with Algebra for College Students Sticker Package Preface For the past half century many introductory differential equations courses for science and engineering students have emphasized the formal solution of standard types of differential equations using a (seeming) grab-bag of mechanical solution techniques. The evolution of the present text is based on experience teaching a new course with a greater emphasis on conceptual ideas and the use of computer lab projects to involve students in more intense and sustained problem-solving experiences. Both the conceptual and the computational aspects of such a course depend heavily on the perspective and techniques of linear algebra. Consequently, the study of differential equations and linear algebra in tandem reinforces the learning of both subjects. In this book we have therefore combined core topics in elementary differential equations with those concepts and methods of elementary linear algebra that are needed for a contemporary introduction to differential equations. The availability of technical computing environments like Maple, Mathematica, and MATLAB is reshaping the current role and applications of differential equations in science and engineering, and has shaped our approach in this text. New technology motivates a shift in emphasis from traditional manual methods to both qualitative and computer-based methods thatrender accessible a wider range of more realistic applications; permit the use of both numerical computation and graphical visualization to develop greater conceptual understanding; and encourage empirical investigations that involve deeper thought and analysis than standard textbook problems. Major Features The following features of this text are intended to support a contemporary differential equations course with linear algebra that augments traditional core skills with conceptual perspectives:The organization of the book emphasizes linear systems of algebraic and differential equations. Chapter 3 introduces matrices and determinants as needed for concrete computational purposes. Chapter 4 introduces vector spaces in preparation for understanding (in Chapter 5) the solution set of an nth order homogeneous linear differential equation as an n-dimensional vector space of functions, and for realizing that finding a general solution of the equation amounts to finding a basis for its solution space. (Students who proceed to a subsequent course in abstract linear algebra may benefit especially from this concrete prior experience with vector spaces.) Chapter 6 introduces eigenvalues and eigenvectors in preparation for solving linear systems of differential equations in Chapters 7 and 8. In Chapter 8 we may go a bit further than usual with the computation of matrix exponentials. These linear tools are applied to the analysis of nonlinear systems and phenomena in Chapter 9. We have trimmed the coverage of certain seldom-used topics and added new topics in order to place throughout a greater emphasis on core techniques as well as qualitative aspects of the subject associated with direction fields, solution curves, phase plane portraits, and dynamical systems. To this end we combine symbolic, graphic, and numeric solution methods wherever it seems advantageous. A healthy computational flavor should be evident in figures, examples, problems, and projects throughout the text. Discussions and examples of the mathematical modeling of real-world phenomena appear throughout the book. Students learn through modeling and empirical investigation to balance the questions of what equation to formulate, how to solve it, and whether a solution will yield useful information. Students also need to understand the role of existence and uniqueness theorems in the subject. While it may not be feasible to include proofs of these fundamental theorems along the way in a elementary course, students need to see precise and clear-cut statements of these theorems. We include appropriate existence and uniqueness proofs in the appendices, and occasionally refer to them in the main body of the text. Computer methods for the solution of differential equations and linear systems of equations are now common, but we continue to believe that students need to learn certain analytical methods of solution (as in Chapters 1 and 5). One reason is that effective and reliable use of numerical methods often requires preliminary analysis using standard elementary techniques; the construction of a realistic numerical model often is based on the study of a simpler analytical model. We therefore continue to stress the mastery of traditional solution techniques (especially through the inclusion of extensive problem sets). Computational Flavor The following features highlight the computational flavor that distinguishes much of our exposition. About 250 computer-generated graphics—over half of them new for this version of the text, and most constructed using MATLAB—show students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life. Over 30 computing projects follow key sections throughout the text. These "technology neutral" project sections illustrate the use of computer algebra systems like Maple, Mathematica, and MATLAB, and seek to actively engage students in the application of new technology. A fresh numerical emphasis that is afforded by the early introduction of numerical solution techniques in Chapter 2 (on mathematical models and numerical methods). Here and in Section 7.6, where numerical techniques for systems are treated, a concrete and tangible flavor is achieved by the inclusion of numerical algorithms presented in parallel fashion for systems ranging from graphing calculators to MATLAB. A conceptual perspective shaped by the availability of computational aids, which permits a leaner and more streamlined coverage of certain traditional manual topics (like exact equations and variation of parameters) in Chapters 1 and 5. Applications To sample the range of applications in this text, take a look at the following questions:What explains the commonly observed time lag between indoor and outdoor daily temperature oscillations? (Section 1.5) What makes the difference between doomsday and extinction in alligator populations? (Section 2.1) How do a unicycle and a two-axle car react differently to road bumps? (Sections 5.6 and 7.4) Why might an earthquake demolish one building and leave standing the one next door? (Section 7.4) Why might an eartquake demolish one building and leave standing the one next door? (Section 7.4) How can you predict the time of next perihelion passage of a newly observed comet? (Section 7.6) What determines whether two species will live harmoniously together, or whether competition will result in the extinction of one of them and the survival of the other? (Section 9.3) Organization and Content The organization and content of the book may be outlined as follows:After a precis of first-order equations in Chapter 1—with a somewhat stream-lined coverage of certain traditional symbolic methods—Chapter 2 offers an early introduction to mathematical modeling, stability and qualitative properties of differential equations, and numerical methods. This is a combination of topics that ordinarily are dispersed later in an introductory course. Chapters 3 (Linear Systems and Matrices), 4 (Vector Spaces), and 6 (Eigenvalues and Eigenvectors) provide concrete and self-contained coverage of the elementary linear algebra concepts and techniques that are needed for the solution of linear differential equations and systems. Chapter 6 concludes with applications of diagonizable matrices and a proof of the Cayley-Hamilton theorem for such matrices. Chapter 5 exploits the linear algebra of Chapters 3 and 4 to present efficiently the theory and solution of single linear differential equations. Chapter 7 is based on the eigenvalue approach to linear systems, and includes (in Section 7.5) the Jordan normal form for matrices and its application to the general Cayley-Hamilton theorem. This chapter includes an unusual number of applications (ranging from railway cars to earthquakes) of the various cases of the eigenvalue method, and concludes in Section 7.6 with numerical methods for systems. Chapter 8 is devoted to matrix exponentials with applications to linear systems of differential equations. The spectral decomposition method of Section 8.3 offers students an especially concrete approach to the computation of matrix exponentials. Our treatment of this material owes much to advice and course notes provided by Professor Dar-Veig Ho of the Georgia Institute of Technology. Chapter 9 exploits linear methods for the investigation of nonlinear systems and phenomena, and ranges from phase plane analysis to applications involving ecological and mechanical systems. Chapters 10 treats Laplace transform methods for the solution of constant-coefficient linear differential equations with a goal of handling the piecewise continuous and periodic forcing functions that are common in engineering applica Review of HBJ Algebra 1 Reviewed by Tom, from Boston Excellent no nonsense book BUT I did find a couple errors that will drive students nuts...page 303 top, 3 not right.page 150 and 151, numbers 22 and 38 can not be solved. I actually got the author on the phone who said "the answer is the empty set" which in my opinion just means that the author goofed and made problems that couldn't be solved.This far exceeds all other algebra books I've seen. Old but still excellent.

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In the beginning of this school year, I wrote a post called: In Math Class: To Write is Right. It discusses some thoughts on how we use writing to express ourselves, and how we use the written word in today’s world. I also promised to follow-up with some strategies that I have tried, and want to try, in order to help my students practice writing. Here is that follow-up: In class, I have incorporated writing in a few small activities. In the beginning of the school year, I asked each student to write down goals for math class for the upcoming year. Some responses included: “To get better at math.” “I want to not fail and work harder.” “Have fun and enjoy math! 🙂 ” Responses tended to be shorter, and more phrase-like, especially from my Freshmen classes. Students wrote these responses on Post-It notes, and we displayed them on large posters in the room. This way, students are always able to refer back to their original goals. It holds a certain accountability when they can see their own words written in their own handwriting. Phrases, however, aren’t enough. I want students to gain a level of sophistication in how they express themselves in math class (appropriate for each individual, of course). So, I have begun to ask more thought-provoking questions about our work throughout the year, asking students to write down their answers using complete sentences. Furthermore, I have tried to make my questions more specific, rather than vague. I have come to realize that specific questions elicit more specific answers, while vaguer questions elicit more vague answers. Most recently, I asked my Algebra students: “If you check your solution to a system of linear equations, why does the solution work in both equations?” Students know to substitute their coordinate point in each equation to see if it “works,” but I really wanted them to understand why. It also helps bridge the gap between the graphical representation and the algebraic representation of a system of linear equations. Here are some responses: “The x value and the y value work in both equations because they are the same in both equations.” “Because there are three different methods of doing this type of question, and you need to check your work to see if you get this equation/problem right.” “The solution is the point of intersection of the two lines.” “It is the same equation written differently.” “It’s linear, so they have the same slope.” “Both equations share the same x value and y value.” “They both have a point in which they cross.” While some responses are a bit more reasonable than others, I was pleased to see students using decent vocabulary. They were also writing slightly more complex sentences than their phrases from the beginning of the year. It’s clear they have some sense of what the answer means to my question, even if their answers need more fine-tuning. In my Algebra 2 class, I asked the following question: “Which should always have a higher value: sine or cosecant of the same angle? Why?” Here are some responses: “Cosecant, because the hypotenuse is the longest side of the triangle and for cosecant the hypotenuse is on the top.” “The cosecant should be higher because it is the flipped of sine.” “Cosecant has to be higher, because the hypotenuse is the longest side of the triangle.” As we can see, they were trying to express the concept that cosecant is the reciprocal of sine, which brings the hypotenuse, the longest side of the right triangle, to the numerator. They used some good vocabulary, while other vocabulary could be expanded upon (i.e. use “reciprocal” instead of “flipped,” or “numerator” instead of “the top”), but they were definitely trying to describe the differences between sine and cosecant. As the Math Research director at our school, I advise students throughout the year in researching a topic in advanced mathematics. These students also write a 10-15 page research paper and prepare a presentation for a local competition involving dozens of school districts and over 400 students. This is no easy task. These students struggle to begin their papers, because they have found that they have never written an essay about math before. It also takes some practice interweaving equations and mathematical concepts throughout the paper, without sounding like a textbook. I have worked, and continue to work, to help my students use detailed and clear explanations, while using proper vocabulary to describe the mathematics that they are researching. The trend between my classes and my math research students is that students feel more comfortable, and are more adept at, writing about math in a straight-forward, fact generating way, rather than truly analyzing the concepts. We could all use a little more practice in this area. Students can definitely learn to write more analytically about mathematics, and teachers can also learn to assign specific writing prompts in order to elicit detailed, more complex responses. I hope to continue to practice helping students become better, more analytical writers in the field of mathematics for years to come. One thought on “To Write Is Right: A Follow-Up”

|Theme||Visible||Selectable||Appearance||Zoom Range (now: 0)| Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures. Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. Fig 1 illustrates a flow-after-flow test. A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream or tubing pressure only. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the tubing pressure. Proper sizing and selection of an electrical submersible pump (ESP) system is essential to efficient and cost-effective performance. Selection and sizing of proper ESP equipment for a particular application should be based on a nine-step design procedure. This nine-step procedure helps the engineer design the appropriate submersible pumping system for a particular well. Each of the nine steps is explained below, including gas calculations and variable-speed operations. Specific examples are worked through in ESP design. The design of a submersible pumping unit, under most conditions, is not a difficult task, especially if reliable data are available. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation. In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. Energy is the rate of doing work. A practical aspect of energy is that it can be transmitted or transformed from one form to another (e.g., from an electrical form to a mechanical form by a motor). A loss of energy always occurs during transformation or transmission. In drilling fluids, energy is called hydraulic energy or commonly hydraulic horsepower. Rig pumps are the source of hydraulic energy carried by drilling fluids. One of the first mathematical tools a neophyte engineer learns is calculus. Many of the mathematical tools engineers use to evaluate and predict behavior, such as vibrations, require equations that have continuously varying terms. Often, there are many terms regarding the rate of change, or the rate of change of the rate of change, and so forth, with respect to some basis. For example, a velocity is the rate of change of distance with respect to time. Acceleration is the rate of change of the velocity, which makes it the rate of change of the rate of change of distance with respect to time. To quantify formation damage and understand its impact on hydrocarbon production, one must have reasonable estimates of the flow efficiency or skin factor. Several methods have been proposed to evaluate these quantities for oil and gas wells. Multirate tests can be conducted on both oil and gas wells. In these tests, several stabilized flow rates, qi, are achieved at corresponding stabilized flowing bottomhole pressures, pwf. The simplest analysis considers two different stabilized rates and pressures. This article summarizes the fundamental gas-flow equations, both theoretical and empirical, used to analyze deliverability tests in terms of pseudopressure. The four most common types of gas-well deliverability tests are discussed in separate articles: flow-after-flow, single-point, isochronal, and modified isochronal tests. Deliverability testing refers to the testing of a gas well to measure its production capabilities under specific conditions of reservoir and bottomhole flowing pressures (BHFPs). A common productivity indicator obtained from these tests is the absolute open flow (AOF) potential. The AOF is the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the sandface. The Laplace transform of the diffusion equation in radial coordinates yields a modified Bessel's equation, and its solutions are obtained in terms of modified Bessel functions. This page introduces Bessel functions and discusses some of their properties to the extent that they are encountered in the solutions of more common petroleum engineering problems. A solution of Bessel's equation of order v is called a Bessel function of order v. Of particular interest is the case in which λ ki so that Eq. 2 becomes Eq. 3 is called the modified Bessel's equation of order v. A solution of the modified Bessel's equation of order v is called a modified Bessel function of order v.

