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Nonlinear Systems in Function Spaces and Applications in Biomedical Sciences, Control Theory, and EngineeringView this Special Issue Finite-Time Stabilization of Dynamic Nonholonomic Wheeled Mobile Robots with Parameter Uncertainties The finite-time stabilization problem of dynamic nonholonomic wheeled mobile robots with parameter uncertainties is considered for the first time. By the equivalent coordinate transformation of states, an uncertain 5-order chained form system can be obtained, based on which a discontinuous switching controller is proposed such that all the states of the robots can be stabilized to the origin equilibrium point within any given settling time. The systematic strategy combines the theory of finite-time stability with a new switching control design method. Finally, the simulation result illustrates the effectiveness of the proposed controller. Stabilization problem of nonholonomic systems is theoretically challenging and practically interesting. As pointed out in , although every nonholonomic system is controllable, it cannot be stabilized to a point with pure smooth (or even continuous) state feedback law. In order to overcome the difficulty of Brocket's condition , a variety of sophisticated feedback stabilization methods have been proposed which mainly include continuous time-varying feedback control laws [2–4], discontinuous feedback control laws [5–8], and hybrid feedback control laws . A common characteristic of these designs of controllers above is based on kinematic model, where only a kinematic model is considered and the velocities are taken as control inputs. But in fact, for some mechanical systems with nonholonomic constraints, it is more realistic to formulate the control problems at dynamic levels, where the torque and force are chosen as new inputs. Some results can be found in recent papers, for example, the dynamic tracking control of wheeled mobile robots in the presence of both actuator saturations and external disturbances is considered in , where a computationally tractable moving horizon tracking scheme is presented. In [11, 12], the saturated stabilization and tracking control are discussed for simple dynamic nonholonomic mobile robot. For uncertain dynamic nonholonomic systems, Ma and Tso have given a robust control law for the exponential regulation of an uncertain dynamic nonholonomic wheeled mobile robot, in which the authors improved the convergence speed of regulating the state to a desired set point for the first time. In order to drive a system to the equilibrium point with a fast convergence rate, finite-time stability theory has become a studying focus recently, for example, finite-time stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades [14–16]. Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance rejection properties. For the nonholonomic systems, a few researchers have got some excellent results in finite-time control field. In , the relay switching technique and the terminal sliding mode control scheme with finite-time convergence are used for the design of the controller to address the tracking control of the nonholonomic systems with extended chained form. For a class of uncertain nonholonomic chained form systems, Hong et al. have designed a nonsmooth state feedback law such that the controlled chained form system is both Lyapunov stable and finite-time convergent within any given settling time. And the finite-time tracking control for single mobile robots or multiple nonholonomic mobile robots is considered in [19–21]. The previous developed controller for nonholonomic systems can be divided into two categories: one is for the finite-time stabilization problem of chained form systems and the other is for the tracking control problem of mobile robots. However, to the best of our knowledge, there exist no results to deal with the robust finite-time stabilization of uncertain dynamic nonholonomic mobile robots. This paper considers the stabilization problem of dynamic nonholonomic mobile robots with uncertain parameters in a finite time. The main results and contributions can be summarized as the following two respects.(a)An uncertain 5-order chained form system can be obtained under the equivalent coordinate transformation of states, which means the finite-time stabilization of the chained form system is equivalent to the finite-time stabilization of the original dynamic robot system.(b)Applying the theory of finite-time stability and the switching control method, we design a discontinuous robust controller to make the states of the chained form system converge to the equilibrium point in a finite time. The structure of the paper is as follows: Section 2 gives a formalization of the problem considered in the paper. Section 3 states our main results. Section 4 provides an illustrative numerical example and the corresponding simulation results of the proposed methodology. Finally, a conclusion is shown in Section 5. 2. Problem Statement A class of nonholonomic wheeled mobile robots are shown in Figure 1, the two fixed rear wheels of the robot are controlled independently by motors, and a front castor wheel prevents the robot from tipping over as it moves on a plane. Assuming that the geometric center point and the mass center point of the robot are the same and that the radiuses are identical for all the rear wheels, is the length of the fixed two rear wheels, where and are known positive constants. Its kinematic and dynamics model can be described by the following differential equations : where , are the position of the mass center of the robot moving in the plane and , are the mass and inertia of mobile robots; respectively, is the forward velocity, is the steering velocity and denotes its heading angle from the horizontal axis, and , are driving torques on the right and left rear wheels. The geometric and inertia parameters are all assumed to be unknown positive constants, but are bounded by some known positive constants , that is, One of the equilibrium states of systems (1) is . The control objective is to design a discontinuous state feedback law , such that the state trajectory of dynamic nonholonomic mobile robot system (1) starting from an arbitrary initial state converges to the origin equilibrium point in a finite time with the unknown parameters satisfying (2). As pointed out in , take an orthogonal coordinate transformation Then system (1) can be converted to the following equation: where , are new control inputs and , are new unknown parameters with their bounds derived from (2) as follows: Because the coordinate transformation (3) is globally invertible and does not change the origin, it is obvious that the equilibrium point is finite-time stable for its closed-loop system of (4) it is means that is also a finite time stable equilibrium point for the corresponding closed-loop system of (1). Hence, the control task is to design a discontinuous finite-time stabilizing controller for system (4) with the unknown parameters (5). Here, it should be noted that Hong et al. have designed a switching control strategy to discuss the finite-time stabilization of uncertain chained form systems in , however, it is invalid to control the dynamic chained system (4); thus, a new improved discontinuous design method is required. The following definition and lemmas are needed for our controller design later. Definition 1 (see [14–16]). Consider a time-invariant system in the form of where is continuous on an open neighborhood of the origin. The equilibrium of the system is (locally) finite-time stable if (i) it is asymptotically stable, in , an open neighborhood of the origin, with ; (ii) it is finite-time convergent in , that is, for any initial condition , there is a settling time such that every solution of system (6) is defined with for and satisfies , and if . Moreover, if , the origin is globally finite-time stable. Lemma 2 (see [15, 18]). Suppose there exist a positive-definite and proper function , real numbers , , such that is negative semidefinite. Then, the origin is a globally finite-time stable equilibrium point of system (6). Lemma 3 (see ). For the uncertain time-varying chained form system: where , are the state and control input, respectively; are uncertain parameters located in known intervals, that is, is an uncertain function satisfying and satisfies . Let , , and (with and odd integers) be real numbers satisfying Then, finite-time stabilizing control law of (7) can be constructed in the form of where is defined as follows: where denotes the sign function and , are suitable constants. Proof. See for details. 3. Main Results In this section, the main results will be presented. Firstly, we will state the basic idea to design a finite-time switching controller for system (4). Note that system (4) can be decoupled into two subsystems, one of which is -subsystem and the other describes the rest of (4), that is, -subsystem By designing , the state of (13) can be driven to any predetermined point in a finite time, based on which of (14) can be stabilized to by designing the finite-time controller , and the last step is to redesign such that can be driven to in a finite time. Theorem 4. Given , for system (4), take the following switching control law. Step 1. Let , , where , . Until , then go to Step 2. Step 2. Let where Unitll , , then go to Step 3. Step 3. Let , where Until , stop. Then system (4) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1–Step 3. Proof. In the first step, let , , then which means that ; according to the conclusion of Lemma 2, there exists a finite time such that for all , that is, after this time. In Step 2, for the subsystem (14) Let One has Because , where the finite-time stabilization problems of (20) and (22) are equivalent. Set , , , , and from Lemma 3, system (20) can be stabilized to zero by in Step 2 after some finite time ; that is, for all . Similarly, consider subsystem (13) again in the last step Set , , , , , by using Lemma 3, it is clear that there exists a finite time such that system (13) can also be stabilized to zero after . This completes the proof of the theorem. Remark 5. Then control objective can be completed in each step within a finite time, and thus system (4) can be stabilized to zero in a finite time. On the other hand, from (3), we have the following: Therefore, the finite-time switching controller for the original robot system (1) can be stated as follows. Theorem 6. Given , for system (1), take the following switching control law. Step 1′. Let where , . Until , then go to Step 2′. Step 2′. Let where Unitl , , then go to Step 3′. Step 3′. Let where Until , stop. Then system (1) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1′–Step 3′. Proof. According to (25), we have the following: Because the determinant , then it is obvious to see that is equivalent to . On the other hand, the switching controller in Theorem 4 can be used to stabilize the states of system (4) in a finite time; hence, the control task is changed to find the relation between the original controller of system (1) and the controller of system (4). Comparing system (4) with system (1), we have the following: from which, and by using the switching controller of Step 1–Step 3 in Theorem 4, we can solve the corresponding and thus the conclusion can be obtained in Theorem 6. This completes the proof of the theorem. In this section, the discontinuous switching controller proposed in theorems above is used to show how to stabilize the state of (4) and the state of (1) to the zero equilibrium point in a finite time. We will demonstrate the effectiveness of our methods by a numerical example. In the following simulation, we assume that , , , , , , , . The initial condition of system (1) is . From (3) and (5), the initial value of system (4) is , and , , , . Given , the design parameters are taken as follows: , , , , , , , , , , , . The settling time in every step is given in advance; , , . Figures 2–5 show some simulation results with Matlab. From Figures 2 and 3, it can be seen that all the state variables of the closed system (4) are driven to the origin equilibrium point in a given settling time . Observing Figure 2, in time interval 0~5 s, the first step control task is completed, that is, as . Next, from Figure 3, it is clear that can be stabilized to zero by the controller in Step 2 as and remain unchanged. Finally, the controller in Step 3 drives to zero in the settling time . The robust finite-time stabilization problem is discussed in this paper for a class of uncertain dynamic nonholonomic wheeled mobile robot. The contributions of this paper include having applied finite-time control technique and a new switching design method such that all the states can be stabilized to the zero point by the proposed discontinuous controller. And we will work on extending the results to consider the corresponding trajectory tracking control problem in the coming time. This paper was supported by the Natural Science Foundation of China (61304004, 61374040, and 11372097), China Postdoctoral Science Foundation (2013M531263), the Scientific Innovation Program of Shanghai Education Committee (13ZZ115), the National Basic Research Program of China (973 Project, 2010CB832702), the National Science Funds for Distinguished Young Scholars (11125208), the 111 Project (B12032), the R&D Special Fund for Public Welfare Industry (Hydrodynamics Project, 201101014), and the the Natural Science Foundation of Hebei Province (A2014106035). A. Teel, R. Murry, and G. Walsh, “Nonholonomic control systems: From steering to stabilization with sinusoids,” in Proceedings of the IEEE Conference on Decision Control, pp. 1603–1609, 1992.View at: Google Scholar A. M. Bloch and S. Drakunov, “Stabilization of a nonholonomic systems via sliding modes,” in Proceedings of the IEEE Conference on Decision Control, pp. 2961–2963, 1995.View at: Google Scholar H. Chen, C. Wang, L. yang, and D. Zhang, “Semiglobal stabilization for nonholonomic mobile robots based on dynamic feedback with inputs saturation,” Journal of Dynamic Systems, Measurement, and Control, vol. 134, no. 4, pp. 041006.1–041006.8, 2012.View at: Google Scholar

The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts of the world. Beginning in 2005, the institute started offering joint membership with the Bernoulli Society for Mathematical Statistics and Probability as well as with the International Statistical Institute. The Institute was founded in 1935 with Harry C. Carver and Henry L. Rietz as its two most important supporters. The institute publishes a variety of journals, and holds several international conference every year. The Institute publishes five journals: In addition, it co-sponsors: There are also some affiliated journals: Furthermore, five journals are supported by the IMS: The IMS selects an annual class of Fellows who have demonstrated distinction in research or leadership in statistics or probability. Meetings gives scholars and practitioners a platform to present research results, disseminate job opportunities and exchange ideas with each other. The IMS holds an annual meeting called Joint Statistical Meetings (JSM),and sponsors multiple international meetings, for example, Spring Research Conference (SRC). The American Statistical Association (ASA) is the main professional organization for statisticians and related professionals in the United States. It was founded in Boston, Massachusetts on November 27, 1839, and is the second oldest continuously operating professional society in the US. The ASA services statisticians, quantitative scientists, and users of statistics across many academic areas and applications. The association publishes a variety of journals and sponsors several international conferences every year. Anders Hjorth Hald was a Danish statistician. He was a professor at the University of Copenhagen from 1960 to 1982. While a professor, he did research in industrial quality control and other areas, and also authored textbooks. After retirement, he made important contributions to the history of statistics. Debabrata Basu was an Indian statistician who made fundamental contributions to the foundations of statistics. Basu invented simple examples that displayed some difficulties of likelihood-based statistics and frequentist statistics; Basu's paradoxes were especially important in the development of survey sampling. In statistical theory, Basu's theorem established the independence of a complete sufficient statistic and an ancillary statistic. The COPSS Presidents' Award, along with the International Prize in Statistics, are generally considered the two highest awards in Statistics. Morris Herman DeGroot was an American statistician. The Joint Statistical Meetings (JSM) is a professional conference/academic conference for statisticians held annually every year since 1840. Billed as "the largest gathering of statisticians held in North America", JSM has attracted over 5000 participants in recent years. The following statistical societies are designated as official JSM partners: The Bernoulli Society is a professional association which aims to further the progress of probability and mathematical statistics, founded as part of the International Statistical Institute in 1975. It is named after the Bernoulli family of mathematicians and scientists whose researchers covered "most areas of scientific knowledge". Jayanta Kumar Ghosh or Jaẏanta Kumāra Ghosha was an Indian statistician, an emeritus professor at Indian Statistical Institute and a professor of statistics at Purdue University. Statistics Surveys is an open-access electronic journal, founded in 2007, that is jointly sponsored by the American Statistical Association, the Bernoulli Society, the Institute of Mathematical Statistics and the Statistical Society of Canada. It publishes review articles on topics of interest in statistics. Wendy L. Martinez serves as the coordinating editor. Jun S. Liu is a Chinese-American statistician focusing on Bayesian statistical inference and computational biology. He received the COPSS Presidents' Award in 2002. Liu is a professor in the Department of Statistics at Harvard University and has written many research papers and a book about Markov chain Monte Carlo algorithms, including their applications in biology. He is also co-author of the Tmod software for sequence motif discovery. Xuming He is a Professor of Statistics at the University of Michigan. Sudipto Banerjee is an Indian-American statistician best known for his work on Bayesian hierarchical modeling and inference for spatial data analysis. He is currently Professor and Chair of the Department of Biostatistics in the School of Public Health at the University of California, Los Angeles (UCLA). Marina Vannucci is an Italian statistician, the Noah Harding Professor and Chair of Statistics at Rice University, the past president of the International Society for Bayesian Analysis, and the former editor-in-chief of Bayesian Analysis. Topics in her research include wavelets, feature selection, and cluster analysis in Bayesian statistics. Michael Barrett Woodroofe is an American probabilist and statistician. He is an emeritus professor of statistics and of mathematics at the University of Michigan, where he was the Leonard J. Savage Professor until his retirement. He is noted for his work in sequential analysis and nonlinear renewal theory, in central limit theory, and in nonparametric inference with shape constraints. Maria Eulália Vares is a Brazilian mathematical statistician and probability theorist who is known for her expertise in stochastic processes and large deviations theory. She is a professor of statistics in the Institute of Mathematics of the Federal University of Rio de Janeiro, from 2006 to 2009 was the editor-in-chief of the journal Stochastic Processes and their Applications, publisher by Elsevier for the Bernoulli Society for Mathematical Statistics and Probability, and from 2015 to 2017 was the editor-in-chief of the Annals of Probability, published by the Institute of Mathematical Statistics. The National Institute of Statistical Sciences (NISS) is an American institute that researches statistical science and quantitative analysis. Judith Rousseau is a Bayesian statistician who studies frequentist properties of Bayesian methods. She is a professor of statistics at the University of Oxford, a Fellow of Jesus College, Oxford, a Fellow of the Institute of Mathematical Statistics, and a Fellow of the International Society for Bayesian Analysis.

Last time I set the stage, the mathematical location for quantum mechanics, a complex vector space (Hilbert space) where the vectors represent quantum states. (A wave-function defines where the vector is in the space, but that’s a future topic.) The next mile marker in the journey is the idea of a transformation of that space using operators. The topic is big enough to take two posts to cover in reasonable detail. This first post introduces the idea of (linear) transformations. 13 Comments | tags: linear algebra, matrix transform, QM101, quantum mechanics, vector space, vectors | posted in Math, Physics Whether it’s to meet for dinner, attend a lecture, or play baseball, one of the first questions is “where?” Everything that takes place, takes place some place (and some time, but that’s another question). Where quantum mechanics takes place is a challenging ontological issue, but the way we compute it is another matter. The math takes place in a complex vector space known as Hilbert space (“complex” here refers to the complex numbers, although the traditional sense does also apply a little bit). Mathematically, a quantum state is a vector in Hilbert space. 9 Comments | tags: coordinate system, inner product, QM101, quantum mechanics, vector space, vectors | posted in Math, Physics I’d planned a different first post for May Mind Month, but a recent online conversation with JamesOfSeattle gave me two reasons to jump the gun a bit. Firstly, my reply was getting long (what a surprise), and I thought a post would give me more elbow room (raising, obviously, the possibility of dualing posts). Secondly, I found the topic unusual enough to deserve its own thread. Be advised this jumps into the middle of a conversation that may only be of interest to James and I. (But feel free to join in; the water’s fine.) 55 Comments | tags: neuron, semantic vectors, vector space, vectors | posted in Computers, Philosophy, Science This is a Sideband to the previous post, The 4th Dimension. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes involved with rotation versus axes about which rotation occurs. The latter, rotation about (or around) an axis, is what we usually mean when we refer to a rotation axis. A key characteristic of such an axis is that coordinate values on that axis don’t change during rotation. Rotating about (or on or around) the Y axis means that the Y coordinate values never change. In contrast, an axis involved with rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices. 5 Comments | tags: 2D, 3D, 4D, column vector, matrix math, matrix transform, rotation, rotation matrix, unit vector, vectors | posted in Math, Sideband In the last installment I introduced the idea of a transformation matrix — a square matrix that we view as a set of (vertically written) vectors describing a new basis for a transformed space. Points in the original space have the same relationship to the original basis as points in the transformed space have to the transformed basis. When we left off, I had just introduced the idea of a rotation matrix. Two immediate questions were: How do we create a rotation matrix, and how do we use it. (By extension, how do we create and use any matrix?) This is where our story resumes… 3 Comments | tags: 3Blue1Brown, column vector, linear algebra, matrix math, matrix multiplication, rotation, unit vector, vectors | posted in Math, Sideband For me, the star attraction of March Mathness is matrix rotation. It’s a new toy (um, tool) for me that’s exciting on two levels: Firstly, it answers key questions I’ve had about rotation, especially with regard to 4D (let alone 3D or easy peasy 2D). Secondly, I’ve never had a handle on matrix math, and thanks to an extraordinary YouTube channel, now I see it in a whole new light. Literally (and I do mean “literally” literally), I will never look at a matrix the same way again. Knowing how to look at them changes everything. That they turned out to be exactly what I needed to understand rotation makes the whole thing kinda wondrous. I’m going to try to provide an overview of what I learned and then point to a great set of YouTube videos if you want to learn, too. Continue reading 19 Comments | tags: 3Blue1Brown, column vector, complex numbers, linear algebra, matrix math, matrix multiplication, rotation, trigonometry, unit vector, vectors | posted in Math, Sideband Put on your arithmetic caps, dear readers. Also your math mittens, geometry galoshes and cosine coats. Today we’re venturing after numeric prey that lurks down among the lines and angles. There’s no danger, at least not to life or limb, but I can’t promise some ideas won’t take root in your brain. There’s a very real danger of learning something when you venture into dark territory such as this. Even the strongest sometimes succumb, so hang on to your hats (and galoshes and mittens and coats and brains). Today we’re going after vectors and scalars (and some other game)! 15 Comments | tags: 2D, 3D, azimuth, coordinates, declination, dimensions, direction, elevation, location, scalars, speed, technology, vectors, velocity | posted in Math Throwing like a girl! I’ve introduced the idea of an inertial frame of reference. This is when we, and objects in our frame, are either standing still or moving with constant (straight-line) motion. In this situation, we can’t tell if we’re really moving or standing still relative to some other frame of reference. In fact, the question is meaningless. I’ve also introduced the idea that objects moving within our frame — moving (or standing still) along with us, but also moving from our perspective — move differently from the perspective of other frames. Specifically, the speed appears different. Now I’ll dig deeper into that and introduce a crucial exception. 11 Comments | tags: Albert Einstein, Emmy Noether, Galilean invariance, Galileo Galilei, Mo'ne Davis, motion, scalars, Special Relativity, speed, vectors, velocity | posted in Physics Last time I talked about opposing pairs: Yin and Yang, light and dark, north and south. I mentioned that some pairs are true opposites of each other (for example, north and south), whereas other pairs are actually a thing and the lack of that thing (for example, light and dark). Such pairs are only opposites in the sense that an empty cup is the opposite of a full cup. However in both cases, the opposites stand for opposing ideas; two poles of polarity, and it is polarization that I address today. Specifically I want to discuss a way of thinking that helps avoid it. It’s easy to divide the world into sides. Many sayings begin with, “There’s two kinds of…” It seems easier to break things down into opposing points of view than to consider a variety of views. It seems easier to compare features between two things than twenty. Our court system has two sides and so does our political system (despite many attempts to create a viable third party). 3 Comments | tags: debate, discussion, parameter space, thinking, vectors, worldview | posted in Basics, Philosophy

0 of 21 questions completed Click on Start Quiz button to take the Online test. Below are some articles to help you prepare for GATE exams. You can install GATE Mobile App for unlimited online tests with answers. Read my article on GATE exams for details of exam (eligibility criteria, format etc) and also post your questions or doubts. Online tests and Books My article on GATE Mechanical Engineering provides details to crack the exam with ease ( free Online tests, recommended books, detailed syllabus). You have already completed the quiz before. Hence you can not start it again. Quiz is loading... You must sign in or sign up to start the quiz. You have to finish following quiz, to start this quiz: 0 of 21 questions answered correctly Time has elapsed You have reached 0 of 0 points, (0) - Not categorized 0% |Table is loading| |No data available| - Question 1 of 21 For a fully developed flow of water in a pipe having diameter 10 cm, velocity 0.1 m/s and kinematic viscosity 10−5 (10 to power -5) m2/s, the value of Darcy friction factor isCorrectIncorrect - Question 2 of 21 In a simple concentric shaft-bearing arrangement, the lubricant flows in the 2 mm gap between the shaft and the bearing. The flow may be assumed to be a plane Couette flow with zero pressure gradient. The diameter of the shaft is 100 mm and its tangential speed is 10 m/s. The dynamic viscosity of the lubricant is 0.1 kg/m.s. The frictional resisting force (in newton) per 100 mm length of the bearing isCorrectIncorrect - Question 3 of 21 In an air-standard Otto cycle, air is supplied at 0.1 MPa and 308 K. The ratio of the specific heats (γ) and the specific gas constant (R) of air are 1.4 and 288.8 J/kg.K, respectively. If the compression ratio is 8 and the maximum temperature in the cycle is 2660 K, the heat (in kJ/kg) supplied to the - Question 4 of 21 A reversible heat engine receives 2 kJ of heat from a reservoir at 1000 K and a certain amount of heat from a reservoir at 800 K. It rejects 1 kJ of heat to a reservoir at 400 K. The net work output (in kJ) of the cycle isCorrectIncorrect - Question 5 of 21 Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is 40 minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will beCorrectIncorrect - Question 6 of 21 A metal rod of initial length L0 is subjected to a drawing process. The length of the rod at any instant is given by the expression, L(t) = Lo(1+t*t) , where t is the time in minutes. The true strain rate (in per min) at the end of one minute isCorrectIncorrect - Question 7 of 21 During pure orthogonal turning operation of a hollow cylindrical pipe, it is found that the thickness of the chip produced is 0.5 mm. The feed given to the zero degree rake angle tool is 0.2 mm/rev. The shear strain produced during the operation isCorrectIncorrect - Question 8 of 21 For the given assembly: 25 H7/g8, match Group A with Group B Group A Group B P. H I. Shaft Type Q. IT8 II. Hole Type R. IT7 III. Hole Tolerance Grade S. g IV. Shaft Tolerance GradeCorrectIncorrect - Question 9 of 21 If the Taylor’s tool life exponent n is 0.2, and the tool changing time is 1.5 min, then the tool life (in min) for maximum production rate isCorrectIncorrect - Question 10 of 21 An aluminium alloy (density 2600 kg/m3) casting is to be produced. A cylindrical hole of 100 mm diameter and 100 mm length is made in the casting using sand core (density 1600 kg/m3). The net buoyancy force (in newton) acting on the core is (m3 is meter cube)CorrectIncorrect - Question 11 of 21 Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is a.The standard deviation for this distribution is given byCorrectIncorrect - Question 12 of 21 Solve the equation x = 10 cos(x) using the Newton-Raphson method. The initial guess is x = pi/4 . The value of the predicted root after the first iteration, up to second decimal, is (pi is constant 3.14)CorrectIncorrect - Question 13 of 21 A cantilever beam having square cross-section of side a is subjected to an end load. If a is increased by 19%, the tip deflection decreases approximately byCorrectIncorrect - Question 14 of 21 A car is moving on a curved horizontal road of radius 100 m with a speed of 20 m/s. The rotating masses of the engine have an angular speed of 100 rad/s in clockwise direction when viewed from the front of the car. The combined moment of inertia of the rotating masses is 10 kg-m2. The magnitude of the gyroscopic moment (in N-m) isCorrectIncorrect - Question 15 of 21 A single degree of freedom spring mass system with viscous damping has a spring constant of 10 kN/m. The system is excited by a sinusoidal force of amplitude 100 N. If the damping factor (ratio) is 0.25, the amplitude of steady state oscillation at resonance is ________mm.CorrectIncorrect - Question 16 of 21 The spring constant of a helical compression spring DOES NOT depend onCorrectIncorrect - Question 17 of 21 For a floating body, buoyant force acts at theCorrectIncorrect - Question 18 of 21 A plastic sleeve of outer radius r0 = 1 mm covers a wire (radius r = 0.5 mm) carrying electric current. Thermal conductivity of the plastic is 0.15 W/m-K. The heat transfer coefficient on the outer surface of the sleeve exposed to air is 25 W/m2-K. Due to the addition of the plastic cover, the heat transfer from the wire to the ambient willCorrectIncorrect - Question 19 of 21 Which of the following statements are TRUE with respect to heat and work? (i) They are boundary phenomena (ii) They are exact differentials (iii) They are path functionsCorrectIncorrect - Question 20 of 21 Propane (C3H8) is burned in an oxygen atmosphere with 10% deficit oxygen with respect to the stoichiometric requirement. Assuming no hydrocarbons in the products, the volume percentage of CO in the products isCorrectIncorrect - Question 21 of 21 Consider two hydraulic turbines having identical specific speed and effective head at the inlet. If the speed ratio (N1/N2) of the two turbines is 2, then the respective power ratio (P1/P2) isCorrectIncorrect

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Can you complete this jigsaw of the multiplication square? Use the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards? Use the interactivities to complete these Venn diagrams. In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time? Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark? Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots! Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time? If you have only four weights, where could you place them in order to balance this equaliser? An environment which simulates working with Cuisenaire rods. Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time. Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights? Factors and Multiples game for an adult and child. How can you make sure you win this game? What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares? Given the products of adjacent cells, can you complete this Sudoku? Given the products of diagonally opposite cells - can you complete this Sudoku? Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line. In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square? Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters. The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse? Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light? Can you explain the strategy for winning this game with any target? A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?" Got It game for an adult and child. How can you play so that you know you will always win? A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6. The clues for this Sudoku are the product of the numbers in adjacent squares. In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest? A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3. A collection of resources to support work on Factors and Multiples at Secondary level. Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit? How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction? Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3... Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice. The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box. Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst? A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target. There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements? Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers? Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all? Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice? Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on? An investigation that gives you the opportunity to make and justify predictions. List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it? Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it? Can you work out some different ways to balance this equation? Have a go at balancing this equation. Can you find different ways of doing it? A game in which players take it in turns to choose a number. Can you block your opponent? Does this 'trick' for calculating multiples of 11 always work? Why or why not? Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65. Play this game and see if you can figure out the computer's chosen number.

In my last post, I considered Wisdom of Crowds as a way to increase team strength from a macro perspective. On the other hand, communication with each person is also very important to improve team performance. No matter how many structural improvements you make, if you do not communicate well with each person, it does not seem to lead to good results. In that communication, when I work as a PM or team manager, there are so many situations where I ask questions. I ask a variety of questions in different situations, such as to learn something, to get an opinion, or to check on a situation. Nonetheless, I am always nervous when I ask questions. I am not very used to it. Why is that? The reason is that asking a question itself can easily create a pressure, depending on how I ask the question, and then affect the quality of communication, which in turn greatly affects the relationship beyond that. Therefore, I am very careful when I ask a question about something and I make up my words very carefully. Especially if it is an online chat. Of course, the specific question will naturally vary depending on the content and the situation. However, regardless of the situation or content of the question, I think there is something fundamentally important. I would like to write about that. Tell the intent of the question at first I see this practiced very often by people who are aware of it. For example, when checking on the progress of a project, whether you just want to know, whether you want to know in order to report to someone, whether you are aware that something doesn’t seem to be going well and want to be sure, whether you already have an idea for improvement that might be useful, etc., the context of the question varies. The background of the question can vary. On the other hand, if you simply ask, “What is the progress of the project?” the person being asked the question will not know from what perspective to answer. To avoid this ambiguity, when asking a question, I discuss the intent of the question and the assumptions I will make after hearing the answer “before asking the question”. This simplifies the conversation and avoids questions and answers like “finding out each other’s real intentions”. This is also true when you are on the receiving end of a question. As a matter of course, if you are unsure of the intent of a question, you should check the intent before answering. It may be even better to assume the intent of the question, such as “If this is the intent, then the answer is this way,” or “If this is a different intent, then the answer is that way,” and then present the answers to each assumption as a choice. Asking and answering questions without sharing the intent of the question can create a gap where you do not hear what you wanted to hear or communicate what you wanted to communicate. And if the gap is left unresolved, it will hinder the creation of a trusting relationship. It is very basic, but in any situation, it is a good idea to politely share the purpose and intent of the question. But on the other hand, no matter how carefully you share the intent of the question, in many cases the question can create a certain pressure. In the case of a question about the status of an earlier project, it is an obsession with the person answering: “I’m sure you want me to answer that the project is going well.” This is the sense of pressure. Sharing the purpose of the question seems to have the effect of reducing gaps in understanding, but not so much of reducing the sense of pressure. Therefore, we take a different approach to reducing the sense of pressure. Direct the question to things, not to the person Often I see people practicing this. The structure is to direct questions to things and think about them together. For example, instead of “How is the project progressing?” instead of “How do you think the project is progressing?” Instead of directing the question at the person, they direct the question at the thing. This method is very easy to put into practice because it requires only a slight change in word choice to change the impression. In the previous example, the question is simply changed from “How is it?” to “What do you think about it?” It is a small thing, but by directing the question to things rather than the person, you can expect to reduce the psychological pressure to some extent. However, if the person you are talking to is a stakeholder in the thing, it is up to the person to determine whether or not it is effective to direct questions to the thing. This is because it is quite difficult to put things in an objective position. If the person is used to separating “things” from “himself/herself,” it can be effective, but if that is difficult, it is difficult to avoid the feeling of being “attacked. Also, directing the question indirectly at the thing may sound like a indirect “sarcasm”. In other words, while the wording is easy to create, it is a risky method that can have negative effects if used in the wrong situation. Therefore, unexpectedly, there are not many situations in which it can be used effectively. So, let me consider a different method. Direct the question to yourself I do not see many people practicing this. I, on the other hand, often do this method consciously. The way I do this is to direct the question of what I want to confirm with the person to myself, not to them. In other words, instead of asking the other person about the content, I tell them my assumptions and then ask them to point out whether my understanding is correct and what is different. Continuing to use the previous example, when asking about the progress of a project, instead of asking the person to confirm progress, you could say, for example, “I feel like we’re about a week behind in my assumptions, and the reason I think that is…. I think this is the kind of idea we might have to try to solve this, what do you think about this understanding and idea?” And so on. By making the question a question about your own understanding, rather than directed at the person you are asking, you change the direction of the question to you. The good thing about this method is that the question is directed to you, which more certainly reduces the pressure they feel. As is often said, asking “Why?” is a very strong pressure. However, it is not as easy as saying, “I think this is the reason. Am I right?” If the answer is “yes”, the answer will be “yes”, and if the answer is wrong, the answer will be “no, actually, this is the reason that..”. In other words, instead of answering the question, they should be teaching you. This will reduce the pressure on the respondent to answer the question, who may feel judged. Another benefit from a different accuracy perspective is that talking about what you assume and understand beforehand will help the person understand the extent of your understanding. By communicating your understanding of the question to the partner in advance, it is easier for the person to confirm that the content of the answer matches the questioner’s expectations. If the person understands your expectations, they will naturally be more likely to give you an answer from the perspective you expect. On the other hand, there are difficulties. If you want to talk about your own assumptions, you need to be observant on a daily basis. You cannot talk about assumptions if you do not know anything about them. You need to imagine what is happening based on daily communication, information obtained, and changes in the atmosphere. Without assumptions, it becomes very difficult to direct questions to yourself. Thus, it is not as easy as it may seem to direct questions to yourself. Conversely, if you can have a conversation in which you direct the questions to yourself, it seems to confirm that you are making appropriate observations. However, it is not necessary that the assumptions you make be 100% correct. If they are wrong, you can simply ask them to correct them, so it is perfectly functional even if they are not correct. In fact, it is sometimes better to be wrong occasionally so that the conversation is less pressure. Of course, you must avoid a situation where you are so wrong or constantly wrong that the person is hesitant to point it out. In summary, directing questions to oneself is a less risky way to remove unnecessary pressure, but it is also more difficult because it tests your ability to observe and understand. At the beginning of this article, I said that I get nervous when I ask questions because I often ask questions in this pattern, and I am always nervous about whether my assumptions are too far off the mark. And this method is very effective in many situations, especially when talking to a person with whom you do not yet have a trusting relationship. The difficulty of coming up with assumptions is even higher when the relationship of trust is still very weak, since most of the time you don’t have much information, but you often manage to find assumptions based on little information or past patterns. At the end I have introduced three ways to ask questions that avoid pressure. 1. Tell the intent of the question at first 2. Direct the question to things, not to the person 3. Direct the question to yourself The first was very basic, the second was easy but risky and required consideration of the situation in which it would be used, and the third was effective but more difficult. Of course, the purpose of paying attention to the way you ask questions is to get them to tell it as it is as much as possible. If there is a relationship of trust and it is perfectly acceptable to ask straightforwardly, it is quicker to omit redundant expressions and ask questions in a direct way, such as “Why?” If I choose my words carefully when asking questions, I am sometimes asked to be more straightforward. If they say so, I feel comfortable because I can ask straight questions without hesitation. On the other hand, however, it is also true that once the pressure is given, it does not go away. Therefore, if there is even the slightest uncertainty, make the question as polite as possible, and if the person feels troublesome, gradually change to more direct expressions. I think it is better to take steps in this way, so that the quality of communication can be higher without risk. Asking questions is a regular, everyday act that I do countless times. Therefore, it seems to me that the accumulation of small differences can have a significant impact on the performance of a team. I would like to be even more careful myself.

