The First Condition for Equilibrium

Phan Chu Trinh Primary School Ms. JennyMy Name : Class : Mark: Number : The 1 st English Test I. The Alphabet: __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ II. Numbers: ._________ three : ._________ one : ._________ six : ._________ ten : ._________ five : III. Colors: Màu cam : purple : Màu xanh lá : brown : Màu trắng : yellow : The First Condition for Equilibrium The First Condition for Equilibrium Bởi: OpenStaxCollege The first condition necessary to achieve equilibrium is the one already mentioned: the net external force on the system must be zero Expressed as an equation, this is simply net F = Note that if net F is zero, then the net external force in any direction is zero For example, the net external forces along the typical x- and y-axes are zero This is written as net Fx = and Fy = [link] and [link] illustrate situations where net F = for both static equilibrium (motionless), and dynamic equilibrium (constant velocity) This motionless person is in static equilibrium The forces acting on him add up to zero Both forces are vertical in this case 1/3 The First Condition for Equilibrium This car is in dynamic equilibrium because it is moving at constant velocity There are horizontal and vertical forces, but the net external force in any direction is zero The applied force Fapp between the tires and the road is balanced by air friction, and the weight of the car is supported by the normal forces, here shown to be equal for all four tires However, it is not sufficient for the net external force of a system to be zero for a system to be in equilibrium Consider the two situations illustrated in [link] and [link] where forces are applied to an ice hockey stick lying flat on ice The net external force is zero in both situations shown in the figure; but in one case, equilibrium is achieved, whereas in the other, it is not In [link], the ice hockey stick remains motionless But in [link], with the same forces applied in different places, the stick experiences accelerated rotation Therefore, we know that the point at which a force is applied is another factor in determining whether or not equilibrium is achieved This will be explored further in the next section An ice hockey stick lying flat on ice with two equal and opposite horizontal forces applied to it Friction is negligible, and the gravitational force is balanced by the support of the ice (a normal force) Thus, net F = Equilibrium is achieved, which is static equilibrium in this case 2/3 The First Condition for Equilibrium The same forces are applied at other points and the stick rotates—in fact, it experiences an accelerated rotation Here net F = but the system is not at equilibrium Hence, the net F = is a necessary—but not sufficient—condition for achieving equilibrium PhET Explorations: Torque Investigate how torque causes an object to rotate Discover the relationships between angular acceleration, moment of inertia, angular momentum and torque Torque Section Summary • Statics is the study of forces in equilibrium • Two conditions must be met to achieve equilibrium, which is defined to be motion without linear or rotational acceleration • The first condition necessary to achieve equilibrium is that the net external force on the system must be zero, so that net F = Conceptual Questions What can you say about the velocity of a moving body that is in dynamic equilibrium? Draw a sketch of such a body using clearly labeled arrows to represent all external forces on the body Under what conditions can a rotating body be in equilibrium? Give an example 3/3 Annals of Mathematics The Hopf condition for bilinear forms over arbitrary fields By Daniel Dugger and Daniel C. Isaksen Annals of Mathematics, 165 (2007), 943–964 The Hopf condition for bilinear forms over arbitrary fields By Daniel Dugger and Daniel C. Isaksen Abstract We settle an old question about the existence of certain ‘sums-of-squares’ formulas over a field F, related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of charac- teristic 0, then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic p > 2. 1. Introduction Fix a field F . A classical problem asks for what values of r, s, and n do there exist identities of the form  r  i=1 x 2 i  ·  s  i=1 y 2 i  = n  i=1 z 2 i (1.1) where the z i ’s are bilinear expressions in the x’s and y’s. Equation (1.1) is to be interpreted as a formula in the polynomial ring F [x 1 , . . . , x r , y 1 , . . . , y s ]; we call it a sums-of-squares formula of type [r, s, n]. The question of when such formulas exist has been extensively studied: [L] and [S1] are excellent survey articles, and [S2] is a detailed sourcebook. In this paper we prove the following result, solving Problem C of [L]: Theorem 1.2. If F is a field of characteristic not equal to 2, and a sums- of-squares formula of type [r, s, n] exists over F, then  n i  must be even for n − r < i < s. We now give a little history. It is common to let r ∗ F s denote the smallest n for which a sums-of-squares formula of type [r, s, n] exists. Many papers have studied lower bounds on r ∗ F s, but for a long time such results were known only for fields of characteristic 0: one reduces to a geometric problem over R, and then topological methods are used to obtain the bounds (see [L] for a summary). In this paper we begin the process of extending such re- sults to characteristic p, replacing the topological methods by those of motivic homotopy theory. 