Start studying Calculus 1 Final Exam Study Guide. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Final Exam Study Guide for Calculus III Vector Algebra 1. The length of a vector and the relationship to distances between points 2. Addition, subtraction, and scalar multiplication of vectors, together with the geometric interpretations of these operations 3. Basic properties of vector operations (p.774) 4. B Veitch Calculus 2 Study Guide This study guide is in no way exhaustive. As stated in class, any type of question from class, quizzes, exams, and homeworks are fair game. Study Guide for the Advanced Placement Calculus AB Examination By Elaine Cheong. 1 Table of Contents INTRODUCTION 2 TOPICS TO STUDY 3 • Elementary Functions 3 • Limits 5 • Differential Calculus 7 • Integral Calculus 12 SOME USEFUL FORMULAS 16 CALCULATOR TIPS AND PROGRAMS 17 ... Advanced Placement Calculus AB Exam tests students on ... CLEP Precalculus: Study Guide & Test Prep Final Free Practice Test Instructions Choose your answer to the question and click 'Continue' to see how you did. Then click 'Next Question' to answer the ... Calculus 2 Final Exam Review study guide by michael_owens3 includes 55 questions covering vocabulary, terms and more. Quizlet flashcards, activities and games help you improve your grades. 12/14/2013 · This video is about the correction of questions that were put in college Calculus 1 Final exam. The solution of each question was explained in detail and accompanied with the concept. Final Exam Study Guide for Calculus II The nal exam will be a 2.5 hour CUMULATIVE exam, and you are re-sponsible for everything that we have covered this term. There will certainly be an emphasis on power series, Taylor series, and analytic functions, so make sure to focus on those topics. As a study aid, I have listed below the major 5/11/2015 · Solutions to a previous final exam for a multivariable calculus course. Download exam at: https://drive.google.com/open?id=0BzoZ-FzkrMLdRFRiV28yY3NDY28 Downl... 1 GPS Pre-Calculus Final Exam Study Guide Unit 3 Rational Functions 1. Give the degree, the number of real zeros, the number of (nonreal) complex zeros, for this polynomial function. Test and improve your knowledge of CLEP Calculus: Study Guide & Test Prep with fun multiple choice exams you can take online with Study.com math. We make the study of numbers easy as 1,2,3. From basic equations to advanced calculus, we explain mathematical concepts and help you ace your next test. Our study guides are available online and in book form at barnesandnoble.com. B Veitch Business Calculus Final Exam Study Guide This study guide is in no way exhaustive. As stated in class, any type of question from class, quizzes, exams, and homeworks are fair game. There’s no information here about the word problems. 1. Some Algebra Review (a) Factoring and Solving i. Quadratic Formula: ax2 + bx+ c = 0 x = b p b2 4ac ... Calculus II, Study Guide for Final Exam Page 3 Difference quotients, definition of the derivative: You should be able to set up a difference quotient, and simplify it to the point where there is no longer an “ ” in the denominator. This is the first step in proving the various short-cut rules for computing derivatives. View Test Prep - Final Exam Study Guide on Calculus 1 from MATH 180 at University of Illinois, Chicago. Math 180, Final Exam, Study Guide Problem 1 Solution 1. Dierentiate with respect to x. Write Calculus - Semester 1 Final Exam Study Guide Questions - Fall 2010-11 Name_____ Solve each problem completely on your notebook paper. Record your answers on this sheet. Find domain and range of the function. 1)f(x) = 9 - x Determine if the function is even, odd, or neither. 2)f(x) = -2x5 - 3x3 Review AP Calculus AB by watching and listening to over 11 hours worth of videos carefully coordinated to the AP Calculus AB syllabus. 182 AP Calculus AB practice questions Test your understanding of each concept without having to take an entire AP Calculus AB practice exam. Calculus I | Final Study Guide Exam Date: Friday June 30, 9:00-10:35 The following is a list of topics that we will have covered over the second three weeks of the course. It is not necessarily a complete list, but I believe it does hit all of the points that I stressed most in class. Accompanying each topic on the list are the sections of the ... AP Calculus – Final Review Sheet When you see the words …. This is what you think of doing 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator 2. Show that f() x is even Show that (−)= ( ) symmetric to y-axis 3. 1/9/2018 · I don’t take calculus but I can give you the most basic study tips. 1. organized all of the notes you have taken in class in a specific order ( easiest to hardest, first to last etc) 2. Take a blank sheet and write down everything you remember lea... Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018 PART 1: Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer

Time Dilation and Quasars In May of 2001 Hawkins published a paper called “Time Dilation and Quasar Variability”. Part of the Abstract reads as follows. “We find that the timescale of quasar variation does not increase with redshift as required by time dilation. Possible explanations of this result all conflict with widely held consensus in the scientific community.” http://xxx.lanl.gov/abs/astro-ph/0105073 The conflict arises since this indicates that space-time is not expanding. This is contrary to the evidence of type 1a super novas that confirms the time dilation effect due to the expansion of space. Initially this topic was posted by Dunash on this BB on January 10, 2002, but there was no follow up discussion of his posting. http://www.badastronomy.com/phpBB/viewtopic.php? I am appreciative for dgruss23 bringing up the paper in the course of a poll discussion called “Is the expansion of space-time accelerating or decelerating?”. http://www.badastronomy.com/phpBB/vi...2&start=50 (Page 3) I believe reference to this paper may also have been found in a discussion about the Red Shift but I could not find it but I think I remember reading it there. Hopefully someone will provide additional links to preserve the reference value of this BB. I thought that a more through discussion of this topic is in order on its own since it provides evidence that something is wrong with current cosmological models. I will attempt a “layman’s” description of the report. Hopefully someone with more expertise will provide a more explicit description. Time dilation generally refers to an increase in the observed time a physical process occurs. There are at least two possible physical interpretations or descriptions for time dilation. The most common is the application of special relativity. Time progresses comparatively slower for a moving object, so an object observed in the past with a high velocity (indicated by red shift) will have physical processes occur at a slower rate. The decay of a muon entering the earth’s atmosphere is a classic example of how a physical process is slowed when an object is moving at near light speed velocities. The time scale of rapidly moving objects can be described by how long a physical process takes to occur, as predicted by special relativity. Specifically the time scale, Ts, can be described by the red shift proportion z as follows. Ts =Tm/Tl =1+z. Tm = interval of time moving, Tl = time interval of time local or “at rest”, z = ratio of wavelength. The other physical interpretation is that the expansion of space-time itself results in a time dilation. Lets say that we are at a bowling alley and we roll two balls down the ally separated by 1 second of time. The distance between the two balls remains essentially constant while traveling down the alley. (Ignoring friction effects). The two balls will arrive at the end of the ally one second apart. Now lets throw the two balls again with a 1 second separation, but this time the bowling ally is “stretched” while the balls roll down the alley. This will physically increase the distance between the two balls. For example, Instead of the balls being 2 meters apart, they can end up being 4 meters apart. When the balls reach the end of the alley, in this example, the separation in time for when they reach the end will now be 2 seconds. (Ignoring the effect of the expansion on the velocity and energy of the balls, at least for this posting since the possible variance in the speed of light and the loss of energy of a photon (instead of a bowling ball), with the expansion of space-time is a whole other issue). I prefer this explanation of the cosmological red shift since it keeps galaxies “at rest” locally, allowing them to be carried by the expansion of space-time. Regardless of the model, the basic general effect of time dilation will be the same. The time dilation will be Td = 1+z. A process that took 1 second to occur in a “rest” frame, will take 2 seconds to occur as measured by an observer if the red shift of the observed object producing the effect has a cosmological red shift of 1. I am sure some will provide a better explanation of time dilation, and different interpretations, but I hope it gives the reader a general idea. (In the application of my uniform expansion hypothesis (www.uniformexpansion.com) both special relativity and expansion result in time dilation, but one of the effects is unobservable due to the specific geometric rate the expansion occurs. This would alter the assumed distance of 1asn’s and the assumed “acceleration” (really deceleration) indicated by such. It also addresses the issue involved with no observed time dilation effects noticed in the variation of energy output of quasars. This is merely an aside for now. It is hoped that the postings of others will provide additional explanations and perspectives. ) The time variance of Quasars The time variance of quasars, while not described in the Hawkins paper, is based upon observed variation in the energy output from quasars. It is the extreme variance of energy output of quasars in short periods of time that has helped determine the size of quasars. Quasars put out about 1,000 times the amount of energy of an entire galaxy, in a region of space 100,000 times smaller. Of course this is based upon the assumption that the cosmological red shift correlates not only to a velocity measure describing the expansion of space but to a distance measure. (v = Ho x D and v causes the red shift). (Some will take issue with this assumption arguing that quasars are much closer, “tired light proponents”). I regret not being able to find a link with a graph illustrating the time variance of the energy output of quasars. I will try to explain verbally a graph of quasar 3c 279, which is in one of my texts. One of the most dramatic peak cycles of energy output shows that the increase in luminosity varies by a magnitude of 7 over a period of about 1200 days (rise, peak to fall) . There are a number of smaller cycles (rise peak and fall), with a variation of magnitude 2 over about 800 days. Amongst this variation there are additional variations in magnitude of about 1 or perhaps a bit more times over the passage of a just a 50 or so days. There is also some variation with a magnitude of 1 over periods of only a few days. A very “noisy” graph. While there is great variation in the cycles of energy output from quasars, there is a discernable pattern. Large energy peaks last longer than short energy peaks. Large peaks tend to last a thousand days, etc. Mathematically, it is possible to extract frequency relationships utilizing a Fourier based transformation with what is called a power spectrum analysis. It allows a statistical manipulation of cyclic processes with a “noise” component. It works best if even numbers of cycles are in the mix, but if there are sufficient numbers of cycles within the analysis, this restraint is not that critical. Categorizing cycle events helps in the statistical evaluation, “large” energy output events last over 1000 days, etc. The anticipated result It was anticipated that the further away a quasar was observed, as indicated by the red shift, the greater the time dilation of the cycles observed in the energy output of quasars. The increase in the period of the cycles should correspond to an increasing red shift. Specifically it was anticipated that the cycle length should vary by 1+z. For example, the period of “averaged” cycles should be two times greater than another quasar if the red shift for one quasar has a z of 1 while the other quasar had a z of 3. No such effect was observed. This is opposite to the results found with type 1a supernovas. It is assumed that Type 1a supernovas are always the result of a supernova explosion with a white dwarf star with a mass of about 1.44 masses involved. (Baring the variation induced by rotational effects of the two stars involved and the mass of the sister star losing mass to the white dwarf star). (This also assumes that high red shift 1asn’s are the same as “local”, which is an assumption I have issues with). Since the mass involved in the supernova is assumed to be the same, the duration of the event should be generally the same. Time dilation should increase the observed duration of the 1asn by a factor of (1 +z). This time dilation is observed in that the light curves of high red shift supernovas; the “explosion” takes longer to occur the greater the red shift. (Generally). How can one process associated with Supernovas indicate time dilation associated with red shift, while another process associated with quasars indicate no time dilation associated with red shift?