By Elizabeth Louise Mansfield This publication explains contemporary leads to the speculation of relocating frames that drawback the symbolic manipulation of invariants of Lie crew activities. particularly, theorems about the calculation of turbines of algebras of differential invariants, and the family they fulfill, are mentioned intimately. the writer demonstrates how new rules bring about major development in major purposes: the answer of invariant usual differential equations and the constitution of Euler-Lagrange equations and conservation legislation of variational difficulties. The expository language used this is basically that of undergraduate calculus instead of differential geometry, making the subject extra obtainable to a scholar viewers. extra refined principles from differential topology and Lie idea are defined from scratch utilizing illustrative examples and workouts. This ebook is perfect for graduate scholars and researchers operating in differential equations, symbolic computation, functions of Lie teams and, to a lesser quantity, differential geometry. Read or Download A Practical Guide to the Invariant Calculus PDF Similar topology books A. N. Kolmogorov (b. Tambov 1903, d. Moscow 1987) used to be the most remarkable mathematicians that the area has ever identified. tremendously deep and inventive, he used to be in a position to process each one topic with a very new viewpoint: in a number of superb pages, that are versions of shrewdness and mind's eye, and which astounded his contemporaries, he replaced significantly the panorama of the topic. Spencer Bloch's 1979 Duke lectures, a milestone in glossy arithmetic, were out of print nearly given that their first booklet in 1980, but they've got remained influential and are nonetheless the simplest position to benefit the guiding philosophy of algebraic cycles and factors. This version, now professionally typeset, has a brand new preface by means of the writer giving his standpoint on advancements within the box during the last 30 years. Das challenge der Unterscheidung und Aufzählung verschiedener Formen von Knoten gehört zu den ältesten Problemen der Topologie. Es ist anschaulich und doch überraschend schwierig, und es stand und steht im Zentrum eines reichhaltigen Geflechts von Beziehungen zu anderen Zweigen der Mathematik und der Naturwissenschaften. Descriptive Set concept is the learn of units in separable, entire metric areas that may be outlined (or constructed), and so might be anticipated to have particular houses no longer loved through arbitrary pointsets. This topic was once begun by means of the French analysts on the flip of the twentieth century, so much prominently Lebesgue, and, first and foremost, was once involved basically with setting up regularity homes of Borel and Lebesgue measurable features, and analytic, coanalytic, and projective units. - Laminations and Foliations in Dynamics, Geometry and Topology: Proceedings of the Conference on Laminations and Foliations in Dynamics, Geometry and ... at Stony Brook - An introduction to contact topology - Symplectic Topology and Floer Homology: Volume 1, Symplectic Geometry and Pseudoholomorphic Curves - Algebraic Topology (EMS Textbooks in Mathematics) - Harmonic maps, conservation laws, and moving frames Additional resources for A Practical Guide to the Invariant Calculus 1 A function f : M → R is said to be an invariant of the action G × M → M if f (g ∗ z) = f (z), for all z ∈ M. 2 Consider the conjugation action of a real matrix group M on itself, given by (A, B) → A−1 BA. Show that the functions trn : M → R given by trn (B) = trace(B n ), for n ∈ N, are invariants for this action. 22) is an example of the following construction. Let G act on both M and N and let M(M, N) be the set of maps from M to N . If both these actions are left actions then there is an induced right action of G on M(M, N ) given by (g · f )(x) = g −1 · f (g · x). The group acts on this element as v = a1 e1 + a2 e2 + · · · + an en . 38) above, we obtain by collecting terms, v = a1 e1 + a2 e2 + · · · + an en . Then a = (a1 , . . , an ) → a = (a1 , . . , an ) is a right action. 15 Show that if g has matrix A with respect to the basis ei , Aij ej , then a = aA. i = 1, . . , n, so that ei = j Similarly, we have actions induced on the dual of S n (V ). A typical element in S n (V ) is written as a symbolic polynomial in the ei ; since the products are symmetric, this makes sense. 13 A set T of invertible maps taking some space X to itself is a transformation group, with the group product being composition of mappings, if, (i) for all f , g ∈ T , f ◦ g ∈ T , (ii) the identity map id : X → X, id(x) = x for all x ∈ X, is in T , and (iii) if f ∈ T then its inverse f −1 ∈ T . † More technically, a submanifold. 18 Actions galore The associative law holds automatically for composition of mappings, and thus does not need to be checked. Matrix groups are groups of linear transformations since matrix multiplication and composition of linear maps coincide.

For mathematicians and computer scientists, 2020 was full of discipline-spanning discoveries and celebrations of creativity. Several long-standing problems yielded to sustained collaboration, sometimes answering other important questions as a happy byproduct. While some results had immediate applications, with researchers improving on the findings or incorporating them into other work, others served for now as inspiration, suggesting that progress is within reach. Early in the year, Quanta described how five computer scientists established limits on the ability of entangled quantum computers to verify problems. As part of their work, the team also answered long-standing questions in physics and mathematics — much to the surprise of the researchers who had been working on those problems. Another set of collaborations strengthened a far-reaching bridge connecting distant areas of mathematics. Known as the Langlands correspondence, the conjectured bridge offers hope of deepening our understanding of many subfields of mathematics. This year we also explored mathematicians’ growing familiarity with geometric constructs, examined how computer programs are helping mathematicians with their proofs, and surveyed the current state of mathematics and its problems. But not all the news this year was welcome: the spread of COVID-19 complicated the research of working mathematicians, who increasingly rely on collaboration to push the field forward. The pandemic also claimed the life of the great mathematician John Conway about a month before we broke the news that a graduate student had solved a famous problem involving his signature knot. Occasionally, a scientific result is so important, multiple disciplines are forced to take notice. Such was the case in January with a landmark proof simply titled “MIP* = RE.” Written by five computer scientists, the paper establishes that quantum computers calculating with entangled qubits can theoretically verify the answers to an enormous set of problems. Along the way, the researchers also answered two other major questions: Tsirelson’s problem in physics, about models of particle entanglement, and a problem in pure mathematics called the Connes embedding conjecture. Of course, for researchers whose work involved this conjecture — which states that infinite-dimensional matrices can always be approximated with finite ones — abruptly learning from an outside paper that it’s false was quite a shock. Mathematicians must now revisit other assumptions relating to these matrices, while hurriedly learning enough computer science to understand the paper. Computer scientists also triumphed this year in dealing with the famous traveling salesperson problem, which concerns how to find the shortest round trip for any collection of cities. In July, three computer scientists used a mathematical discipline called the geometry of polynomials to show that a modern algorithm is guaranteed to be at least infinitesimally more efficient than the long-standing best method. A difference of at least “0.2 billionth of a trillionth of a trillionth of a percent” might not sound like much, but it proved that progress is indeed possible on a problem that’s lingered for decades. The proof of Fermat’s Last Theorem nearly three decades ago was lauded by math journals and newspapers around the world. But it was also just the beginning of a larger effort. The theorem established a kind of bridge between distant mathematical continents, with certain algebraic equations on one side and a kind of symmetric organization of geometric tilings on the other. Known as the Langlands correspondence, this bridge received major upgrades when two papers dramatically expanded the kinds of equations and tilings that are now connected and eliminated long-standing barriers to further expansions. “There are some fundamental number-theoretic phenomena that are being revealed, and we’re just starting to understand what they are,” said Matthew Emerton of the University of Chicago. In other number news, Vesselin Dimitrov used another well-known bridge — connecting polynomials to power series — to quantify exactly how certain numerical solutions to polynomials work to geometrically repel each other. Quanta also explored the power of representation theory, which shows the links connecting complicated objects called groups with the much simpler concept of matrices. All of these results show the importance of considering existing mathematical ideas in new contexts in order to figure out if a problem is even currently solvable. The Oxford mathematician James Maynard, for example, regularly spends time attacking famously difficult problems and has stubbornly refused to accept defeat, instead wresting new insights from the gaps between prime numbers. Many mathematical questions don’t have real-life consequences, but in March, Quanta took on no less than the geometry of the universe itself. Our exploration of the mind-bending prospects of living in flat, spherical and hyperbolic geometries (the most likely options given current data) revealed a hall-of-mirrors existence where you see infinite copies of yourself, or a world where companions grow larger as they move away. While certain clues suggest our universe is likely to be a flat one, it could be that it only seems flat, just as the Earth appears self-evidently flat when you’re standing on it. On a less cosmic front, two mathematicians finally cracked an old problem in May about what kind of rectangles can be found by connecting points on a smooth and continuous closed loop. By reimagining the possible rectangles as collections of points within a special version of four-dimensional space, the pair found that all such loops contain sets of points that define rectangles of any desired proportion. Also in May, a trio of mathematicians resolved a basic question about the dodecahedron (a 12-sided object that, to our columnist Robbert Dijkgraaf, exemplifies a form of mathematical beauty). They showed that it is indeed possible to trace a round trip over the surface of the shape starting at one of the corners without passing through any others — in fact, they found that an infinite number of such paths exist. For decades, mathematicians have used computer programs known as proof assistants to help them write proofs — but the humans have always guided the process, choosing the proof’s overall strategy and approach. That may soon change. Many mathematicians are excited about a program called Lean, an efficient and addictive proof assistant that could one day help tackle major problems. First, though, mathematicians must digitize thousands of years of mathematical knowledge, much of it unwritten, into a form Lean can process. Researchers have already encoded some of the most complicated mathematical ideas, proving in theory that the software can handle the hard stuff. Now it’s just a question of filling in the rest. One major test for the software could come next year at the International Mathematical Olympiad, or IMO. Daniel Selsam of Microsoft Research has founded the IMO Grand Challenge, which hopes to develop an artificial intelligence, using Lean, that can win a gold medal at the math competition. We’re still a long way from computers replacing humans, of course, and many mathematicians still don’t fully embrace these programs. But computers are now mainstream in mathematical research, with their sheer computing power proving essential for answering certain kinds of big questions, such as whether multidimensional squares must precisely share edges. If you’re interested in math, but you can’t tell a Calabi-Yau manifold from a finite field, perhaps our Map of Mathematics can help. Organized around three starting points — numbers, shapes and change — the map provides a crash course in the current state of mathematics, as understood and practiced by mathematicians. While it’s obviously not a comprehensive look at the subject, our biggest goal was to illustrate not just the most important mathematical concepts, but also their relationships to one another. Other potentially helpful dives into fundamental mathematical ideas include an explanation of Gödel’s incompleteness theorems, which proved that all mathematical systems have some unprovable statements, and a discussion of how the alternative number systems known as the p-adic numbers work and why they’ve proved so helpful for understanding rational numbers. Successful mathematicians are often creative individuals, capable of spotting new connections and finding new approaches to old problems. In February, the Annals of Mathematics published a proof by Lisa Piccirillo, who had dusted off some long-known but little-utilized mathematical tools to answer a decades-old question about knots while still a graduate student. A particular knot named after the legendary mathematician John Conway had long evaded mathematical classification in terms of a higher-dimensional property known as “sliceness.” But by developing a version of the knot that yielded to traditional knot analysis, Piccirillo finally determined that the Conway knot is not “slice.” Unfortunately, Conway himself died of COVID-19 in April, and his wife confirmed in our comments section that he didn’t know about Piccirillo’s result. Conway’s own contributions extended far beyond knot theory — he enriched group theory, number theory, analysis and more, while always delighting in games and puzzles. Quanta paid tribute to him with our October Insights puzzle, which included a numerical riddle he invented and other games based on or inspired by his work.

How to perform a Kaplan-Meier survival analysis in a statistics exam? The way I did it: And now to summarize it, I wanted to list the most used in the statistic exam: I’ve read this question out every few years. Originally they said “HACK” In a similar thread to this one, I tried to write a little test: And now to summarize it: 3 Survival – Prob Note how the times are different – 1 is not perfect – 5 is. Of course this is interesting, as Ive never done it before. Every you need to compare your survival rate to estimate t < 10, if the t is smaller. I would like you to go fix your t = 2 and compare your t --100, this will give you the probability that your t is above 10. I just wanted to point out that the t in this question is in the range: 05201051013 So if my answer is 0130, then my answer is 0130. If my answer is 0150, then my answer is 0140. If my answer is 0150, then my answer is 0130. The kind of thing I would recommend is to run f5 for 5 things within a few seperate steps to identify the most crucial parts of the statistic exam. If you chose as your t (so you don't forget to write it before reading it, and that t and 5 are the groups you wouldnt like to enter), now you do that. And f5 is a simple way with many f5 variables. Use the most convenient f5 for smaller tests. Perhaps it might be enough to write it out as: Here I have a function f5 which does what I want it to do: and I think you can write it as 7 instead of 6 or maybe 6 6 It is then just to show how you have worked it out. With my program I write it like this: double f5 = 5 How to perform a Kaplan-Meier survival analysis in a statistics exam? We discuss about a topic about the Kaplan-Meier survival analysis in a statistics exam, where we define the number of years following a diagnosis of lung cancer on survival. Actually there is a reason before our exam. About such individuals, the Kaplan-Meier results are normally tested by means of Kaplan-Meier curve by means of the estimation curve, when we use the distance, the likelihood and the probability function to calculate the Kaplan-Meier curves of date to death, we call the p-values the summary and the hazard. check it out is a wide discussion whether these results can be generalized to analyze the most characteristics of a population. For survival prediction we need to consider several questions. For example, whether it is possible before the age of 60 years to recognize the time of the tumor in a cohort? To this end, we present some results regarding the Kaplan-Meier curves, the curves for the survival, the prognosis data and a few survival prediction analyses based on the data for age and type of tumor in our field. 1. How Can I Get People To Pay For My College? 1 Is the p-value for many such studies valid instead of using p-values in the Kaplan-Meier analysis? Many of the most complex and important investigations into the statistics exam, for example the results of some classic publications for survival predicting the date and time of death to death etc, have been performed, but the shape of the prognosis curve has generally not been considered. In studies of a prognosis effect is differently classified. When the prognosis is established by means of Kaplan-Meier curve, when we calculate the following three values : a; b; c, we get: The a-value is used to conduct a Kaplan-Meier curve prediction. There is a broad discussion whether one should use a-value to classify data. For the survival prediction given here use the following data A data b ifHow to perform a Kaplan-Meier survival analysis in a statistics exam? (The title of this article is wrong; I would only link it here if that makes any difference. It might help some browse around here to read the piece if there are a lot of rules and not a lot of guidelines.) Shared Clinical Medical Product Overview Introduction I am a healthcare professional and I have been an exercise in clinical statistics for 35 years and I have a clinical experience. I have practiced for 15 years starting and trying to go into the field and gained some knowledge from going into clinical statistics. I am committed to reading anything I can important source an opinion right now and what they are really interested in reading. Here are some of the questions I have: 1- What are some of the most high-profile healthcare news stories in the past decade? 2- How big has clinical statistics become? 3- is it actually gaining a lot this popularity? 4- Do they really want to know what is going on over in clinical statistics? 5- How did I get started and what seems to happen right now? Can I keep my eyes and ears open? What are the risks of doing clinical statistics nowadays? What are the benefits of going into clinical statistics? The most important thing are your eyes and ears. What is the right answer to the most important questions? what does it take to cover all of these major issues? 1- Has anyone ever faced a death due to clinical statistics? 2- How many cases of death have in body compared to what is expected on the day they die? 3- If we are going to have to make decisions over weeks, months, etc., are we just going to spend weeks or months doing clinical statistics looking at possible outcomes to make the most out of it? 4- Do clinical statistics help us with our understanding of the things that must be done to be a good clinical