944 DANIEL DUGGER AND DANIEL C. ISAKSEN The most classical result along these lines is Theorem 1.2 for the particular case F = R, which leads to lower bounds for r ∗ R s. It seems to have been proven in three places, namely [B], [Ho], and [St]; but in modern times the given condition on binomial coefficients is usually called the ‘Hopf condition’. The paper [S1] gives some history, and explains how K. Y. Lam and T. Y. Lam deduced the condition for arbitrary fields of characteristic 0. Problem C of [L, p. 188] explicitly asked whether the same condition holds over fields of characteristic p > 2. Work on this question had previously been done by Adem [A1], [A2] and Yuzvinsky [Y] for special values of r, s, and n. In [SS] a weaker version of the condition was proved for arbitrary fields and arbitrary values of r, s, and n. Stiefel’s proof of the condition for F = R used Stiefel-Whitney classes; Behrend’s (which worked over any formally real field) used some basic inter- section theory; and Hopf deduced it using singular cohomology. Our proof of the general theorem uses a variation of Hopf’s method and motivic cohomol- ogy. It can be regarded as purely algebraic—at least, as ‘algebraic’ as things like group cohomology and algebraic K-theory. These days it is perhaps not so clear that there exists a point where topology ends and algebra begins. We now explain Hopf’s proof, and our generalization, in more detail. Given a sums-of-squares formula of type [r, s, n], one has in particular a bi- linear map φ: F r × F s → F n given by (x 1 , . . . , x r ; y 1 , . . . y s ) → (z 1 , . 1 So, you want to go to grad school in Economics? A practical guide of the first years (for outsiders) from insiders Ceyhun Elgin Mario Solis-Garcia University of Minnesota April 2007 2 Introduction You may be wondering about our intentions in writing this document. A bit of background information may be useful here. We, along with many other young, bright students across the U.S. and other countries, were fascinated with the idea of enrolling in a Ph.D. program in economics, but were missing the big picture of the process, as well of the outcome! For our case, it is true that we are still working for the degree, but after one year and a half in the program (as of December 2006), we think that the experience may not be for everyone (but may be more exciting for some than for others!). This document is intended to provide, to the best of our knowledge, an accurate picture of the life before and during a Ph.D. program in economics. Our hope is that potentially interested students may benefit from our experience. As could be expected from a theme such as this one, a usual disclaimer arises. The conclusions expressed within are based from our experience in the economics program at the University of Minnesota; we expect many of the features discussed below to be fairly similar among schools. However, the experience might be different in other programs, within the U.S. or across the world! Our paper first will discuss what you should do before going to the Ph.D. Then we will try to give you some flavor of the life in the first year of the program, followed by our conclusion. So you’ve decided to go for a Ph.D. program Getting a Ph.D. degree is easier said than done. Your undergraduate or master’s advisor is right: It’s going to be difficult. In fact, we can agree and go even further: It’s going to be the most difficult experience of your life. However, should you decide to go for it, the rewards are easily greater than the costs. The first moral of the story: Think twice before you decide! 3 In our view, graduate work in economics (or in any other discipline, for that matter) is an exercise in discipline, endurance, hard work, and patience. It is not so easy having all 4 at the same time… but you’ll get used to it. The early work: tests So, suppose you are really decided to go on the adventure of a Ph.D. program. Congratulations! But now you have to get to work. You should have heard from everyone: “Take the GRE test.” Well, we’ll also say it: Take the GRE test. And take it early. Even though most universities suggest taking the test no later than the December before the year to enter the program, we suggest taking it no later than June or July of the year before entering the program. Why? For starters, bad luck happens. You can take special courses, study for a couple of months, score 800s on your practice tests, but it all comes down to one particular test. And here’s where the bad luck comes into play. If you don’t get the score you expect, you Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 798319, 20 pages doi:10.1155/2009/798319 Research Article On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings Gang Cai and Chang song Hu Department of Mathematics, Hubei Normal University, Huangshi 435002, China Correspondence should be addressed to Gang Cai, caigang-aaaa@163.com and Chang song Hu, huchang1004@yahoo.com.cn Received 17 April 2009; Accepted 9 July 2009 Recommended by Tomonari Suzuki We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others. Copyright q 2009 G. Cai and C. S. Hu. 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. 1. Introduction Let C be a nonempty closed convex subset of a Hilbert space H. A mapping T : C → C is said to be nonexpansive if for all x, y ∈ C we have Tx−Ty≤x−y. It is said to be asymptotically nonexpansive 1 if there exists a sequence {k n } with k n ≥ 1 and lim n →∞ k n  1 such that T n x − T n y≤k n x − y for all integers n ≥ 1andforallx, y ∈ C. The set of fixed points of T is denoted by FT. Let φ : C × C → R be a bifunction, where R is the set of real number. The equilibrium problem for the function φ is to find a point x ∈ C such that φ  x, y  ≥ 0 ∀y ∈ C. 1.1 The set of solutions of 1.1 is denoted by EPφ. In 2005, Combettes and Hirstoaga 2 introduced an iterative scheme of finding the best approximation to the initial data when EPφ is nonempty, and they also proved a strong convergence theorem. 