[Most of the quotes in this piece are taken from Lee Smolin's Time Reborn: From the Crisis in Physics to the Future of the Universe, which was published in 2013.] This piece focuses on Lee Smolin's position on what he takes to be Platonism in (mathematical) physics. Smolin's words are also used as a springboard for discussing other issues and positions (including my own) within this general debate. Firstly, Platonism in physics is tackled as it was explicitly stated by the physicists John Wheeler and Stephen Hawking. Max Tegmark (as a Platonist) is also featured. The position advanced by Tegmark is that mathematics can perfectly describe the world/reality because the world/reality is itself mathematical. Wheeler and Hawking argued against such a position (or at least they appeared to). Then there's a section on a position best described as “the-map-is-not-the-territory”. This too inevitably focuses on Platonism in physics. It also asks the question as to how, exactly, (mathematical) models relate to the world/reality. There's also discussion of the relation between mathematical objects and mathematical concepts as this is brought out within the specific context of Platonism in physics. An old problem is then discussed: the precise relation between our world and the Platonic world. The issue of (as it were) “causal closure” was the traditional focus of this particular debate; though other aspects are tackled in the following. Finally, mathematical structuralism - and how it relates to Platonism in physics - is discussed. This leads naturally on to the final section which discusses what Smolin calls “intrinsic essences” (or what philosophers call “intrinsic properties”). Lee Smolin puts a psychological and sociological slant on the issue of Platonism in physics when he discusses the personal motivations of Platonic philosophers and mathematicians. He writes: “Does the seeking of mathematical knowledge make one a kind of priest, with special access to an extraordinary form of knowledge?” It can safely be said that this was true of Pythagoras, Plato and their followers. Whether or not it's also true of an everyday mathematician or philosopher ensconced in a university department in Nottingham or Oxford, I don't know. Having said that, Smolin does speak about a friend of his in this respect. Smolin tells us that he “sometimes wonder[s] if his belief in truths beyond the ken of humans contributes to his happiness at being human”. In any case, it's probably best to leave the personal psychologies of Platonists there. After all, if Smolin argues that Platonists are Platonists for reasons of personal psychology, then Platonists can also argue that Smolin is an anti-Platonist for reasons of personal psychology. And where does that get us? Fire In the Equations The physicist John Archibald Wheeler provided the most powerful riposte to Platonism in physics. In an oft-quoted story, we're told that Wheeler used to write many arcane equations on the blackboard and stand back and say to his students: “Now I'll clap my hands and a universe will spring into existence.” According to Max Tegmark and others, however, the equations are the universe - at least in a manner of speaking - and perhaps not even in a manner of speaking! (More of which later.) Then Steven Hawking (in his A Brief History of Time) nearly trumped Wheeler with an even better-known quote. He wrote: “Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?” The science writer Kitty Ferguson (in her The Fire in the Equations) offers a possible Platonist answer to Hawking's question by saying that “it might be that the equations are the fire”. Alternatively, could Hawking himself have been “suggesting that the laws have a life or creative force of their own?”. Again, is it that the “equations are the fire”? The theoretical physicist Lee Smolin, on the other hand, explains why the idea that “mathematics is prior to nature” is unsupportable. He writes: “Math in reality comes after nature. It has no generative power.” More philosophically, Smolin continues when he says that “in mathematics conclusions are forced by logical implication, whereas in nature events are generated by causal processes in time”. The Platonist will simply now say that mathematics fully captures those “causal processes”. Or, in Max Tegmark's case, the argument is that the maths and the causal processes are one and the same thing. More relevantly to the position of people like Tegmark, Smolin says that “logical relations can model aspects of causal processes, but they're not identical to causal processes”. What's more, “[l]ogic is not the mirror of causality”. Yet according to Tegmark: i) Because the models of causal processes are identical to those processes, ii) then they must be one and the same thing. More precisely, Tegmark's argument is as follows: i) If a mathematical structure is identical (or “equivalent”) to the physical structure it “models”, ii) then the mathematical structure and the physical structure must be one and the same thing. Thus if that's the case (i.e., that structure x and structure y are identical), then it makes little sense to say that x “models” (or is “isomorphic with”) y. That is, x can't model y if x and y are one and the same thing. So Tegmark also applies what he deems to be true about the identity of two mathematical structures to the identity of a mathematical structure and a physical structure. He offers us an explicit example of this: electric-field strength = a mathematical structure Or in Tegmark's own words: “' [If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure – indeed, that same mathematical structure.” In any case: i) If x (a mathematical structure) and y (a physical structure) are one and the same thing, ii) then one needs to know how they can have any kind of relation at all to one another. [Gottlob Frege's “Evening Star” and “Morning Star” story may work here.] In terms of Leibniz's law (Smolin is a big fan of Leibniz and frequently mentions him), that must also mean that everything true of x must also be true of y. But can we observe, taste, kick, etc. mathematical structures? (Yes, if they're identical to physical structures!) In addition, can't two structures be identical and yet separate (i.e., not numerically identical)? Well, not according to Smolin's Leibniz. All this is perhaps easier to accept when it comes to mathematical structures being compared to other mathematical structures (rather than to something physical). Yet if the physical structure is a mathematical structure, then that qualification doesn't seem to work either. All this is also problematic in the following sense: i) If we use mathematics to describe the world, ii) and maths and the world are the same thing, iii) then we're essentially either using maths to describe maths or using the world to describe the world. What's more, maths can't be the “mirror” of anything in nature if the two are identical in the first place. In other words, any mathematical models which are said to “perfectly capture nature” (or causality) can only do so because nature (or causality) is already mathematical. If that weren't the case, then no perfect modelling (or perfectly precise equations) could exist. Thus, again, that perfect symmetry (or isomorphism) can only be explained (according to Tegmark) if nature and maths are one and the same thing. A sharp and to-the-point anti-Platonist position is also put by the science writer, Philip Ball. He writes: “... equations purportedly about physical reality are, without interpretation, just marks on paper”. In other words, what exactly (as Hawking put it above) “breathes fire into the equation [to] make a world”? The Philip Ball quote above also highlights two problems. i) The fact that we can make mistakes about physical reality. ii) That even if the equations are about physical reality, they're not one and the same as physical reality. Indeed, even Ball's “interpretation” won't make the equations equal physical reality. So let's go all the way back to Galileo (as Smolin himself does). Surely we must say that “Nature's book” isn't written in the language of mathematics. We can say that Nature's book can be written in the language mathematics. Indeed it often is written in the language of mathematics. Though Nature's book is not itself mathematical because that book - in a strong sense - didn't even exist until human beings began to write (some of) it. Perhaps I'm doing Galileo a disservice because he did say that “we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”. Yet Galileo was talking about our understanding of Nature here - not just Nature as it is “in itself”. Nonetheless, Galileo also said that the the “book is written in mathematical language”. So was he also talking about Nature as it is in itself being mathematical? Perhaps Galileo wasn't only saying that mathematics is required to understand Nature. There is, therefore, an ambivalence here between the idea that Nature itself is mathematical and the idea that mathematics is required to understand Nature. Sure, ontic structural realists and other structural realists (in the philosophy of physics) would say that this distinction (i.e., between maths and the world) hardly makes sense when it comes to physics generally - and it doesn't make any sense at all when it comes to quantum physics. Nonetheless, surely there's still a distinction to be made here. The Map is Not the Territory Philip Ball (who's just been quoted) puts the main problem of Platonism perfectly when he says that “[i]t's not surprising, the, that some scientists want to make maths itself the ultimate reality, a kind of numinous fabric from which all else emerges”. Thus, in more concrete terms, such mathematical Platonists fail to see that the “[r]elationships between numbers are no substitute” for the world/reality. Indeed, adds Ball, “[s]cience deserves more than that”. This is the mistaking-the-map-for-the-territory problem. As the semanticist Alfred Korzybski once put it: “A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.” Indeed we can take this further and say that “all models are wrong”. This the-map-is-not-the-territory idea is put by Smolin himself when he tells us that “[m]athematics is one language of science”. In other words, the maths (in mathematical physics) isn't self-subsistent: it needs to be tied to reality: it isn't reality itself. Thus, “[maths] application to science is based on an identification between results of mathematical calculations and experimental results, and since the experiments take place outside mathematics, in the real world, the link between the two must be stated in ordinary language”. More directly, Smolin tells us that “the pragmatist will insist that the mathematical representation of a motion as a curve [for example] does not imply that the motion is in any way identical to the representation”. “By succumbing to the temptation to conflate the representation with the reality and identify the graph of the records of the motion with the motion itself...” Then Smolin tells us about one Platonist (or Tegmarkian) conclusion to all this. He writes: “Once you commit this fallacy [i.e., of mistaking the map for the territory], you're free to fantasise about the universe being nothing but mathematics.” Finally, Smolin puts his particular slant on the importance of time in all of this. He writes: “The very fact that the motion takes place in time whereas the mathematical representation is timeless means they aren't the same thing.” How Can Maths Model Nature? To put it at its most simple and - perhaps - extreme. The Platonic mistake is to move from the fact that mathematics can be (almost) perfect for describing or modelling the world to the conclusion that the world must therefore be intrinsically mathematical itself. Smolin captures this position when he discusses the work of Isaac Newton. According to Smolin, Newton's world was “infused with divinity, because timeless mathematics was at the heart of everything that moved, on Earth and in the sky”. Slightly earlier, Smolin had also written that “[w]hen Galileo discovered that falling bodies are described by a simple mathematical curve, he captured an aspect of the divine”. We can of course ask if Galileo thought in these terms himself: even if only at the subconscious level. However, would that even matter to Smolin's take on this? In any case, is mathematics “at the heart of everything that move[s]” or is it simply a tool for description or modelling? Max Tegmark (again) may argue the following: i) If mathematics is “at the heart of everything that moves”, ii) and it's also a perfect tool for description and modelling, iii) then in what sense is the world not itself mathematical? Indeed Smolin himself goes way beyond Galileo and Newton and says that “the whole history of the world” [in general relativity] is “represented by a mathematical object”. Now if we turn to quantum mechanics and the words of Philip Ball, he says that superposition is “considered only as an abstract mathematical thing”. It's also the case the the/a wavefunction is also a “mathematical object”. Now if we turn to quantum mechanics and the words of Philip Ball, he says that superposition is “considered only as an abstract mathematical thing”. It's also the case the the/a wavefunction is also a “mathematical object”. If we get back to mathematical models. It was said earlier that mathematics can describe (or even perfectly model) nature and that the physicists who aren't Platonists have no problem with this. How could they? Indeed Smolin himself