Cosmic Ray Anisotropy Analysis with a Full-Sky Observatory High Energy Astrophysics Institute University of Utah Physics Department 115 S 1400 E, Room 201 Salt Lake City, UT 84112-0830 A cosmic ray observatory with full-sky coverage can exploit standard anisotropy analysis methods that do not work if part of the celestial sphere is never seen. In particular, the distribution of arrival directions can be fully characterized by a list of spherical harmonic coefficients. The dipole vector and quadrupole tensor are of special interest, but the full set of harmonic coefficients constitutes the anisotropy fingerprint that may be needed to reveal the identity of the cosmic ray sources. The angular power spectrum is a coordinate-independent synopsis of that fingerprint. The true cosmic ray anisotropy can be measured despite non-uniformity in celestial exposure, provided the observatory is not blind to any region of the sky. This paper examines quantitatively how the accuracy of anisotropy measurement depends on the number of arrival directions in a data set. The origin of the highest energy cosmic rays is a problem that has persisted for 4 decades since the pioneering measurements at Volcano Ranch [1, 2]. There is some consensus that, above the spectrum’s ankle at about eV, they originate outside the disk of the Galaxy. For particles of such high magnetic rigidity, sources in the Galaxy’s disk would presumably cause an obvious anisotropy in arrival directions that is not observed. Evidence for a composition changing to lighter particles at the ankle strengthens this argument (particles having even greater rigidity because of lesser charge) and supports the view that cosmic rays with energies above the ankle are of extragalactic origin. The sources of those particles remain to be identified. The observations of cosmic rays with energies above the expected GZK cutoff [4, 5] should be a powerful clue to the nature of the sources. The Fly’s Eye and AGASA measured air showers with energies well above the GZK threshold. Recent reports [8, 9, 10] suggest that the spectrum might continue without a strong GZK effect. These super-GZK results have posed several related, but distinguishable, puzzles: How are particles produced with such prodigious energy? Why do the arrival directions of those particles not point back to recognizable sources in our local part of the universe? Why is the intensity of particles above eV not more strongly suppressed? New attempts to improve the observational data include the recently commissioned High Resolution Fly’s Eye (HiRes) , the Pierre Auger Observatory , the proposed Telescope Array Project , and Airwatch/OWL plans for future space-based detectors of atmospheric air showers. The Auger Observatory and the potential space-based detectors will have exposure to the entire sky, which will open new possibilities for anisotropy analysis. These methods will be explored here in the context of the better controlled exposure of the Auger surface arrays. Hillas pointed out years ago that there are few astrophysical sites that can produce a large enough “electrical potential” to accelerate even highly charged nuclei to eV. (Here is the relative velocity of moving media with magnetic field strength and size . In the case of statistical acceleration, including Fermi shock acceleration, the same product governs the maximum particle rigidity even though particles do not pass through a monotonic change in electrical potential.) Puzzle #1 is exacerbated in many contexts by synchrotron radiation and/or pion photoproduction. Puzzle #2 can be resolved by invoking stronger-than-expected extragalactic magnetic fields, but that does not readily simplify puzzle #3. The expected suppression of particles above the GZK cutoff is based on travel time (or total distance traveled) rather than the straight-line distance to the sources. Particles below the GZK threshold have been accumulating over billions of years, whereas the mean age of particles well above the threshold cannot be greater than tens of millions of years . One possible inference from the lack of an observable GZK spectral break is that sub-GZK particles may not be much older than 30 million years either, in which case the GZK effect would not significantly suppress the cosmic rays above the threshold relative to those below. It is not reasonable to suppose that the sources of high energy cosmic rays turned on so recently, so the young age of sub-GZK cosmic rays would require an intergalactic mechanism for dissipating their energy. Known mechanisms (e.g. nuclear interactions, synchrotron radiation, production, pion photoproduction via infrared and visible background photons, etc.) do not rob energy from sub-GZK particles rapidly enough. Perhaps high energy cosmic rays are attenuated through interactions with the unknown dark matter of the universe . More conventional approaches to puzzle #3 are to defeat the GZK cutoff with a very hard extragalactic spectrum (e.g. from topological defect annihilation ) or to evade it by invoking neutrinos [19, 20] or non-standard particles [21, 22] that are immune to the microwave background radiation. Others [23, 24] conclude that the sources must be localized to the Galaxy, but distributed in a halo large enough that galactic anisotropy has not become obvious. The cumulative cosmic ray observations at this time are not sufficient to sort out the possibilities. AGASA and HiRes are currently building up the world’s total exposure at the highest energies. With better statistics and better measurements, the observations could soon lead to a breakthrough that identifies the sources of the highest energy cosmic rays. This same hope has been expressed for decades through the course of numerous experiments, however, and the puzzles have only become deeper mysteries. The answers may not come easily, and we should prepare the best possible analyses of the energy spectrum, particle mass distribution, and arrival directions. Careful determinations of the energy spectrum and mass composition can be used to weed out classes of theories, but these tools are not likely to yield a clear signature for picking out a unique theory. A positive identification of the cosmic ray sources requires seeing their fingerprint in the sky. This may come in the form of arrival direction clusters [25, 26] that identify discrete sources, or it may come as a large-scale celestial pattern that characterizes a particular class of potential sources. In the worst case, we might discover that the arrival directions are isotropic and the sources still elude positive identification. In that case, observers must strive for the best possible upper limit on anisotropy. The Auger Project’s surface arrays will provide the best search for anisotropy fingerprints. Their combined exposure function on the celestial sphere will be unambiguous because they operate continuously and are not sensitive to atmospheric variability. Continuous operation means the celestial exposure function is uniform in right ascension. By having observatories appropriately located in both the southern and northern hemispheres, the exposure does not vary strongly with declination either. The methods described in this paper are applicable to any observatory with full-sky coverage. They are not limited to the Auger Project, although the specific Auger site locations are used in the example simulations that are reported here. Coverage of the full sky could be achieved piecemeal by combining results from different experiments. There is serious risk of spurious results from such meta-analyses, however, unless the exposures, energy resolutions, and detector systematics are perfectly understood and correctly incorporated in the analysis. The reliable approach is to use identical detectors in both hemispheres or the same (orbiting) detector for both hemispheres. This paper seeks to evaluate the sensitivity of a full-sky observatory to large-scale anisotropy patterns and how that sensitivity depends on the number of arrival directions in a data set. Large-angle fingerprinting will be needed if there are many contributing sources or if the flux from each single source is diffused over a large solid angle due to magnetic deflection of the charged cosmic rays. If, instead, there are point sources to be detected, then the advantage of a full-sky observatory is in mapping the entire celestial sphere with comparable sensitivity in all regions. Full-sky coverage is crucial for large-scale anisotropy analysis. It makes it possible to do integrals over the sky, so the powerful tools of multipole moments and angular power spectra are available. With full sky coverage, cosmic ray anisotropy analysis will be similar to gamma ray burst anisotropy analysis. The numbers of events will be comparable, the direction error boxes will be comparable, the exposure non-uniformities will be comparable, and in both cases events come from all parts of the sky. All of the techniques that were employed to search for anisotropy in the BATSE data [27, 28] can be applied to a full-sky cosmic ray data set. With a cosmic ray detector in only one hemisphere, there is a solid angle hole in the sky where the detector has zero exposure despite the Earth’s rotation. A zero-exposure hole makes it impossible to do integrals over the whole celestial sphere. No matter how many events the detector collects overall, it will never determine any multipole moment. A single-hemisphere detector can test hypotheses like, “Does the observed distribution match better what would be accepted from the clustering of radio galaxies toward the supergalactic plane or what would be accepted from an isotropic distribution?” It can also make qualified measurements like, “Assuming the anisotropy is a perfect quadrupole with axial symmetry, fit for the axis orientation that best explains the observed celestial distribution.” The role of an observatory, however, should be to map the sky and make results available in a form which is readily usable without knowledge of the detector properties and which is independent of any theoretical hypothesis. Low-order multipole tensors (or spherical harmonic coefficients) can summarize the large-scale information. The angular power spectrum reveals if there is clumpiness on smaller scales. These results can be tabulated so that theorists can test arbitrary models quantitatively without privileged access to the data. With approximately uniform exposure, even eyeball inspection of arrival direction scatter plots can show large-scale patterns that are hidden when steep exposure gradients dominate the scatter plots. While the role of an observatory should be to map the sky and determine the patterns without preconceived expectations, it is nevertheless worthwhile to consider what might be learned by measuring the low order multipoles or the angular power spectrum. Monopole. There is no information about anisotropy patterns in the monopole scalar by itself. It is simply the sky integral of the cosmic ray intensity. That is information already present in the energy spectrum. A pure monopole intensity distribution is equivalent to isotropy. The strength of other multipoles relative to the monopole is a measure of anisotropy. Dipole. A pure dipole distribution is not possible because the cosmic ray intensity cannot be negative in half of the sky. A “pure dipole deviation from isotropy” means a superposition of monopole and dipole, with the intensity everywhere . A predominantly dipole deviation from isotropy might be expected if the sources are distributed in a halo around our Galaxy, as has been suggested [23, 24]. In this case, there is a definite prediction that the dipole vector should point toward the galactic center. An approximate dipole deviation from isotropy could be caused by a single strong source if magnetic diffusion or dispersion distributes those arrival directions over much of the sky. In general, a single source would produce higher-order moments as well. A dipole moment is measurable in the microwave radiation due to Earth’s motion relative to the universal rest frame . If we are moving relative to the cosmic ray rest frame, a dipole moment should exist also in the cosmic ray intensity (the Compton-Getting effect). At lower energies, this may occur if the sun and Earth are moving relative to the galactic magnetic field or if the cosmic rays are not at rest with respect to the galactic field. For extragalactic cosmic rays, a Compton-Getting dipole is expected if the Galaxy is moving relative to the intergalactic field or if the cosmic rays themselves are streaming in intergalactic space. In any case, the expected velocities would be small (), and the Compton Getting anisotropy () should be . (Here is the differential spectral index, which is roughly 3.) An anisotropy of one-half percent would require high statistics for detection (see section 3). A larger dipole anisotropy might be produced by a cosmic ray density gradient. If the magnetic field is disorganized, the gradient produces streaming by diffusion and the Compton-Getting dipole vector is parallel to the density gradient. If there is a regular magnetic field, however, the expected dipole vector can be perpendicular to both the gradient and the field direction, . The direction of strongest intensity corresponds to the arrival direction of particles whose orbit centers are located in the direction of increased density. Quadrupole. An equatorial excess in galactic coordinates or supergalactic coordinates would show up as a prominent quadrupole moment. A measurable quadrupole is expected in many scenarios of cosmic ray origins, and is perhaps to be regarded as the most likely result of a sensitive anisotropy search. In general, a quadrupole tensor is characterized by 3 relative eigenvalues with associated orthogonal eigenvectors. In the case of axial symmetry, there is a single non-degenerate eigenvector that gives the symmetry axis. An axisymmetric “prolate” distribution would be hot spots at antipodal points of the sky, whereas an “oblate” distribution has the excess concentrated toward the equator that is perpendicular to the symmetry axis. The axis of an oblate quadrupole distribution might differ from the galactic axis or the supergalactic axis if we are embedded in a magnetic field that systematically rotates the arrival directions. The angular power spectrum. Spherical harmonic coefficients for a function on a sphere are the analogue of Fourier coefficients for a function on a plane. Variations on an angular scale of radians contribute amplitude in the modes just as variations of a plane function on a distance scale of contribute amplitude to the Fourier coefficients with . For cosmic ray anisotropy, we might look for power in modes from (dipole) out to , higher order modes being irrelevant because the detector will smear out any true variations on scales that are smaller than its angular resolution. For charged cosmic rays, magnetic dispersion will presumably smear out any point source more than the detector’s resolution function. Even at the highest observed particle energies, there is unlikely to be any structure in the pattern of arrival directions over angles smaller than . The interesting angular power spectrum is therefore probably limited to . For a cosmic ray observatory, exposure is a function on the celestial sphere. Measured in units , it gives the observatory’s time-integrated effective collecting area for a flux from each sky position. In this paper, the relative exposure is usually the function of interest. That will be a dimensionless function on the sphere whose maximum value is 1. In other words, at any point of the sky is a fraction between 0 and 1 given by the exposure at that point divided by the largest exposure on the sky. In other contexts, the term “exposure” refers to the total exposure integrated over the celestial sphere. It then has units . For example, in determining the cosmic ray energy spectrum, one divides the number of cosmic rays observed in each energy bin by the total exposure for that energy. (In general, an observatory’s exposure is energy dependent.) If there were evidence that the energy spectrum were not uniform over the sky, then we would need to use the exposure’s dependence on celestial position to map the spectrum over the sky. Since the spectrum is defined by the number of observed events divided by total exposure, one can use the measured spectrum to get the expected number of cosmic rays for any given total exposure. In the case of the Auger surface arrays, the continuous acceptance is approximately , independent of energy above eV. After operating for 5 years, they will have a total exposure of . The integral cosmic ray intensity above eV is approximately , and it falls roughly like (perhaps less rapidly, but the energy dependence is not well determined above eV.) Using this simple dependence gives the following estimates for Auger cosmic ray counts after 5 years: 35,000 above eV (believed to be mostly extragalactic) 2200 above eV (compared to 47 in the AGASA cluster analysis) 350 above eV (above the GZK threshold region) 35 above eV (highest energy measured so far) How any number of detected cosmic rays are distributed on the sky depends on both the true celestial anisotropy and the observatory’s relative exposure . The relative exposure can be calculated as follows for a detector at a single site with continuous operation. Full-time operation means that there is no exposure variation in sidereal time and therefore constant exposure in right ascension. Suppose the detector is at latitude and that it is fully efficient for particles arriving with zenith angles less than some maximum value . (Full efficiency means the zenith angle acceptance depends on zenith angle only due to the reduction in the perpendicular area given by .) This results in the following dependence on declination : where is given by The upper left plot of figure 1 shows the resulting declination dependence for a site at and another site at , which are the latitudes for the two Auger observatories. The detectors are assumed to be fully efficient out to , and no arrival directions are counted from larger zenith angles. The combined exposure is also shown. The maximum is at the north pole direction, which is always detectable at the northern site, although the effective detector area is reduced by for flux arriving from the north pole direction. The lower plots in figure 1 are scatter plots of the accepted cosmic rays for each site, where directions have been sampled from an isotropic distribution but accepted according to each detector’s exposure. (A sampled cosmic ray direction is accepted if a randomly sampled number between 0 and 1 is less than the relative exposure for that direction.) There are 10,000 accepted arrival directions in each of those two plots. Shown in the upper right plot is the superposition of all 20,000 events from the combined observatory. The combined distribution is not uniform, but has the modest declination dependence indicated in the upper left plot. 3 Dipole sensitivity The objective here is to study the sensitivity of a full-sky observatory to a dipole deviation from isotropy. How well can the dipole be measured? How does that accuracy depend on the number of arrival directions in the data set? How does it depend on the amplitude of the dipole anisotropy? For a dipole deviation from isotropy, the cosmic ray intensity varies over the sky as Here is a unit vector defining the celestial direction, is the average intensity, is the dipole direction unit vector, and is its (non-negative) amplitude. In order for the cosmic ray intensity to be nowhere negative, must lie in the range . The amplitude gives the customary measure of anisotropy amplitude: . The dipole can be recovered from the celestial intensity function by In our case, the observed intensity function consists of discrete arrival directions, each associated with a relative exposure . The components of the dipole vector are then estimated by where denotes a component of the th vector, and is the simple sum of the weights . (These dipole components are linear combinations of the three spherical harmonic coefficients with .) To test this method’s sensitivity to a dipole of amplitude when there are N directions in the data set, one can produce an ensemble of artificial data sets of this type (with random dipole directions ). For each data set, use the above formula to estimate the dipole vector, and record the difference of the estimated from the input alpha and also the angle between the estimated direction and the input dipole direction. These error distributions describe the measurement accuracy. The RMS deviation from the true is a single number to characterize the amplitude measurement accuracy, and the average space angle error summarizes the accuracy of determining the dipole direction. This procedure can be repeated for different values of and different values of . For any pair () the ensemble of simulation data sets yields the amplitude resolution and direction resolution as above. To generate an individual simulation data set, one samples arrival directions on the celestial sphere. First, a direction is sampled from the assumed celestial distribution with dipole deviation from isotropy. Then the detector acceptance is applied by rejecting the sampled direction if a random number is greater than the relative exposure for that direction. This continues until the data set has arrival directions. The data set then reflects both the presumed celestial anisotropy and the detector’s non-uniform exposure. These methods yield the results summarized in figure 2. The number of arrival directions was increased by factors of two: = 250, 500, 1000, 2000, 4000, 8000, 16000, 32000. For each , amplitudes were studied at = 0.1, 0.2,…, 1.0. The upper left plot of figure 2 shows that the amplitude is determined to an accuracy of about 0.1 with 250 directions, improving to approximately 0.01 with 32,000 directions. The upper right plot shows the dipole direction resolution as a function of the number of arrival directions. The mean error is less than 10 degrees for all cases (250 directions or more) if the amplitude is nearly 1, and it is less than 10 degrees regardless of the amplitude if the number of directions is 16,000 or more. The lower left plot shows that the mean dipole direction error decreases as the dipole amplitude increases. A strong amplitude yields a good direction determination even for a small number of directions in the data set. For example, with only 250 cosmic ray arrival directions, the dipole direction is determined to better than 20 degrees if the amplitude exceeds 0.5. To get that same resolution with =0.1, you need a data set with more than 4000 directions. For a fixed number of arrival directions, the RMS error in the amplitude has little dependence on the amplitude. That is to say, you can distinguish amplitudes 0.85 and 0.90 as well as you can distinguish 0.10 and 0.15. For the purpose of detecting an anisotropy (as opposed to measuring it), the relevant quantity is the amplitude divided by the RMS error, which is the number of sigmas deviation from isotropy. That quantity increases with the amplitude . It can also be expected to increase in proportion to as the number of arrival directions increases. The lower right plot shows that For =250, for example, the deviation from isotropy increases from 1-sigma to 10-sigma as increases from 0.1 to 1.0. For =8000, the range is from 6-sigma to 60-sigma. Etc. To achieve a 5-sigma detection of a Compton-Getting anisotropy amplitude of 0.005 would require 2.4 million arrival directions. An anisotropy amplitude of 0.2 from a galactic halo distribution of sources, however, could be detected at the 5-sigma level with 1500 arrival directions. 4 Quadrupole sensitivity A quadruople deviation from isotropy is characterized by an intensity function on the celestial sphere given by where is an arbitrary direction unit vector and is a symmetric 2nd order tensor. Its trace gives the monopole moment. Its other 5 independent components in any coordinate basis are determined from the spherical harmonic coefficients . Denoting by the three eigenvalues of and the three (unit) eigenvectors by , the intensity function has the form To keep the number of studied variables manageable, consideration will be limited to axisymmetric oblate intensity functions, as might be expected from sources in the galactic disk or near the supergalactic plane. Let denote the eigenvector that defines the symmetry axis, and let be the ratio of its eigenvalue to those in the symmetry plane, so the intensity function on the sphere is of the form where is the part of perpendicular to the symmetry axis, and . The anisotropy amplitude is related to by The objective here is to test how accurately the anisotropy amplitude and the symmetry axis direction can be determined from a data set of arrival directions. How does the accuracy depend on the number of directions and on the amplitude ? The method of investigation is the same as for the dipole sensitivity study. For each pair (,), an ensemble of simulation data sets are produced, each with a randomly chosen direction for its symmetry axis. For each data set, the arrival directions are sampled from the relative intensity function (with quadrupole anisotropy), and each direction is accepted with probability equal to the relative exposure evaluated at that direction. The anisotropy amplitude and symmetry axis direction are estimated for each simulation data set. The tensor with components ( denoting a component of ) has the same eigenvectors as and the components can be estimated by The eigenvectors and eigenvalues of this symmetric matrix are then found. The symmetry axis is taken to be defined by the eigenvector with the smallest eigenvalue. Let be that smallest eigenvalue subtracted from the average of the other two (which should be equal, corresponding to directions in the symmetry plane). The eigenvalue of the intensity tensor is given in terms of by Then the anisotropy amplitude is gotten by . Results for ensembles with different () values are presented in figure 3 in complete analogy with the dipole results presented in figure 2. The RMS error in the amplitude and the average space-angle error in the direction of the symmetry axis both decrease as increases. The symmetry axis is also seen to be better determined as the anisotropy amplitude increases for any fixed number of arrival directions . The sensitivity for detecting anisotropy, as shown in the lower right plot, is given by With the definition of anisotropy amplitude , detecting a quadrupole anisotropy requires more data than for a dipole anisotropy of the same amplitude. Twice as many cosmic ray arrival directions () are needed for the same resolution. 5 Spherical harmonics For any data set of arrival directions (with full-sky exposure), the anisotropy patterns can be fully characterized by the set of spherical harmonic coefficients , in terms of which the intensity function over the sphere is given by The coefficients are given by Real-valued spherical harmonics are used in this paper, so the coefficients are real. The real-valued functions are obtained from the complex ones by substituting For a set of discrete arrival directions with non-uniform relative expousre , the estimate for is given by where is the relative exposure at arrival direction and is the simple sum of the weights . The upper left plot in figure 4 shows a scatter plot of 2921 directions to extragalactic infrared sources with , obtained from the NASA/IPAC Extragalactic Database (NED) . There are certainly selection effects in these directions, but they are used here only as an example of an anisotropic celestial distribution. The spherical harmonic coefficients for this distribution are plotted to the right of that scatter plot. There are 440 coefficients plotted for . Each set of coefficients is plotted for each over an interval of 0.4 units on the abscissa. These constitute a “fingerprint” of the anisotropy. They define a celestial intensity function that is a smoothed version of the scatter plot. The prominent coefficients and in this example result from the strong excess of the Virgo Cluster seen in the left central part of the scatter plot. Virgo is at declination 12.7 degrees and right ascension 187 degrees, and the coefficients are derived here using that equatorial coordinate system (not the plotted supergalactic coordinate system). To illustrate how well the coefficients characterize the anisotropy, arrival directions can be sampled from the intensity function that they define. The lower left plot is a scatter plot with the same number of directions (2921) based on the relative intensity function The Auger exposure function has also been imposed. The lower left plot should not be identical to the upper left plot because of the Auger exposure simulation as well as the random sampling from the smoothed celestial anisotropy function. It clearly does have the same primary features, however. The lower right plot in figure 4 summarizes how well the anisotropy is determined this way as a function of the number of arrival directions. The filled circles in that plot correspond to the displayed scatter plot with 2921 arrival directions. The coefficients were derived from that simulation data set (using the relative exposure weights) and compared with those derived from the infrared source distribution. For each , the RMS difference in the values is plotted. One can see that the typical error in is small compared to the significant coefficients in the upper right plot that characterize the anisotropy. The asterisks in the lower right plot are derived in the same way using a simulation with 1000 arrival directions. The open circles are the RMS coefficient differences resulting from a simulation with just 250 sampled arrival directions. The anisotropy fingerprint in this example is still measurable with 250 directions, although the RMS uncertainty in the coefficients grows as as decreases. 6 The angular power spectrum The angular power spectrum is the average as a function of : The power in mode is sensitive to variations over angular scales near radians. The angular power spectrum provides a quick and sensitive method to test for anisotropy and to determine its magnitude and characteristic angular scale(s). As an example, consider the distribution of galaxies shown in the upper left plot of figure 5. These are all galaxies with redshift (also obtained from NED), and they are plotted in the supergalactic coordinate system. The Virgo cluster is the highest density region toward the left in this plot. There are 7321 galaxy positions plotted. The angular power spectrum (with no exposure correction) is shown out to in the upper right plot of that figure. There is excess power at all -values, but especially for the dipole and quadrupole moments ( = 1, 2) due to the high intensity from Virgo and other parts of the supergalactic plane. The lower left plot in figure 5 indicates how sensitivity to the power spectrum is affected by the number of arrival directions (with non-uniform exposure as is expected for the Auger Observatory). Open circles in that plot are the power spectrum derived using a data set of 250 arrival directions. Those directions were obtained by randomly sampling from the 7321 galaxy directions (without replacement) and rejecting a sampled direction if a random number between 0 and 1 fell above the relative exposure evaluated at that direction. The 250 directions therefore represent a simulation Auger data set if each galaxy direction were to have equal probability of being a cosmic ray arrival direction. The open circles in the lower left plot are a decent approximation to the power spectrum (filled circles) of the “true” power spectrum defined by all 7321 directions with uniform exposure. The triangles in that plot represent the power spectrum for 1000 arrival directions sampled in the same way from the 7321, and the squares are obtained from 4000 sampled with the Auger exposure. It is clear that the approximation to the true power spectrum improves as the data set gets richer, but the gross information is already present with 250 arrival directions. The lower right plot of figure 5 indicates how much power is expected due to fluctuations when directions are sampled from an isotropic intensity (and biased for the non-uniform expousre). The power is the same for all -values and decreases like as the number of arrival directions increases. The power spectrum of the galaxy distribution is well above this noise level for all -values for the cases = 4000 or 16,000. For 250 directions, only the first prominent harmonics in the lower left plot are clearly above the noise level indicated in the lower right plot. Any class of candidate objects (e.g. active galaxies, or active galaxies with giant radio hot spots) has a celestial distribution that can be compared with a cosmic ray map when the whole sky has been surveyed with adequate sensitivity. Full information about the celestial distribution is provided by the set of coefficients . They can be tabulated out to in a list of 441 numbers (including the monopole). The angular power spectrum is a coordinate-independent gross summary of the features present in the celestial distribution. For example, you may learn from it that there is a large quadrupole moment, but you do not learn if the quadrupole has axial symmetry or the orientations of its principal axes. Full anisotropy information is given by the 441 coefficients, not the 20 powers. The magnitude of the angular power for larger -values may contain useful information in the case that cosmic rays come from a limited number of discrete sources. The solid angle extent of the typical source affects the power at large values of . Figure 6 displays an example in which there are 50 sources of equal flux with positions sampled randomly on the sky. Three different sky plots are shown, corresponding to different hypotheses about how much the arrival directions are dispersed from the source direction. In the upper left plot, sampled arrival directions are accepted only if they lie within of one of the sources. In the upper right plot, they are accepted only within 5 degrees of a source, and only within 1.5 degrees in the lower left plot. The graph in the lower right shows the power spectra for the three different simulations. The power at low -values is governed by the chance pattern in the distribution of source positions. For , however, the power clearly increases as the amount of source smearing decreases. The high end of the measurable angular power spectrum is sensitive to anisotropy structure on that finer scale. There is great advantage in a cosmic ray observatory having exposure to the entire celestial sphere, especially if the relative exposure is nearly uniform. In that case, scatter plots of arrival directions are immediately interpretable, and eyeball evaluations can readily identify discrete sources or large-scale patterns. Discrete sources will be identified with equal sensitivity anywhere in the sky. If no such sources are found, the flux upper limits will be uniform over the sky. At the highest energies there is no proven anisotropy. Unlike the COBE anisotropy analysis, it is not necessary to subtract a large known dipole pattern and a myriad of uninteresting foreground sources. Any cosmic ray deviations from isotropy will be of immediate interest. The search for cosmic ray anisotropy is more similar to the case of gamma-ray bursts, where expectations and early claims of anisotropy were not supported by additional data. The role of an observatory is to map the sky and make the results available to the scientific community. This is highly challenging for an observatory without full-sky coverage. Measurements in that case are made with different sensitivity in different parts of the sky, and nothing at all can be said about a large hole where the exposure is zero. Certainly it is not possible to perform the full-sky integrations that are required to measure the multipoles of the celestial cosmic ray intensity. In this paper, frequent use is made of the inverse of the relative exposure, . Such methods obviously fail if the relative exposure anywhere becomes infinitesimal or zero. To underscore the difficulty of anisotropy analysis without full-sky coverage, one can cite the work by Wdowczyk and Wolfendale . In that paper, the authors argue that the cosmic ray intensity measurements support a model of excess arrivals from equatorial galactic latitudes. The argument is based on the same data that two experimental groups had previously used in support of a gradient in galactic latitude that suggested an excess from southern latitudes relative to northern latitudes. In effect, because those northern detectors had poor exposure for southern galactic latitudes, Wdowczyk and Wolfendale were able to argue that the data supported a quadrupole distribution rather than a dipole distribution. Neither a dipole moment nor a quadrupole moment can be measured without full-sky coverage. The techniques outlined in this paper pertain to any full-sky detector. Non-uniformity in celestial exposure is not hard to handle, provided it is well determined and there is adequate exposure to all parts of the sky. The true cosmic ray intensity is mapped with a sensitivity that depends primarily on the total number of detected arrival directions. This number is related to particle energy and observing time for any detector of known acceptance. This relationship for the Auger Observatory was given in the Introduction. While complete information about anisotropy is encoded in the coefficients (tied to some specified coordinate system), important gross properties of the anisotropy are characterized by the (coordinate independent) angular power spectrum . One can tell, for example, if there is a strong dipole or quadrupole moment. Such large-scale patterns are expected in many theories. It should be noted, however, that gives the dipole moment but not its direction. It is obviously important whether the dipole points toward the galactic center, toward Virgo, toward Cen A, or in some unexpected direction. Similarly, all components of the quadrupole tensor are of interest, not just the average of their squares, . Nevertheless, the angular power spectrum provides a powerful tool for discriminating between viable and non-viable theories without detailed investigation. Also, the higher-order moments of the angular power spectrum can quantitatively characterize whatever clumpiness may exist in a map of arrival directions. The techniques and examples mentioned in this paper are only representative of the powerful analysis methods that become possible with full-sky observatories. Data sets from such observatories will open a rich field of anisotropy study. The primary goal seen from the present time is the discovery of the highest energy cosmic ray origins. If that objective is accomplished (perhaps even before full-sky data sets are available), then the observed patterns from a known source distribution will be analyzed to infer properties of magnetic fields in the galactic halo and in intergalactic space. A full-sky observatory has the ability to summarize completely the anisotropy information with the use of the spherical harmonic expansion coefficients . A table of coefficients (perhaps with multiple columns for different energy cuts) will provide the whole story. As has been done in this paper, those coefficients will be reliably corrected for the observatory’s non-uniform exposure. Detailed anisotropy analysis will no longer require privileged access to detector data. The published anisotropy fingerprint encoded in the spherical harmonic coefficients can be matched against any theoretical suspect by any interested investigator. Acknowledgement. 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Imagine we have four bags containing numbers from a sequence. What numbers can we make now? Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice. When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . . Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . . Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . . Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why? Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal? A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . . Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. Which set of numbers that add to 10 have the largest product? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades? Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct. Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . . Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . . Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why! I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour? Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . . If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . . Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . . Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely? Can you see how this picture illustrates the formula for the sum of the first six cube numbers? Which hexagons tessellate? A huge wheel is rolling past your window. What do you see? Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total. Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article. Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number? How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest? Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions. Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits. Here are some examples of 'cons', and see if you can figure out where the trick is. Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. Can you discover whether this is a fair game? Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

explanation: dipole moment is the product of magnitude of charge and seperation between charges . In such a molecule the dipole moment of the bond also gives the dipole moment of the molecule. HI 0.42. This is done by measuring relative permittivity … Dipole Moments. Dipole moment of HCl = 1.03 D, HI-0.38 D. Bond length of `HCI-1.3 A^(@)` and `HI= 1.6A^(@)` The ratio of fraction of electric charge, delta, existing on each atom in HCI and HI is Calculated electric dipole moments for HCl (Hydrogen chloride). CO2 is a linear molecule, so our dipoles are symmetrical; the dipoles are equal in magnitude but point in opposite directions. The dipole moment of HCl molecule is 3.4 × 10-30 cm. but the bond distance order between H-X (X=F,Cl,Br,I) is H-I>H-Br>H-Cl>H-F. When molecules have an even charge distribution and no dipole moment, then they are nonpolar molecules. The dipole moment is defined as the product of the partial charge Q on the bonded atoms and the distance r between the partial charges.. CH4 has very low electronegativity difference between C & H ,so we should think of a dipole but here we have to see the tetrahedral structure of CH4 ,the resultant moment of the 4 bonds is zero as the moment from one bond nullifies that from three … it has magnitude as well as direction. By signing up, you'll get thousands of step-by-step solutions to your homework questions. As a polar diatomic molecule possesses only one polar bond, the dipole moment of that molecule is equal to the dipole moment of the polar Bond. Permanent dipole-dipole is the 2 nd strongest intermolecular force, with Hydrogen bonds being the strongest, and Van der Waals being the weakest. Electric quadrupole moment . The dipole moment of Nitrosyl chloride has been determined numerically as 1.84 ± 0.1 Debye. Quadrupole (D Å) Reference comment Point Group Components; xx yy zz dipole (a) Calculate the dipole moment, in D, that would result if the charges on the H and Cl atoms were 1+ and 1-, respectively. here compound is HCl,you know as well, HCl is ionic compound in which one electron transfers from hydrogen to chlorine. is abbreviated by the Greek letter mu (µ). Note also that HF has a greater dipole moment than H 2 O, which is in turn greater than that of NH 3 . The answer that The Hydrogen Halides, being ionic do not have a D-moment is wrong. Hopefully this video helps some of you guys conceptualize how dipole moments work and how to determine polarity! State Config State description Conf description Exp. In this experiment dipole moments of some polar molecules in non-polar solvents are determined. The electronegativity order in halogens F>Cl>Br>I . i.e., P = q × d . Measuring dipole moments thus provides information about the electron distribution inside a molecule. HBr 0.80. Hydrogen chloride is a diatomic molecule, consisting of a hydrogen atom H and a chlorine atom Cl connected by a polar covalent bond.The chlorine atom is much more electronegative than the hydrogen atom, which makes this bond polar. In HCl, the molecular dipole moment is equal to the dipole moment of H-Cl bond i.e 1.07 D. H2 has Zero Dipole Moment whereas water (H2O) has 1.85D. Dipole moment is a vector quantity i.e. The asymmetrical charge distribution in a polar substance such as HCl produces a dipole moment where \( Qr \) in meters (m). We can conclude that fluorine atoms have a greater electronegativity than do chlorine atoms, etc. Mathematically, it is defined as the product of charges of two atoms forming a covalent polar bond and the distance between them. Due to their different three-dimensional structures, some molecules with polar bonds have a net dipole moment (HCl, CH2O, NH3, and CHCl3), indicated in blue, whereas others do not because the bond dipole moments cancel (BCl3, CCl4, PF5, and SF6). How well does your estimated bond length agree with the bond length in Table 5.4? Made for a UCLA Chemistry course 2015. The dipole moment of a compound is the measure of its polarity. (b) The experimentally measured dipole moment of HCl(g) is 1.08 D. What magnitude of charge, in units of e, on the H and Cl atoms would lead to this dipole moment? HCl 1.08. so, magnitude of charge on dipole , q = 1.6 × 10^-19C A diatomic molecule has two atoms bonded to each other by covalent bond. Because of the force of attraction between oppositely charged particles, there is a small dipole-dipole force of attraction between adjacent HCl … It only acts between certain types of molecules; Molecules with a permanent dipole will experience dipole-dipole forces; It is found between molecules with a differing electronegativity, such as HCl. HCl (due to electronegativity difference) H2 or N2 are made of similar atoms with no polarity difference between them,so no question of dipole. DIPOLE MOMENT AND MOLECULAR STRUCTURE (a) Diatomic molecules. = 1.03 D). Click hereto get an answer to your question ️ * dipole moment of HCl is 1.03 D. 20. Use the dipole moments of HF and HCl (given at the end of the problem) together with the percent ionic character of each bond (Figure 5.5) to estimate the bond length in each molecule. Generally, when dipole distribution is symmetrical, there is no dipole moment. Transition dipole moment, the electrical dipole moment in quantum mechanics; Molecular dipole moment, the electric dipole moment of a molecule. Dipole moment may refer to: . Individual bond dipole moments are indicated in red. Figure 9: Molecules with Polar Bonds. The observed dipole moment of HCIS Bond length is 1.275 A then the percentage of ionic character is (1) 16.83% (3) 30.72% (4) 14.21% (2) 21% -Mila Xof HCl is 17.6%. CH3Cl has larger dipole moment than CH3Fbecause dipole moment is based on the product of distance and charge, and not just charge alone. Debye is the SI unit of the dipole moment. Where the unit is Debye. The distance between its ions will be: (a) 2.12 × 10-11 m (b) zero (c) 2 mm (d) 2 cm Answer to: Does HCl have a dipole moment? HCl molecules, for example, have a dipole moment because the hydrogen atom has a slight positive charge and the chlorine atom has a slight negative charge. This organic chemistry video tutorial provides a basic introduction into dipole moment and molecular polarity. Note that each hydrogen halide (HF, HCl, HBr, and HI) has a significant dipole moment. D = Q * R min. Purely rotational transition energies are obtained with an accuracy of about 0.1 cm -1 , and vibrational transition energies agree within 10-20 cm-1 with the experimental values. A compound doesn't need to be polar covalent to have a dipole moment… We know dipole moment generally vary with separated charge and bond distance. Electric dipole moment, the measure of the electrical polarity of a system of charges . For example, dipole moment of HCl molecule is the same as that of H-Cl bond (µ. Because of the force of attraction between oppositely charged particles, there is a small dipole - dipole force of attraction between adjacent HCl … Potential energy and dipole moment functions of the HF, HCl, and HBr molecules in their electronic ground states have been calculated from highly correlated SCEP/CEPA ab initio wave functions. HCl molecules, for example, have a dipole moment because the hydrogen atom has a slight positive charge and the chlorine atom has a slight negative charge. Moreover, the dipole moments increase as we move up the periodic table in the halogen group.

ANOVA was performed to determine which parameters were significantly affecting glass durability. Table 3 indicated that the longitudinal velocity square affected all the longitudinal modulus (p ≤ 0.05). The finding also illustrated that the shear velocity square affected the shear modulus 2 (p = 0.039) and shear modulus 3 (p = 0.004). Hence, it could be concluded that all the longitudinal modulus, shear modulus 2 and shear modulus 3 were the parameters that produced quality glass durability. Duncan's post hoc tests revealed that there was a significant difference between some pairs of means of glass properties (Tables 3, 4 and 5). For longitudinal velocity square factor, the values of longitudinal modulus 1 have been observed to have ranged from (61.17 ± 1.42) to (66.97 ± 2.52) between erbium concentrations (Table 2). The highest mean value of longitudinal modulus 1 was found at 0.05 erbium concentration and the lowest at 0.02. The longitudinal modulus 1 was significantly different between concentrations of (0.02, 0.03, 0.01); (0.01, 0.04) and 0.05 (Tables 3, 4 and 5). Longitudinal modulus 2 showed a statistically significant increase in erbium concentrations from (60.90 ± 1.41) to (65.99 ± 2.48) (p≤0.05) (Table 2). The longitudinal modulus 2 was significantly different between (0.01, 0.02); (0.02, 0.03, 0.04) and (0.03, 0.04, 0.05 (Tables 3, 4 and 5). The same trend can be seen for longitudinal modulus 3. The erbium concentrations increased from (62.46 ± 1.45) to (68.09 ± 2.56) (p≤0.05) (Table 2). The highest mean erbium concentration for longitudinal modulus 3 was recorded at 0.05. The value of 0.05 concentration (subset 3) was not in the same subset as subsets 1 and 2. Therefore, 0.05 concentration was significantly different from 0.01, 0.02, 0.03 and 0.04 concentrations (Tables 3, 4 and 5). The mean values between erbium concentrations for longitudinal modulus 4 and 5, from (60.73 ± 1.41) to (55.62 ± 2.09) and from (58.86 ± 1.36) to (59.73 ± 2.24) respectively (P≤0.05) (Table 2). The lowest mean value was recorded at 0.05 concentration (subset 1) for longitudinal modulus 4 and 0.05 concentration was significant different from 0.01, 0.02, 0.03 and 0.04 concentrations as in subsets 2 and 3 (Tables 3, 4 and 5). For Longitudinal modulus 5, the erbium concentrations were significantly different between (0.02, 0.01, and 0.05); (0.05, 0.04) and (0.04, 0.03) (Tables 3, 4 and 5). For shear velocity square factors, the mean values for shear modulus 2 and 3 were observed to be fluctuated between erbium concentrations from (20.41 ± 0.47) to (20.42 ± 0.77) and from (20.17 ± 0.47) to (20.39 ± 0.77) respectively (P≤0.05) (Table 2). The 0.02 concentration value for shear modulus 2 was significantly different from 0.01, 0.03, 0.04 and 0.05 concentrations. Therefore, it was placed in one subset (subset 1) while other concentrations in another subset (subset 2) (Tables 3, 4 and 5). For shear modulus 3, the erbium concentrations were significantly different between (0.03, 0.02); (0.02, 0.04, 0.01) and (0.04, 0.01, 0.05) (Tables 3, 4 and 5). Therefore, there were 3 subsets produced (Tables 3, 4 and 5). The analysis extended to the means plot in determining the optimum or the best condition for produced a quality glass durability. Figure 2 depicted that the average Longitudinal Modulus 1 was maximized when the Longitudinal Velocity Square 1 was 12.202 and the erbium concentration was 0.05. Figure 3, on the other hand showed that the optimum level of average Longitudinal Modulus 2 was reached at Longitudinal Velocity Square 2 of 12.025 and erbium concentration of 0.05. It was also found that the best condition that could maximize the average of Longitudinal Modulus 3 was when the Longitudinal Velocity Square 3 was 12.406 and the erbium concentration was 0.05 (Figure 4). Meanwhile, in Figure 5, it could be seen that the Longitudinal Velocity Square 4 of 11.586 (erbium concentration – 0.03) could maximize the average of Longitudinal Modulus 4. Based on Figure 6, the best condition that could maximize the average of Longitudinal Modulus 5 was Longitudinal Velocity Square 5 of 11.637 with erbium concentration of 0.03. In additional, the finding presented that the best condition that could maximize the average of Shear Modulus 2 was Shear Velocity Square 5 of 3.805 with erbium concentration of 0.03 (Figure 7). Figure 8 revealed that the best condition that could maximize the average of Shear Modulus 3 was when the Shear Velocity Square 5 was 3.715 at erbium concentration of 0.05 . As has been discussed, (p≤0.05) in statistics sight provide the most significant data to be applied. Effective prediction made for all longitudinal modulus, shear modulus 2 and shear modulus 3 can be strongly related to the respect of longitudinal and shear velocities, originally. Before longitudinal velocities been ruled by two, the original longitudinal modulus has acted as initial data to produce good values for longitudinal modulus. All the values of longitudinal velocities in the materials perspective based on data statistic provided are the values to produce glass that can withstand pressure longitudinally or called as longitudinal modulus. Meanwhile, shear modulus 2 is predicted having the most significant value when amount of shear velocity 2 has been applied to the glass samples. Therefore, glass samples are able to withstand the pressure in a shear and longitudinal directions as compared and depicted in Figure 9 among all the elastic moduli. Figures 9 and 10 depict the image of the exerted force of elastic moduli and Poisson’s ratio act on the glass samples. The longitudinal and shear velocities are the other name of ultrasonic velocities. It is strongly related to the creation of non-bridging and bridging oxygen within the glass system. Kannapan et al., (2009) has determined that the small values of ultrasonic velocity are attributed to the small amount of electronegativity of element that causes the network to form a weak bond within the glass structure and allow easy creation of non-bridging oxygen. In this case, erbium oxide with the smallest electronegativity, 1.24 has less capability for the attraction of the atoms as compared to zinc (1.65) and tellurium (2.1). This occurrence will eventually create weaker bond, and this is predicted to happen during execution of shear wave throughout the experiment when erbium concentration is 0.01, 0.03, 0.04 and 0.05. Consequently, the glass network will loosen up and create more free spaces between the atoms which will cause the ultrasonic wave to be transmitted slower and decrease its velocity. Besides that, the presence of erbium oxide in the glass series proves the ability of the elements to act as a glass modifier that modify the glass structure in the glass system. This condition will cause a decrease in shear velocity. Saddeek (2004) had mentioned that the inclusion of erbium oxide would modify the glass structure by splitting the Te-O-Te bond and promote the conversion of bridging oxygen into non-bridging oxygen by forming a trigonal bipyramid into a trigonal pyramid . Furthermore, the addition of erbium oxide into the glass interstices enables more ions to be opened up which will weaken the glass structure. These explain the reason for the insignificant values of all shear modulus except for shear modulus 2. In the meantime, the replacement of lighter molecular weight of tellurium dioxide and zinc oxide by heavier molecular weight of erbium oxide will cause changes in the overall weight of the glass and promotes stronger connection between the bonds in the glass. This indirectly will be a strong indicator to conclude that all values of longitudinal modulus are significant as listed in Table 3 . They have also reported that the formation of glass network with large concentration of dopants in the interstices space would increase the molar mass of the glass sample and improve the compactness as well . Furthermore, the increasing compactness of the glass can also cause by the close distance between the molecules where it allows the transmission of the ultrasonic wave to pass through the glass sample easier. Closer distance between the molecules will result in formation of bridging oxygen in the glass system and contributes to the improvement of the connectivity within the glass network. This is relevant and can be inferred to follow all the significant values of longitudinal modulus that can be used for the fabrication of fiber optic. Other than that, large values of ultrasonic velocities can also be supported by large packing density as mentioned by Elokr and AbouDeif, (2016) . Parameters such as density, molar volume, longitudinal velocity, shear velocity, shear modulus 1, shear modulus 3, shear modulus 4 and shear modulus 5 were not statistically significant at 0.05 level of significant (P≥0.05) as tabulated in Table 3. The insignificant difference of means between groups can be explained by several reasons. Factor of erbium oxide, initially have affected the significance of the parameters including density, molar volume and longitudinal and shear velocities. Density and molar volume are interrelated to each other in this work where both parameters are theoretically related. Nevertheless, in this work, erbium concentration has played a role in the glass sample. The insignificance of the parameters can first be attributed to the presence of erbium oxide the glass samples. The bond length, inter-atomic spacing within the atoms, the presence of non-bridging oxygen atoms and the rearrangement of the lattice [19–21] might affect the vicinity of the glass structure. The bond length of Er atoms which is 2.26 Å is longer than the bond length of Te atoms (1.6 Å) and Zn atoms (1.42 Å). The increase in the bond length of the dopants will enhance the inter-atomic spacing between the atoms which can influence the escalation of the molar volume in the glass sample that produce numbers of non-bridging oxygen that causes the bond to break. Therefore, the spaces between the glasses are growing exponentially and more excess free volume are formed . Numbers of bridging oxygen can be formed by the formation of more tellurite networks of trigonal pyramid compared to trigonal bipyramid . In addition, the increment of the molar volume can also be predicted by large d-spacing obtained by XRD spectra. Proper thickness with flat parallel surface of the glass sample is another crucial factor to be discussed. For elastic measurement, the thickness is required to be thicker and both surface of the glass sample must be as parallel as possible. This is to ensure the transmission of ultrasonic wave can propagate smoothly in the glass in order to obtain the smooth wave form. Due to the thinning of the glass sample and less parallel surface of the glass which actually should be equal or more than 5 mm, the ultrasonic wave was most probably can transmit in the glass sample unevenly making the obtained outcome not significant to be used. Therefore, based on the statistical data been compared with physical data, some values of the parameters were highlighted and predicted to be used as indicator for the application. Table 6 list parameter estimates for the significant factors that will produce an efficient fiber optic durability. Values of significant parameters in materials perspective based on statistics sight Longitudinal velocity square (m2/s2) Longitudinal modulus (GPa) Shear velocity square (m2/s2) Shear modulus (GPa) 10130.0 – 12410.0 50.45 – 76.16 3480.0 – 3840.0 17.50 – 24.05