2 Fixed Point Theory and Applications For a bifunction φ : C × C → R and a nonlinear mapping A : C → H, we consider the following equilibrium problem: Find z ∈ C such that φ  z, y    Az, y − z  ≥ 0, ∀y ∈ C. 1.2 The set of such that z ∈ C is denoted by EP,thatis, EP   z ∈ C : φ  z, y    Az, y − z  ≥ 0, ∀y ∈ C  . 1.3 In the case of A  0, EP  EPφ. In the case of φ ≡ 0, EP is denoted by VIC, A. The problem 1.2  is very general i n the sense that it includes, as special cases, some optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others see, e.g., 3, 4. Recall that a mapping A : C → H is called monotone if  Au − Av, u − v  ≥ 0, ∀u, v ∈ C. 1.4 A mapping A of C into H is called α-inverse strongly monotone, see 5–7, if there exists a positive real number α such that  x − y, Ax − Ay  ≥ α   Ax − Ay   2 1.5 for all x, y ∈ C. It is obvious that any Tran Quy Cap Senior High School TEST ON ENGLISH Class: 11/ Time: 45 minutes 124 Full name : Term II - Academic year 2009-2010 I. Reading: Read the paragraphs below then choose True ( F) or False (F) WHAT ARE RENEWABLE AND NONRENEWABLE RESOURCES? Renewable resources can be replaced. Plants are renewable resources. Plants might get cut down, but they can grow back. Animals are renewable resources. They can reproduce. Solar energy is a renewable resource. Solar energy comes from the Sun. No matter how much solar energy you use, there will always be more. Wind, water, and soil are also renewable resources. Nonrenewable resources cannot be replaced easily. Fossil fuels are nonrenewable resources. Coal, oil, and natural gas are fossil fuels. Fossil fuels come from plants and animals that died millions of years ago. We are using up fossil fuels much faster than Earth can replace them. Ores come from rocks that formed millions of years ago. We use ores to make metals. Ores cannot be replaced. T F 01. Plants, animals, wind, water are renewable resources. √ 02. Renewable (used about sources of energy) that will always exist. √ 03. Coal, oil, and natural gas are also renewable resources. √ 04. The word “soil” in the passage means land used to build houses. √ 05. Ores are nonrenewable. √ 06. We are not using up fossil fuels as quickly as Earth can create them. √ II. Language focus and writing A. Circle the letter A, B, C or D that you think it the best option to finish each of the following sentences. Which of the underlined sounds is pronounced differently from the rest? 07. A. sandwich B. sure C. sorry D. same 08. A. delivered B. telephoned C. subscribed D. developed 09. A. animals B. consequences C. organizations D. pesticides 10. It was a kind of computer with I was not familiar. A. which B. whom C. that D. who 11. The number of rare animals is decreasing quickly. "quickly" means: A. seriously B. gradually C. rapidly D. eventually 12. A/An …………………. is someone who pays rent to live in a house or flat. A. tenant B. coward C. burglar D. shoplifter 13. Human beings are responsible the changes in the environment. A. with B. to C. for D. on 14. They buried hundreds of wild animals ………… by forest fires just three weeks ago. A. kill B. killed C. to kill D. which killed 15. I cannot forget the house in …………… I spent my childhood. A. which B. who C. where D. whose 16. The song was interesting. We listened to it last night. The two sentences can be rewritten like this: A. The song to whom we listened to it last night was interesting. B. The song to which we listened last night was interesting. C. The song to that we listened it last night was interesting. D. The song to which we listened to it last night was interesting. 17. When I arrived, the film show ………………… A. had already performed B. already performed C. has performed D. are performing Which of the words is stressed differently from the rest? 18. A. plentiful B. potential C. environment D. consumption 19. A. convenient B. technology C. fortunately D. development 20. Toyota, ……… has ten thousand employees, is an international car company. A. that B. who C. where D. which 21. ‘The shoplifters were arrested’. The word “arrested” means ………………………. A. captured B. released C. jailed D. imprisoned 22. The children ……………………. to be homeless. A. reported B. were reported C. was reported D. reporting 23. The first person ……………… is Mr. Smith. A. to see B. seeing C. saw D. has been seen 24. Three new roads ……………… in my town at the moment. A. have been built B. are being built C. is being built D. are building 25. Which word does not belong to its part of speech? A. dangerous B. expensive C. harmful D. convenience 26. I Tom since he Da Nang ... and the gravitational force is balanced by the support of the ice (a normal force) Thus, net F = Equilibrium is achieved, which is static equilibrium in this case 2/3 The First Condition for Equilibrium. . .The First Condition for Equilibrium This car is in dynamic equilibrium because it is moving at constant velocity There are horizontal and vertical forces, but the net external force in... with the same forces applied in different places, the stick experiences accelerated rotation Therefore, we know that the point at which a force is applied is another factor in determining whether