tells us that “[i]t's impossible to state these laws [i.e., Newton's laws] without mathematics”. This is often said about quantum mechanics. Yet Smolin is going beyond that and saying that it's also true “the first two of Newton's laws”. More specifically, Smolin says that “[a] straight line is an ideal mathematical concept”. That is, “it lives not in our world but in the Platonic world of ideal curves”. In terms of “acceleration” and the “rate of change of velocity” (to take just two examples), it was the case that “Newton needed to invent a whole new branch of mathematics: the calculus” in order to “describe it adequately”. But here again we mustn't conflate the maths with what the maths describes (or models). Philip Ball (again) puts this position as it applies specifically to Hilbert space. He tells us that “a Hilbert space is a construct – a piece of maths, not a place”. He then quotes the physicist Asher Peres stating the following: “The simple and obvious truth is that quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.” Ball also mentions Max Tegmark's position. He writes: “If the Many Worlds are in some sense 'in' Hilbert space, then we are saying that the equations are more 'real' than what we perceive: as Tegmark puts it, 'equations are ultimately more fundamental than words' (an idea curiously resistant to being expressed without words). Belief in the MWI seems to demand that we regards the maths of quantum theory as somehow a fabric of reality.” Mathematical Objects and Mathematical Concepts Smolin has a problem with such mathematical objects. He (implicitly) argues against this Platonic position when he says (in a note) that “[m]athematicians like to speak of curves, numbers, and so forth as mathematical 'objects', which implies a kind of existence”. However, it's fairly clear that Smolin has a problem with this position. He says that you may want to call these “mathematical objects” by the name of “concepts”. That, on one interpretation, surely takes mathematical objects out of the Platonic world and places them in the realm of human minds. (Except for the fact the concepts too can be seen as “abstract objects”.) Stephen Hawking (for one) certainly didn't believe that maths and nature are one - and he too talked about “concepts”. He once wrote that “mental concepts are the only reality we can know”. Furthermore he stated: “There is no model-independent test of reality.” This seems to mean that Hawking went further than simply saying that mathematics describes (or perfectly models) nature. After all, he stresses the importance of “mental concepts”. However, it can still be said that the models of physics are of course mathematical and accurate. Thus even if we require mental concepts to get at these mathematical models, the models can still perfectly capture “reality”. So whichever way we interpret Hawking's words, he certainly doesn't seem to put a Platonic position (or replicate Tegmark's stance) on mathematical physics. Smolin himself distinguishes mental concepts from mathematical objects when (in a note) he writes: “If you aren't comfortable adopting a radical philosophical position [i.e., of believing in mathematical objects] by a habit of language, you might want to call them [mathematical objects] concepts instead.” In that passage Smolin doesn't seem to explicitly commit himself to mathematical concepts (rather than mathematical objects); though elsewhere he is more explicit when he also talks about “inventing” (i.e., not “discovering”) mathematical objects. It's also interesting to note that Smolin puts a Wittgensteinian position. Wittgenstein, for example, once wrote that “a cloud of philosophy condenses into a drop of grammar”. Smolin, on the other hand, talks about “adopting a radical philosophical position [because of] a habit of language”. In any case, Smolin defines a “mathematical object” thus: “Mathematical objects are constituted out of pure thought. We don't discover the parabolas in the world, we invent them. A parabola or a circle or a straight line is an idea. It must be formulated and then captured in a definition... Once we have the concept, we can reason directly from the definition of a curve to its properties.” Of course there are a couple of words in the passage above which a Platonist may have a problem with. Firstly, the word “invent” (as in “we invent [mathematical objects]”. And then there's the use of the word “concept” (i.e., rather than “object”). In Fregean style, we can have a “concept of an object”. Thus an abstract mathematical object can generate (as it were) various mental concepts. In terms of “[o]nce we have the concept”, then certain things logically and objectively follow from that concept. So it's the philosophical nature of the concept which raises questions. How Do We Get to the Platonic Realm? Even if the Platonic mathematical realm does indeed exist, then it' still the case that we still need to gain (causal) access to it. This is a problem that's often been commented upon. Smolin himself puts it this way: “One question that Jim [a friend of Smolin] and other Platonists admit is hard for them to answer is how we human beings, who live bounded in time, in contact only with other things similarly bounded, can have definite knowledge of the timeless realm of mathematics.” Plato himself answered Smolin's question when he argued that we have “intuitive” (or even genetic) access to this abstract realm from birth. (He elaborated on this in his slave boy story.)This doesn't seem to solve the problem of causal access to a Platonic realm. Thus, as a addendum to this argument about causal access (or the “causal closure” of both the human world and the Platonic world), Smolin says that “[b]ecause we have no physical access to the imagined timeless world, sooner or later we'll find ourselves just making stuff up”. In other words, even if the Platonic realm does exist and we can also gain access to it, that doesn't mean that we can't get things wrong or make mistakes about it. Smolin himself says that “[w]e get the truths of mathematics by reasoning, but can we really be sure our reasoning is correct?”. What's more: “Occasionally errors are discovered in the proofs published in textbooks, so it's likely that errors remain.” I suppose Plato himself might have argued that we can't get things wrong because our intuition somehow guarantees access to the truths found in this realm. Or, more correctly, if we use our reason (or intuition) correctly (as Descartes also argued), then we simply can't go wrong. So now here's Smolin quoting Roger Penrose (who's a personal friend of Smolin) putting the Platonic/Cartesian position just mentioned: “You're certainly sure that one plus one equals two. That's a fact about the mathematical world that you can grasp in your intuition and be sure of. So one-plus-one-equals-two is, by itself, evidence enough that reason can transcend time. How about two plus two equals four? You're sure about that, too! Now, how about five plus five equals ten? You have no doubts, do you? So there are a very large number of facts about the timeless realm of mathematics that you're confident you know?” It's of course the case that many philosophers and mathematicians will say that one doesn't need a “timeless realm” to explain all that's argued in Penrose's words above. It can, for example, be given a Wittgensteinian explanation in terms of rules and our knowledge of the rules. Our “intuitive grasp” (as it's sometimes put) of basic arithmetic can also be partly explained by cognitive scientists, evolutionary psychologists or philosophers. It's also interesting that Penrose gives basic arithmetical examples as demonstrations of our Platonic intuition. So what about higher or more complex maths? Do mathematicians have immediate intuitions about such equations or do they need to work at them? And if they do need to work at them, then surely intuition must have a minimal role to play. In one of his notes, Smolin gives another argument as to why the Platonic realm and the human realm can never be split asunder. He writes: “It's also not quite true to say that the truths of mathematics are outside time, since, as human beings, our perception and thoughts take place at specific moments in time – and among the things we think about are mathematical objects.” The Platonist would say that Smolin is conflating the Platonic realm with the fact that we can gain access to that realm. That is, one realm can still be abstract and timeless even if we concrete and time-bound human beings can gain access to it. But here we have a analogue of the mind-body problem. That is, what is the precise relation between the time-bound and concrete world and the timeless and abstract world? Smolin himself explains the Platonic position in terms of human psychology. He continues: “It's just that those mathematical objects don't seem to have any existence in time themselves. They are not born, they do not change, they simply are.” Smolin uses the word “seem” in the above (as in “seem to have any existence in time”). That implies that what seems to be the case may not actually be the case. Yet Smolin does then say that mathematical objects “are not born, they do not change, they simply are”. Here he may simply be putting the position of the Platonist. Again, even if mathematical objects aren't born, we still need to explain our access to them and acknowledge the possibility of getting things wrong about them - even systematically getting things wrong! Interestingly, Smolin offers us a kind of “conventionalist” middle way when he states that “[w]e invent the curves and numbers of mathematics, but once we have invented them we cannot alter them”. A Platonist would have a profound problem with the word “invent”. However, even though we may indeed invent numbers (or functions), once they're invented or created, then they become (as it were) de facto Platonic objects. That is, they're then set in stone and other things must necessarily follow from them. This is something that a philosopher like the late Wittgenstein might have happily accepted. That is, that rules and symbol-use themselves create the “objectivity” (or at least the “intersubjectivity”) of maths - and also, perhaps, even the timelessness of mathematics. Interestingly enough, Smolin puts his anti-Platonist position by adopting the position of mathematical structuralism. (There are also types of mathematical structuralism which are Platonist - see here.) Firstly (in a note) he expresses the essence (as it were) of mathematical structuralism when he says that “relationships are exactly what mathematics expresses”. He then makes the ontological point that “[n]umbers have no intrinsic essence, nor do points in space; they are defined entirely by their place in a system of numbers or points – all of whose properties have to do with their relationships to other numbers or points”. Moreover, “[t]hese relationships are entailed by the axioms that define a mathematical system”. It can be said that Platonists believe that numbers do have an "intrinsic essence". In other words, a system doesn't gain its nature because of the relations between numbers: the relations between numbers are parasitical on the nature of numbers themselves. After all, the following can be argued: i) If numbers didn't have an intrinsic essence, ii) then they couldn't engender the precise relations to other numbers which they have in each system. i) If numbers have intrinsic essences, ii) then those essences can't be dependent on the systems to which they belong (or, indeed, to any system). iii) Therefore those intrinsic essences must come before all systems of relations. Of course the obvious point to put against that position was put by Paul Benacerraf in 1965. The French philosopher wrote: “For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that.” In simple terms, we can say that the number 1 is (partly) defined by being the successor of 0 in the structure determined by a theory of natural numbers. In turn, all other numbers are defined by their respective places in the number line. So, again, it can of course be said that the “essence” of, say, the number 2 is that it comes after 1 and before 3. But surely then its intrinsic essence is determined by its relations to 1, 3 and to other numbers. Perhaps, then, relations and numbers are two sides of the same coin. Having said that, it's still hard to understand what the intrinsic essence of a number could be when that essence is taken separately to that number's relations to other numbers, functions, etc. Of course this foray into the philosophy of mathematics completely ignores the precise relation between mathematical structuralism and the world. Despite saying that, Smolin does make an explicit philosophical commitment to mathematical structuralism. He writes: “If there's more to matter than relationships and interactions, it is beyond mathematics.” Thus Smolin firstly began by articulating the/a position of mathematical structuralism and ends up stating a position that's very close to ontic structural realism. However, the ontic structural realist argues that there's no “beyond mathematics” – or at least that there's nothing beyond the “relationships and interactions” of physics which are described by mathematics. Yet Smolin himself appears to leave it open that there may well actually be a beyond mathematics. And elsewhere in his writings Smolin seems to state that there are “intrinsic properties” (qualia, etc.) beyond mathematics and even beyond physics itself. Intrinsic