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- Research article - Open Access A simulation study of sample size for multilevel logistic regression models BMC Medical Research Methodology volume 7, Article number: 34 (2007) Many studies conducted in health and social sciences collect individual level data as outcome measures. Usually, such data have a hierarchical structure, with patients clustered within physicians, and physicians clustered within practices. Large survey data, including national surveys, have a hierarchical or clustered structure; respondents are naturally clustered in geographical units (e.g., health regions) and may be grouped into smaller units. Outcomes of interest in many fields not only reflect continuous measures, but also binary outcomes such as depression, presence or absence of a disease, and self-reported general health. In the framework of multilevel studies an important problem is calculating an adequate sample size that generates unbiased and accurate estimates. In this paper simulation studies are used to assess the effect of varying sample size at both the individual and group level on the accuracy of the estimates of the parameters and variance components of multilevel logistic regression models. In addition, the influence of prevalence of the outcome and the intra-class correlation coefficient (ICC) is examined. The results show that the estimates of the fixed effect parameters are unbiased for 100 groups with group size of 50 or higher. The estimates of the variance covariance components are slightly biased even with 100 groups and group size of 50. The biases for both fixed and random effects are severe for group size of 5. The standard errors for fixed effect parameters are unbiased while for variance covariance components are underestimated. Results suggest that low prevalent events require larger sample sizes with at least a minimum of 100 groups and 50 individuals per group. We recommend using a minimum group size of 50 with at least 50 groups to produce valid estimates for multi-level logistic regression models. Group size should be adjusted under conditions where the prevalence of events is low such that the expected number of events in each group should be greater than one. The idea that individual action is shaped by macro-level forces was evident in sociological theories of psychiatric illness and delinquency arising out of the Chicago School [1, 2]. These theories suggest that while individual risk factors can affect individual health and delinquent behavior, so also can the structure of the social environment in which we live. It is only in the last 20 years that these theories could be truly tested, when statistical models were developed that allowed researchers to examine the additive and interactive effects of individual-level and contextual features that affect sociological outcomes at the individual level. In the last ten years the use of multilevel models has burgeoned in epidemiology. These models are highly appropriate in assessing how context affects individual-level health risks and outcomes . Many kinds of data, including national surveys, have a hierarchical or clustered structure. For example, respondents in a complex large survey are naturally clustered in geographical units (e.g., health regions) and may be grouped into smaller units (e.g. census tracts). Over the last two decades, researchers have developed a class of statistical models designed for data with hierarchical structure. These models are variously known as mixed, hierarchical linear, random coefficient, and multilevel models. Hierarchical data routinely arise in many fields where multilevel models can be used as an extended version of the more traditional statistical techniques either to adjust for the dependency of the observations within clusters by using variables at higher levels or assessing the impact of higher level characteristics on the outcome after controlling for individual characteristics at the base level. An important feature of this class of models is the ability to estimate the cross-level interaction which provides a measure of the joint effect of a variable at the individual level in conjunction with a variable at the group level. The robustness issue and the choice of sample size and power in multilevel modeling for continuous dependent variables has been studied by several authors [4–13]. Austin used Monte Carlo simulation to assess the impact of misspecification of the distribution of random effects on estimation of and inference about both the fixed effects and the random effects in multilevel logistic regression models. He concluded that estimation and inference concerning the fixed effects were insensitive to misspecification of the distribution of the random effects, but estimation and inferences concerning the random effects were affected by model misspecification. Simulation studies indicate that a larger number of groups is more important than a larger number of individuals per group [4, 5]. The overall conclusion from these studies is that the estimates of the regression coefficients are unbiased, but the standard errors and the variance components tend to be biased downward (underestimated) when the number of level 2 units is small (e.g. less than 30) [4, 11]. Outcomes of interest in many fields do not only reflect continuous measures. Binary outcomes such as depression, presence or absence of a disease, and poor versus good self-reported general health are also of interest. Few studies have examined the accuracy of estimates, sample size or power analysis in binary multilevel regression [5, 15]. Although Sastry et al. calculate power and sample size in multilevel logistic regression models for their survey of children, families and communities in Los Angeles, they used a test of proportions between two comparison groups to calculate preliminary total sample size for a given baseline proportion and minimum detectable differences. After adjusting the calculated preliminary sample size for design effect, a total sample size of 3,250 was adopted. Finally based on simulation studies with total sample size of 3,250 and group sizes of 51, 66, 75, and 81 they decided to sample 65 groups (tracts) each of size 50. We are unaware of any studies to date that have focused on these issues in multilevel logistic regression in a more comprehensive manner. In this paper simulation studies based on multilevel logistic regression models are used to assess the impact of varying sample size at both the individual and group level on the accuracy of the estimates of the parameters and their corresponding variance components. We focus on the following multilevel logistic model with one explanatory variable at level 1 (individual level) and one explanatory variable at level 2 (group level): Here P ij is the probability that individual i in group j will experience the outcome, x ij is an explanatory variable on the respondent level, and z j is a group level explanatory variable. Model (1) can be written in the following single equation: logit(p ij ) = γ 00 + γ 10 x ij + γ 01 z j + γ 11 x ij z j + u 0j + u 1j x ij In equation (2) the segment γ 00 + γ 10 x ij + γ 01 z j + γ 11 x ij z j is the fixed effect part and the segment u 0j + u 1j x ij is the random part of the model. An important feature of equation (2) is the presence of a cross-level interaction term represented by γ 11 z j x ij in which the coefficient γ 11 shows how π 1j , the slope of equation (1), varies with z j , the group level variable. The size of the intra-class correlation coefficient (ICC) may also affect the accuracy of the estimates . The ICC for the logistic model is defined as where and is the variance of the random intercept in a fully unconditional multilevel logistic model logit(p ij ) = γ 00 + u 0j where u 0j ~ N(0, ) . The accuracy of the parameter estimates is quantified by the percentage relative bias . Let stand for the estimate of the population parameter θ, then indicates the percentage relative bias for parameter θ. The accuracy of the standard error of the parameter estimate is assessed by analyzing the observed coverage of the 95% confidence interval created by using the asymptotic standard normal distribution . Following the simulation conditions used by Maas and Hox we set the following conditions for our simulation studies: (i) the number of individuals per group, j, n j , was set at 5, 30, and 50, (ii) the number of groups was set at 30, 50, and 100, and (iii) the variances of the random intercept were set at 0.13, 0.67, and 2.0, corresponding to intra-class correlation coefficients (ICC) of 0.04, 0.17, and 0.38, respectively. The individual and group explanatory variables x ij and z j are generated from the standard normal distribution. The group random components u 0j and u 1j are independent normal variables with mean zero and standard deviations σ 0 and σ 1where σ 1 = 1 in all simulations and σ 0 follows from the ICC and is set to 0.36, 0.82, and 1.42. We set the fixed effect parameters for all simulated models as: γ 00 = -1.0, γ 01 = 0.3, γ 10 = 0.3, and γ 11 = 0.3. To generate the outcome, a Bernoulli distribution with probability is used. The overall prevalence of the outcome is close to 30 percent. For practical purposes we generated 1000 data sets for each combination since a larger number of replications would have substantially increased processing time. The software SAS 9.1 (SAS Institute, North Carolina, US) was used for simulating observations and estimating the parameters. The SAS procedure NLMIXED with default options was used for estimation. This procedure only allows full maximum likelihood estimation. If convergence was not achieved the estimated parameters were not included in calculating summary statistics. We set initial values as the "true" values of each parameter. Distributions for random effects were normal, the optimization technique was Dual Quasi-Newton, and the integration method was Adaptive Gaussian Quadrature (AGQ). The number of quadrature points in AGQ was selected automatically. The absolute value for parameter convergence criterion was 10-8 and the maximum number of iterations was 200. The overall rate of model convergence varied from 56% to 100%. There were no negative variance estimates in converged models. Logistic regression was used to investigate the impact of ICC, number of groups and group size on the convergence. The rate of convergence (percent converged) significantly improved with either an increase in the number of groups or an increase in the group size. The overall rate of convergence for groups of sizes 5, 30, and 50 was 80.4%, 99.3, and 99.9%, respectively. For group of sizes 30, 50, and 100, the rate of convergence was 87.7%, 93.8%, and 98.1%. For the three ICC conditions of 0.04, 0.17, and 0.38 the rate of convergence was 89.2%, 94.5% and 95.9%. When we compared the samples that did and did not converge findings indicate no significant differences in prevalence (30.4% vs. 30.0%), mean and standard deviation of z j (0.01 and 1.01 versus 0.02 and 1.00), or mean and standard deviation of x ij (0.00 and 1.00 vs. 0.00 and 1.00). To further explore the non-convergent samples we examined 168 non-convergent simulated data sets with 30 groups and group size of 5. We first fitted a logistic model with random intercept only and then a logistic model with random slope only to each of these data sets. The estimated random intercepts and random slopes were classified as significant if the corresponding p-value was less that 0.05; otherwise each was classified as non-significant. Both the random intercept and random slope were statistically significant in only a small proportion of these data sets (2.4%). A closer investigation showed that when both random intercept and random slope were statistically significant either the random slope or the random intercept was severely underestimated. This suggests that non-convergence result from lack of sufficient variation in both the intercept and slope and further suggests that simplifying the model is appropriate; for example either a random intercept or a random slope is estimated, but not both. Distribution of parameter estimates P-values and confidence intervals given by the NLMIXED procedure are based on asymptotic normality which may not be accurate for small sample sizes. The Shapiro-Wilk test calculates a W statistic that tests whether a random sample of size n comes from a normal distribution. The Shapiro-Wilk test for normality was used to test the normality of the distribution of fixed effect estimates for different combinations. Logistic regression was used to assess the effects of each factor on normality. The ICC was not associated with normality of the parameter estimates. The number of groups was associated with normality of the estimates for γ 10 and γ 01 group size was associated with normality of the estimates for γ 00, γ 10, and γ 11. The majority of estimates from simulations with a group size of 5 were non-normal even with 100 groups. For simulations with a group size of 30 a few estimates were non-normal even with 50 groups. All estimates were normally distributed with 100 groups and group size of 50. In simulation studies of multilevel regression with continuous outcomes, Maas and Hox found negligible bias for the fixed effect parameter estimates. They reported an average bias less than 0.05% for the fixed parameter estimates, intercept and the regression slopes. Our simulations show that the overall biases for the fixed effect parameters γ 00, γ 01, γ 10, and γ 11 were 0.6%, 2.6%, 1.4%, and 3.7% respectively (data not shown). The cross-level interaction parameter (γ 11) had the largest overall bias. Table 1 shows the percent relative bias and rate of convergence (percent converged) for different simulation conditions. For the fixed effect parameters, the largest biases (8.8%, 11.1%, 15.8%, and 13.3% for γ 00, γ 01, γ 10, and γ 11) were found under conditions where of the smallest variance for the random intercept (0.13), the smallest group size (5), and the lowest number of groups (30). When the size of the group was increased to 30 with 30 groups, the bias was reduced to less than 6%. These biases were reduced to less than 4% when the size of the group was 30 and the number of groups was 50. Even further reductions occurred (bias of 1% or less) when the size of the group was 30 and there were 100 groups. The estimates of the random intercept and random slope have larger biases compared to the fixed effect parameters. The overall biases (data not shown) for σ 0 and σ 1 were 6.9% and 5.0%. The bias for σ 1 remained at the level of 5% for different values of σ 0, however the estimates for σ 0 had the largest bias (21.2%) for σ 0 = 0.36. The relative bias for the variance components was less than 4% when the size of the group was 50 and there were 100 groups. The variance-covariance parameter estimates are positively biased in all cases when the group size was 5 regardless of the number of groups (some exceeded 100%). The variance components were consistently underestimated when with a group size of 30 or more regardless of the number of groups. This problem of underestimation has been noted previously in simulation studies of multilevel models for continuous outcomes . The overall relative bias for the random intercept was 21%, 0.5%, 0.1% for ICC 0.04, 0.17, 0.38, respectively. For the random slope, the overall relative biases for the three ICC conditions were not statistically different, ranging from 4% to 6%. There were no statistically significant differences in bias for the fixed effect parameters for any of the ICC conditions. We adopted the method used by Maas and Hox to assess the accuracy of the standard errors. For each parameter in each simulated data set the 95% Wald confidence interval is established. For each parameter a non-coverage indicator variable is set to zero if the confidence interval contains the true value, otherwise if the true value lies outside the 95% confidence interval it is set to 1. The effect of number of groups, group size, and ICC on the non-coverage is presented in Tables 2 and 3, respectively. Logistic regression was used to assess the effect of the different simulated conditions on non-coverage. As shown in Table 2 the effect of number of groups on the standard errors of the fixed effect parameters is small with non-coverage rates ranging from 5% and 6%. The nominal non-coverage rate is 5%. The effect of number of groups on the standard errors of the variance component was larger than the nominal 5%, with non-coverage ranging from 7% to 11%. With 30 groups the non-coverage rate was 11% for the random intercept and 10% for the random slope. These non-coverage rates were reduced to 9% and 7% percent, respectively, for 100 groups. The extent of non-coverage implies that the standard error for the variance components is underestimated, a phenomenon reported by Maas and Hox in their simulation studies of two-level linear regression models. The rate of non-coverage decreased as number of groups increased however, the non-convergence cannot be ignored. The rates of non-coverage for the fixed effect parameters varied between 4 to 6% which is close to 5% nominal (Table 2). The effect of group size on the standard error of the estimates of the random intercept (close to 10%) was not significant; however the rate of non-coverage for the random slope increased as the group size increased. Table 2 shows that ICC had no effect on the non-coverage rates for the fixed effects or the random slope. Similar to findings for the number of groups and group size, the rate of non-coverage is close to 5% for the fixed effect parameters and over 5% for the random effect parameters. The rate of non-coverage for the random intercept decreased as ICC increased. Table 3 shows the rates of non-coverage for each simulation condition. The minimum and maximum rates of non-coverage for the fixed effect parameters, γ 00, γ 01, γ 10, and γ 11, range from 3% and 7%. The rates of non-coverage for the variance-covariance components range from 7% and 17%. These findings indicate that the estimates of the standard errors are acceptable for the fixed effect parameters but not acceptable for the variance covariance components. The accuracy of the estimates of the parameters at the individual level depends on the prevalence of the outcome. To assess the relationship between the prevalence of the outcome and the sample size we repeated our simulations with prevalence rates of 0.10, 0.34, and 0.45. We set the parameters, γ 00, γ 01, γ 10, and γ 11, at -3.0, 0.3, 0.3, and 0.3 for prevalence of 10%, at -1.0, 0.3, 0.3, and 0.3 for prevalence of 34% and -0.3, -0.3, -0.3, and -0.3 for prevalence of 45%. The variances of the random intercepts and random slopes were 1 for all simulations. Table 4 shows that for both fixed and random effect parameters the simulated data with 10% prevalence had the largest bias. The overall effect of prevalence on the non-coverage rates was not significant (data not shown). As shown in Table 4 the rate of non-coverage for all fixed effect parameter estimates ranged from 5 to 6%. The rate of non-coverage for the random intercept and random slope variance estimates ranged from 8 to 11%. This suggests that a larger sample size is necessary to minimize bias for low-prevalent outcomes. The largest bias was observed under conditions when the size of the group was 5 and the prevalence of the outcome was 10% (12% for fixed effect and 50% for random effect). Similarly, with 30 groups and a 10% prevalence the largest bias was 9% for the fixed effect parameters and 15% for the random slope. The rate of convergence was lowest with 10% prevalence. Discussion and conclusion In this paper we investigated the impact of varying sample size at both the group and individual level on the accuracy of the parameter estimates and variance components using multilevel modeling for logistic regression. We also examined the effect of prevalence of the outcome on the accuracy of the estimates. The number of replications was restricted to 1000 due to extensive computer processing time. Previous research has indicated that a sample of 50 groups and 30 units per group is sufficient to produce reliable parameter estimates for linear multilevel regression models . Our findings suggest this may not be the case for logistic regression. Simulations presented in this paper suggest that the number of level two groups and the number of individuals in each group should be adjusted for prevalence of the outcome. Low prevalent events require a larger number of individuals per group. We did not study the effect of different estimation procedures on the accuracy of the parameter estimates. However Rodriguez and Goldman [18, 19] showed that the marginal quasi likelihood with first order Taylor expansion underestimates both the fixed effect and the variance-covariance components. The data set that formed the basis for these conclusions was extreme in the sense that the variance components were large and the sample size at the lowest level was quite small. In less extreme cases it appears that predictive quasi likelihood with second order Taylor expansion usually provides accurate estimates for both fixed and random parameters . Simulations by Callens and Croux compared penalized quasi-likelihood (PQL) with adaptive Gaussian quadrature (AGQ) and non-adaptive Gaussian quadrature (NGQ) and showed that PQL suffers from large bias but performs better in terms of mean-squared error (MSE) than standard versions of quadrature methods. They also showed that automatic selection of the number of quadrature points in AGQ (the default of the NLMIXED procedure) might be inadequate and lead to a loss in MSE. Thus, numerical results may change slightly depending on the statistical package, number of iterations, or algorithm used to estimate the parameters. In multilevel analysis non-convergence can occur when estimating too many random components that are close to zero. Hox suggests a solution to this problem which is to remove some random components, thereby simplifying the model. In our case non-convergence was a significant problem when group size was 5 and the number of groups was 30. There were no significant differences in the prevalence of the outcome or in the distribution of the explanatory variables among the converged versus non-converged samples. When the sample size is small there may not be sufficient variation to estimate a random effect, thus leading to non-convergence. Simulation studies come with their own set of limitations. This said, our results are comparable with simulation results for multilevel regression models as reported by other researchers . We focused on the impact of sample size at the individual and group level on the bias and accuracy of parameter estimates. We did not consider the impact of varying the distribution and variance of the individual and group level explanatory variables for practical reasons, specifically due to the large number of conditions that would have to be considered and which would result in extensive computer processing time. For a discussion of the impact of misspecification of the distribution of random effects on estimation of and inference about both the fixed effects and the random effects in multilevel logistic regression models see Austin . Our results and recommendations are based on extensive simulation studies from data which are generated from normal distributions. Since the normal distribution assumption may be violated in real study applications we conducted further simulations with 100 replications for each model and relaxed the normal distribution assumption. This allowed us to compare convergence, coverage, and bias of the simulated non-normal models with the simulated normal model. Comparisons were done by each parameter, number of groups, and group size. The distributions for generating z j , u 0j , u 1j , and x ij for 8 simulated models are as follows: Model 1: N(0,1), N(0,1), N(0,1), N(0,1); Model 2: N(0,1), N(0,1), N(0,4), N(0,4); Model 3: N(0,4), N(0,4), N(0,4), N(0,4); Model 4: U(-0.5,0.5), N(0,1), N(0,1), U(-0.5,0.5); Model 5: U(-0.5,0.5), U(-2,2), U(-2,2), U(-0.5,0.5); Model 6: N(0,1), t(df = 3), t(df = 3), N(0,1); Model 7: t(df = 3), t(df = 3), t(df = 3), t(df = 3); Model 8: t(df = 5), t(df = 5), t(df = 5), t(df = 5) where N(a, b) stands for a normal distribution with mean a and variance b, U(a, b) represents uniform distribution over the interval (a, b), and t(df = k) stands for t-student distribution with k degrees of freedom. Table 5 shows the convergence, coverage and relative bias for above 8 models. Logistic regression was used to compare the rate of convergence of models 2 to 8 with model 1 (the model with 4 normal standard distributions). Models 3, 6, 7, and 8 were more likely to converge while models 4 and 5 were less likely to converge (p-value < 0.01) when group size was 5. When group size was greater than 5 there was no significant difference between the rates of convergence for all models. The comparisons (Ttest) between the fixed effect parameter estimates of models 2–8 with model 1 did not show any significant differences. For models 6, 7, and 8 the random intercept and random slope were underestimated compare to model 1 (p < 0.01). This phenomenon was also observed and reported by Austin for a logistic model with random intercept. Fisher exact test was used to test the rate of coverage of models 2–8 with model 1 for each parameter, number of groups, and group size. There was no significant difference between the rates of coverage for the fixed effect parameters when group size was greater than 5 using groups of 50 or more. The rates of coverage for the random components of models 6 and 7 were significantly lower than the coverage rates of model 1 (p < 0.01). Austin reported similar conclusions for a logistic model with random intercept. Although the misspecification of random components significantly affected the estimates and standard errors of the random intercepts and random slopes when either group size or number of groups was small, the estimates and standard errors of all models were statistically the same as those estimates and standard errors for model 1 when the number of groups and group size was 50 or more; the exception being for a t-distribution with 3 degrees of freedom. Despite the limitations of simulation studies [see for example 21] our findings can offer some suggestions for sample size selection in multilevel logistic regression. In practice a group size of 30 is often recommended in educational research and a group size of 5 is recommended in family and longitudinal research studies . Based on our findings we recommend a minimum group size of at least 50 and a minimum of 50 groups to produce valid estimates for multilevel logistic regression models. We offer a caveat here such that the group size must be adjusted properly for low-prevalent outcomes; specifically the expected number of outcomes in each group should be greater than one. This caveat is offered as a caution to researchers using multilevel logistic regression in conjunction with small data sets; under these conditions researchers can expect to encounter convergence problems, large biases in their model estimates and inadequate statistical inference procedures. Our findings suggest that when choosing a sample size, researchers should base their decision on the level of bias that they consider acceptable for that particular study. The main findings from this research can be summarized as follows: (i) convergence problems arise when prevalence is low, the number of groups is small, or the group size is small; (ii) the estimates of the fixed effect parameters are unbiased when the number of groups is relatively large (more than 50) and with moderate group size; (iii) when group size is small (e.g. 5) the estimates of the random slope and random intercept are severely overestimated, and; (iv) the standard errors of the variance component estimates are underestimated even with 100 groups and group size of 50. Faris REL, Dunham HW: Mental disorders in urban areas: an ecological study of schizophrenia and other psychoses,. Edited by: Dunham HW. 1939, Chicago, University of Chicago Press Shaw CR, McKay HD: Juvenile delinquency and urban areas;: a study of rates of delinquency in relation to differential characteristics of local communities in American cities. Edited by: McKay HD. 1969, Chicago, University Press O'Campo P: Invited Commentary: Advancing Theory and Methods for Multilevel Models of Residential Neighborhoods and Health. Am J Epidemiol. 2003, 157: 9-13. 10.1093/aje/kwf171. Maas CJM, Hox JJ: Robustness issues in multilevel regression analysis. Statistica Neerlandica. 2004, 58: 127-137. 10.1046/j.0039-0402.2003.00252.x. Hox JJ: Multilevel analysis: techniques and applications. 2002, Mahwah, N.J., Lawrence Erlbaum Publishers Snijders TAB, Bosker RJ: Standard Errors and Sample Sizes for 2-Level Research. Journal of Educational Statistics. 1993, 18: 237-259. 10.2307/1165134. Leyland AH, Goldstein H: Multi-Level Modeling of Health Statistics. 2001, London, John Wiley Raudenbush SW, Liu XF: Effects of study duration, frequency of observation, and sample size on power in studies of group differences in polynomial change. Psychological Methods. 2001, 6: 387-401. Bingenheimer JB, Raudenbush SW: Statistical and substantive inferences in public health: Issues in the application of multilevel models. Annual Review of Public Health. 2004, 25: 53-77. 10.1146/annurev.publhealth.25.050503.153925. Atkins DC: Using multilevel models to analyze couple and family treatment data: Basic and advanced issues. Journal of Family Psychology. 2005, 19: 98-110. 10.1037/0893-318.104.22.168. Maas CJM, Hox JJ: Sufficient Sample Sizes for Multilevel Modeling. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences. 2005, 1: 85-91. 10.1027/1614-2241.1.3.86. Shieh YY, Fouladi RT: The Effect of Multicollinearity on Multilevel Modeling Parameter Estimates and Standard Errors. Educational and Psychological Measurement. 2003, 63: 951-985. 10.1177/0013164403258402. Dickinson LM, Basu A: Multilevel Modeling and Practice-Based Research. Ann Fam Med. 2005, 3: S52-S60. 10.1370/afm.340. Austin PC: Bias in Penalized Quasi-Likelihood Estimation in Random Effects Logistic Regression Models When the Random Effects are not Normally Distributed. Communications in Statistics: Simulation & Computation. 2005, 34: 549-565. 10.1081/SAC-200068364. Sastry N, Ghosh-Dastidar B, Adams J, Pebley AR: The Design of a Multilevel Survey of Children, Families, and Communities: The Los Angeles Family and Neighborhood Survey. 2003, California, RAND, Working Paper Series 03-21: 1-55. Goldstein H: Multilevel statistical models. 2003, New York, Distributed in the United States of America by Oxford University Press, 3rd ed. Guo G, Zhao H: Multilevel Modeling for Binary Data. Annual Review of Sociology. 2000, 26: 441-462. 10.1146/annurev.soc.26.1.441. Rodriguez G, Goldman N: An Assessment of Estimation Procedures for Multilevel Models with Binary Responses. Journal of the Royal Statistical Society Series A (Statistics in Society). 1995, 158: 73-89. 10.2307/2983404. Rodriguez G, Goldman N: Improved estimation procedures for multilevel models with binary response: a case-study. Journal of the Royal Statistical Society: Series A (Statistics in Society). 2001, 164: 339-355. 10.1111/1467-985X.00206. Callens M, Croux C: Performance of likelihood-based estimation methods for multilevel binary regression models. Journal of Statistical Computation and Simulation. 2005, 75: 1003-1017. 10.1080/00949650412331321070. Kreft IGG: Are multilevel techniques necessary? An overview, including simulation studies. 1996 The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2288/7/34/prepub From the Centre for Research on Inner City Health, The Keenan Research Centre in the Li Ka Shing Knowledge Institute of St. Michael's Hospital. The authors gratefully acknowledge the support of the Ontario Ministry of Health and Long-Term Care. The views expressed in this manuscript are the views of the authors and do not necessarily reflect the views of the Ontario Ministry of Health and Long-Term Care. The authors are also grateful to the reviewers for their thoughtful and constructive comments. The author(s) declare that they have no competing interests. RM performed simulation studies and drafted the paper. RM, FIM and RHG conceptualized the research and revised the manuscript for intellectual content. All authors read and approved the final manuscript. Rights and permissions This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. About this article Cite this article Moineddin, R., Matheson, F.I. & Glazier, R.H. A simulation study of sample size for multilevel logistic regression models. BMC Med Res Methodol 7, 34 (2007). https://doi.org/10.1186/1471-2288-7-34 - Group Size - Relative Bias - Random Slope - Multilevel Logistic Regression - Fixed Effect Parameter

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1. Earth in Motion * Rotation: 24 hrs > CCW about north pole > 750 mi/hr at State College > Earth bulges due to rotation * Revolution: 365.25 days > CCW about north pole > Avg speed: 67,000 mi/hr * Orientation of Earth in Space > 23.5o tilt of polar axis * Precession: 26,000 yrs > 'Wobble' of Earth's rotation (polar) axis > Due to gravity of Moon & Sun * Motion relative to nearby stars > Video: Jack Horkheimer describes motion of Arcturus * Orbit about the Milky Way: 230 million yrs > Orbit radius: 28,000 ly > Speed: 500,000 mi/hr > Vertical oscillation * Motion within Local Group > Expected collision with M31 (Andromeda Spiral) * Expansion of the Universe > Space itself is expanding (raisin-cake model) 2. Latitude, Longitude & All That * Latitude: angle between equator plane & your location (0o < Lat < 90o) * Latitudes of tropic & polar circles are related to orientation of Earth * Longitude: Measured east/west of Prime Meridian (0o to 180o E/W) * State College: Lat = 41o N, Long = 78o W 3. Celestial Sphere: A Model of the Sky * Horizon (plane), Meridian, Zenith * Celestial poles, Equator 4. Rising & Setting * Due to rotation of Earth * Stars follow circles centered about a celestial pole * Circumpolar stars 5. The Sky & Your Latitude * Altitude (elevation) of celestial pole = your latitude * Daily motion of sky at various latitudes * Drift of celestial pole relative to stars (Precssion) 6. Different Stars in Different Seasons * Due to orbit of Earth about sun * Traditional and Modern definitions * Constellation stars distributed in space 1. Ques. #11, pg. 26. 2. All points on Earth rotate with the same period, namely 24 hrs. So, why does a point on Earth's equator rotate about Earth's axis with a speed greater than the rotation speed of State College? 3. Ques. #2, pg. 48. 4. Ques. #4, pg. 48. 5. Ques. #6, pg. 48. 6. Ques. #7, pg. 48. [Hint: Have a look at FIG 10 above.] 7. Prob. # 2, pg. 49. [Hint #1: As reported in class, for State College: Lat = 41o N; Long = 78o W. Hint #2: A study of FIG 2 (above) may prove useful in responding to this question.] 8. In FIG 2 above, what is the value of the angle (in degrees) between celestial equator and celestial pole (either pole will do). 9. Suppose the angle between Earth's equatorial and ecliptic (orbit) planes were only 16.5o.What would be the latitude of the Tropic of Cancer and the Arctic Circle? [Hint: Take a look at FIG 1 above & note that angle I is identical to the angle between equatorial and ecliptic planes.] 10. Your spaceship lands on the mysterious planet Xenon at a locale where all stars appear to circle a point elevated 57o above your northern horizon. What's your latitude on Xenon? [Hint #1: Assume Xenonians measure latitude in exactly the same way we do; i.e., equator lies at 0 lat & north pole lies at 90o lat. Hint #2: Take a look at FIG 6 above.] 11. As observed from State College, a star that rises directly East must set where on the horizon? A star that sets in the southwest must have risen where? [Hint: Have a look at FIG 5 above.] 12. The image in FIG 7 above was taken in the northern hemisphere. Are we looking east or west? 13. [Multiple-Choice] A certain star appears on your zenith at midnight on a certain night. About how much time must pass before this same star appears again at your zenith? a) One hour b) One day c) One month d) One year 14. Where on Earth does the celestial equator coincide with the horizon? 15. Where would you be standing on Earth if no visible stars are circumpolar? 16. Take a look at Figure 2.13 (pg. 34) in your textbook. (a) The center of all of the arcs of circles traced by stars in this picture is what special point on the sky? (b) Is the star Polaris found in this picture? Explain. (c) How would this picture change if the viewer were standing at Earth's south pole, instead of in Utah? (Apart from the obvious change in topography!) 17. The constellation Pegasus appears directly overhead at midnight. One month later, at midnight, will Pegasus be found west of your meridian, or east of your meridian? Explain. [Hint: Take a look at FIG 10 above.] 18. Precession has what effect on the appearance of the sky? 19. Ques. #1, pg. 48. 20. Ques. #3, pg. 48. 1. The motions of Earth (from rotation to participation in the expansion of the Universe) were well summarized in lecture; consult your notes. 2. Have a look at Fig 1.16 on pg. 19 in your text. A person standing on Earth is carried around on a circle once every day (24 hours). The size of that circle depends on latitude (the smaller the circle as we get closer to either pole). Now, the larger the circle, the greater the distance that must be covered in 24 hours. So, as you move further from the equator, your speed will decrease. 3. The celestial sphere is an imaginary sphere, essentially infinite in radius, surrounding any oberver on Earth (or any place else, for that matter). Except for the celestial meridian, the main features of the celestial sphere are nicely identified in FIG 2 above. 4. Your sky is really the collection of all possible directions you can look outward from your location on Earth. This collection of directions adds up to a hemisphere (or, dome). Horizon, zenith and meridian are defined very satisfactorily on pg. 31 in your text. 5. Here's one way to think about the changing sky: Extend a line from Earth's center through you and on out into space. This line runs along an Earth radius and points toward your zenith. As Earth is spherical, the direction toward the zenith thus changes continuously as you move around Earth. So you see different stars in the zenith depending on where you are. As your sky is cut-off by the plane (horizon plane) perpendicular to the zenith direction, your whole sky thus varies as you move around Earth. 6. It's a matter of Earth moving in an orbit about the sun, giving those on Earth's dark side a constantly changing view of the universe through the year. 7. a) State College: lat = 41o North; long = 78o West. b) The NCP lies about 41o elevation above the northern horizon, as viewed from State College (SC). The south celestial pole lies below the horizon, as viewed from SC. c) Polaris is circumpolar in the SC sky; it lies less than 1o from the NCP. d) As it does everywhere, the meridian extends from north to south on the horizon, while passing through the zenith. e) Celestial equator passes from east to west on horizon, crossing the meridian 49o above the horizon in SC. (See FIG 3 under Sun & Seasons). 8. The angle between celestial pole and celestial equator is 90o. 9. Tropic of Cancer: 16.5o North. Arctic Circle: 73.5o North. 10. Your latitude on Xenon is 57o North. 11. A star that rises directly east must set directly west. A star observed to set in the southwest must have rises in the southeast. 12. We're looking west in this picture. (North must be off to your right, as the stars in this picture are evidently circling a point elevated and to the right - this point must be the NCP.) 14. The celestial equator coincides with horizon at either of Earth's poles. 15. If no visible stars are circumpolar, you must be standing on Earth's equator. 16. a) The center of the circles traced by the stars is the north celestial pole. (As the caption indicates, the picture was taken in Utah.) b) The star Polaris is represented by the short, very bright arc very near the center of all of the arcs. We know Polaris is a moderately bright star near, but not at the NCP. c) In order to see pattern at Earth's south pole, you'd have to be looking toward the zenith, in which case the landscape would likely lie outside your field of view. 17. Pegasus will lie west of your meridian one month after your initial observation. Have a look at FIG 10, which clearly shows that Pegasus lies to the west of directly overhead in September, whereas it was located directly overhead in July. The general rule illustrated here is a follows: At the same clock time, all objects in our sky appear to drift westward as we proceed through the year. 18. Precession is the very slow (but very real) wobble of Earth's rotation axis; 26,000 years is required to complete one full wobble. The orientation of Earth's rotation axis defines the directions to the north and south celestial poles. Thus, if the axis is changing direction, the particular stars that lie in thte direction of the poles changes continuously through the precession cycle. As it happens, the north end of the Earth's rotation axis points toward a moderately bright star (Polaris). This coincidence has not always been so, nor will it be so in the future. 19. By modern definition, a constellation is a fixed, well-defined piece of the celestial sphere. All traditional constellation figures (e.g., Ursa Major) are enclosed by thet boundaries of a modern constellation. 20. The Milky Way in our sky is the projection on the celestial sphere of the disk of the Milky Way galaxy; we're viewing our galaxy from a vantange point inside of it, of course!