Essences, Qualia, Etc. Smolin makes it explicit that he (at the very least) acknowledges the possibility of “intrinsic properties” as they occur in both minds (i.e., qualia) and in inanimate objects. For example, he writes: “We don't know what a rock really is, or an atom, or an electron. We can only observe how they interact with other things and thereby describe their relational properties. The external properties are those that science can capture and describe – through interactions, in terms of relationships.” The passage above might well have been written by someone like David Chalmers or Philip Goff – both of whom are advocates of panpsychism. In the case of panpsychists, the “what is” (or “what it is like to be”) of a rock can be explained by referring to its experiences (or to its “proto-experiences”). These experiences are therefore the “intrinsic essences” (to use Smolin's own term) of rocks for panpsychists (if not for Smolin himself). Clearly, according to the passage above, philosophical relationalism (or relationalism in physics itself – which Smolin thoroughly endorses) doesn't capture these intrinsic properties. So it's no surprise that Smolin continues on the theme of intrinsic essence. He writes: “The internal aspect is the intrinsic essence; it is the reality that is not expressible in the language of interactions and relations.” We can of course ask why Smolin accepts the very existence (or reality) of an “internal aspect” of anything when many philosophers and other physicists reject this idea. What's more, Smolin ties all this to both consciousness and qualia. Firstly he writes the following: “What's missing when we describe a color as a wavelength of light or as certain neurons lighting up in the brain is the essence of the experience of perceiving red. Philosophers give these essences a name: qualia.” Again (like Smolin's “intrinsic aspect” earlier), why does Smolin need to use the somewhat archaic word “essence” (archaic at least according to certain philosophers) at all? Why believe in essences? Finally, Smolin writes: “Consciousness, whatever it is, is an aspect of the intrinsic essence of brains.” So clearly Smolin has been reading some contemporary (analytic) philosophers. It's just a little odd that he begins with the words “consciousness, whatever it is”; and then goes on to tell us exactly what it is: “the intrinsic essence of brains”.

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Section: New Results Newtonian fluid flows simulations and their analysis Simulations of water distribution systems :Water losses may constitute a large amount of the distributed total water volume throughout water distribution systems. Here, a new model method is proposed that intends to minimize the total water volume distributed through leakage reduction. Our group has worked on the derivation of advection-reaction-diffusion type equations with an explicit relationship between the local pressure and the leakage rate. An original splitting technique to solve this type of hydraulic problem was then achieved. This technique allows pressure-dependent leakage to be taken into account, whereas in most models leakage is assumed to be uniform along a pipe. Finally, a constrained optimization problem was formulated for leakage reduction in WDS. The control variable had the mean of a local head loss and is considered in the Boundary Conditions to avoid dealing with discontinuities in the governing equations. The objective function to minimize was a regularization of the total water volume distributed. Specific operational constraints were added to ensure enough pressure at consumption points. The direct solution for this minimization problem was sought with a Gradient type method. The leakage reduction was proven to be significant in a case study. The percentage of leakage reduced from 24% to 10% in the linear relationship between pressure and leakage flow rate. With other leakage exponents, the same rate of reduction was achieved . The method was applied on a real network in the South-West of France. Controlling the pressure at two different strategic points permits a significant amount of the total distributed water to be saved (5%). This work was performed in collaboration with Cemagref Bordeaux . Future work will consist of applying a sensibility analysis of control location points to optimize the method. Incompressible flows : modeling and simulation of moving and deformable bodies. The incompressible Navier-Stokes equations are discretized in space onto a fixed cartesian mesh. The deformable bodies are taken into using a first order penalization method and/or second order immersed boundary method. The interface between the solid and the fluid is tracked using a level-set description so that it is possible to simulate several bodies freely evolving in the fluid. A turbulence model based on Samgorinsky model has been added to the numerical code. The numerical code written in the C langage is massively parallel. The large linear systems (over than 100 millions of dofs) are solved using the Petsc Library. As an illustration of the methods, fish-like locomotion is analyzed in terms of propulsion efficiency. Underwater maneuvering and school swimming are also explored. We were able to simulate the three-dimensional flow about a swimmer for realistic physical configurations. Another application is the turbulent 3D flow around complex wind turbine (see http://www.math.u-bordeaux1.fr/~mbergman and http://www.math.u-bordeaux1.fr/MAB/mc2/analysis.html for simulation movies). Wake flows generated by boat propellers are also modeled and simulated. We recently take in account a simplified elasticity model of the swimmer (elastic caudal tail of a fish). Some elastic parameters allows to increase the swimming efficiency around 20%-30%. Recent developments on multiphase flows have been performed. We are able to simulate water/air interactions with interface regularization. The interface with a boat is also taken into account. See http://www.math.u-bordeaux1.fr/~mbergman for simulations. ANR Cyclobulle lead by Hamid Kellay Soap hemi-bubble film experiments have shown some links between the formation of vortices when the hemi-bubble is heated at the equator and the formation of tornados in the earth atmosphere. Two-dimensional simulations using a stereographic map are used to compare to these experimental results and confirm the results when Coriolis force and heat source terms are added. Compressible flows: Immersed boundary methods. We are concerned with immersed boundary methods, i.e., integration schemes where the grid does not fit the geometry, and among this class of methods, more specifically with cartesian grid methods, where the forcing accounting for the presence of boundaries is performed at the discrete level. We have developed a simple globally second order scheme inspired by ghost cell approaches to solve compressible flows, inviscid as well as viscous. In the fluid domain, away from the boundary, we use a classical finite-volume method based on an approximate Riemann solver for the convective fluxes and a centered scheme for the diffusive term. At the cells located on the boundary, we solve an ad hoc Riemann problem taking into account the relevant boundary condition for the convective fluxes by an appropriate definition of the contact discontinuity speed. This method can easily be implemented in existing codes and is suitable for massive parallelization. It has been validated in two dimensions for Euler and Navier-Stokes equations, and in three dimensions for Euler equations. The order of convergence is two in norm for all variables, and between one and two in depending on the variables. The 3D code has been parallelized with MPI. The case of a moving solid has been tested (flapping wing) and gives results for the drag and the lift in agreement with the references in the literature. The Oldroyd B constitutive model is used to study the role of the viscoelasticity of dilute polymer solutions in two-dimensional flows past a bluff body using numerical simulations. This investigation is motivated by the numerous experimental results obtained in quasi two dimensional systems such as soap film channels. The numerical modeling is novel for this case and therefore a comprehensive comparison is carried out to validate the present penalization method and artificial boundary conditions. In particular we focus on flow past a circular object for various values of the Reynolds number, Weissenberg number, and polymer viscosity ratio. Drag enhancement and drag reduction regimes are discussed in detail along with their flow features such as the pattern of vortex shedding, the variation of lift as well as changes in pressure, elongational rates, and polymer stress profiles. A comprehensive study of the flow behavior and energy balance are carefully carried out for high Reynolds numbers. Flow instabilities in both numerical and experimental results are discussed for high Weissenberg numbers . Elliptic problems: We have developed a new cartesian method to solve elliptic problems with immersed interfaces. These problems appear in numerous applications, among them: heat transfer, electrostatics, fluid dynamics, but also tumour growth modelling, or modelling of electric potential in biological cells This method is second order accurate in the whole domain, notably near the interface. The originality of the method lies on the use of additionnal unknows located on interface points, on which are expressed flux equalities. Special care is dedicated to the discretization near the interface, in order to recover a stable second order accuracy. Actually, a naive discretization could lead to a first order scheme, notably if enough accuracy in the discretization of flux transmission condtions is not provided. Interfaces are represented with a distance level-set function discretized on the grid points. The method has been validated on several test-cases with complex interfaces in 2D. A parallel version has been developed using the PETSC library. Simulations of fluid-solid interactions : The interaction of an elastic structure and an fluid occurs in many phenomena in physics. To avoid the difficulty of coupling lagrangian elasticity with an eulerian fluid we consider a whole eulerian formulation. The elasticity of the structure is computed with retrograde caracteristics which satisfy a vectorial transport equation. We derive the associated fluid-structure models for incompressible and compressible media. The equations are discretized on a cartesian mesh with finite differences and finite volumes schemes. The applications concern the bio-locomotions and the study of air-elastic interaction. Vortex methods : The aim of this work is to couple vortex methods with the penalization methods in order to take advantage from both of them. This immersed boundary approach maintains the efficiency of vortex methods for high Reynolds numbers focusing the computational task on the rotational zones and avoids their lack on the no-slip boundary conditions replacing the vortex sheet method by the penalization of obstacles. This method that is very appropriate for bluff-body flows is validated for the flow around a circular cylinder on a wide range of Reynolds numbers. Its validation is now extended to moving obstacles (axial turbine blades) and three-dimensional bluff-bodies (flow around a sphere). See . Moreover, using the global properties of the penalization method, this technique permits to include porous media simultaneously in the flow computation. We aim to adapt the porous media flows to our new method and to apply it in order to implement passive control techniques using porous layers around bluff-bodies. Domain decomposition : Domain decomposition methods are a way to parallelize the computation of numerical solutions to PDE. To be efficient, domain decompositions methods should converge independently on the number of subdomains. The classical convergence result for the additive Schwarz preconditioner with coarse grid is based on a stable decomposition. The result holds for discrete versions of the Schwarz preconditioner, and states that the preconditioned operator has a uniformly bounded condition number that depends only on the number of colors of the domain decomposition, and the ratio between the average diameter of the subdomains and the overlap width. Constants are usually non explicit and are only asserted to depend on the "shape regularity" of the domain decomposition. two years ago, we showed the result holds the additive Schwarz preconditioner can also be defined at the continuous level and provided completely explicits estimates. Last year, we established that a similar result also holds for non shape regular domain decompositions where the diameter of the smallest subdomain is significantly smaller than the diameter of the largest subdomain. The constants are also given explicitely and are independent of the ratio between the diameter of the largest sudomain and the diameter of the smallest subdomain. This year, we have studied explored new coarse spaces algorithms for domain decomposition methods. Coarse spaces are necessary to get a scalable algorithm whose convergence speed does not deteriorate when the number of subdomains increases. For domains decomposition methods with discontinuous iterates, we showed that continuous coarse spaces can never be an optimal choice. As an alternative, we introduced both the use of discontinuous coarse spaces(DCS) and a new coarse space algorithm using these discontinuous coarse spaces.