International Journal of Modern Physics D, Vol. 0, No. 0 (1993) 000–000 c World Scientific Publishing Company TOPOLOGICAL MEASURE AND GRAPH-DIFFERENTIAL GEOMETRY ON THE QUOTIENT SPACE OF CONNECTIONS ***This work was supported in part by the National Science Foundation grant PHY91-07007 and Polish Committee for Scientific Research (KBN) through grant no. 2 0430 9101. JERZY LEWANDOWSKI†††Permanent address: Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoza 69, 00-689 Warszawa, Poland Department of Physics, University of Florida Gainesville, FL 32611, USA Received September 1, 1993ABSTRACT Integral calculus on the space of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of . The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly The space of gauge equivalent connections plays a central role in gauge theories as well as in the connection-dynamics formulation of general relativity. The problem of constructing quantum kinematics of these theories can therefore be reduced to: i)Introducing the algebra of a complete set of manifestly gauge invariant observables; and, ii) Finding a suitable representation of this algebra. For the first step, one can use the loop-strip variables introduced by Rovelli, Smolin and Ashtekar. A general strategy to complete the second step was developed by Ashtekar and Isham using the Gel’fand theory of representations of algebras. We are able to carry this strategy to completion by introducing a diffeomorphism invariant measure on (an appropriate completion of) , using the resulting space of square-integrable functions on this space as quantum states and introducing techniques from differential geometry on this space to represent the strip operators. These results provide a rigorous kinematical framework for the quantum version of these theories. Graphs, loops, connections and holonomies are used as the main tools. The measure is introduced via a non-linear generalization of the theory of cylindrical functions on topological vector spaces. In addition to being diffeomorphism invariant, to our knowledge, it is the first faithful measure on (the completion of) . Similar techniques enable us to do differential geometry on this space in terms of a family of projections onto finite dimensional manifolds. The resulting family of strip operators (representing momenta) are shown to be symmetric with respect to this measure. This framework, in particular, enables one to define a Rovelli-Smolin loop transform in a rigorous fashion, thereby leading to a loop representation for quantum states. Furthermore, the loop (i.e. holonomy) and strip (i.e. conjugate momentum) operators are well defined also in the loop representation. Finally, the framework offers a new and rigorous avenue to regularization of several physically interesting operators. Mathematically, it opens a new avenue to a graph-theory of measures, differential geometry on the space of (gauge equivalent) connections, and a relation between knot and link invariants and measures on this space. We consider here the space of -connections defined over a manifold . is the quotient with respect to the group of gauge transformations on . In application to gravity, is a Cauchy 3-surface and a gauge group is taken to be . Nonetheless, most of our results remains true if we assume only that is a compact Lie group. Below we give an outline of our technics. This research was carried out in colaboration with Abhay Ashtekar. Details will appear in a joined paper.. 2 Graph-manifold structure on . We introduce on a family of projections where takes all the natural values depending on and an element of is the class of conjugated elements of ( ). The projections which we construct are labelled by trivialised graphs. We fix in a reference point and consider the set of all the piecewise analytic loops which begin at . Let be a graph analytically embedded into . It consists of two sets: of edges and of vertexes. Connect each vertex with by an analytic path which does not overlap any of the edges. A trivialization of a graph is the map where the vertex and is the beginning and the end of respectively. A trivialized graph defines the projection with denoting the parallel transport with respect to a connection around a loop ( is the number of edges in ). Below we present two examples of geometrical objects of our theory: cylindrical functions and a topological measure. Given a map (1) we can lift any continuous function defined on to a function defined on . A function which can be obtained in this way by using an analytically embedded graph will be called cylindrical function. We consider only such functions for which does not depend on trivialization of . Denote by the set of cylindrical functions. First important property of is that if then both . Hence is an algebra. We can equip it with a norm . A principal example of a cylindrical function is a ”Wilson loop”, the function defined by any loop in : The second important result is that ”Wilson loops” span a vector space dense in . Therefore, the completion of , (the *-operation being a complex conjugation) is isomorphic to the Ashtekar-Isham holonomies algebra. Our equivalent construction of the A-I algebra provides us automatically with a natural functional defined on . Indeed, it turns out that for every cylindrical function the quantity is independent of which projection compatible with we use, where was naturally extended from to and is the normalized Haar measure on . The functional , as the notation suggests, may be thought of as a generalized measure on . On the other hand, since is strictly positive and continuous on it defines a regular and faithful measure on the A-I extension for , the spectrum of the algebra. In addition, for our definition of the measure we did not use any extra structure on . This is reflected in the topological character of the measure: is invariant with respect to the diffeomorphisms acting on . In this way the Hilbert space of quantum states for a topological theory on has been constructed. It is worth noting, that our measure, although arises naturally from the Haar measure on , is not the only measure which fits the projections (1). Recently, Baez has exploited this general strategy to introduce other, more involving the graph theory, examples of graph measures. 3 The momentum operator Unlike the measure, a strip operator which represents momentum was introduced before directly on . Our graph-differential geometry technics however are successfully applied for an evaluation of the Hermitian conjugation of this operator. To begin with, consider a momentum variable canonically conjugate to the connection variable ( is a space index and is a Lie algebra index). In gravity is a sensitized tried. Equivalently, we represent by a matrix valued 2-form such that A strip means an analytical embedding . To every point we assign the valued function of , where is the loop passing through . The strip variable is a gauge invariant integral of on , The strip operator corresponding to is the ”distributional vector field” defined on by the usual Poisson bracket: What is important for us, is that carries every function into a linear superposition of other s. Hence, from our point of view, the operator is is densely defined in . Now, we are coming back to the graph-differential geometry approach. A graph is compatible with a strip if the following two conditions are satisfied (i) every edge either intersects at most at its ends or is entirely contained in , (ii) if a vertex belongs to then the loop is contained in the graph . Here are our results: (a) Suppose is compatible with a strip ; then there exists on the corresponding a vector field such that for every cylindrical function compatible with (and such that is differentiable) (b) for every strip operator and a cylindrical function there exists a graph simultaneously compatible with both of them. We shall not write here the full expression for but we summarize below its properties which are relevant. The component on the copy of corresponding to an edge is zero if does not intersect ; otherwise is a conformal Killing vector in . The total divergence of vanishes. Finally, the map preserves the Hermitian conjugations and extends naturally to the commutators. Therefore, since , This extends to the full Lie algebra generated by all the strip operators. - A. Ashtekar, Non-perturbative canonical quantum gravity (Notes prepared in collaboration with R.S. Tate), (World Scientific, Singapore, 1991). - A. Ashtekar, in: Proceedings of the 1992 Les Houches summer school on ‘‘Gravitation and Quantization’’ - A. Ashtekar and C. Isham, Class. Quantum Grav. (1992) 9, 1433-85. - C. Rovelli and L. Smolin, Nucl. Phys. (1992)B331, 80-152. - A. Ashtekar and J. Lewandowski Representation Theory of Analytic Holonomy Algebras to appear in the Procedings of the Conference on ‘‘Quantum Gravity and Knot Theory’’, edited by J. Baez (Oxford U. Press, Oxford 1993), - J. Baez, In: Proceedings of the conference on quantum topology (to appear, 1993)

12th International Conference on Domain Decomposition Methods Editors: Tony Chan, Takashi Kako, Hideo Kawarada, Olivier Pironneau, c 2001 DDM.org 41. Application of the Domain Decomposition Method to the Flow around the Savonius Rotor Testuya Kawamura1 , Tsutomu Hayashi2 , Kazuko Miyashita3 Introduction In this study, we focus on the Savonius Rotor and try to compute the flow field under its operation and make clear the running performance by means of the numerical simulation. Our final objective is to simulate the flow field around the whole rotor and estimate the effect of the sidewall or the other rotor. Incompressible Navier-Stokes equations are solved in a few regions separately where the fixed coordinate and the rotating coordinate are used respectively. We employ domain decomposition method in order to connect these regions with adequate accuracy. The basic equations in each region are solved by using standard MAC method[HW65]. The physical quantities such as the velocity and the pressure in each region are transferred through the overlapping region, which is common in each domain. Reasonable results are obtained in the present calculations. Recently, the wind force is widely recognized as the environmentally friendly energy and attracts public attention. The wind power plant using windmills is the typical example. In order to make an effective windmill, it is very important to analyze the flow field around a windmill. In this case, numerical simulation becomes a promising method. The most important part of the investigation is to analyze the flow field near the rotating rotor of the windmill. On the other hand, it is also very important to investigate the interaction among the windmills if they are placed without long distance. For the numerical simulation of rotating body, it is convenient to use the rotating coordinate system, which rotates with the same speed. However, if there is another body which is not rotating or if there are many rotors which rotate at different position and with different speed, it is very difficult to choose one special rotating coordinate system. In these situations, it is natural to use many coordinate systems separately, which are suitable for the flow simulation around each rotor and connect these co- ordinates adequately. We focus on these points and simulate the flow fields around a windmill by using domain decomposition method in which the whole computation region is divided into several domains and they are connected adequately. The Savonius rotor[Sav31] is chosen for the simulation since in this case the ro- tating bluff body generates the complex flow field with large separation and it is very interesting to investigate such flow from the fluid dynamical point of view. Figure 1 is the schematic figure of the Savonius rotor. The features of this windmill are easy to make, independent of the direction of the wind, low speed and high torque. The Savonius rotor is usually used as the pump. 1 Ochanomizu University, email@example.com 2 Tottori University, firstname.lastname@example.org 3 Ochanomizu University, email@example.com 394 KAWAMURA, HAYASHI, MIYASHITA Figure 1: Savonius rotor There are several experimental and numerical works concerning with Savonius rotor [RESF78] [Oga83] [IST94]. Among them, Ishimatsu et al.[IST94] calculated the flow around a Savonius rotor. Their objective is to compute running performance of one windmill. Therefore, they ignore the effect of the sidewall, ground and other windmills. Their numerical method is based on the finite volume method with unstructured grids. As is discussed above, one of the important objectives of the present study is to investigate the effect of the obstacles. Therefore, we employ the domain decomposition method in this study. Numerical Method Since the rotational frequency is low enough, the flow around the Savonius rotor is assumed as incompressible. The basic equations are ∇v = 0 ∂v 1 2 + (v · ∇v) = −∇p + ∇ v ∂t Re where Re is the Reynolds number. We use both Cartesian coordinate system (x,y) and R: radius of rotation D: bucket diameter ω: angular velocity θ: attack angle Figure 2: Savonius Rotors without walls & obstacle the rotating coordinate system (X,Y) which rotates around vertical axis with constant angular velocity ω. If we use the symbols indicated in Figure 2, the relation between two coordinate systems is X = x cos θ − y sin θ, FLOW AROUND SAVONIUS ROTOR 395 Y = x sin θ + y cos θ, where θ is the angle between two coordinate systems. The basic equations are ex- pressed in the rotating coordinate system as ∂U ∂V + =0 ∂X ∂Y ∂U ∂U ∂U ∂P 1 ∂2U ∂2U +U +V − ω 2 X + 2ωV = − + + ∂t ∂X ∂Y ∂X Re ∂X 2 ∂Y 2 ∂V ∂V ∂V ∂P 1 ∂2V ∂2V +U +V − ω 2 Y − 2ωU = − + + ∂t ∂X ∂Y ∂Y Re ∂X 2 ∂Y 2 where (U,V) are the velocity components in (X,Y) direction while (u,v) are those in the fixed Cartesian coordinate system. These velocity components are connected to each other through the following relations: U = u cos θ − v sin θ − ωY, V = u sin θ + v cos θ + ωX. We use two computational domains. One domain(region1) includes the rotating rotor and another(region2) includes the fixed walls. Since the shape of the Savonius rotor is semicircular, it is convenient to use a semicircular region. The region including rotors consists of two semicircular regions whose centers are located at different positions. These two regions are connected by one line which passes two centers as is shown in Figure 3. Clearly, it is convenient to use the grid system based on the cylindrical co- ordinate. Another domain(region2) is rectangular and includes the fixed walls(Figure 4). The Cartesian coordinate system is used and the non-uniform rectangular grid is Figure 3: Inner region(region1). Figure 4: Outer region(region2). The bold lines indicate two blades The bold lines indicate the sidewalls employed in this region. The grid points do not coincide with each other in both x(X) and y(Y) direction. The computations in the two domains, which have the overlapping region are performed alternatively at every time step. Figure 5 indicates the whole computational region. The physical quantities (velocity and pressure) are exchanged through the common overlapping region as is shown in Figure 6. When we compute 396 KAWAMURA, HAYASHI, MIYASHITA Figure 5: Whole computational region Figure 6: Domain decomposition by the overlapping region the flow field of region1, the boundary conditions are required on each boundary. If the boundary locates outside of the region, the boundary conditions are determined by the usual way, i.e. free stream condition or something like this. If the boundary locates inside of the region2, the boundary values can be obtained from the computa- tional results of the region2. In this case, some interpolations are required since the grid systems in both regions are different. In this study, the interpolation shown in Figure 7 is used. Since this formula requires only the distance from the four corners 1 1 1 1 1 fP = fQ + fR + fS + fT R rQ rR rS rT 1 1 1 1 where R = + + + rQ rR rS rT Figure 7: Interpolation in the overlapping region in one grid cell, it can be used even if the grid cell is highly deformed. Similar technique is used for determining the boundary conditions of region2 from the computational results of region1. In region1, two regions of semicircular shape are connected through one line without overlapping region. The boundary conditions on FLOW AROUND SAVONIUS ROTOR 397 this line are given by the average value of the nearest grid points in each region(Figure 8) as follows: The numerical method to solve incompressible Navier-Stokes equation fP = (fQ + (1 − r)fR + rfS )/2. Figure 8: Interpolation along the line is the standard MAC method. All the spatial derivatives except the nonlinear term of the Navier-Stokes equation is approximated by the second order central difference. Nonlinear terms are approximated by the third order upwind scheme[KK84] due to the numerical stability. Euler explicit scheme is employed for the time integration. Result Typical results obtained by the present study are shown here. The dimensionless gap width(=S/D, see Figure 2) is chosen to 0.15 and tip speed ratio λ(= Rω/u∞) is changed from 0.25 to 1.25. Figure 9 indicates the initial position of the rotor. In this Figure 9: Initial position of the rotor case, the rotational angle θ is defined as zero and the rotor begins to rotate clockwise from this position. Figure 10 is an example of the instantaneous velocity vectors. Both the vectors in the inner region and the outer region are plotted in the same figure. The vectors vary continuously from the inner region to the outer region, which indicates the interpolation works well in this calculation. Figure 11 is time history of the torque coefficient. The torque coefficient Cr (= T /qRA where T is the torque, q is the dynamic pressure, R is the radius of the rotor, and A is the projection area). The tip speed ratio is 0.25 and no walls exist. Clearly, it has a period of 180 degree. The torque becomes maximum and minimum around 30 and 150 degree respectively and becomes zero around 120 and 180 degree. Figure 12 is also time history of the 398 KAWAMURA, HAYASHI, MIYASHITA Figure 10: An example of the instantaneous velocity fields in the whole region torque coefficient but the tip speed ratio is 0.5 and 0.75. As tip speed ratio becomes large, the negative part of the curve becomes large indicating the total torque becomes small. Figure 13 is the result of the calculation with walls. It corresponds to Figure 11 and Figure 12. Although the shape of each curve is similar, the absolute value becomes large for the latter case. Figure 14 is the time-averaged torque coefficient Figure 11: Time history of the torque coefficient without walls(Tip speed ratio is 0.25) for various tip speed ratio λ. Both the results of the calculations with and without walls are indicated in the same figure. Torque coefficients decrease nearly linear as the tip speed ratio increases and become negative around 0.8. They become almost twice when the walls exist. Figure 15 is the time-averaged power coefficients Cp (= λCr ) for various tip speed ratio. Both the results with and without walls are shown. The power coefficient has its maximum value around λ = 0.5 and λ = 0.4 for the case with and without walls respectively. The maximum value is almost twice for the case with FLOW AROUND SAVONIUS ROTOR 399 Figure 12: Time history of the torque coefficient without walls(Tip speed ratio is 0.5 and 0.75) Figure 13: Time history of the torque coefficient with walls(Tip speed ratio is 0.25, 0.5 and 0.75) Figure 14: Time-averaged torque coeffi- Figure 15: Time-averaged power coeffi- cient for various tip speed ratios for the cient for various tip speed ratios for the cases with and without walls cases with and without walls 400 KAWAMURA, HAYASHI, MIYASHITA walls. Summary In this study, the flow field around the windmill is computed by using domain decom- position method. Although the Savonius rotor is chosen for the present computation, this method can be applied for the computations of other windmills. Two compu- tational domains are used and connected to each other. One domain contains the rotating rotor and rotational coordinate system is employed. Another contains the fixed walls and the Cartesian coordinate system is used. Both regions have common overlapping region. The physical quantities on the boundary of one domain in the over- lapping region are calculated by interpolating the physical values at the grid points in another region which are located inside of the region. The running performance of the Savonius rotor such as the torque coefficient and the power coefficient is obtained for various tip speed ratios. The effect of the walls on the running performance is also investigated. It is found that torque coefficient and the power coefficient become almost twice when the walls are placed adequately. References [HW65]F. H. Harlow and J. E. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids, 8(12):2182–2189, December 1965. [IST94]K. Ishimatsu, T. Shinohara, and F. Takuma. Numerical simulation for savonius rotors(running performance and flow fields). JSME(B), 60(569):154–160, 1994. [KK84]T. Kawamura and K. Kuwahara. Computation of high Reynolds number flow around a circular cylinder with surface roughness. AIAA paper, 84(0340), 1984. [Oga83]Ogawa. The study on the savonius wind turbine (1st. report; theoretical anal- ysis). JSME, 49(441), 1983. [RESF78]B. F. Blackwell R. E. Sheldahl and L. V. Feltz. Wind tunnel performance data for two- and three-bucket savonius rotors. J. Energy, 1978. [Sav31]S. J. Savonius. The s-roter and its application. Mech. Eng., 53:333, 1931. Pages to are hidden for "41. 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A Shortcut is the Longest Distance Between Two Points The two critical regions left and right are determined by the left and right critical values tcrit. In the middle, under the heading Ha: diff! You wish to maximize your ability to detect the improvement, so you opt for a one-tailed test. Figure 2 — Box plot for sample data Column E of Figure 1 contains all the formulas required to carry out the t test. But how do you choose which test? Is this a one or two tailed test? Again, the test can be one-tailed if the researcher has good reason to expect the difference goes in a particular direction. In doing so, you fail to test for the possibility that the new drug is less effective than the existing drug. t-Test in Excel - Easy Excel Tutorial The specific test that was conducted did not reject the null hypothesis, but we also see that such a test would only have found a very small effect of size. When using a two-tailed test, regardless of the direction of the relationship you hypothesize, you are testing for the possibility of the relationship in both directions. Our help write papers hypothesis is that the mean is equal to x. The test is comparing the mean male score to the mean female score. Some Basic Null Hypothesis Tests Look at the below example to understand the concept practically. We then briefly consider some other versions of the ANOVA that are used for within-subjects and factorial research designs. Because this p value is less than. P value is denoted by decimal points but it is always a good thing to tell the result of the P value in percentage instead of decimal points. NOTICE2: The Ho is the null hypothesis and so always contains the equal sign as it is the case for which there is no significant difference between the two groups. In particular, we see that the Effect size cell B10 contains the value 0. Step 2: Set the Level of Significance Although you can calculate the P-value of a z-score by hand, the formula is extremely complex. For example, we may wish to compare the cover letter sample for job abroad of a sample to a given value x using a t-test. However, there are many applications that run such tests. Since Excel only displays the values of these formulas, we show each of the formulas in text format in column G so that you can see how the calculations are performed. - One-Tailed z-test Hypothesis Test By Hand | Learn Math and Stats with Dr. G - So when is a one-tailed test appropriate? Asymmetrical distributions like the F and chi-square distributions have only one tail. If the sample mean differs from the hypothetical population mean in the expected direction, then we one tailed hypothesis test excel a better chance academic and creative writing rejecting the null hypothesis. This is because only one tail of the distribution is used for the test. Now imagine further that the pretest estimates are,and that the posttest estimates for the same participants in the same order are,one tailed hypothesis test excel One-Tailed z-test Hypothesis Test By Hand Example: Suppose it is up to you to determine if a certain state receives significantly more public school funding per student than the USA average. We double the value of alpha since we are considering the one-tailed test. He has been writing for one tailed hypothesis test excel years and has been published in ubc creative writing requirements academic journals in fields such as psychology, drug addiction, epidemiology and help write papers. This means that. For all three p-values, the test statistic is 2. In this example, it makes sense to subtract the pretest estimates academic and creative writing the posttest estimates so that positive difference scores mean that the estimates went up after the training and negative difference scores mean the estimates went down. Here is how it works. Using the dataset, you would need to first calculate the sample mean. The consequences in this example are extreme, but they illustrate a danger of inappropriate use of a one-tailed test. Statistical significance addresses the question: "If, in the entire population from which this sample was drawn, the parameter estimate was 0, how likely are results as extreme as this or more extreme? - P Value or Probability Value is a popular concept in the statistical world. - Assignment writing help services essay editing for making nursing cover letter - Basic cover letter for college student buy resume for writer position graduate thesis - Write my essay uk cheap - When you conduct a test of statistical significance, whether it is from a correlation, an ANOVA, a regression or some other kind of test, you are given a p-value somewhere in the output. Is the p-value appropriate for your test? Step 1: What is Ho and Ha? trofeoaccademianavale.com function - Office Support It is a ratio of two estimates of the population variance based on the sample data. A dialog box will appear as in Figure 5a. Figure 1 — One sample t test A negative value in column B indicates that the subject gained weight. In this example, the two-tailed p-value suggests rejecting the null hypothesis of no difference. Can we conclude that the program is effective? Because this is less than. All the aspiring analysts should know about the P Value and its purpose in data science. What is a one-tailed test? What is a two-tailed test? Step 2: Set the Level of Significance Decide if you want the P-value to be higher than this z-score or lower than this z-score. This test makes sense when we have good reason to expect the sample mean will differ from the hypothetical population mean in a particular direction. The tail refers to the end of the distribution of the test statistic for the particular restaurant at new york botanical garden that you are conducting. It is a ratio of two estimates of the population variance based on the sample data. If your z-score is negative, you cover letter sample for job abroad certainly want a more negative P-value;if it is positive, you almost certainly want a more positive P-value. Statistical functions in Excel As in Example 5, we can then calculate the power of the test to be Academic and creative writing do not care if it dissertation help service dissertationhelpservice.co.uk significantly more effective. Except for the sign, this is the same result that was obtained using the T Test help write papers Non-parametric Equivalents data analysis tool see cell V51 of Figure 5. In this scenario, a one-tailed test would be appropriate. However, in this example, we will run the two-tailed test. He holds a Ph. However, the p-value presented is almost always for a two-tailed test. Because he has no real sense of whether the students will underestimate or overestimate the number of calories, he decides to do a two-tailed test.

Page 122 (4,11,12,13) Page 144 (2, 4, 6, 9, 10,11) 6-4 What is the difference between gross private domestic investment and net private domestic investment? If you were to determine net domestic product (NDP) through the expenditures approach, which of these two measures of investment spending would be appropriate? Explain. Gross private domestic investment less depreciation is net private domestic investment. Depreciation is the value of all the physical capital—machines, equipment, buildings—used up in producing the year’s output. Since net domestic product is gross domestic product less depreciation, in determining net domestic product through the expenditures approach it would be appropriate to use the net investment measure that excludes depreciation, that is, net private domestic investment. 6-11 (Key Question) Suppose that in 1984 the total output in a single-good economy was 7,000 buckets of chicken. Also suppose that in 1984 each bucket of chicken was priced at $10. Finally, assume that in 1996 the price per bucket of chicken was $16 and that 22,000 buckets were purchased. Determine the GDP price index for 1984, using 1996 as the base year. By what percentage did the price level, as measured by this index, rise between 1984 and 1996? Use the two methods listed in Table 7-6 to determine real GDP for 1984 and 1996. X/100 = $10/$16 = .625 or 62.5 when put in percentage or index form (.625 x 100) 100 62.5 16 10 6 .60 or 60% (Easily calculated .6 60% ) 62.5 10 10 Method 1: 1996 = (22,000 x $16) ÷ 1.0 = $352,000 1984 = (7,000 x $10) ÷ .625 = $112,000 Method 2: 1996 = 22,000 x $16 = $352,000 1984 = 7,000 x $16 = $112,000 6-12 (Key Question) The following table shows nominal GDP and an appropriate price index for a group of selected years. Compute real GDP. Indicate in each calculation whether you are inflating or deflating the nominal GDP data. Nominal GDP, Price index Real GDP, Year Billions (1996 = 100) Billions 1960 $527.4 22.19 $ ______ 1968 911.5 26.29 $ ______ 1978 2295.9 48.22 $ ______ 1988 4742.5 80.22 $ ______ 1998 8790.2 103.22 $ ______ Values for real GDP, top to bottom of the column: $2,376.7 (inflating); $3,467.1 (inflating); $4,761.3 (inflating); $5,911.9 (inflating); $8,516 (deflating). 6-13 Which of the following are actually included in this year’s GDP? Explain your answer in each case. a. Interest on an AT&T bond. b. Social security payments received by a retired factory worker. c. The services of a family member in painting the family home. d. The income of a dentist. e. The money received by Smith when she sells her economics textbook to a book buyer. f. The monthly allowance a college student receives from home. g. Rent received on a two-bedroom apartment. h. The money received by Josh when he resells his current-year-model Honda automobile to Kim. i. Interest received on corporate bonds. j. A 2-hour decrease in the length of the workweek. k. The purchase of an AT&T corporate bond. l. A $2 billion increase in business inventories. m. The purchase of 100 shares of GM common stock. n. The purchase of an insurance policy. (a) Included. Income received by the bondholder for the services derived by the corporation for the loan of money. (b) Excluded. A transfer payment from taxpayers for which no service is rendered (in this year). (c) Excluded. Not a market transaction. If any payment is made, it will be within the family. (d) Included. Payment for a final service. You cannot pass on a tooth extraction! (e) Excluded. Secondhand sales are not counted; the textbook is counted only when sold for the first time. (f) Excluded. A private transfer payment; simply a transfer of income from one private individual to another for which no transaction in the market occurs. (g) Included. Payment for the final service of housing. (h) Excluded. The production of the car had already been counted at the time of the initial sale. (i) Included. The income received by the bondholders is paid by the corporations for the current use of the “money capital” (the loan). (j) Excluded. The effect of the decline will be counted, but the change in the workweek itself is not the production of a final good or service or a payment for work done. (k) Excluded. A noninvestment transaction; it is merely the transfer of ownership of financial assets. (If AT&T uses the money from the sale of a new bond to carry out an investment in real physical assets that will be counted.) (l) Included. The increase in inventories could only occur as a result of increased production. (m) Excluded. Merely the transfer of ownership of existing financial assets. (n) Included. Insurance is a final service. If bought by a household, it will be shown as consumption; if bought by a business, as investment—as a cost added to its real investment in physical capital. Page 144 (2, 4, 6, 9, 10,11) 7-2 (Key Question) Suppose an economy’s real GDP is $30,000 in year 1 and $31,200 in year 2. What is the growth rate of its real GDP? Assume that population was 100 in year 1 and 102 in year 2. What is the growth rate of GDP per capita? Growth rate of real GDP = 4 percent (= $31,200 - $30,000)/$30,000). GDP per capita in year 1 = $300 (= $30,000/100). GDP per capita in year 2 = $305.88 (= $31,200/102). Growth rate of GDP per capita is 1.96 percent = ($305.88 - $300)/300). 7-4 (Key Question) What are the four phases of the business cycle? How long do business cycles last? How do seasonal variations and secular trends complicate measurement of the business cycle? Why does the business cycle affect output and employment in capital goods and consumer durable goods industries more severely than in industries producing nondurables? The four phases of a typical business cycle, starting at the bottom, are trough, recovery, peak, and recession. As seen in Table 8-2, the length of a complete cycle varies from about 2 to 3 years to as long as 15 years. There is a pre-Christmas spurt in production and sales and a January slackening. This normal seasonal variation does not signal boom or recession. From decade to decade, the long-term trend (the secular trend) of the U.S. economy has been upward. A period of no GDP growth thus does not mean all is normal, but that the economy is operating below its trend growth of output. Because capital goods and durable goods last, purchases can be postponed. This may happen when a recession is forecast. Capital and durable goods industries therefore suffer large output declines during recessions. In contrast, consumers cannot long postpone the buying of nondurables such as food; therefore recessions only slightly reduce nondurable output. Also, capital and durable goods expenditures tend to be “lumpy.” Usually, a large expenditure is needed to purchase them, and this shrinks to zero after purchase is made. 7-6 (Key Question) Use the following data to calculate (a) the size of the labor force and (b) the official unemployment rate: total population, 500; population under 16 years of age or institutionalized, 120; not in labor force, 150; unemployed, 23; part-time workers looking for full-time jobs, 10. Labor force 230 500 - 120 150 ; official unemployme nt rate 10% 23 / 230 100 7-9 Explain how an increase in your nominal income and a decrease in your real income might occur simultaneously. Who loses from inflation? Who loses from unemployment? If you had to choose between (a) full employment with a 6 percent annual rate of inflation or (b) price stability with an 8 percent unemployment rate, which would you choose? Why? If a person’s nominal income increases by 10 percent while the cost of living increases by 15 percent, then her real income has decreased from 100 to 95.65 (= 110/1.15). Alternatively expressed, her real income has decreased by 4.35 percent (= 100 - 95.65). Generally, whenever the cost of living increases faster than nominal income, real income decreases. The losers from inflation are those on incomes fixed in nominal terms or, at least, those with incomes that do not increase as fast as the rate of inflation. Creditors and savers also lose. In the worst recession since the Great Depression (1981-82), those who lost the most from unemployment were, in descending order, blacks (who also suffer the most in good times), teenagers, and blue-collar workers generally. In addition to the specific groups who lose the most, the economy as a whole loses in terms of the living standards of its members because of the lost production. The choice between (a) and (b) illustrates why economists are unpopular. Option (a) spreads the pain by not having a small percentage of the population bear the burden of employment. There is the risk, however, that inflationary expectations will give rise to creeping inflation and ultimately hyperinflation; or that the central bank will raise interest rates to reduce inflation, stalling economic growth. If one chooses (b) the central bank will have no cause to raise interest rates and cut off the economic expansion needed to get unemployment down from the unforgivable 8 percent. However, the weakness in spending resulting from an 8% unemployment rate might push the economy into deflation, which would ultimately exacerbate the weak economic conditions. 7-10 What is the Consumer Price Index (CPI) and how is it determined each month? How does the Bureau of Labor Statistics (BLS) calculate the rate of inflation from one year to the next? What effect does inflation have on the purchasing power of a dollar? How does it explain differences between nominal and real interest rates? How does deflation differ from inflation? The CPI is constructed from a “market basket” sampling of goods that consumers typically purchase. Prices for goods in the market basket are collected each month, weighted by the importance of the good in the basket (cars are more expensive than bread, but we buy a lot more bread), and averaged to form the price level. To calculate the rate of inflation for year 5, the BLS subtracts the CPI of year 4 from the CPI of year 5, and then divides by the CPI of year 4 (percentage change in the price level). Inflation reduces the purchasing power of the dollar. Facing higher prices with a given number of dollars means that each dollar buys less than it did before. The rate of inflation in the CPI approximates the difference between the nominal and real interest rates. A nominal interest rate of 10% with a 6% inflation rate will mean that real interest rates are approximately 4%. Deflation means that the price level is falling, whereas with inflation overall prices are rising. Deflation is undesirable because the falling prices mean that incomes are also falling, which reduces spending, output, employment, and, in turn, the price level (a downward spiral). Inflation in modest amounts (<3%) is tolerable, although there is not universal agreement on this point. 7-11 (Key Question) If the price index was 110 last year and is 121 this year, what is this year’s rate of inflation? What is the “rule of 70”? How long would it take for the price level to double if inflation persisted at (a) 2, (b) 5, and (c) 10 percent per year? This year’s rate of inflation is 10% or [(121 – 110)/110] x 100. Dividing 70 by the annual percentage rate of increase of any variable (for instance, the rate of inflation or population growth) will give the approximate number of years for doubling of the variable. (a) 35 years ( 70/2); (b) 14 years ( 70/5); (c) 7 years ( 70/10). p163-164 (3-10) 8-3 Explain how each of the following will affect the consumption and saving schedules or the investment schedule: a. A large increase in the value of real estate, including private houses. b. A decline in the real interest rate. c. A sharp, sustained decline in stock prices. d. An increase in the rate of population growth. e. The development of a cheaper method of manufacturing computer chips. f. A sizable increase in the retirement age for collecting Social Security benefits. g. The expectation that mild inflation will persist in the next decade. (a) If this simply means households have become more wealthy, then consumption will increase at each income level. The consumption schedule should shift upward and the saving schedule shift downward. The investment schedule may shift rightward if owners of existing homes sell them and invest in construction of new homes more than previously. (b) The decline in the real interest rate will increase interest-sensitive consumer spending; the consumption schedule will shift up and the saving schedule down. Investors will increase investment as they move down the investment-demand curve; the investment schedule will shift upward. (c) A sharp decline in stock prices can be expected to decrease consumer spending because of the decrease in wealth; the consumption schedule shifts down and the saving schedule upward. Because of the depressed share prices and the number of speculators forced out of the market, it will be harder to float new issues on the stock market. Therefore, the investment schedule will shift downward. (d) The increase in the rate of population growth will, over time, increase the rate of income growth. In itself this will not shift any of the schedules but will lead to movement upward to the right along the upward sloping investment schedule. (e) This innovation will in itself shift the investment schedule upward. Also, as the innovation starts to lower the costs of producing everything using these chips, prices will decrease leading to increased quantities demanded. This, again, could shift the investment schedule upward. (f) The postponement of benefits may cause households to save more if they planned to retire before they qualify for benefits; the saving schedule will shift upward, the consumption schedule downward. This impact is uncertain, however, if people continue to work and earn productive incomes. (g) If this is a new expectation, the consumption schedule will shift upwards and the saving schedule downwards until people have stocked up enough. After about a year, if the mild inflation is not increasing, the household schedules will revert to where they were before. 8-4 Explain why an upward shift in the consumption schedule typically involves an equal downshift in the saving schedule. What is the exception to this relationship? If, by definition, all that you can do with your income is use it for consumption or saving, then if you consume more out of any given income, you will necessarily save less. And if you consume less, you will save more. This being so, when your consumption schedule shifts upward (meaning you are consuming more out of any given income), your saving schedule shifts downward (meaning you are consuming less out of the same given income). The exception is a change in personal taxes. When these change, your disposable income changes, and, therefore, your consumption and saving both change in the same direction and opposite to the change in taxes. If your MPC, say, is 0.9, then your MPS is 0.1. Now, if your taxes increase by $100, your consumption will decrease by $90 and your saving will decrease by $10. 8-5 (Key Question) Complete the accompanying table. Level of Output and income (GDP = DI) Consumption Saving APC APS MPC MPS $240 $ _____ $-4 _____ _____ _____ _____ 260 $ _____ 0 _____ _____ _____ _____ 280 $ _____ 4 _____ _____ _____ _____ 300 $ _____ 8 _____ _____ _____ _____ 320 $ _____ 12 _____ _____ _____ _____ 340 $ _____ 16 _____ _____ _____ _____ 360 $ _____ 20 _____ _____ _____ _____ 380 $ _____ 24 _____ _____ _____ _____ 400 $ _____ 28 _____ _____ _____ _____ Data for completing the table (top to bottom). Consumption: $244; $260; $276; $292; $308; $324; $340; $356; $372. APC: 1.02; 1.00; .99; .97; .96; .95; .94; .94; .93. APS: -.02; .00; .01; .03; .04; .05; .06; .06; .07. MPC: .80 throughout. MPS: .20 throughout. a. Show the consumption and saving schedules graphically. b. Find the break-even level of income. How is it possible for households to dissave at very low-income levels? c. If the proportion of total income consumed (APC) decreases and the proportion saved (APS) increases as income rises, explain both verbally and graphically how the MPC and MPS can be constant at various levels of income. (a) See the graphs. Question 9-5a 420 400 380 C 360 Consumption 340 Question 9-5a 320 300 Break-Even Income 30 S 280 25 260 240 20 45 220 15 220 240 260 280 300 320 340 360 380 400 420 Savings 10 Real GDP 5 0 220 240 260 280 300 320 340 360 380 400 420 -5 -10 Real GDP (b) Break-even income = $260. Households dissave borrowing or using past savings. (c) Technically, the APC diminishes and the APS increases because the consumption and saving schedules have positive and negative vertical intercepts, respectively. (Appendix to Chapter 1). MPC and MPS measure changes in consumption and saving as income changes; they are the slopes of the consumption and saving schedules. For straight-line consumption and saving schedules, these slopes do not change as the level of income changes; the slopes and thus the MPC and MPS remain constant. 8-6 What are the basic determinants of investment? Explain the relationship between the real interest rate and the level of investment. Why is investment spending unstable? How is it possible for investment spending to increase even in a period in which the real interest rate rises? The basic determinants of investment are the expected rate of return (net profit) that businesses hope to realize from investment spending, and the real rate of interest. When the real interest rate rises, investment decreases; and when the real interest rate drops, investment increases—other things equal in both cases. The reason for this relationship is that it makes sense to borrow money at, say, 10 percent, if the expected rate of net profit is higher than 10 percent, for then one makes a profit on the borrowed money. But if the expected rate of net profit is less than 10 percent, borrowing the money would be expected to result in a negative rate of return on the borrowed money. Even if the firm has money of its own to invest, the principle still holds: The firm would not be maximizing profit if it used its own money to carry out an investment returning, say, 9 percent when it could lend the money at an interest rate of 10 percent. Investment is unstable because, unlike most consumption, it can be put off. In good times, with demand strong and rising, businesses will bring in more machines and replace old ones. In times of economic downturn, no new machines will be ordered. A firm can continue for years with, say, a tenth of the investment it was carrying out in the boom. Very few families could cut their consumption so drastically. New business ideas and the innovations that spring from them do not come at a constant rate. This is another reason for the irregularity of investment. Profits and the expectations of profits also vary. Since profits, in the absence of easy access to borrowed money, are essential for investment and since, moreover, the object of investment is to make a profit, investment, too, must vary. As long as expected rates of return rise faster than real interest rates, investment spending may increase. This is most likely to occur during periods of economic expansion. 8-7 (Key Question) Suppose a handbill publisher can buy a new duplicating machine for $500 and that the duplicator has a 1-year life. The machine is expected to contribute $550 to the year’s net revenue. What is the expected rate of return? If the real interest rate at which funds can be borrowed to purchase the machine is 8 percent, will the publisher choose to invest in the machine? Explain. The expected rate of return is 10% ($50 expected profit/$500 cost of machine). The $50 expected profit comes from the net revenue of $550 less the $500 cost of the machine. If the real interest rate is 8%, the publisher will invest in the machine as the expected profit (marginal benefit) from the investment exceeds the cost of borrowing the funds (marginal cost). 8-8 (Key Question) Assume there are no investment projects in the economy that yield an expected rate of return of 25 percent or more. But suppose there are $10 billion of investment projects yielding an expected rate of return of between 20 and 25 percent; another $10 billion yielding between 15 and 20 percent; another $10 billion between 10 and 15 percent; and so forth. Cumulate these data and present them graphically, putting the expected rate of net return on the vertical axis and the amount of investment on the horizontal axis. What will be the equilibrium level of aggregate investment if the real interest rate is (a) 15 percent, (b) 10 percent, and (c) 5 percent? Explain why this curve is the investment-demand curve. See the graph below. Aggregate investment: (a) $20 billion; (b) $30 billion; (c) $40 billion. This is the investment-demand curve because we have applied the rule of undertaking all investment up to the point where the expected rate of return, r, equals the interest rate, i. 8-9 (Key Question) What is the multiplier effect? What relationship does the MPC bear to the size of the multiplier? The MPS? What will the multiplier be when the MPS is 0, .4, .6, and 1? What will it be when the MPC is 1, .9, .67, .5, and 0? How much of a change in GDP will result if firms increase their level of investment by $8 billion and the MPC is .80? If the MPC is .67? The multiplier effect describes how an initial change in spending ripples through the economy to generate a larger change in real GDP. It occurs because of the interconnectedness of the economy, where a change in Haslett’s spending will generate more income for Davidic, who will in turn spend more, generating additional income for Grimes. The MPC is directly (positively) related to the size of the multiplier. The MPS is inversely (negatively) related to the size of the multiplier. The multiplier values for the MPS values: undefined, 2.5, 1.67, and 0. The multiplier values for the MPC values: undefined, 10, 3 (approx. actually 3.03), 2, 0. If MPC is .80, change in GDP is $40 billion (5 x $8 = $40) If MPC is .67, change in GDP is $24 billion (approximately) (3 x $8 = $24) 8-10 Why is the actual multiplier for the U.S. economy less than the multiplier in this chapter’s simple examples? The actual multiplier (estimated to be about 2) is smaller because it includes other leakages from the spending and income cycle besides just saving. Imports and taxes reduce the flow of money back into spending on domestically produced output, reducing the multiplier effect.