Maximum power transfer 3 learn more about 37: maximum power transfer theorem on globalspec. Aim of experiment: to prove maximum power transfer theorem practically apparatus 1 dc circuit training system 2 set of wires 3 dc power supply 4 digital avo meter theory the power transferred from a supply source to a load is at its maximum when. Consider an ac voltage source is placed in series with a complex impedance (z) and a load impedance the maximum power transfer to the load impedance occurs when the load impedance equals the complex. 1 maximum power transfer problem statements network of sources and resistors i l r l o o a b + v l- what's the maximum power that can be delivered to a load. The technology interface/spring 2008 cartwright 1 non-calculus derivation of the maximum power transfer theorem by kenneth v cartwright, phd. Hello, i'd just like to know why does the maximum power occur when the load resistance equals the source resistance, can it be explained verbally without matching or calculus also, why does maximum power transfer occur when the voltage across the load is equal to half the value. Suppose a 12 volt lead-acid battery has an internal resistance of 20 milli-ohms (20 m ): if a short-circuit were placed across the terminals of this large battery, the fault current would be quite large: 600 amps now suppose three of these batteries were connected directly in parallel with one. Th venin's and norton's theorems can be used to analyze complex circuits by focusing on the source and load circuits one application of th venin's and norton's theorems is to calculate the maximum power for a load circuit the power p coming from the source circuit to be delivered to. Maximum power transfer maximize the power delivered to a resistive load ece 201 circuit theory i ece 201 circuit theory i consider the general case a resistive network contains independent and dependent sources. Ab82 introduction ab82 is a compact, ready to use thevenin's theorem maximum power transfer theorem experiment board this is useful for students to study. Maximum power transfer theorem (r bolton - 2012) physics 1553: introduction to electricity and magnetism 1 maximum power transfer theorem physics, 7th edition, cutnell & johnson. Ece 202 - experiment 8 - lab report transformers and maximum power transfer your name_____ gta's signature_____ lab meeting time_____. Maximum power transfer is another useful analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source. For the purpose of this laboratory experiment, the given complex dc circuit should be constructed and analyzed by taking measurements as well as by doing calculations. Th evenin's, norton's, and maximum power transfer theorems this worksheet and all related les are licensed under the creative commons attribution license. The power that can be taken from a homogeneous magnetic field b is dependent on the induced voltage uind in the used receiver coil. 1 eece202 network analysis i dr charles j kim class note 13: maximum power transfer a maximum power transfer 1 in many practical situations, a circuit is designed to provide power to a load. In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Maximum power transfer Answer: c explanation: the maximum power is delivered from a source to its load when the load resistance is equal to the source resistance the maximum power transfer theorem can be applied to both dc and ac circuits. The maximum power-transfer problem for iz ports when the load impedance matrix is required to be passive, and it is further required that the network obtained by inter- connecting z, and be solvable for our purposes, an n x n. Back to basics: impedance matching (part 1) lou frenzel 1 | oct 24, 2011 the maximum power-transfer theorem says that to transfer the maximum amount of power from a source to a load, the load impedance should match the source impedance. - Maximize power delivered to circuits the first derivative is used to maximize the power delivered to a load in electronic circuits so in order to have maximum power transfer from the electronic circuit to the load r, the resistance of r has to be equal to r. - Maximum power transfer, and the load voltage v r at which this occurs (hint: recall the maximum power transfer theorem from your basic circuits course) medium line (50-150 mi) ee 340 - transmission lines author: yahia created date. - Solved example of maximum power transfer theorem for ac & dc circuits application and limitation of maximum power transfer theorem max power to load trans. - Te 2 s - total electrical energy solution also known as maximum power transfer solution (mpts) or industrial power optimization control (ipoc. - Maximum power transfer theorem suppose we have a voltage source or battery that's internal resistance is r i and a load resistance r l is connected across this battery.

The first scientific evidence for a non-static universe came with the formulation of the theory of General Relativity by Einstein in 1915, following a completely new interpretation of the nature of the gravitational force. The basic idea is that the presence of matter alters the geometry of space-time. Test-particles follows the geodesics in space-time, which determine their orbits. Quantitatively this idea is expressed by equating a geometry tensor Gµ (which incorporates the ten curvature terms required to specify the geometry) to a matter tensor Tµ which contains all the information (ten terms) about the mass-energy present in the manifold. (G is Newton's constant) In a simplified model in which the universe is represented as an homogeneous isotropic fluid (an approximation suggested by the homogeneous distribution of galaxies in space) this equation reduces to a much simpler equation for R: the distance between any two objects (galaxies for instance) in the universe. (Here (R0) is the time derivative of R). The space-time curvature terms are on the left-hand side of the equation. The first term: (R0/R)2 is the time curvature while the second one: k/R2 is the space curvature. The value of k is normalized to take one of the three values: k = 0 (flat universe), +1 (closed universe) or -1 (open universe). The gravitational effect of matter is represented by the energy density . The last term is the famous cosmological constant. Mathematically it is a sort of a priori undetermined constant of integration. It is this equation which historically gave the first indication that our universe may not be static. It equates the time derivative of the scale factor R(R0) to a sum of terms which has no obvious reason to be zero. Einstein decided to give to the exact numerical value required to obtain R0 = 0. This decision was most unfortunate. Einstein failed to make the important discovery that the universe is not static. Furthermore it was soon shown that his solution is not stable. Even if is (arbitrarily) chosen to neutralize the time derivative of R today, any density perturbation would suffice to bring the fluid out of this unstable equilibrium and induce a general motion one way or the other (contraction or expansion). After 1930, thanks to the work of Edwin Hubble, the systematic motion (recession) of the galaxies was detected. The observations gave the relation R0/R = cst = H, where H is the Hubble constant. The value of H is still uncertain by a factor of two. It lies between 50 and 100 km/sec/megaparsec. The inverse of this parameter has the dimension of a time: the time scale of the universal expansion. Its value lies between 10 and 20 billion years. Soon after this detection, models of the expanding universe were presented by Friedman, Lemaitre, and Alpher, Herman, Gamov. Assuming an homogeneous fluid of ideal gases, the Tµ simplifies to its diagonal form containing (, - P, - P, - P) where P is the pressure and the density of the fluid. The dynamics is given by eq. 2. The energy-momentum conservation expression, Tµ; = 0, simplifies to the following equation: The equation Tµ; = 0 implies that the entropy per comoving volume S = sR3 remains constant which, in turn, requires that: The entropy per unit volume s, given by s = ( + P)/T, decreases with R-3, just as the number-density of particles. One should be careful about these two last equations. They assume that entropy generating processes are operating in the fluid. In the standard mode this is guaranteed by the hypothesis of a fluid of pure non-interacting particles (except by gravitational interactions). In the real world these equations are valid only when entropy generating processes are negligibly small. This is correct most of the time. There are important chapters of cosmic expansion when this is not true, and appropriate corrections have to be made. Eqs. (3, 4) need to be completed by an equation of state for the fluid. Three regimes are important during cosmic expansion: radiation, matter, and vacuum. A) When the universe is radiation dominated, i.e., when the largest contribution to the energy density is due to relativistic particles (kT >> Mc2), we have T4 and the pressure term is one-third the density term (P = 1/3). From eqs. (3, 4) we get: R-4 or R 1/T as expected from entropy conservation. The entropy density s is proportional to the number-density of relativistic particles, with a numerical factor of 7.1. This regime applies to the early hot universe. Consider eqs. ((3, 4) again. Through the work of Hubble we know that R increases with time. At early moments, the density term R-4 dominates the k/R2 and the cosmological constant. Thus we have (R0/R) R-2 or R t1/2 where t is the timescale of the universe. B) For a matter dominated (cold) universe the pressure of the non-relativistic particles is negligible compared to their mass density. P = 0. Using eqs. (3, 4) again, we find R-3 T3. This time we have R t2/3. C) In recent works, a third regime has been shown to be of great importance in the early universe. At certain times, the energy-density is dominated by the so-called vacuum energy terms associated with various physical phase transitions. These terms, which do not appear in the standard classical fluid model described before, are related to the quantum field description of matter. The undetermined cosmological constant of Einstein has become a convenient way to introduce these terms in the formalism. The equation of state of this regime is (P = -), hence = cst from eqs. (3, 4) (note the strange result that the vacuum density does not change during the expansion ...). And we have R exp(t/t0); an exponential rate of expansion called inflation.

ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 5, с. 846-878 = ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ INTRODUCTION TO COSMOLOGY ©2010 A. D. Dolgov* Institute for Theoretical and Experimental Physics, Moscow, Russia; University of Ferrara, Italy Received August 28, 2009 An introductory review on cosmology for students specializing in particle physics is presented. Many important subjects are not covered because of lack of space but hopefully the review may serve as a starting point for further studies. Modern cosmology is vast interdisciplinary science and it is impossible to cover it in any considerable detail in a short review. Planning this review, I have prepared the following short list of subjects, which is surely will be made much shorter at this course, but hopefully it may be useful for the students who would like to continue studying this field. So the idealistic content could be the following: 1. A little about general relativity and its role in cosmology. 2. Four basic cosmological equations and expansion regimes. 3. The Universe today and in the past. 4. Kinetics in hot expanding world and freezing of species. 5. Inflation: kinematics, models, the Universe heating, and generation of density perturbations and gravitational waves. 6. Big Bang nucleosynthesis. 7. Field theory at nonzero temperature and cos-mological phase transitions. 8. Baryogenesis and cosmological antimatter. 9. Neutrino in cosmology (bounds on mass, oscillations, magnetic moment, and anomalous interactions). 10. Dark matter and large scale structure. 11. Vacuum and dark energies. 12. Cosmic microwave radiation and cosmological parameters. In reality only a half of this plan was fulfilled. Nevertheless, the list presented above may be helpful as a guiding line for further studies. We will start from some nontechnical introduction to general relativity and relations between the latter and cosmology in Section 2. Next, in Section 3 we will derive the basic cosmological equations in a rather naive way studying motion of nonrelativistic test body in spherically symmetric gravitational field. There we also talk about realistic regimes of the Universe expansion and basic cosmological parameters. In the next Section 4, the Universe history is very briefly presented. Section 5 is dedicated to thermodynamics and kinetics in the early Universe. Section 6 is dedicated to freezing of species and cosmological limit on neutrino mass. Big Bang nucleosynthesis (BBN) is presented in Section 7. In Section 8 the role of neutrinos in BBN is described. Neutrino oscillations in the early Universe are considered in Section 9. In Section 10 inlationary cosmology is discussed, and the last Section 11 is dedicated to cosmological baryogenesis. 2. GRAVITY AND COSMOLOGY Two simple observations that the sky is dark at night and that there are shining stars lead to the conclusion that the Universe is finite in space and time. The first one is the well known Olbers' paradox, based on the estimate of the sky luminosity, which in infinite homogeneous static Universe must be infinitely high. Shining stars should exhaust their fuel in finite time and thus cannot exist in the infinitely old Universe — thermal death of the Universe. General relativity (GR) successfully hit both targets leading to the notion of expanding Universe of finite age, but created instead its own very interesting problems which we discuss in what follows. Newtonian theory of gravity has an evident shortcoming that it has action-at-a-distance property. In other words, gravitation acts instantaneously, at any distance. On the other hand, in the spirit of contemporary wisdom interactions are always mediated by some bosonic fields and are relativistically invariant. If we wished today to generalize Newtonian theory of gravity to relativistic theory we could take, a priori as a mediator of interactions scalar, vector, or tensor intermediate bosons, confining ourselves to lower spins. Since we know that gravity operates at astronomically large distances, the mass of the intermediate boson should be zero or very small. Indeed, massless bosons create static Coulomb type potential U ~ 1/r while massive bosons lead to exponentially cutoff Yukawa potential U ~ exp(-mr)/r. Interactions mediated by vector field are odd with respect to charge parity transformation, C-transformation, and as one can see from the vector boson propagator, such interactions induce matterantimatter attraction and matter—matter repulsion, recall electromagnetic interactions. Hence, vector field cannot mediate attractive gravitational force. Scalar and tensor mediators lead to attraction of matter—matter and matter—antimatter and both are a priori allowed. According to nonrelativistic Newtonian theory the source of gravity is mass. Possible relativistic generalisation for scalars should be a scalar quantity coinciding in nonrelativistic limit with mass. The only known such source is the trace of the energy—momentum tensor of matter Ttf. The relativistic equation of motion for scalar gravity should have the form: d2$ = 8nGN Tff, where GN is the Newtonian gravitational coupling constant. Such theory is rejected by the observed light bending in gravitational field, since for photons: T^ = 0. A small admixture of scalar gravity to tensor one, i.e. Brans—Dicke theory , is allowed. There remains massless tensor theory with the source which may be only the energy—momentum tensor of matter T^v. In first approximation the equation of motion takes the form: d2V =8nGNT^ • (2) This equation is valid in the weak field approximation because the energy—momentum of h¡v itself should be included to ensure conservation of the total energy—momentum. Massless particles, as, e.g., gravitons, must interact with a conserved source. Otherwise theory becomes infrared pathological. The energy—momentum tensor of matter is conserved only if the energy transfer to gravitational field is neglected. Taking into account energy leak into gravity leads to nonlinear equations of motion and allows to reconstruct GR order by order. For a discussion of this approach see papers . Historically, Einstein did not start from field-theoretical approach but formulated GR in an elegant and economical way as geometrical theory postulating that matter makes space—time curved and that the motion of matter in gravitational field is simply free fall along geodesics of this curved manifold. This construction is heavily based on the universality of gravitational action on all types of matter — the famous equivalence principle, probably first formulated by Galileo Galilei. The least action principle for GR was formulated by Hilbert with the action given by J R + An where R is the curvature scalar of 4D space—time and Am is the matter action, written in arbitrary curved coordinates. Gravitational field is identified with the metric tensor g¡v of the curved space—time. The curvature is created by matter through equations of motion: Rfiv — — 8irG^Tf where R^V is the Ricci tensor. There is no space here to stop on technicalities of Riemann geometry. A good introduction can be found, e.g., in where one can find definition and properties of the Christoffel symbols , Riemann tensor R^av?, Ricci tensor R^V = gafiR^av?, scalar curvature R = g^VR¡v, co-variant derivatives in curved space—time D^, etc. The source of gravity is the energy—momentum tensor of matter taken in this curved space—time: The impact of gravity on matter is included into T V due to its dependence on metric and in some more complicated cases on the curvature tensors. Let us repeat that the motion of matter in the gravitational field is simply the free fall, i.e. motion along geodesics. Classical tensor theory of gravity agrees with all available data and is a self-consistent, very beautiful and economic theory. It is essentially based on one principle of general covariance, which is a generalization of Galilei principle of relativity to arbitrary coordinate frames. Invariance with respect to general coordinate transformation (which is called general covariance) is a natural framework which ensures vanishing of the graviton mass mg. Even if the underlying classical theory is postulated to be massless, quantum corrections should generally induce nonzero mass if they are not prevented from that by some symmetry principle. This is another advantage of tensor gravity with respect to scalar one for which no principle which forbids nonzero mass is known. Though quantum gravity is not yet understood, it is natural to expect that quantum corrections should induce mg = 0 in absence of general covariance. An important property of equations of motion (4) is that their r. h. s. is covariantly conserved: Dß ( RßV - \gßVR I = o Accordingly, the energy—momentum tensor must be conserved too: Dß = 0. DßVv = OßVv - r«vV«. in the early stage, as indicated by isotropy of cosmic microwave background radiation (CMB), and even now at large scales. Correspondingly the metric can be taken as homogeneous and isotropic one (Friedmann—Robertson—Walker (FRW) metric ): ds2 = dt2 - a2(t) f (r)dr2 + r2dti\, Here, D^ is covariant derivative, as we have already mentioned. To those not familiar with Riemann geometry it may be instructive to mention that covariant derivative appears when one differentiates in curved coordinate system, e.g., in spherical one, even in flat space—time. From another point of view, covariant derivative in curved space—time, which, e.g., is acting on vector field, looks as It is similar to covariant derivative in gauge theories, because the latter includes gauge field, A^ analogous to ra According to the Noether theorem, the conservation of T1V follows from the least action principle if the matter action is invariant with respect to general coordinate transformation. So the gravitationa Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.

What are the 3 formulas for the area of a triangle? The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h. Hence, to find the area of a tri-sided polygon, we have to know the base (b) and height (h) of it. What is triangular formula? The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles, whether it is a scalene triangle, an isosceles triangle, or an equilateral triangle. What are the 5 ways to prove the Pythagorean Theorem? Other proofs of the theorem - Proof using similar triangles. - Euclid’s proof. - Proofs by dissection and rearrangement. - Einstein’s proof by dissection without rearrangement. - Proof by area-preserving shearing. - Algebraic proofs. - Proof using differentials. Does a2 b2 c2 work for all triangles? Pythagorean theorem: If a triangle is a right triangle (has a right angle), then a2+b2=c2. Converse: If a2+b2=c2 in a triangle with c is the longest side, then a triangle is a right triangle. How many different triangle formulas are there? Types of Triangle |Based on the Sides||Based on the Angles| |Scalene Triangle||Acute angled Triangle| |Isosceles Triangle||Right angle Triangle| |Equilateral Triangle||Obtuse-angled Triangle| How many types of triangle and its formula? Six Types of Triangles |Based on their Sides||Based on their Angles| |Scalene Triangle||Acute Triangle| |Isosceles Triangle||Obtuse Triangle| |Equilateral Triangle||Right Triangle| What are the first 7 triangular numbers? List Of Triangular Numbers. 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, and so on. How many formulas are in a triangle? A triangle is a 3-sided closed shape. There are two important formulas related to triangles, i.e., the Heron’s formula and Pythagoras theorem. The sum of the interior angles of a triangle is 180° and is expressed as ∠1 + ∠2 + ∠3 = 180°. Does Pythagorean Theorem work on all triangles? Pythagoras’ theorem only works for right-angled triangles, so you can use it to test whether a triangle has a right angle or not. What is a Pythagorean triple give 3 examples? Integer triples which satisfy this equation are Pythagorean triples. The most well known examples are (3,4,5) and (5,12,13). Notice we can multiple the entries in a triple by any integer and get another triple. For example (6,8,10), (9,12,15) and (15,20,25). Why is Pythagorean Theorem a2 b2 c2? The Pythagorean Theorem describes the relationship among the three sides of a right triangle. In any right triangle, the sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse: a2 + b2 = c2. How do you find 2 in Pythagorean Theorem? 2.Pythagorean Theorem find a or b – YouTube What are the 7 types of triangles? To learn about and construct the seven types of triangles that exist in the world: equilateral, right isosceles, obtuse isosceles, acute isosceles, right scalene, obtuse scalene, and acute scalene. What are the 12 types of triangles? Is 666 a triangular number? , i.e. 1 + 2 + 3 + + 34 + 35 + 36 = 666), and thus it is a triangular number. Because 36 is also triangular, 666 is a doubly triangular number. Also, 36 = 15 + 21; 15 and 21 are also triangular numbers, and 152 + 212 = 225 + 441 = 666. Is 0 A triangular number? Why Pythagoras theorem is used in right angle triangle? No, the Pythagorean theorem can only be applied to a right-angled triangle since the Pythagorean theorem expresses the relationship between the sides of the triangle where the square of the two legs is equal to the square of the third side which is the hypotenuse. Why Pythagoras theorem is used in right triangle? Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. What is the formula of Pythagorean triples Class 8? Pythagorean triples are a2+b2 = c2 where a, b and c are the three positive integers. What are the first 10 Pythagorean triples? , are (3, 4, 5), (6, 8,10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34), (21, 28, 35). Which formula is this A² B² C² *? The Pythagorean theorem helps find the lengths of the sides of a right triangle. It states that a²+b²=c², where a and b are the sides around the right angle and c is the hypotenuse. What is c2 a2 b2 2abcosC? The law of cosines (often abbreviated LoC) states: For △ABC with sides a,b,c, the following relationships hold: c2=a2+b2−2abcosC a2=b2+c2−2abcosA b2=c2+a2−2cacosB This gives us a way to find the third side of a triangle, given two sides and the angle measure between them. How do you find C in a triangle? The hypotenuse is opposite the right angle and can be solved by using the Pythagorean theorem. In a right triangle with cathetus a and b and with hypotenuse c , Pythagoras’ theorem states that: a² + b² = c² . To solve for c , take the square root of both sides to get c = √(b²+a²) . How do you find Pythagorean theorem with 3 numbers? Pythagorean Triples – MathHelp.com- Geometry Help – YouTube What is 180 degree triangle called? Angles that are 180 degrees (θ = 180°) are known as straight angles. Angles between 180 and 360 degrees (180°< θ < 360°) are called reflex angles. Who discovered Heron’s formula? Heron of Alexandria Heron’s formula, formula credited to Heron of Alexandria (c. 62 ce) for finding the area of a triangle in terms of the lengths of its sides. In symbols, if a, b, and c are the lengths of the sides: Area = Square root of√s(s – a)(s – b)(s – c) where s is half the perimeter, or (a + b + c)/2. What does S mean in Heron’s formula? The s in Heron’s formula denotes the semi-perimeter of a triangle, whose area has to be evaluated. Semi-perimeter is equal to the sum of all three sides of the triangle divided by 2. How accurate is Heron’s formula? Heron’s formula computes the area of a triangle given the length of each side. If you have a very thin triangle, one where two of the sides approximately equal s and the third side is much shorter, a direct implementation Heron’s formula may not be accurate. How do you find the missing side of a triangle? How To Calculate The Missing Side Length of a Triangle – YouTube What is square formula? The Formula for the Area of A Square The area of a square is equal to (side) × (side) square units. The area of a square when the diagonal, d, is given is d2÷2 square units. For example, The area of a square with each side 8 feet long is 8 × 8 or 64 square feet (ft2). Who invented pi? Archimedes of Syracuse The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Who invented zero? “Zero and its operation are first defined by [Hindu astronomer and mathematician] Brahmagupta in 628,” said Gobets. He developed a symbol for zero: a dot underneath numbers. Who invented maths? But Archimedes is known as the father of mathematics. Why area of square is a2? Why is the area of a square a side square? A square is a 2D figure in which all the sides are of equal measure. Since all the sides are equal, the area would be length times width, which is equal to side × side. Hence, the area of a square is side square. Can herons formula be used for all triangles? Heron’s formula can be applied to any type of triangle. Since, Heron’s formula is used to calculate the area of a triangle and every type of triangle in this world will have some area. So, for a triangle, if the three sides are known, you can directly calculate the area using Heron’s formula. How do I find the third side of a triangle? How to Determine the Length of the Third Side of a Triangle When You How do you find the third side of a triangle without angles? Pythagorean Theorem for the Third Side of a Right Angle Triangle. The Pythagorean Theorem is used for finding the length of the hypotenuse of a right triangle. So, as long as you are given two lengths, you can use algebra and square roots to find the length of the missing side. What are the 7 properties of rectangle? The fundamental properties of rectangles are: - A rectangle is a quadrilateral. - The opposite sides are parallel and equal to each other. - Each interior angle is equal to 90 degrees. - The sum of all the interior angles is equal to 360 degrees. - The diagonals bisect each other. - Both the diagonals have the same length. What is side formula? The side angle side formula is the SAS area formula which means we can find the area of a triangle if the length of two sides of a triangle and its included angle is known. The SAS formula is expressed as: Area of a triangle = (1/2) × side1 × side2 × sin (included angle) What is the 100th digit of pi? The value of pi starts with a 3 followed by a decimal point. Since pi is an irrational number, the digits after the decimal point are infinite. The 100th digit after the decimal point is 9. What is the 1000000 digits of pi? 3.14159265358979323846264338327950288419716939937510 etc. Before you click remember – it’s a byte a digit! The first 1000000 decimal places contain: 99959 0s, 99758 1s, 100026 2s, 100229 3s, 100230 4s, 100359 5s, 99548 6s, 99800 7s, 99985 8s and 100106 9s. Who is the father of maths? The Father of Math is the great Greek mathematician and philosopher Archimedes. Perhaps you have heard the name before–the Archimedes’ Principle is widely studied in Physics and is named after the great philosopher. Who created math? Archimedes is considered the Father of Mathematics for his significant contribution to the development of mathematics. His contributions are being used in great vigour, even in modern times. Who found zero? Who is the father of Math? What is area formula? Area = length × lengthArea = l2. Area = length x breadthArea = l × b. Area = π × radius × radiusArea = π × r2(π = 3.14) Using these formulas, area for different quadrilaterals (a special case of polygons with four sides and angles ≠ 90o) is also calculated. Application. What is S in a triangle? Another is Heron’s formula which gives the area in terms of the three sides of the triangle, specifically, as the square root of the product s(s – a)(s – b)(s – c) where s is the semiperimeter of the triangle, that is, s = (a + b + c)/2. How do you find the unknown side of a triangle? What is the sine of θ? Formulas for right triangles If θ is one of the acute angles in a triangle, then the sine of theta is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.

All we do is draw little arrows… A couple of weeks ago I read somewhere that we have got enough coal left for over 200 years. So despite the greenhouse gas emissions that it entails, we needn’t worry about our immediate energy supply. Coal can be converted to oil through the Fischer-Tropsch process , securing even oil for the next decades. Good news I thought, there is more time to develop renewables without jeopardizing an already frail economy. Yet then the inquisitive merry-go-round in my brain got going: how did they get that number? For over 200 years of coal? Is this a lower bound? An upper bound? What if the coal price rises? What if there are new discoveries? How can they predict how much coal we will use for the next 200 years? In short, they can’t. In long, they still can’t. But hey, let’s give it a try. Energy companies, governments and since short also bloggers around the world estimate how much fossil fuels are left. The remaining technically extractible volume of a certain fossil fuel, be it oil, coal or natural gas, is called the remaining resource. Some resources are proven (90% certainty), others just probable (50%) or possible (10%). The reserves constitute that part of the resources that is not only technically but also economically extractible. Hence the proven reserves are those resources available profitably with a 90% probability. Changing economical and technical conditions impact the amount of fuel in each category. Once you know the size of the reserves (R) and production (P), you are just a division away from the number of years those reserves would last if production wouldn’t change. This is called the R/P-ratio. The R/P-ratios at the end of 2009 were: The effect of growth The numbers above are for the proven reserves. Accordingly, today’s proven reserves of oil would be depleted in 2055 if today’s production of oil wouldn’t change. In fact production does change. In the last 45 years the annual growth rate of oil production has swayed between 10% and -5.8%. Clearly the growth rate affects how long the fossil fuel reserves last. How large is this effect? Knowledge of the growth rate straightforwardly leads to the depletion date, but I’ll spare you the math. The average growth rates since 1981 amount to 1%, 2.6% and 2.2% for oil, natural gas and coal respectively. These points are marked on the graph. If these rates would be constant, the proven reserves would be depleted in 2046 (oil and natural gas) and 2068 (coal), much earlier than if there would be no growth. Luckily (or not), considerable uncertainty surrounds the depletion dates, arising from (1) the uncertainty in growth rate and (2) the uncertainty in reserves. The reserves in the figure displayed above were only the proven reserves. Potential reserves might be up to a factor 4 larger. What happens to the depletion dates if we take this into account? Let’s consider oil. Oil depletion: earliest and latest date For an estimate of when oil reserves might be depleted, take a look at the figure above. To calculate the earliest possible date, I use the proven reserves (the least amount of oil we expect to have, blue curve), with a high growth rate of 4.7% annually. There is nothing to indicate that actual production might grow this fast, but it is warranted as an upper bound as it is the largest growth on record since 1986. Thus the earliest date we arrive at is 2033. For the latest date, we need to know how large potential reserves might be. The U.S. Geological Survey estimates the total remaining oil reserves to be at least twice as big as currently proven reserves with a 95% probability, and even up to 3-4 times as big with just a 5% chance (green curve). Consequently, I assume we can maximally extract about 3 times the currently proven reserves, with on average no growth in production. That’s how I determine the upper bound of 2146. The shaded area indicates all possible depletion dates in the range 2033-2146. The green curve is dashed because it is based on less sound numbers. Except for Fischer-Tropsch, we can be confident the world will stop using oil in 2033-2146. Alas, in return for this confidence we have given up precision. The range 2033-2146 is quite certain and strikingly unsatisfactory. Surely we can do better! If so, it is only by reducing either (1) the uncertainty in growth rate or (2) the uncertainty in reserves. Fighting uncertainty: the world energy model The growth rate depends on complex socioeconomic factors such as policy-making, fuel prices, demographics, technical advancements and so on. Obviously, no one has figured out how to predict these, right? Forecasting one person’s actions is out of reach, let alone the blend of thousands of people’s fears, expectations and decisions. Yet this is precisely what the World Energy Model, developed by the International Energy Agency, tries to do. It is a mathematical construct, comprising nearly 16000 equations, whose main goal is to replicate how energy markets work. Let’s not take the credibility of this model for granted, it produces projections – not predictions. As for any model, its output is no more trustworthy than its input. In this case one inputs some assumptions about economic growth, fuel prices and technological development. In return one gets quantities like energy demand, production growth and carbon dioxide emissions. Putting these remarks aside, let’s just go ahead and use the model’s numbers. For oil it projects an average annual growth rate of 1% in the period 2007-2030. Remarkably this is the same rate as in the last 30 years. Let’s also allow for the possibility that this growth rate is off by a certain margin. Not having a clue how to estimate this margin, I just take the interval 0.7%-1.3% and regard it as reasonable. Over the period 2007-2030 this interval corresponds to a total growth of 17%-35%, compared to the 26% the IEA projects. The figure below illustrates this method. Abusing the world energy model’s numbers, I find some evidence for the conclusion that oil will be depleted in 2070-2105. I get the lower bound by taking a growth rate of 1.3% (assuming the model is 0.3% wrong on the high side), based on a total potential reserve twice as big (purple curve) as the currently proven reserves. According to the USGS, there is a 95% chance that there is at least that much oil, but I still drew the curve dashed since there might be more oil by unconventional means: oil sands, oil shales, deepwater oil, etc. are not included in these graphs. The upper bound is calculated by assuming a 0.7% growth (the model is 0.3% wrong on the low side) with potential reserves three times bigger than currently proven reserves (green curve again). Summary of the numbers Here’s a summary of the numbers. After each date I provide the assumed growth rate and reserves in the form (average growth rate, size of reserves relative to currently proven reserves). For instance “2146 (0%, 3)” means that I calculated the date 2146 assuming a 0% average annual production growth, with remaining reserves 3 times bigger than currently proven reserves. So far I have concentrated on obtaining reliable ranges for the depletion dates. In reality, fossil fuels will probably never be truly depleted. A couple of years, a decade, several decades – who knows – before a depletion date is reached, prices most likely go through the roof. We will run out of cheap oil before we run out of oil. In this sense, the depletion date is irrelevant. Still, it might help us get an idea of the timescale we are faced with. At the moment, little coal is converted to oil because of the large investments needed for the plants. The OPEC ensures that the oil price stays low enough to price Fischer-Tropsch plants out of the market (and high enough to make money). International Energy Agency Statistics Europe’s energy portal U.S. Energy Information Administration U.S. Geological Survey: World Petroleum Assessment U.K. Energy Research Centre BP Statistical Review