APPENDIX 6: Doppler Shifts and the ZPE A varying ZPE, with an inversely varying c, calls into question what is being measured in the cases where there are genuine Doppler shifts involved rather than the cosmological redshifts of distant galaxies discussed above. It is expected that these Doppler effects will usually be non-relativistic at their point of origin. Consequently, the basic Doppler formula becomes Here, λ is the laboratory wavelength and Δλ is the change in wavelength compared with the laboratory standard. The velocity producing the Doppler shift is given by v, while the speed of light is c. The relativistic counterpart can be found in reference . It has the same primary term v/c as in (125) but also includes higher order terms. Now it has been shown above that wavelengths remain unchanged in transit through space. Thus, apart from any ZPE induced red-shifting, the term Δλ remains unchanged in transit. The laboratory wavelength, λ, also remains unchanged. Thus the left hand side of equation (125) is independent of changing conditions in transit and depends entirely on conditions at the time of emission. If we now designate the speed of light at the time of emission as c1, and the velocity involved as being v1, then, retaining c as the velocity of light now and v the inferred velocity at reception, we can write (125) as From (126) it follows that Therefore, the actual velocity at the point of emission that we are measuring is (c1/c) times greater than the velocity v that we are inferring from the measurements. The practical outcome of this conclusion may be assessed by two examples. This conclusion is of some importance in view of data from supernova SN 1993J in M81 (NGC 3031). The rate of expansion of the supernova was observed in optical, radio and ultraviolet wavelengths. For example, optical spectroscopy using Doppler shifts of the blue edge of the hydrogen alpha line absorption trough determined the expansion rate based on a constant speed of light. These data gave a distance to M81 that closely agreed with Cepheid data [192-195]. But if c was higher at emission and slowed in transit, then we, the observers, were seeing those events in slow motion. However, the Doppler shift calculated on the current speed of light would exactly correspond with the observed sequence of events. This result is obtained since the actual velocity of expansion, v1, and the actual velocity of light, c1, were both proportionally greater. Because of this proportionality, the ratio (v1/c1) at the point of emission is still the same as the inferred (v/c) at reception, so measured Doppler velocities will always be in agreement with observed phenomena under conditions of varying ZPE & c. The other important conclusion is that galaxy rotation rates will be faster than suspected from the Doppler measurements. One aspect of this problem has already given rise to the idea of the ‘missing mass’ or ‘dark matter’, and has been dealt with in Appendix 3. However, this development with Doppler shifts has been seen as a distinct problem by some. There is an effective answer to this, and it comes from the Plasma Model of galaxy formation which was elaborated by Anthony Peratt in two major articles in the IEEE Journal of Plasma Physics. There, Peratt published photographs of experiments in the laboratory with plasma filaments and Birkeland currents . The photographs reveal that every form of galaxy can be reproduced as a sequence starting with a double radio galaxy and a quasar and ending up with a spiral galaxy simply by the interaction of two or more plasma filaments. The interaction time governs the final form of the object . The experiments demonstrate that the rotation rates of spiral arms in galaxies are not controlled by gravitation, but rather by the strength of the Birkeland current in the filaments . The animated versions of the experimental photographs reveal this clearly . As the current strength drops off, so, too, does the rotation rate of the spiral arms about the galaxy center. It has nothing to do with gravity and orbital mechanics. Thus when the Birkeland current strength is greater, the galaxy rotation rate is faster . It is to be expected that the strength of the Birkeland currents would decline with time after the formation of the universe. For this reason alone, it would be expected that actual galaxy rotation rates would be faster the closer they are to the inception of the cosmos. This situation approximates to what we are seeing if the Doppler velocities are corrected for higher lightspeed at the time of emission. However, we can go further than this. It can be shown that when the ZPE strength is lower and lightspeed is higher, then electric current strengths will be intrinsically greater. Equation (7A) shows that e2/ε = constant. Since the permittivity of space, ε is shown to be proportional to the ZPE strength, U, and hence to 1/c, then we have the strength of the electronic charge, e, being proportional to √U and hence to √(1/c). Thus the Hall resistance h/e2 is a constant from (2) and (7A). It can be shown that all resistances, R, generally follow this constancy with ZPE variation. Importantly, the expression for electrostatic force, F, is given by (e2/ε)[1/(4π r2)]. Now from (7A) it can be stated that F is a constant with varying ZPE since r is unchanged. But we also have F = eE = constant, where E is the electric field strength. Therefore the field strength, E, is proportional to √c. Equation (4) indicates that we have symmetry between the electric and magnetic properties of space. Therefore the force between two parallel currents is constant. If those currents are of equal magnitude I, then the force is proportional to (μI2) and is constant. So from (4), the electric current, I, is proportional to √c. Furthermore, since F is proportional to IB, where B is the magnetic flux density or the magnetic induction, then it follows that B is proportional to √U and hence √(1/c). Now since electric currents, I, will be intrinsically higher with higher c values and lower ZPE strengths, then it follows that galaxy rotation rates will also be higher on the various Plasma Models for galaxies, This supports the evidence from galaxy Doppler shifts with changing c and ZPE. Intrinsically stronger currents in earlier epochs of the Universe have implications for other aspects of the Plasma Model that can only be touched on here. Along with stronger currents goes a higher voltage or potential, since voltage, V = IR, where R is resistance and I is current as above. Since we have shown that R is unchanged with changing ZPE strengths, then it follows that V is proportional to I. But since I is proportional to √(1/U), or the √c, so too is V. In addition, since power, P, in an electrical circuit is given by the relation P = IV, it necessarily follows that this power must be proportional to 1/U or c. This treatment means that, if the Plasma Model for the Sun is followed , then for a Birkeland current, I, and potential difference, V, this process has a power output, P, which is proportional to c. This is the same result as that obtained for the thermonuclear case treated in Appendix 2 (ii). Now equation (77) in Appendix 2 points out that the energy density of all radiation, ρ, is proportional to 1/c. This is still true on the Plasma Model. Furthermore, the radiation intensities (and hence stellar luminosities) are given by the quantity ρc [148-150] as in (86A). But ρc is constant for changing ZPE strength on both models, so stellar luminosities must also be constant in these conditions for both models. Thus the only two models which have been proposed for the Sun’s light output, the thermonuclear and the plasma approach, both give the same result. Note that a full discussion of how Plasma Models behave with a varying ZPE requires another paper. F.A. Jenkins & H.E. White, “Fundamentals of Optics” 3rd Edition, (McGraw-Hill, 1957). pp. 412-414. F.A. Jenkins & H.E. White, op.cit. p.403 (see ref.#148) A.L. Peratt, IEEE Transactions on Plasma Science, Vol. PS-14 No.6 (Dec. 1986), pp. 639-778.

The first difficulty in RI mapping will only be mentioned briefly without a detailed analysis. At best, a new gene can be localized to an approximate position between two previously mapped loci, and quite frequently the new gene can only be mapped near another locus without knowing which side it is on. Because the number of recombinations in any region is so low, distances between loci often do not add linearly. Also, the determination of gene order in classical gene mapping depends on examining flanking markers. With RI mapping, there are numerous cases where several recombinations have occurred on a chromosome in any given line, and so the examination of nearby markers in the RI lines is not a reliable indication of gene order. The second difficulty in RI mapping concerns what could be called "ectopic localization": unlinked loci can appear to be linked, and unknown genes can appear to be located in more than one region of the genome. It is this problem that I wish to address in more detail here. The RI mapping scheme compares the allelic distribution of 205 loci (database from early 1988) distributed over the maize genome, using two independent families of recombinant-inbreds. The COXTx family has 48 RI's, and the TXCM family has 41 RI's. As explained by Burr et al., the RI method allows a direct calculation of an R value, which is related to recombination frequency by the formula, r = R/(2-2R). Some of the lines show heterozygosity for some loci even after 7 generations of selfing: Burr et al. report a residual heterozygosity of 7.5%. The database does not contain all possible data points for every locus: 13.7% of the potential data points are missing. Most of these missing points are for probes which gave no usable data for one of the RI families, presumably due to lack of detectable polymorphisms. I devised a computer program to compare the allelic distribution of each locus with that of every other mapped locus, except for the nearest 5 loci on either side. This exclusion eliminated most of the tightly linked loci. I found that, on the average, each locus was 0.481 R units (46.4 map units) away from every other locus, with a range of 0.434 to 0.512. That is, except for nearby loci, the RI method shows that every locus is essentially unlinked to the bulk of other loci. This result is exactly as expected. However, the distance to the "nearest" locus on a different chromosome is quite variable, with an average of 0.317 R units (23 map units), and a range of 0.207 to 0.400 (13.0 to 33.3 map units). This means that, with the use of RI mapping, every locus is between 13 and 33 map units from another locus which is definitely unlinked. The lower number is especially significant, because in a normal mapping experiment, loci 13 map units apart are clearly linked. Also, some of the adjacent loci mapped by Burr et al. are more than 13 map units apart. Thus, it seems possible that an unknown locus mapped by the RI method could by chance appear to be located quite far from its actual location. To take a specific example, the region between 8.05 and Pl on chromosome 1S is apparently close to a region on 7L between 7.61 and 8.37. The closest approach is between 10.38 on 1S and 7.61 on 7L, which are separated by 0.207 R units (13.0 map units). In comparison, the loci flanking 10.38 on 1S are 9.4 and 1.8 map units away. It is clear that the loci on 1S and 7L have been properly located, because they are linked in a chain to previously mapped genes. However, an unknown gene could fall at an ambiguous position, equally close to loci on different chromosomes. This is especially true if the unknown gene falls in a relatively sparsely mapped area. It was not difficult for me to create an artificial set of data that was equally close to 7.61 and 10.38, and more distant from every other locus. This example is by no means unique: many regions of the genome are apparently close to one another when mapped by the RI method. After seeing that the RI mapping method produces apparent linkages between loci on different chromosomes, I decided to see how well random data sets could be mapped. These data sets were created by assigning the two parental alleles to the different RI lines at random. After a number of trials it became clear that random numbers for both the COXTx and TXCM families rarely produced any apparent linkages. That is, since the two families are independent, using both of them to map a locus is quite likely to yield a good, unique location. However, not all probes will give polymorphic bands for both families: 52 of the 205 probes in the RI data base have data for only one family. Also, some probes, such as those from transposable elements, will not map to the same locations in both families. For these reasons, I attempted to map random data into the COXTx family only. To summarize the results, out of 709 random sets of data, 8 contained an R value of less than 0.25, and 2 of these had an R value of less than 0.225. Out of 296 trials, 25 had R < 0.275 for some locus, and out of 192 trials, 39 had an R of less than 0.30. The R value at which there is less than a 5% chance of getting random data to fit would seem to be between 0.275 and 0.25 (i.e. between 16.7 and 19.0 map units). The positions of these R value minima were randomly distributed in the genome. It can be seen that even random data, which might be produced as a result of wishful thinking applied to marginal data, can produce a "locus" for a probe. This problem becomes even more acute for incomplete data sets. To address this issue, I created random data sets for the COXTx family that contained 10-50% missing data points, and compared them with the RI data base. As mentioned above, 1.1% of complete data sets contained an R value of less than 0.25. Using R < 0.25 as a criterion, I found that 1.8% of data sets with 10% of the data missing fit the criterion, 3.1% of data sets with 20% of the data missing fit, 8.2% of data sets with 30% missing fit, 17% of data sets with 40% missing fit, and 50% of data sets with 50% of their data missing fit the criterion of containing an R value of less than 0.25. Thus, small amounts of missing data do not seem likely to give false localizations, but the chance of getting a fit to a random location rises sharply as the amount of missing data increases. This problem is significantly eased if mapping can be performed with both RI families. In conclusion, the RI method is an excellent method for quickly mapping a probe to a genomic position. However, since there are only a limited number of members in the RI families, certain problems arise which are not seen in standard genetic mapping. Specifically, there is a significant chance of mapping to an incorrect location, especially if there are no previously mapped loci near the unknown probe's apparent position. This problem is significantly increased when only one of the RI families is used, and is increased further if the data set is not substantially to the MNL 64 On-Line Index Return to the Maize Newsletter Index Return to the Maize Genome Database Page

Reference no: EM133720 As an Engineer, you are entrusted with the design of an ITU-T G.987 XG- PON FTTB broadband network to provide high bandwidth Internet access to each lecture hall within a university. The university campus is located at 10 Km from the nearest telephone exchange where an optical line terminal (OLT) will be installed. An optical network terminal (ONT) will also be installed in each lecture hall. A single PON port on the OLT will be used to serve 128 lecture halls. A single level of optical split is recommended. All equipment as well as splitter are connectorized. The average distance of the centralised splitter to each ONT is 500m. There are two fusion splices in between the OLT and the splitter; two mechanical splices in between the splitter and each ONT. A Class C optical distribution network (ODN) optics is assumed for this design. Splicing and connector losses are as per TIA 758. The OLT, ONT as well as the optical components specifications are listed below: Optical fiber specifications Optical fiber standard: G652.D Optical fiber channel bandwidth: 20 THz Optical fiber signal to noise ratio at ONT: 15 dB Optical fiber attenuation at 1577 nm: 0.25 dB/Km Optical fiber dispersion coefficient (????) at 1577 nm: 20 ps (nm.Km) Optical fiber dispersion coefficient (????) at 1260 nm: 3.5 ps (nm.Km) ONT optical port specifications Centre RX wavelength: 1577 nm Centre TX wavelength: 1260 nm RX sensitivity: -28 dBm Spectral width (????) of transmitter: 2.0 nm OLT XG-PON optical port specifications Centre RX wavelength: 1260 nm Centre TX wavelength: 1577 nm RX sensitivity: -30 dBm Spectral width (????) of transmitter: 0.1 nm Based on optical distribution network requirements described, fiber and equipment specifications provided, answer the following questions mentioning any assumptions 1. Draw a network diagram of the optical distribution network using TIA 587 symbols 2. Which fiber mode is not used in XG-PON network and why? 3. Which splitter topology is being used in the ODN? 4. Calculate the splitter loss 5. Calculate the maximum carrying capacity of the G.652D fiber using Shannon's theorem 6. What is the maximum allowable optical power loss range in Class C ODN? 7. Determine the ODN optical power loss for the FTTB network 8. What safety margin would you apply for such a network? 9. Determine the minimum OLT transmit power required to allow the ONTs to be registered by the OLT while considering required safety margin 10. Explain why a Fabry Perot laser is generally used in the ONT whereas a DFB laser is used in the OLT 11. Calculate chromatic dispersion for the fiber span in the downstream direction assuming that there is no modal dispersion in the fiber 12. Calculate chromatic dispersion for the fiber span in the upstream direction assuming that there is no modal dispersion in the fiber 13. Calculate maximum achievable throughput due to chromatic dispersion in the downstream direction 14. Calculate maximum achievable throughput due to chromatic dispersion in the upstream direction 15. Why is upstream throughput lower than downstream throughput on same optical fiber 16. Which wavelength would you use to test the ODN without affecting live data transmission 17. Calculate the maximum downstream throughput that each lecture hall can receive assuming that the OLT downstream throughput is uniformly divided among the lecture halls 18. What is the difference between power and wavelength optical splitters? 19. How do the multiple ONTs access the OLT? 20. How is upstream bandwidth allocated to the ONTs (1) Briefly explain the purpose of synchronization and timing network in a telecommunication network (2) Describe the hierarchy of clocks used in TDM communication networks (3) Briefly explain the difference between the following data streams- a. Synchronous streams b. Asynchronous streams c. Isochronous streams d. Plesiochronous streams (4) The minimum and maximum jitter observed in a 32-channel PCM signal is 463 ns and 512 ns respectively. Express the jitter variation in terms of the signal Unit Interval (UI). Is jitter level as per ITU-T G 811 standard requirements? (1) Describe the Asynchronous Transfer Mode (ATM) protocol (2) Different ATM Adaptation Layers (AALs) are defined for supporting different types of traffic or broadband services. Name the various types of traffic and associated typical services which are related to the various adaptation layers (3) Derive the base synchronous Transport module (STM-1) transport rate (4) Outline the Synchronous Digital Hierarchy (SDH) frame structure, explaining the importance of the section overhead (SOH) (1) Explain the limitations of copper-based broadband access technologies (2) What is the difference between far end and near end crosstalk on copper pairs? Which DSL technology is most affected by crosstalk? (3) Explain how vector DSL technology can support higher data transmission rate using copper pairs. Which type of noise cannot be eliminated by vector DSL? (4) Briefly explain the framing structure of ADSL technology

Classical and Quantum Gravity and Its ApplicationsView this Special Issue Research Article | Open Access Mikhail Z. Iofa, "Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells", Advances in High Energy Physics, vol. 2016, Article ID 5632734, 6 pages, 2016. https://doi.org/10.1155/2016/5632734 Kodama-Schwarzschild versus Gaussian Normal Coordinates Picture of Thin Shells Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems: Kodama-Schwarzschild coordinates and Gaussian normal coordinates. We find explicit coordinate transformation between the Kodama-Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications to the Israel junction conditions are discussed. Dynamics of domain walls was studied by Israel , Poisson , Ipser and Sikivie , Berezin et al. , Blau et al. , Chowdhury , Gladush , Kraus and Wilczek , and many other authors. There are two natural settings to study geometry of the spacetime with a spherical shell: that based on Kodama-Schwarzschild coordinates and that employing the Gaussian normal coordinate system. Kodama found that in any (possibly time-dependent) spherically symmetric spacetime there exists a conserved vector which is timelike in the exterior of the shell . Although the Kodama vector does not reduce to the Killing vector even in the static spacetime, it can be used to define a preferred “time coordinate” and to construct a geometrically preferred coordinate system for a spherically symmetric spacetime [11–15]. Because the Kodama vector is orthogonal to , one can construct the time coordinate so that is orthogonal to . Using the Schwarzschild radial coordinate , one arrives at the diagonal, time-dependent spherically symmetric metric which in this parametrization in the sector has the metric components and . The Gaussian normal coordinate system in the neighborhood of the shell [7, 16] is constructed by using a family of nonintersecting geodesics orthogonal to the surface swept by the shell. Coordinates of a point outside of the shell are introduced as the geodesic distance from the point to the shell along the geodesic orthogonal to the surface and coordinates of the intersection point of the geodesic with . The aim of the present paper is to study connection between two approaches. We find explicit coordinate transformation between the Kodama -Schwarzschild and Gaussian normal coordinate systems. We show that projections of the metrics on the surface swept by the shell in the 4D spacetime in both cases are identical. In the general case of time-dependent metrics we calculate extrinsic curvatures of the shell in both coordinate systems and show that the results are identical. Applications of the above results to the Israel junction conditions are discussed. 2. Kodama-Schwarzschild Coordinates The () dimensional hypersurface swept by a spherically symmetric shell divides 4D spacetime in two regions . Any spherically symmetric metric in spacetime has the general formHere , where are coordinates in the base space and and are coordinates on the spherically symmetric fibers. For any spherically symmetric spacetime it is possible to introduce a vector (Kodama vector) [11–15, 17], which lies in the radial-temporal plane, where By construction Kodama vector is orthogonal to . Choosing the time coordinate so that , one obtains the metric in the diagonal form because and are orthogonal to . In the parametrization through the time coordinate and Schwarzschild radial coordinate the metric can be expressed aswhere . In this parametrization is interpreted as the quasi-local mass (Misner-Sharp-Hernandez mass) [18, 19]. Note that in this parametrization the metric is diagonal. Position of the surface is defined by parametric equations , . The metrics induced on the shell from the regions are From the requirement that the metrics induced on from both regions coincide (first Israel condition) it follows that By choosing as the proper time on the surface, one obtainsand projection of the metric on is 3. Gaussian Normal Coordinates Gaussian normal coordinate system in 4D spacetime in which a hypersurface swept by the spherical shell divides into two regions is introduced starting from a certain coordinate system with a metric . The surface is parametrized by coordinates Consider a neighborhood of with a system of geodesics orthogonal to . The neighborhood is chosen so that the geodesics do not intersect; that is, any point in the neighborhood is located on one and only one geodesic. Let us consider a point in the neighborhood of with the geodesic orthogonal to which goes through this point. The new coordinate system is introduced in the following way. Three coordinates of the point coincide with the coordinates of the point of intersection of the geodesic with . The fourth coordinate of a point is equal to the proper geodesic distance along the geodesic from the point to . The proper length along the geodesic iswhere is the affine parameter along the geodesic. Expression (8) is invariant under the coordinate transformations with Jacobian equal to unity, and we can rewrite (8) through the new coordinates and the metric . Taking the derivative over from both sides of (8) over , one hasor . Orthogonality condition of the tangent vector to geodesic to the tangent surface to isor . The tangent vector is orthogonal to and the vector is in the plane tangent to . The metrics in are (below, to simplify formulas, we omit the subscript everywhere, where it does not lead to confusion)Because of condition (10), on the surface the interval reduces toOn the surface reparametrization of allows setting , which is assumed in the following. It is seen that one can identify with and with . In the following we use the variable . 4. Transformation between the Coordinate Systems Coordinate transformation and from Kodama-Schwarzschild coordinates to Gaussian normal coordinates yields the following relations between the components of the metrics (6) and (11):where prime and dot denote derivatives over and . On the surface transformations (13)–(15) are of the same form with the substitution , and , . It is straightforward to obtain solution of systems (13)–(15) in the spacetime regions as , , and . Instead of writing this cumbersome and not instructive general solution, we consider the restriction of the transformation to the surface which we use as follows:Because is orientable, on a normal vector can be defined. In Kodama-Schwarzschild coordinates tangent, , and normal, , vectors to the surface at either side of the surface areNormalizing to unity, we obtainTransformations of the components of the tangent vector from Kodama-Schwarzschild coordinates to Gaussian coordinates areThe corresponding transformations of the components of the normal vector areIn (22) and (23) we used expressions (16), where all the square roots for , and are taken with the same signs. The upper sign in (23) corresponds to the square roots taken with the sign (+). Next, we consider another method to construct the explicit form of the coordinate transformation from Kodama-Schwarzschild coordinates to Gaussian normal coordinates. The problem can be solved in principle by solving the geodesic equations:In the general case with metric (3) depending on the system of nonlinear differential equations is not tractable. Explicit relations can be obtained in the case of metric (3) with the components independent of (in this case it is possible to set : introducing new variable by the relation and denoting , we obtain the metric (3) with .). In this case the geodesic equations in Kodama -Schwarzschild coordinates arewhere is affine parameter. The first integrals of the system of equations areHere , . To maintain spherical symmetry, we take . By construction the vector is tangent to the geodesic. Let us consider the geodesics orthogonal to . In this case the affine parameter can be identified with the parameter . From (26) it follows that the vector is normalized to unity. At the surface the vector up to the sign coincides with the normal vector (19). Thus, at the surface we have . For we obtainOn the surface solution (27) coincides with formulas (16). In Kodama-Schwarzschild parametrization the variables and have a clear geometrical meaning: at the plane varies along the trajectory of the shell , and varies along the geodesics orthogonal to the surface swept by the shell. 5. Extrinsic Curvature The extrinsic curvatures at either side of are where are coordinates on , , and (;) denote covariant derivative with respect to . In Kodama-Schwarzschild parametrization . In Kodama-Schwarzschild parametrization the nonzero components of the extrinsic curvature areUsing the identity , we have . From the identity it follows that or . Direct calculation yields is expressed asUsing the expressions of Section 4 for and with the upper signs, we obtain the extrinsic curvature in Kodama-Schwarzschild coordinatesHere transforming from (31) to (32) we substituted which follows from (6). In Gaussian normal coordinates the components of the extrinsic curvature areIn the general case of the functions depending on calculation is straightforward but cumbersome. Below we perform calculation for the case of independent of . Using solutions (27), we haveUsing (13), we obtainSubstituting expressions (38), we obtainIt is seen that extrinsic curvatures in both parametrizations coincide. 6. Israel Junction Conditions Next, we consider the Einstein equations and the Israel junction conditions. The energy-momentum tensor is taken in the formBecause the values of the extrinsic curvatures at the opposite sides of are different, the derivative of the extrinsic curvature through the surface contains -singularity. From the singular part of the component of the Einstein equations projected on , follow the relations,where and . The component of the Einstein equations, (vertical bar stands for covariant derivative with respect to metric (7)) yieldsFrom the component of the Einstein equations, it follows thatFurther restrictions on follow from the conservation equations of the energy-momentum tensor . Projections of the components of the bulk metric defined as are Projections of the tangent vector are Assuming that the energy-momentum tensor has the form , from (43), we obtainFrom Israel conditions (45) written as we haveIn the case of metric (3) with the function independent of from (40) one obtainsIsrael conditions take a simple form in the case . From the Israel conditions it follows thatSolving this system of equations, we find that andRelation (55) can be rewritten as (cf.) Geometry and connection between the two coordinate systems used to study dynamics of thin shells, Kodama-Schwarzschild coordinates and Gaussian normal coordinate system, were studied. Transformation between the coordinate systems is studied and explicitly constructed for the case of Kodama-Schwarzschild metric independent of time. Extrinsic curvatures of the surface swept by the shell in the ambient space are calculated for a general time-dependent metric in both Kodama-Schwarzschild and normal Gaussian parametrizations and are shown to give the same result. Application to the Israel junction conditions is discussed. The author declares that there is no conflict of interests regarding the publication of this paper. This work was partially supported by the Ministry of Science and Education of Russian Federation under Project 01201255504. - R. Brout, S. Massar, R. Parentani, and P. Spindel, “A primer for black hole quantum physics,” Physics Reports, vol. 260, no. 6, pp. 329–446, 1995. - A. Paranjape and T. Padmanabhan, “Radiation from collapsing shells, semiclassical backreaction, and black hole formation,” Physical Review D, vol. 80, no. 4, Article ID 044011, 2009. - W. Israel, “Singular hypersurfaces and thin shells in general relativity,” Il Nuovo Cimento B Series, vol. 44, no. 1, pp. 1–14, 1966, Erratum in Il Nuovo Cimento B Series, vol. 48, p. 463, 1967. - E. Poisson, A Relativist Toolkit, Cambridge University Press, Cambridge, UK, 2004. - J. Ipser and P. Sikivie, “Gravitationally repulsive domain wall,” Physical Review. D. Particles and Fields. Third Series, vol. 30, no. 4, pp. 712–719, 1984. - V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, “Dynamics of bubbles in general relativity,” Physical Review. D. Particles and Fields. Third Series, vol. 36, no. 10, pp. 2919–2944, 1987. - S. K. Blau, E. I. Guendelman, and A. H. Guth, “Dynamics of false-vacuum bubbles,” Physical Review D, vol. 35, no. 6, pp. 1747–1766, 1987. - B. D. Chowdhury, “Problems with tunneling of thin shells from black holes,” Pramana, vol. 70, no. 1, pp. 3–26, 2008. - V. D. Gladush, “On the variational principle for dust shells in general relativity,” Journal of Mathematical Physics, vol. 42, no. 6, pp. 2590–2610, 2001. - P. Kraus and F. Wilczek, “Self-interaction correction to black hole radiance,” Nuclear Physics B, vol. 433, no. 2, pp. 403–420, 1995. - H. Kodama, “Conserved energy flux for the spherically symmetric system and the backreaction problem in the black hole evaporation,” Progress of Theoretical Physics, vol. 63, no. 4, pp. 1217–1228, 1980. - S. A. Hayward, “General laws of black-hole dynamics,” Physical Review. D. Third Series, vol. 49, no. 12, pp. 6467–6474, 1994. - S. A. Hayward, “Gravitational energy in spherical symmetry,” Physical Review D, vol. 53, no. 4, pp. 1938–1949, 1996. - S. A. Hayward, “Quasi-local gravitational energy,” Physical Review D, vol. 49, no. 2, p. 831, 1994. - G. Abreu and M. Visser, “Kodama time: geometrically preferred foliations of spherically symmetric spacetimes,” Physical Review D, vol. 82, no. 4, Article ID 044027, 2010. - J. Smoller and B. Temple, “Shock-wave solutions of the Einstein equations: the Oppenheimer-Snyder model of gravitational collapse extended to the case of non-zero pressure,” Archive for Rational Mechanics and Analysis, vol. 128, no. 3, pp. 249–297, 1994. - V. Faraoni, “Cosmological apparent and trapping horizons,” Physical Review D, vol. 84, no. 2, Article ID 024003, 15 pages, 2011. - C. W. Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse,” Physical Review, vol. 136, no. 2, pp. B571–B576, 1964. - W. C. Hernandez Jr. and C. W. Misner, “Observer time as a coordinate in relativistic spherical hydrodynamics,” The Astrophysical Journal, vol. 143, p. 452, 1966. Copyright © 2016 Mikhail Z. Iofa. 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. The publication of this article was funded by SCOAP3.

In the past, thermal engineers were content to calculate a single junction temperature in the thermal characterization of an integrated circuit chip in a package. In recent years, as power levels have increased, more attention has been paid to the calculation of local temperature variations on a chip. This is necessary to adequately account for high-heat-flux regions of the chip, often called “hot spots.” This Calculation Corner extends the use of techniques explored in previous installments to perform these calculations very quickly and with reasonable accuracy. Radial Thin Fin Model In previous columns, the radial thin fin (RTF) model has been used successfully to calculate the thermal resistance of a heat spreader or circuit board surrounding a single centrally-located heat source or sink [1, 2, 3]. An alternative and algebraically simpler method is found in . In this article, analytical expressions are used to calculate the temperature of a square plate, both in the region under the heat source as well as the surrounding area . The RTF model does an excellent job of calculating in-plane thermal gradients in a plate, resulting from heat spreading. However, it neglects through-thickness thermal gradients. This will be a source of error that should be considered. However, it is used in these calculations because of its relative simplicity and because it can readily provide the local temperature in any region of a plate due to an arbitrarily located heat source. The RTF model can, of course, represent a fin cooled by convection and radiation assuming a constant ambient temperature. However, it can also deal with a mathematically equivalent situation of a plate which conducts heat through a lower conductivity layer to an isothermal surface . The situation envisioned here would be a heat source on a silicon chip attached to an isothermal surface using a thermal interface material (TIM). The power map on an IC typically consists of rectangular regions of different heat flux, each representing a different functional block on the chip. Single Heat Source As its name implies, the RTF model applies only to situations with circular symmetry. Hence, the first step in using it would be to circularize a rectangular heat source. The principle applied here is that the circularized heat source should have the same area as the original rectangular source. Accordingly, where a is the radius of the transformed source. The next step is to calculate the value of an effective heat transfer coefficient due to heat flow through the TIM using the following equation: The following equations can be used to calculate the temperature everywhere in a circular plate with a centrally located heat source, in which the constants C1 and C2 are calculated from where: P is the power dissipated at the source; k and t are the thermal conductivity and thickness of the plate, respectively; r is the radial distance from the location of interest to the center of the source; T0 is the temperature of the isothermal surface; α = (h/kt)1/2 ; and I0 and I1 are modified Bessel functions of the 1st kind, K0 and K1 are modified Bessel functions of the 2nd kind, and the subscripts, 0 and 1, indicate the order. Note, in Eqn. 4, the far-field term, proportional to I0(α r), has been dropped. This is due to the fact that in the physical situation explored, the temperature field drops off rapidly with distance from the edge of the heat source. Additionally, this fact permits the use of the above equations in calculating the temperature field of a source not located at the geometrical center of the plate. This is critical to our use of these equations in a power map calculation. Multiple Heat Sources It is a relatively simple matter to calculate the temperature map for a number of heat sources. Using the principle of superposition, one need only calculate the temperature rise at a given x,y location for each heat source considered separately, and, then, take the sum of all the individual temperature rises . To do this, one need only calculate the radial distance from a particular x,y location to each source and plug it into equations 3-6 along with the specific values of a and P characterizing the source. In contrast, k, h, t, and α can be considered global parameters that are used in the calculations for all the heat sources. This process is captured by the following equation in which Ti and xi and yi represent the temperature and the location of source #i, respectively. When a source is sufficiently close to an edge or a corner of the plate, the heat flow path is truncated. Equations 3-6 do not account directly for these boundary effects. However, by including image sources that are symmetrically located across the boundary with respect to the real sources, the boundary effects can be readily accounted for. The image sources should have the same size and power as the real source . The above techniques are used to calculate the temperature distribution in a 20 mm square silicon chip for various heat source configurations. Table 1 lists the dimensions and properties assumed for the physical components in this example. Table 1. Dimensions and Properties of Physical Components in Examples The analytical method discussed above was applied to this problem using a spreadsheet having a fairly elaborate construction. Details of its construction are presented below. This problem was also solved using a commercial finite element analysis tool . The accuracy of the analytical method was determined by comparison with the FEA results. The table in the spreadsheet in which the parameters specific to each source are input is depicted in Figure 1a. A total of nine sources are listed. Of them, five are real sources and are located within the chip area. Four of them (highlighted in yellow) are image sources which create the same boundary conditions for source #1 as a physical corner. Figure 1. a) Input table used in spreadsheet. Image sources are highlighted. b) Output table of calculated temperatures in spreadsheet with temperature-dependent color coding. c) 3D plot of temperature array in output table. d) FEA calculation of temperature contours of chip using image sources. White rectangle represents lateral dimensions of 20 mm sq. chip. The spreadsheet is constructed so that each of the ten possible heat sources has its own dedicated sheet for calculating its own temperature field over the entire chip area. Each of these sheets is linked to the appropriate parameters in the input table for its assigned source and also to the global input parameters. Figure 1b depicts a table in the spreadsheet containing the calculated temperatures for the entire chip. It contains an 80 x 80 matrix of calculated temperatures. With a 20 mm square die, a temperature is sampled at intervals of 0.25 mm. Each cell in the table represents a summation of the temperatures in the same row-column location on all the source-specific sheets. The insets zoom in on selected regions of the matrix to show the temperatures calculated for two of the sources. The temperature-dependent color coding is accomplished by using the “conditional formatting” capability of the spreadsheet. One should note that source #5 located at the lower left corner of the table does not have image sources included in the model. Hence, its temperature distribution is the same as it would have been had it been far away from the corner. Sources 6 � 9, located near the center of the chip, are in the same relative position as source #1 and its off-chip image sources. Note that each quadrant of this 4-source array has the same temperature contours as source #1. Figure 1c contains a 3D plot of the temperature distribution as generated by the spreadsheet. It illustrates the complex temperature field represented by the superposition of the temperature distributions from all the sources. It clearly illustrates that, due to the interaction with its image sources, the peak temperature at source #1 is: 1) greater than that at source #5, and 2) equal to that of the centrally located sources. Figure 1d shows a temperature contour plot as calculated by FEA for the same situation as the analytical calculation. The white square in the middle of the plot represents the physical boundaries of the chip. One notes that the temperature contours within the chip outline show the same overall distribution as the analytical calculation due to the explicit inclusion of the same image sources in the model. Since there are no image sources near source #5, its temperature distribution differs from the one that would have been obtained had the lateral dimensions of the model corresponded to the actual chip dimensions. In this example the temperature distribution along a 1-dimensional path is calculated as a 2 mm x 2 mm source is successively relocated from the center of the chip toward either the edge of the chip or the chip corner. FEA is used to calculate the temperature distribution at both the top and bottom surfaces of the chip. In all cases, the appropriate image sources were present in the analytical model. Figure 2. Temperature distribution along a 1-dimensional path calculated for square source successively relocated along the path: along x-axis from center to edge and b) along diagonal from center to corner. Upper and lower curves represent FEA calculations for top and bottom chip surfaces. Dotted curve represents the analytical model results. Figures 2a and 2b contain graphs of temperature distributions along the x-axis and chip diagonal, respectively. One notes that the temperature distribution calculated at each location by both methods have the same lateral extent. Also, the peak temperature at each location calculated from the analytical method is between the temperatures calculated by FEA for the top and bottom surfaces. In general, the analytical result is 23% lower than the FEA value for the top surface and 13% higher than the value for the bottom surface. The analytical method clearly demonstrates an increase in the temperature for a source located near a corner or an edge of the same magnitude as that in the FEA model. The error in the analytical calculation will be smaller in situations in which the ratio of the temperature difference in the chip to the total temperature difference is smaller. Reductions in either the chip thickness or in the value of hEff associated with the TIM tend to reduce the error. A quick estimate of the error in the analytical model can be obtained by taking the ratio of the thermal resistance for the chip alone to the combined thermal resistance for the chip plus that for the TIM, assuming 1-dimensional heat flow: Applying this formula to the current situation yields an estimated error of 18%, which is comparable to the actual error reported above. It is recommended that the error be estimated in this manner when using the analytical method described here. In fact, in critical situations, the estimated error could be used to make a first-order correction to the calculated peak temperature values using the following expression: Applying this simple correction to the current results would reduce the error to around 6%. A third study was performed on the representation of rectangular heat sources by the analytical method. It was found that when a rectangular heat source is represented by a coarse array of appropriately-sized square sources, the temperature contours were accurately accounted for. The error in calculating the peak temperature was similar to that for the square sources. An analytical method was demonstrated for the calculation of temperature distributions for a silicon chip with an arbitrary power map. The method captured well the lateral distribution and relative magnitude of the resultant hot spots. In the examples evaluated, the error in calculating peak temperatures was on the order of 20% and on the low side. The error would be reduced for situations with thinner chips and/or higher thermal resistance TIMs. A method for estimating the error was demonstrated. In critical cases it can be used to make a first-order correction to the peak temperature calculation. In the present examples, its use would reduce the error to around 6%. - Guenin, B., “Heat Spreading Calculations Using Thermal Circuit Elements,” ElectronicsCooling, Vol. 14, No. 3, August, 2008. - Guenin, B., “Convection and Radiation Heat Loss from a Printed Circuit Board,” ElectronicsCooling, Vol. 4, No. 3, September, 1998. - Guenin, B., “Thermal Vias � A Packaging Engineer’s Best Friend,” ElectronicsCooling, Vol. 10, No. 3, August, 2004. - Lasance, C., “Heat Spreading: Not a Trivial Problem,” ElectronicsCooling, Vol. 13, No. 3, August, 2008 - Lall, B., Ortega, A., and Kabir, H., “Thermal Design Rules for Electronic Components on Conducting Boards in Passively Cooled Enclosures,” Proceedings, Fourth InterSociety Conference on Thermal Phenomena in Electronic Systems (iTherm), May, 1994, pp. 50-61. - Guenin, B., “Device Temperature Prediction in Multi-Chip Packages,” ElectronicsCooling, Vol. 12, No. 3, August, 2006. - Mart�n H�riz, V., et al., “Method of Images for the Fast Calculation of Temperature Distributions in Packaged VLSI Chips,” Proceedings, THERMINIC Conference, Budapest, September, 2007, pp. 18-25 - ANSYS™, Version 11.0.

IonSource Significant Figures & Our policy with respect to significant figures and rounding at the IonSource.Com web site. Scientists routinely attempt to describe the world with numbers. If you are a mass spectroscopist you had better love numbers, because in many instances they are all you have. As a good friend once told me, "Every credible scientific study should be reducible to a table filled with meaningful significant numbers." It is important to establish a a policy with which you treat numbers. Some companies go so far as to create a document called an SOP, standard operating procedure. Then when a regulatory agency comes to call, the officers at the company can show the investigators the policy. You do not want to be in a situation where you barely pass a test because the analyst always rounds up, but the regulatory agency finds an instances where another analyst, or worse the same analyst rounded down in a different situation. This can lead the agency to the conclusion that you only round up when you need to pass a test. Even if you are not answerable to a regulatory agency you will gain respect from your peers by treating numbers with respect, and by reporting only significant figures and by rounding properly. There are two types of significant figures, measured and exact. A) All non-zero numbers are significant. B) All zeros between significant numbers are significant, for example the number 1002 has 4 significant figures. D) Zeros to the left of a significant figure and not bounded to the left by another significant figure are not significant. For example the number 0.01 only has one significant figure. E) Numbers ending with zero(s) written without a decimal place posses an inherent ambiguity. To remove the ambiguity write the number in scientific notation. For example the number 1600000 is ambiguous as to the number of significant figures it contains, the same number written 1.600 X 106 obviously has four significant figures. 1) It is important to know the accuracy and precision of the measuring device one is using and it is important to report only those digits that have significance. To reiterate, your electrospray mass spectrometer may be able to spit out 10 numbers past the decimal place but you should only use the digits that have significance in reporting or in a calculation. 2) It is generally accepted that the uncertainty is plus or minus 0.5 unit at the level of the uncertainty, for example the "true value" for the number 0.003 can be described as being bounded by the numbers 0.0025 and 0.0035. It is important to note that in some instances scientists will want to express an uncertainty that exceeds 1 at the level of the uncertainty and this should be noted explicitly in the following fashion, 0.003 ± 0.002 Exact numbers are those that are counted without ambiguity, for example the number of mass spectrometers in the lab is exactly three, or the number of cars in the parking lot is exactly four. These numbers carry no ambiguity and can be considered to have an infinite number of significant figures. When using these numbers in a calculation the restriction on reporting is borne by the measured number if any. As far as we can tell rounding of significant figures carries a certain degree of controversy and people will argue with you based on what they were taught at some point in their education. For example I learned from my "Biostatistics" course in college that when rounding a number that is followed by a 5, for example 1.1150, one should round up to the even number, 1.12 or not round up if the number was already even. The explanation that the professor gave was that even numbers are easier to deal with in a calculation, which now seems to me like a a bad reason. More recently I have been told from statisticians, that I respect, that this procedure removes the rounding bias. They explain, that without bias half of the time the number is rounded up. To me this makes more sense, after all as scientists we want to be as unbiased as humanly possible. Others always round up in this situation regardless of whether the number is even or odd. Our position on this subject is that we don't care what you do, but be consistent. Another painful detail that can cause controversy is that if the number following the 5 is not a zero, for example 1.1151, the number should be rounded up. This is the policy that we follow. Again set your own policy, or if you are working with a larger group follow that policy. Be consistent. Rounding policies that everyone agrees with: If you are rounding a number to a certain degree of significant digits, and if the number following that degree is less than five the last significant figure is not rounded up, if it is greater than 5 it is rounded up. A) 10.5660 rounded to four significant figures is 10.57 B) 10.5640 rounded to four significant figures is 10.56 We agree that we have not addressed every controversy on this subject but we hope that you understand how we deal with numbers at IonSource.Com. For a quality easy to follow tutorial on rounding and significant figures visit Dr. Stephan Morgan at the University of South Carolina. If you need to find a consultant to teach a course on statistics at your company we suggest Statistical Designs , they also have several tutorials on-line. The people at Statistical Designs teach statistics, and experimental design for the American Chemical Society. For an interesting paper on significant figures and rounding visit Dr.Christopher Mulliss at his web site. Other significant figure and rounding sites we have found:

Classical and quantum statistics Classical Maxwell–Boltzmann statistics and quantum mechanical Fermi–Dirac statistics are introduced to calculate the occupancy of states. Special attention is given to analytic approximations of the Fermi–Dirac integral and to its approximate solutions in the nondegenerate and the highly degenerate regime. In addition, some numerical approximations to the Fermi–Dirac integral are summarized. Semiconductor statistics includes both classical statistics and quantum statistics. Classical or Maxwell–Boltzmann statistics is derived on the basis of purely classical physics arguments. In contrast, quantum statistics takes into account two results of quantum mechanics, namely (i) the Pauli exclusion principle which limits the number of electrons occupying a state of energy E and (ii) the finiteness of the number of states in an energy interval E and E + dE. The finiteness of states is a result of the Schrödinger equation. In this section, the basic concepts of classical statistics and quantum statistics are derived. The fundamentals of ideal gases and statistical distributions are summarized as well since they are the basis of semiconductor statistics. Probability and distribution functions Consider a large number N of free classical particles such as atoms, molecules or electrons which are kept at a constant temperature T, and which interact only weakly with one another. The energy of a single particle consists of kinetic energy due to translatory motion and an internal energy for example due to rotations, vibrations, or orbital motions of the particle. In the following we consider particles with only kinetic energy due to translatory motion. The particles of the system can assume an energy E, where E can be either a discrete or a continuous variable. If Ni particles out of N particles have an energy between Ei and Ei + dE, the probability of any particle having any energy within the interval Ei and Ei + dE is given by where f(E) is the energy distribution function of a particle system. In statistics, f(E) is frequently called the probability density function. The total number of particles is given by where the sum is over all possible energy intervals. Thus, the integral over the energy distribution function is In other words, the probability of any particle having an energy between zero and infinity is unity. Distribution functions which obey are called normalized distribution functions. The average energy or mean energy of a single particle is obtained by calculating the total energy and dividing by the number of particles, that is In addition to energy distribution functions, velocity distribution functions are valuable. Since only the kinetic translatory motion (no rotational motion) is considered, the velocity and energy are related by The average velocity and the average energy are related by and is the velocity corresponding to the average energy In analogy to the energy distribution we assume that Ni particles have a velocity within the interval vi and vi + dv. Thus, where f(v) is the normalized velocity distribution. Knowing f(v), the following relations allow one to calculate the mean velocity, the mean square velocity, and the root-mean-square velocity Up to now we have considered the velocity as a scalar. A more specific description of the velocity distribution is obtained by considering each component of the velocity v = (vx, vy, vz). If Ni particles out of N particles have a velocity in the ‘volume’ element vx + dvx, vy + dvy, and vz + dvz, the distribution function is given by Since Σi Ni = N, the velocity distribution function is normalized, i. e. The average of a specific propagation direction, for example vx is evaluated in analogy to Eqs. (13.11 – 13). One obtains In a closed system the mean velocities are zero, that is . However, the mean square velocities are, just as the energy, not equal to zero. Ideal gases of atoms and electrons The basis of classical semiconductor statistics is ideal gas theory. It is therefore necessary to make a small excursion into this theory. The individual particles in such ideal gases are assumed to interact weakly, that is collisions between atoms or molecules are a relatively seldom event. It is further assumed that there is no interaction between the particles of the gas (such as electrostatic interaction), unless the particles collide. The collisions are assumed to be (i) elastic (i. e. total energy and momentum of the two particles involved in a collision are preserved) and (ii) of very short duration. Ideal gases follow the universal gas equation P V = R T (13.19) where P is the pressure, V the volume of the gas, T its temperature, and R is the universal gas constant. This constant is independent of the species of the gas particles and has a value of R = 8.314 J K–1 mol–1. Next, the pressure P and the kinetic energy of an individual particle of the gas will be calculated. For the calculation it is assumed that the gas is confined to a cube of volume V, as shown in Fig. 13.1. The quantity of the gas is assumed to be 1 mole, that is the number of atoms or molecules is given by Avogadro’s number, NAvo = 6.023 × 1023 particles per mole. Each side of the cube is assumed to have an area A = V 2/3. If a particle of mass m and momentum m vx (along the x-direction) is elastically reflected from the wall, it provides a momentum 2 m vx to reverse the particle momentum. If the duration of the collision with the wall is dt, then the force acting on the wall during the time dt is given by where the momentum change is dp = 2 m vx. The pressure P on the wall during the collision with one particle is given by where A is the area of the cube’s walls. Next we calculate the total pressure P experienced by the wall if a number of NAvo particles are within the volume V. For this purpose we first determine the number of collisions with the wall during the time dt. If the particles have a velocity vx, then the number of particles hitting the wall during dt is (NAvo / V) A vx dt. The fraction of particles having a velocity vx is obtained from the velocity distribution function and is given by. Consequently, the total pressure is obtained by integration over all positive velocities in the x-direction Since the velocity distribution is symmetric with respect to positive and negative x-direction, the integration can be expanded from – ∞ to + ∞. Since the velocity distribution is isotropic, the mean square velocity is given by The pressure on the wall is then given by Using the universal gas equation, Eq. (13.19), one obtains The average kinetic energy of one mole of the ideal gas can then be written as The average kinetic energy of one single particle is obtained by division by the number of particles, i. e. where k = R / NAvo is the Boltzmann constant. The preceding calculation has been carried out for a three-dimensional space. In a one-dimensional space (one degree of freedom), the average velocity is and the resulting kinetic energy is given by (per degree of freedom) . (13.29) Thus the kinetic energy of an atom or molecule is given by (1/2) kT. Equation (13.29) is called the equipartition law, which states that each ‘degree of freedom’ contributes (1/2) kT to the total kinetic energy.

What is sin of 90 minus theta?Space and Astronomy What is sin 90 minus theta? Sin (90 – Theta) is equal to Cos Theta. What does 90 minus theta mean? Trigonometric ratios of 90 degree minus theta is one of the branches of ASTC formula in trigonometry. Trigonometric-ratios of 90 degree minus theta are given below. sin (90° – θ) = cos θ cos (90° – θ) = sin θ tan (90° – θ) = cot θ What is tan 90 degree minus theta? tan(90° – θ) = cot θ What is sin minus theta? For any acute angle of θ, the functions of negative angles are: sin(-θ) = – sinθ cos (-θ) = cosθ What is the formula of sin 90 Theta? All triangles have 3 angles that add to 180 degrees. Therefore, if one angle is 90 degrees we can figure out Sin Theta = Cos (90 – Theta) and Cos Theta = Sin (90 – Theta). What is sin theta? As per the sin theta formula, sin of an angle θ, in a right-angled triangle is equal to the ratio of opposite side and hypotenuse. The sine function is one of the important trigonometric functions apart from cos and tan. Why does sin theta equal Cos 90 Theta? Video quote: Hi guys so I was wondering why sine of 90 90 minus theta is equal to cos of theta. Is it correct to say that sin theta is equal to cos 90 minus theta by? Answer. Yes, the value of sinΘ and cos(90-Θ) are the same. What is the value of sin 90 in fraction? Hence, Sin 90° will be equal to its fractional value i.e. 1/1. How do you calculate sin theta? The sine of an angle of a right-angled triangle is the ratio of its perpendicular (that is opposite to the angle) to the hypotenuse. The sin formula is given as: sin θ = Perpendicular / Hypotenuse. sin(θ + 2nπ) = sin θ for every θ How do you calculate sin? Video quote: So sine is going to be the opposite over the hypotenuse cosine will be the adjacent. Over the hypotenuse. And tangent will be the opposite. What is the sin of 60 in degrees? The value of sin 60 degrees is 0.8660254. . .. Sin 60 degrees in radians is written as sin (60° × π/180°), i.e., sin (π/3) or sin (1.047197. . .). How do you calculate sin 30? The value of sin 30 degrees can be calculated by constructing an angle of 30° with the x-axis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of sin 30° is equal to the y-coordinate (0.5). ∴ sin 30° = 0.5. Why is the sin of 90 degrees 1? Start measuring the angles from the first quadrant and end up with 90° when it reaches the positive y-axis. Now the value of y becomes 1 since it touches the circumference of the circle. Therefore the value of y becomes 1. Therefore, sin 90 degree equals to the fractional value of 1/ 1. How do you solve sin 45 without a calculator? Video quote: So that's put it down 1 over square root of 2 and in this case once again with how to rationalize. The denominator. So this is going to be square root of 2 over 2. Is the value of cos 60? The value of cos 60 is 1/2. What is the exact value of sin 45? Sin 45 degrees is the value of sine trigonometric function for an angle equal to 45 degrees. The value of sin 45° is 1/√2 or 0.7071 (approx). What does tan 45 mean? The value of tan 45 degrees is 1. Is the value of sin 45? = 1 / 2 |Sine 45° or Sine π/4||1 / 2| |Sine 60°or Sine π/3||3 / 2| |Sine 90° or Sine π/2||1| |Sine 120° or Sine 2π/3||3 / 2| How do you solve tan 30? The value of tan 30 degrees is 1/√3. The value of tan π/6 can be evaluated with the help of a unit circle, graphically. In trigonometry, the tangent of an angle in a right-angled triangle is equal to the ratio of opposite side and the adjacent side of the angle. How do you find the value of sin 50? The value of sin 50 degrees can be calculated by constructing an angle of 50° with the x-axis, and then finding the coordinates of the corresponding point (0.6428, 0.766) on the unit circle. The value of sin 50° is equal to the y-coordinate (0.766). ∴ sin 50° = 0.766. How do you find the value of sin 15? Value of Sin 15 in Decimal Since, we know, Sin 15° = (√3–1)/2√2. How do you solve sin 18? The value of sin 18° is equal to the y-coordinate (0.309). ∴ sin 18° = 0.309. How do you find sin 20? The value of sin 20 degrees can be calculated by constructing an angle of 20° with the x-axis, and then finding the coordinates of the corresponding point (0.9397, 0.342) on the unit circle. The value of sin 20° is equal to the y-coordinate (0.342). ∴ sin 20° = 0.342. - Compaction in the Rock Cycle: Understanding the Process Behind Sedimentary Rock Formation - Crystallization in the Water Cycle: A Fundamental Process in Water Distribution and Purification - Understanding Crystallization in the Rock Cycle: A Fundamental Process in Rock Formation - SQL Server to Google Maps - Stereo-pair Image Registration - Extracting Lat/Lng from Shapefile using OGR2OGR/GDAL - Constructing query in Nominatim - In Ogr2OGR: what is SRS? - Identifying port numbers for ArcGIS Online Basemap? - Remove unwanted regions from map data QGIS - Waiting for Vector & WFS loading - Adding TravelTime as Impedance in ArcGIS Network Analyst? - Listing total number of features into an ArcGIS Online feature pop-up - Criteria for cartographic capacity

The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena). This is the proverbial sixty-four thousand dollar question for fundamental physics. It may be helpful to split it down into steps. - What are the possible consistent theories of quantum gravity? - Which of these can (or must) be extended to include matter and guage fields? - Which of these can be made to include the standard model as a low energy limit? Once we have answered these questions the theoretical program to understand the foundations of physics is essentially complete and the rest is stamp collecting and experiment. That is not going to happen today but let's see where we are. String Theory works very well as a perturbative theory of gravitons that appears to be finite at all orders, but there is no full proof that it is a complete theory of quantum gravity. It requires matter and gauge fields with supersymmetry to avoid anomalies. The size of gauge groups suggests that it could potentially include the standard model. It is too strong a claim to say that it does incorporate the standard model. A popular view is that it has a vast landscape of solutions which is sufficiently diverse to suggest that the standard model is covered, but crucial elements such as supersymmetry breaking and the cosmological constant problem are not yet resolved. Supergravity theories are potentially alternative non-string theories that could provide a perturbative theory of quantum gravity. Indications are that they are finite up to about seven loops due to hidden E7 symmetry but they are likely to have problems at higher loops unless there are further hidden symmetries. These theories have multiplets of gauge groups and matter. The 4D theories do not have sufficiently large gauge groups for the standard model but compactified higher dimensional supergravity does. A more subtle problem is to include the right chiral structure and this may be possible only with the methods of M-theory. It has long been the conventional wisdom that supergravity theories can only be made complete by adding strings. Recent work using twistor methods on 4D supergravity seems to support this idea (e.g. Skinner etc.) Loop Quantum Gravity is an attempt to quantise gravity using the canonical formualism and it leads to a description of quantum gravity in terms of loops and spin network states which evolve in time. Although this is regarded as an alternative to string theory and supergravity it does not give a picture of a purtabative limit which would make it possible to compare with these approaches. It is possible that ST/SUGRA and LQG are looking at similar things from a different angle. In fact the recent progress on N=8 supergravity as a twistor string theory has some features that are similar to LQG. Both involve 2D worldsheet objects and network like objects. The main distinctions are that LQG does not have supersymmetry and N=8 SUGRA does not use knots. Even then there has been some progress on a supersymmetric version of LQG and the Yangian symmetries used in N=8 SUGRA should be amenable to a q-deformation that brings in knots. It remains to be seen if these theories can be unified. It is worth saying that all these approaches involve trying to quantise gravity in different ways. Although quantisation is not a completely unique procedure it is normal to expect that different ways of quantising the same thing should lead to related results, If something like supersymmetry or strings or knots are needed to get consistency in one approach the chances are that they will be needed in another. I have not mentioned other approaches to quantum gravity such as spin foams, group field theory, random graphs, causal sets, shape dynamics, non-commutative geometry, ultra-violet fixed points etc. Some of these are related to the other main approaches but are less well developed. It should also be mentioned that there are always attempts to unify gravity and the standard model classically e.g. Garrett's E8 TOE, Weinstein's Geometric Unity etc. These may tell us something interesting or not, but it is only when you try to quantise gravity that strong constraints apply so there is no reason to think they should be related to the attempts to quantise gravity. So in conclusion all approaches that have made any kind if real progress with quantising gravity look like they may be related. Much more has been revealed so far from this need to quantise consistently than from directly trying to unify gravity with the standard model. This may not be so surprising when you consider the enormous difference in energy scales between the two. So far, the answer seems to be no, but there is no mathematical proof. The main reason to believe that string theory is essentially unique is that it incorporates the holographic principle, the idea that the spacetime near and inside a black hole is emergent from the degrees of freedom of the black hole, and this idea is so difficult to imagine working, that it is hard to see some other solution. Within string theory, the standard model emerges from either some matter, or from the Horava-Witten orbifold which produces an E8 gauge group in a circular compactification of M-theory. The E8 gauge group can naturally break to E6 and contains the standard model in a way as natural as SO(10) or U(5) (it is just a supergroup). So there is no difficulty embedding the standard model, but it is not predicted, just happens to work. In other approaches, not only does the gravity not work well, the other stuff is not so natural as it is in string theory, where the total amount of stuff, like fields, gauge-groups, is constrained to be (of the right order but a few times bigger than) what we see. The Loop Quantum Gravity folks have not been able to get elementary particle theory into their picture very well. They have been working with braids which have twists, which start to sound a bit like string to me. There seems to be a trend where all roads lead to string theory.

This set of Wireless & Mobile Communications Multiple Choice Questions & Answers (MCQs) focuses on “Linear Modulation Techniques”. 1. In linear modulation technique, ______ of transmitted signal varies linearly with modulating digital signal. Explanation: In linear modulation technique, the amplitude of transmitted signal varies linearly with modulating digital signal. It is a form of digital modulation technique. 2. Linear modulation techniques are not bandwidth efficient. State whether True or False. Explanation: Linear modulation techniques are bandwidth efficient. They are used in wireless communication systems when there is an increasing demand to accommodate more and more users within a limited spectrum. 3. Which of the following is not a linear modulation technique? b) π/4 QPSK Explanation: OQPSK, π/4 QPSK and BPSK are the most popular linear modulation techniques. They have very good spectral efficiency. However, FSK is an non-linear modulation technique. 4. In BPSK, the _____ of constant amplitude carrier signal is switched between two values according to the two possible values. Explanation: In binary phase shift keying (BPSK), the phase of a constant amplitude carrier signal is switched between two possible values m1 and m2. These two values corresponds to binary 1 and 0 respectively. 5. By applying cos(2πft), BPSK signal is equivalent to ________ a) Double sideband suppressed carrier amplitude modulated waveform b) Single sideband suppressed carrier amplitude modulated waveform c) Frequency modulated waveform d) SSB amplitude waveform Explanation: The BPSK signal is equivalent to a double sideband suppressed carrier amplitude modulated waveform, where cos(2πft) is applied as the carrier. Hence, a BPSK signal can be generated using a balanced modulator. 6. BPSK uses non-coherent demodulator. State whether True or False. Explanation: BPSK uses coherent or synchronous demodulation. It requires the information about the phase and frequency of the carrier be available at the receiver. 7. DPSK uses coherent form of PSK. State whether True or False. Explanation: Differential phase shift keying uses noncoherent form of phase shift keying. Noncoherent form avoids the need for a coherent reference signal at the receiver. Noncoherent receivers are also easy and cheap to build. 8. In DPSK system, input signal is differentially encoded and then modulated using a _____ modulator Explanation: In DPSK system, input binary sequence is first differentially encoded and then modulated using a BPSK modulator. The differentially encoded sequence is generated from input binary sequence by complimenting their modulo-2 sum. 9. The energy efficiency of DPSK is ______ to coherent PSK. Explanation: The energy efficiency of DPSK is inferior to that of coherent PSK by about 3 dB. But, it has an advantage of reduced receiver complexity. 10. QPSK has _____ the bandwidth efficiency of BPSK. d) Four times Explanation: Quadrature phase shift keying (QPSK) has twice the bandwidth of BPSK. It is because two bits are transmitted in a single modulation symbol. The phase of the carrier takes on one of the four equally spaced values, where each value of phase corresponds to a unique pair of message bit. 11. QPSK provides twice the bandwidth efficiency and ____ energy efficiency as compared to BPSK. d) Four times Explanation: The bit error probability of QPSK is identical to BPSK but twice as much data can be sent in the same bandwidth. Thus, when compared to BPSK, QPSK provides twice the spectral efficiency with exactly the same efficiency. 12. What is the full form of OQPSK? a) Optical Quadrature Phase Shift Keying b) Orthogonal Quadrature Pulse Shift Keying c) Orthogonal Quadrature Phase Shift Keying d) Offset Quadrature Phase Shift Keying Explanation: OQPSK stands for offset quadrature phase shift keying. It is a modified form of QPSK which is less susceptible to deleterious effects and supports more efficient amplification. OQPSK is sometimes also called staggered QPSK. 13. The bandwidth of OQPSK is _______ to QPSK. d) Four times Explanation: The spectrum of an OQPSK signal is identical to that of QPSK signal. Hence, both signals occupy the same bandwidth. The staggered alignment of the even and odd bit streams in OQPSK signal does not change the nature of spectrum. 14. QPSK signals perform better than OQPSK in the presence of phase jitter. State whether True or False. Explanation: OQPSK signal perform better than QPSK in the presence of phase jitter. It is due to the presence of noisy reference signal at the receiver. 15. Which of the following is not a detection technique used for detection of π/4 QPSK signals? a) Baseband differential detection b) IF differential detection c) FM discriminator detection d) Envelope detection Explanation: There are various types of detection techniques used for detection of π/4 QPSK signals. They include baseband differential detection, IF differential detection and FM discriminator detection. Sanfoundry Global Education & Learning Series – Wireless & Mobile Communications. To practice all areas of Wireless & Mobile Communications, here is complete set of 1000+ Multiple Choice Questions and Answers.

Voting is important. When your country/province/city/university/bridge club holds an election, they are asking for you to determine who is going to be representing you for the next term. The Governors at the Board are going to be told that the two undergraduates that the U of A student body sends to them were elected from the masses to represent the masses, and so it's important that you vote for the two that best reflect your opinions. However, in the Students' Union, we don't use a first past the post system. Instead of saying, "I like candidate A!", you get the option of deciding your second favourite, third favourite, and so on. At some point, you can even rank None of the Above as if they were a candidate! The options are limitless! The long-lasting problem with systems like this, though, is that most people simply will not understand how their vote is counted. It's a fact that all voting systems can be played strategically, and really die-hard supporters will want to know the ins and outs of how best to vote in order to guarantee their candidate's success. Unfortunately, to most people the process works something like this: I'm going to talk about four different methods of determining the winner of elections. This post is about to get super fun! Before I do that, though, I want to quickly mention a few of my favorite electoral system criteria that "expert political scientists" have come up with: Absolute Winner: May seem obvious, but if one candidate gets more than 50% of the votes on the first round, they should win. Independence of Clones: The election outcome should remain following the addition of an identical candidate with an equal chance of winning. Condorcet Winner: If a candidate wins a head-to-head competition against every other candidate, that candidate must win the election. First Past the Post This one is pretty easy: The person with the most votes wins. Pros: Dead easy. Cons: It kinda sucks in terms of figuring out the best candidate. If five people run, you could expect a candidate to win with only slightly more than 20% of the popular vote. Also, it only examines the first choices of voters. As a voting system, it only satisfies the absolute winner criteria. It fails the independence of clones criteria as vote splitting is a very common issue amongst similar candidates, and it fails the Condorcet criteria because it doesn't even consider the subsequent choices of voters. Each candidate gets a number of points for each first place vote, a smaller number of points for each second place vote, an even smaller number of points for each third place vote, etc. The candidate with the most votes wins! Pros: Still pretty easy. It also takes into account the subsequent ranking of candidates by voters, and tends to give a winner that most people are generally ok with (as opposed to FPTP where a majority is likely to have never voted for a winner). Cons: Fails pretty much every other test in the book. You could have 50.1% of first place votes and still lose if another candidate has a VERY strong second place showing. Instant Runoff Voting A little bit more complicated: 20% of students vote (sad fact). All first-ranked votes are examined. If someone has more than 50%, they win! If they don't, the last-ranked candidate gets kicked off the island, and their second-ranked votes are redistributed to the remaining candidates. This process continues until, at the end of the day, someone finally has more than 50% of the votes (often through borrowed votes from other people). Pros: Vote-splitting doesn't happen in this method: if one candidate pushing for a waterslide in Quad is expected to get 60%, and another candidate comes forward with the same promise, we're still going to get a waterslide in Quad. This assumes, of course, that people would rank both candidates numbers 1 and 2 on their ballot if they really want that slide. By the time that a winner has been determined, more than half the voters will have indicated in some way that they support that candidate more than anyone else who's left, so a majority will be (begrudgingly, at least) satisfied with the results. Cons: It can be complicated. If you didn't understand the SU's system before reading this post, you weren't alone. If you don't understand it after reading this post, please tell me and I'll try to make it clearer. This method satisfies all the above criteria apart form the... Way more complicated: For every possible pair of candidates (that's 10 if there are five candidates in a race), the candidate who is ranked higher more often is declared the winner in that head-to-head contest. If one candidate wins against all the other candidates, then they win the election. If there's a tie in the number of victorious head-to-head contests, then a complicated method based on vote differentials is used to determine the winner. Pros: It's awesome. It's actually by a mile the coolest way to count votes. Cons: It takes a while to explain to people, and for large races it basically can't be done by hand. Now you know! The SU uses an Instant Runoff Voting system for its executive, and a variant iterative IRV system for its council elections. What's especially fun about electoral systems, though, is that different systems often change the results of an election. Take a look at this example: 42 Voters: A, B, C, D 26 Voters: B, C, D, A 15 Voters: C, D, B, A 17 Voters: D, C, B, A Total voters: 100 What's really fun (and I'll let you do the math) is that the winner of this election depends on the method. First past the post simply says that A should win. A Borda Count or the Condorcet Method would let B win, and an Instant-Runoff Vote would elect candidate D. Go figure, right? At the end of the day, what I'm trying to say is that the order in which you rank your candidates in this election really does matter. Stay tuned for an analysis of the election results after they're revealed!

Kondo impurity on the honeycomb lattice at half-filling We consider a Kondo-like impurity interacting with fermions on a honeycomb lattice at half-filling, as in the case of graphene. We derive from the lattice model an effective one-dimensional continuum theory which has, in general, four flavors with angular momentum mixing in the presence of internode scattering processes and six couplings in the spin-isotropic case. Under particular conditions, however, it can be reduced to a single-coupling multichannel pseudogap Kondo model. We finally calculate, in the presence of an energy dependent Fermi velocity, induced by Coulomb interaction, the critical coupling in the large- expansion, the magnetic susceptibility and the specific heat. pacs:71.10.Fd; 72.15.Qm; 75.30.Hx; 71.10.Ay Since the experimental realization of a single monolayer of graphite novoselov , named graphene, a two dimensional crystal made of carbon atoms hexagonally packed, a lot of efforts have been made to study many properties of electrons sitting on a honeycomb lattice castroneto . Also the problem of magnetic impurities in such a system has become a topic of recent investigations in the last few years saremi ; hentschel ; sengpunta ; uchoa ; cornaglia , although a detailed derivation of the effective model is still lacking. The main motivation of the present work is, therefore, that of deriving, from the lattice Hamiltonian, the corresponding continuum model for the Kondo-like impurity, writing the effective couplings from the lattice parameters. From angular mode expansion we get an effective one-dimensional Kondo model which has, in general, four flavors and is peculiar to graphene-like sublattice systems. Strikingly, we find that there is an angular momentum mixing only in the presence of internode scattering processes, being the valleys and the momenta locked in pairs, in each sublattice sector. The complete model has six couplings in the spin-isotropic case, however, thanks to the lattice symmetry, for some particular positions of the impurity, the number of couplings can be reduced to one, obtaining a multichannel pseudogap Kondo model sharing, now, many similarities with other gapless fermionic systems withoff ; ingersent ; ingersent2 , as for example, some semiconductors withoff , -wave superconductors cassanello2 and flux phases cassanello1 . A second issue which is worthwhile being addressed is related to interactions. In real systems logarithmic corrections in the density of states may appears, as a result of many-body effects. In order to include, at some extent, correlation effects we allow the Fermi velocity to be energy dependent. Indeed for a system of electrons in the half-filled honeycomb lattice, like graphene, an effect of Coulomb interaction is that of renormalizing the Fermi velocity gonzalez94 which grows in the infrared limit. This behavior induces in the density of states subleading logarithmic corrections. We plan therefore to analyze the effect of these corrections onto the Kondo effect in order to see how finite coupling constant transition, obtained within the large- expansion technique read and renormalization group approach anderson ; hewson , can be affected by deviations from power law. We find that the critical Kondo coupling becomes non-universal and is enhanced in the ultraviolet by a quantity directly related to the Coulomb screening. Moreover, we find that the impurity contribution to the magnetic susceptibility and the specific heat vanish faster by log than in the free case, as approaching zero magnetic field or zero temperature. Ii The model In this section we will derive the continuum one-dimensional effective model from the microscopic lattice Hamiltonian. ii.1 Lattice Hamiltonian Let us consider a honeycomb lattice which can be divided into two sublattices, A and B. The tight-binding vectors can be chosen as follows where is the smallest distance between two sites. These vectors link sites belonging to two different triangular sublattices. Each sublattice is defined by linear combinations of other two vectors, and . From these values one can derive the reciprocal-lattice vectors in momentum space and draw the Brillouin zone which has an hexagonal shape, i.e. with six corners. We choose two inequivalent corners (the others are obtained by a shift of a reciprocal-lattice vector) at the positions These points are actually the Fermi surface reduced to two dots approaching the zero chemical potential, i.e. at half-filling. We will consider the following Hamiltonian defined on this honeycomb lattice The first contribution is given by the tight-binding Hamiltonian where is the nearest neighbour hopping parameter, () is the creation (annihilation) operator for electrons with spin localized on the site , a vector belonging to the sublattice , while () the creation (annihilation) operator for electrons on the site , belonging to the sublattice . The second contribution to is the Kondo-like impurity term where and are the short-range Kondo couplings, (with ) is the spin of the impurity sitting at the reference position , is the spin operator of the electrons located at from the impurity. can belong to or and we sum over all these vectors. ii.2 Derivation of 1D effective model We now rewrite the fields in the following way Expanding the slow fields around , introducing the multispinor the identities , , and the Pauli matrices , and , , acting respectively on the spin space, , valley space, , and sublattice space, , we get, in the continuum limit, where is the Fermi velocity and a Kondo coupling matrix with the following components, containing the lattice details, where the spin index and the sublattice index. Eq. (15) is a Dirac-Weyl Hamiltonian, constant in spin-space, which, after defining , can be written as to make Lorentz invariance manifest. The spectrum is made of a couple of Dirac cones departing from the two Fermi points, and the density of states vanishes linearly approaching the zero energy, , where . This property plays a fundamental role on the scaling behavior of the Kondo impurity, as we are going to see. Let us rewrite the full effective Hamiltonian in momentum space, where we have parametrized the momenta as follows For the benefits of forthcoming discussions we first notice that the orbital angular momentum operator does not commute with the Hamiltonian in Eq. (15). On the other hand, in order to define proper total angular momenta we introduce the operator which does commute with , and also with the -components of . In particular, given some amplitudes , an eigenstate of with eigenvalue can be written as Performing the following unitary transformation to the fields the Hamiltonian Eq. (21) becomes namely, becomes diagonal, the cost to pay is that the Kondo couplings depends on the angular part of the momenta, Notice that the angular dependence of does not prevent the model to be renormalizable. As one can see by poor man’s scaling procedure anderson ; hewson , being the intermediate momentum in the edge bands, dropping for the moment the spin indices, the contributions which renormalize, for instance, in the particle channel are with . In the same way we can check that the corrections to , with , are Analogous corrections can be verified in the hole channel. In all these corrections always cancels out, recovering the right momentum dependence for the slow modes. From Eqs. (18-20) we actually get access to the renormalization of linear combinations of the original lattice parameters . In order to reduce the problem to one dimension we proceed expanding the fields in angular momentum eigenmodes as follows with . Indeed, due to the gauge in Eq. (28), all the spinor components have the same angular phase. Actually from Eq. (29), one verify that is the eigenvector of , Eq. (27), with eigenvalue , transformed by , and with amplitudes where the subscript replaces the sublattice index and refers to the sign of the energy, , appearing in Eq. (II.2). The original field at position , can be written as where are the Bessel functions of the first kind, and . At the only terms which survive are those with , corresponding to , in terms of eigenvalues of . After integrating Eq. (II.2) over the angles, indeed, we get in the only contributions with , in the following combinations where now are spinors only in spin and energy spaces. Here we are considering the spin-isotropic case, with , to simplify the notation. In the spin-anisotropic case one simply has to replace with . In the free part of the effective model, , we keep only the contributions from the particles with , the only ones which can scatter with the impurity. We now unfold the momenta from to by redefining the fields in the following way where, in order to label the fermions, we choose the index in valley space and the index , the sign of the total angular momenta, eigenvalues of , which are good quantum numbers as soon as there is not internode scattering, i.e. . Introducing for simplicity we finally end up with the following one-dimensional effective Hamiltonian where, in the first term, the indices , and , are summed, and the dispersion relation is . The full model, Eq. (II.2), has six Kondo couplings, in the spin-isotropic case, which are independent for a generic position of the magnetic impurity on the lattice. Moreover Eq. (II.2) exhibits an angular momentum mixing in the presence of internode scattering amplitudes and , namely, when also the nodes are mixed. We are not going to analyze the complete model in full generality but we shall consider only particular cases physically relevant. ii.3 Some particular examples Impurity on a site. If we consider an impurity on top of a site of the honeycomb lattice, belonging to the sublattice , for instance, and consider only the nearest neighbour coupling between the impurity and the electrons located on this site, we have if and assume for . In this case we get Introducing the symmetric combination for the fields the effective Hamiltonian Eq. (II.2) becomes simply which is a single channel Kondo model. Impurity by substitution. If we now consider an impurity sitting on a site of the honeycomb lattice, let us say, belonging to the sublattice , and consider only nearest neighbour couplings between the impurity and the electrons, we have if while , if , , and for . Noticing that Recalling the fields as follows the effective Hamiltonian Eq. (II.2) reduces to where the flavors (the valleys and the momenta are locked in pairs) are decoupled and we realize a two-channel Kondo model. The reduced model Eq. (50) is the same as that found for flux phases cassanello1 . Impurity at the center of the cell. Enumerating the fields as follows, for instance, Iii Large- expansion and the role of Coulomb interaction In this section we solve the model Eq. (50) in the large- approximation, where is the rank of the symmetry group of the impurity, which actually is equal to for spin one-half. Following the standard procedure read ; cassanello2 , within a path integral formalism, we write , introducing additional fermionic fields , with the constraint , the charge occupancy at the impurity site. In the Lagrangian, therefore, a Lagrange multiplier is included to enforce such constraint, which is actually the impurity Fermi level. To decouple the quartic fermionic term one introduces the Hubbard-Stratonovich fields , where , being the number of flavors. For impurity by substitution , as seen before. After integrating over the fermionic fields, and , we end up with the following effective free energy, where is the Fermi function and the phase shift, with , and a positive ultraviolet cut-off which dictates the limit of validity of the continuum Dirac-like model for the free Hamiltonian. For graphene the typical value is eV. So far we have considered a model of free fermions hopping on a lattice and scattering eventually with a magnetic impurity, but in order to get more realistic predictions we should consider, at some extent, interaction effects. In order to do that, we let the Fermi velocity be energy dependent, i.e. . This is not unrealistic since it has been shown gonzalez94 that, due to Coulomb screening in an electronic system defined on the half-filled honeycomb lattice, as in the case of a monolayer of graphene gonzalez99 ; polini , the effective Fermi velocity is renormalized in such a way that flows to higher values in the infrared, and consequently the density of states around the Fermi energy decreases. The low energy behavior for the renormalized velocity is and so the density of states should behave naively as . The aim of the following section is then to study the role of such corrections onto the Kondo effect, neglecting, however, possible renormalization of the Kondo coupling due to Coulomb interaction. The idea is to consider an uncharged magnetic impurity embedded in a cloud of charges dressed by Coulomb interaction. The realistic expression for the Fermi velocity is the following polini where is related to the fine structure constant, for Thomas-Fermi screening it is , being the dielectric constant, and is an energy independent velocity. iii.1 Saddle point equations where is the bandwidth. The Eq. (57) dictates the relation between the singlet amplitude and the impurity level , at fixed occupation charge , and reads For and for a generic value of , we get the following behavior for the impurity level, . In the non-interacting limit, formally, when , it reduces to , in agreement with Ref. cassanello2 . Strikingly, the limit of is finite and equal to , i.e. the two limits do not commute. This means that, in the presence of Coulomb interaction, the occupation charge for an impurity level within the bandwidth is finite only if the singlet is formed and . The energy scale in Eq. (59) does not play the role of an infrared cut-off for , and as a result, in that limit, goes to zero for any value of . This result is different from that found in the free case cassanello2 where the Fermi velocity is constant, . In the latter case vanishes, approaching zero singlet amplitude, for any value of the occupation charge. The second equation, Eq. (58), dropping the indices for simplicity, reads Setting and at , we get the following critical coupling is the Incomplete Gamma function. Sending we recover the standard result withoff . At this point it is worthwhile making a digression. Contrary to the free case, where the limit is trivially finite and equal to , in the interacting case, using Eq. (61), this limit is zero. On the other hand, if we replace with renormalized velocity , the limit is finite and equal to the standard case. This is consistent with the fact that the dimensionless parameter relevant in the Kondo effect is not the bare coupling but the product and that, in the presence of a renormalized Fermi velocity, Eq. (56), the density of states is modified as . In order to validate this result and to get more insights one can address the problem from a renormalization group prospective, as we did in Appendix. From Eq. (61) we find that the critical coupling is not universal, being an increasing function of the ratio , and is larger than the corresponding mean field result in the non-interacting case for any positive . To go beyond the tree level, one should consider quantum fluctuations, i.e. higher orders in large- expansion, which might spoil the critical point obtained in the mean field level, as in the case of strictly power-law pseudogap Kondo systems ingersent2 , if the particle-hole symmetry is preserved. In order to break particle-hole symmetry, however, one can include straightforwardly a gate voltage in the model sengpunta . In any case the role of fluctuations, in the presence of a logarithmic deviation from power-law in the density of states is still an open issue which we are not going to address here. iii.2 Magnetic susceptibility and specific heat The magnetic field can be easily included in our final model introducing a Zeeman term . This term modifies the phase shift in the free energy, Eq. (54), as . We can, therefore, calculate the magnetization and the magnetic susceptibility For and , we have the following magnetization The final result for the magnetization is valid only if . In the same limit the asymptotic behavior of the magnetic susceptibility is, then, given by For , instead, one gets , and for , one recover the result for the non-interacting case cassanello2 . For and , we have, instead, the following magnetic susceptibility

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To download a .zip file containing this book to use offline, simply click here. You are in the checkout line at the grocery store when your eyes wander over to the ice cream display. It is a hot day and you could use something to cool you down before you get into your hot car. The problem is that you have left your checkbook and credit and debit cards at home—on purpose, actually, because you have decided that you only want to spend $20 today at the grocery store. You are uncertain whether or not you have brought enough cash with you to pay for the items that are already in your cart. You put the ice cream bar into your cart and tell the clerk to let you know if you go over $20 because that is all you have. He rings it up and it comes to $22. You have to make a choice. You decide to keep the ice cream and ask the clerk if he would mind returning a box of cookies to the shelf. We all engage in these kinds of choices every day. We have budgets and must decide how to spend them. The model of utility theory that economists have constructed to explain consumer choice assumes that consumers will try to maximize their utility. For example, when you decided to keep the ice cream bar and return the cookies, you, consciously or not, applied the marginal decision rule to the problem of maximizing your utility: You bought the ice cream because you expect that eating it will give you greater satisfaction than would consuming the box of cookies. Utility theory provides insights into demand. It lets us look behind demand curves to see how utility-maximizing consumers can be expected to respond to price changes. While the focus of this chapter is on consumers making decisions about what goods and services to buy, the same model can be used to understand how individuals make other types of decisions, such as how much to work and how much of their incomes to spend now or to sock away for the future. We can approach the analysis of utility maximization in two ways. The first two sections of the chapter cover the marginal utility concept, while the final section examines an alternative approach using indifference curves. Why do you buy the goods and services you do? It must be because they provide you with satisfaction—you feel better off because you have purchased them. Economists call this satisfaction utility. The concept of utility is an elusive one. A person who consumes a good such as peaches gains utility from eating the peaches. But we cannot measure this utility the same way we can measure a peach’s weight or calorie content. There is no scale we can use to determine the quantity of utility a peach generates. Francis Edgeworth, one of the most important contributors to the theory of consumer behavior, imagined a device he called a hedonimeter (after hedonism, the pursuit of pleasure): “[L]et there be granted to the science of pleasure what is granted to the science of energy; to imagine an ideally perfect instrument, a psychophysical machine, continually registering the height of pleasure experienced by an individual…. From moment to moment the hedonimeter varies; the delicate index now flickering with the flutter of passions, now steadied by intellectual activity, now sunk whole hours in the neighborhood of zero, or momentarily springing up towards infinity.”Francis Y. Edgeworth, Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (New York: Augustus M. Kelley, 1967), p. 101. First Published 1881. Perhaps some day a hedonimeter will be invented. The utility it measures will not be a characteristic of particular goods, but rather of each consumer’s reactions to those goods. The utility of a peach exists not in the peach itself, but in the preferences of the individual consuming the peach. One consumer may wax ecstatic about a peach; another may say it tastes OK. When we speak of maximizing utility, then, we are speaking of the maximization of something we cannot measure. We assume, however, that each consumer acts as if he or she can measure utility and arranges consumption so that the utility gained is as high as possible. If we could measure utility, total utilityThe number of units of utility that a consumer gains from consuming a given quantity of a good, service, or activity during a particular time period. would be the number of units of utility that a consumer gains from consuming a given quantity of a good, service, or activity during a particular time period. The higher a consumer’s total utility, the greater that consumer’s level of satisfaction. Panel (a) of Figure 7.1 "Total Utility and Marginal Utility Curves" shows the total utility Henry Higgins obtains from attending movies. In drawing his total utility curve, we are imagining that he can measure his total utility. The total utility curve shows that when Mr. Higgins attends no movies during a month, his total utility from attending movies is zero. As he increases the number of movies he sees, his total utility rises. When he consumes 1 movie, he obtains 36 units of utility. When he consumes 4 movies, his total utility is 101. He achieves the maximum level of utility possible, 115, by seeing 6 movies per month. Seeing a seventh movie adds nothing to his total utility. Figure 7.1 Total Utility and Marginal Utility Curves Panel (a) shows Henry Higgins’s total utility curve for attending movies. It rises as the number of movies increases, reaching a maximum of 115 units of utility at 6 movies per month. Marginal utility is shown in Panel (b); it is the slope of the total utility curve. Because the slope of the total utility curve declines as the number of movies increases, the marginal utility curve is downward sloping. Mr. Higgins’s total utility rises at a decreasing rate. The rate of increase is given by the slope of the total utility curve, which is reported in Panel (a) of Figure 7.1 "Total Utility and Marginal Utility Curves" as well. The slope of the curve between 0 movies and 1 movie is 36 because utility rises by this amount when Mr. Higgins sees his first movie in the month. It is 28 between 1 and 2 movies, 22 between 2 and 3, and so on. The slope between 6 and 7 movies is zero; the total utility curve between these two quantities is horizontal. The amount by which total utility rises with consumption of an additional unit of a good, service, or activity, all other things unchanged, is marginal utilityThe amount by which total utility rises with consumption of an additional unit of a good, service, or activity, all other things unchanged.. The first movie Mr. Higgins sees increases his total utility by 36 units. Hence, the marginal utility of the first movie is 36. The second increases his total utility by 28 units; its marginal utility is 28. The seventh movie does not increase his total utility; its marginal utility is zero. Notice that in the table marginal utility is listed between the columns for total utility because, similar to other marginal concepts, marginal utility is the change in utility as we go from one quantity to the next. Mr. Higgins’s marginal utility curve is plotted in Panel (b) of Figure 7.1 "Total Utility and Marginal Utility Curves" The values for marginal utility are plotted midway between the numbers of movies attended. The marginal utility curve is downward sloping; it shows that Mr. Higgins’s marginal utility for movies declines as he consumes more of them. Mr. Higgins’s marginal utility from movies is typical of all goods and services. Suppose that you are really thirsty and you decide to consume a soft drink. Consuming the drink increases your utility, probably by a lot. Suppose now you have another. That second drink probably increases your utility by less than the first. A third would increase your utility by still less. This tendency of marginal utility to decline beyond some level of consumption during a period is called the law of diminishing marginal utilityThis tendency of marginal utility to decline beyond some level of consumption during a period.. This law implies that all goods and services eventually will have downward-sloping marginal utility curves. It is the law that lies behind the negatively sloped marginal benefit curve for consumer choices that we examined in the chapter on markets, maximizers, and efficiency. One way to think about this effect is to remember the last time you ate at an “all you can eat” cafeteria-style restaurant. Did you eat only one type of food? Did you consume food without limit? No, because of the law of diminishing marginal utility. As you consumed more of one kind of food, its marginal utility fell. You reached a point at which the marginal utility of another dish was greater, and you switched to that. Eventually, there was no food whose marginal utility was great enough to make it worth eating, and you stopped. What if the law of diminishing marginal utility did not hold? That is, what would life be like in a world of constant or increasing marginal utility? In your mind go back to the cafeteria and imagine that you have rather unusual preferences: Your favorite food is creamed spinach. You start with that because its marginal utility is highest of all the choices before you in the cafeteria. As you eat more, however, its marginal utility does not fall; it remains higher than the marginal utility of any other option. Unless eating more creamed spinach somehow increases your marginal utility for some other food, you will eat only creamed spinach. And until you have reached the limit of your body’s capacity (or the restaurant manager’s patience), you will not stop. Failure of marginal utility to diminish would thus lead to extraordinary levels of consumption of a single good to the exclusion of all others. Since we do not observe that happening, it seems reasonable to assume that marginal utility falls beyond some level of consumption. Economists assume that consumers behave in a manner consistent with the maximization of utility. To see how consumers do that, we will put the marginal decision rule to work. First, however, we must reckon with the fact that the ability of consumers to purchase goods and services is limited by their budgets. The total utility curve in Figure 7.1 "Total Utility and Marginal Utility Curves" shows that Mr. Higgins achieves the maximum total utility possible from movies when he sees six of them each month. It is likely that his total utility curves for other goods and services will have much the same shape, reaching a maximum at some level of consumption. We assume that the goal of each consumer is to maximize total utility. Does that mean a person will consume each good at a level that yields the maximum utility possible? The answer, in general, is no. Our consumption choices are constrained by the income available to us and by the prices we must pay. Suppose, for example, that Mr. Higgins can spend just $25 per month for entertainment and that the price of going to see a movie is $5. To achieve the maximum total utility from movies, Mr. Higgins would have to exceed his entertainment budget. Since we assume that he cannot do that, Mr. Higgins must arrange his consumption so that his total expenditures do not exceed his budget constraintA restriction that total spending cannot exceed the budget available.: a restriction that total spending cannot exceed the budget available. Suppose that in addition to movies, Mr. Higgins enjoys concerts, and the average price of a concert ticket is $10. He must select the number of movies he sees and concerts he attends so that his monthly spending on the two goods does not exceed his budget. Individuals may, of course, choose to save or to borrow. When we allow this possibility, we consider the budget constraint not just for a single period of time but for several periods. For example, economists often examine budget constraints over a consumer’s lifetime. A consumer may in some years save for future consumption and in other years borrow on future income for present consumption. Whatever the time period, a consumer’s spending will be constrained by his or her budget. To simplify our analysis, we shall assume that a consumer’s spending in any one period is based on the budget available in that period. In this analysis consumers neither save nor borrow. We could extend the analysis to cover several periods and generate the same basic results that we shall establish using a single period. We will also carry out our analysis by looking at the consumer’s choices about buying only two goods. Again, the analysis could be extended to cover more goods and the basic results would still hold. Because consumers can be expected to spend the budget they have, utility maximization is a matter of arranging that spending to achieve the highest total utility possible. If a consumer decides to spend more on one good, he or she must spend less on another in order to satisfy the budget constraint. The marginal decision rule states that an activity should be expanded if its marginal benefit exceeds its marginal cost. The marginal benefit of this activity is the utility gained by spending an additional $1 on the good. The marginal cost is the utility lost by spending $1 less on another good. How much utility is gained by spending another $1 on a good? It is the marginal utility of the good divided by its price. The utility gained by spending an additional dollar on good X, for example, is This additional utility is the marginal benefit of spending another $1 on the good. Suppose that the marginal utility of good X is 4 and that its price is $2. Then an extra $1 spent on X buys 2 additional units of utility ( ). If the marginal utility of good X is 1 and its price is $2, then an extra $1 spent on X buys 0.5 additional units of utility ( ). The loss in utility from spending $1 less on another good or service is calculated the same way: as the marginal utility divided by the price. The marginal cost to the consumer of spending $1 less on a good is the loss of the additional utility that could have been gained from spending that $1 on the good. Suppose a consumer derives more utility by spending an additional $1 on good X rather than on good Y: The marginal benefit of shifting $1 from good Y to the consumption of good X exceeds the marginal cost. In terms of utility, the gain from spending an additional $1 on good X exceeds the loss in utility from spending $1 less on good Y. The consumer can increase utility by shifting spending from Y to X. As the consumer buys more of good X and less of good Y, however, the marginal utilities of the two goods will change. The law of diminishing marginal utility tells us that the marginal utility of good X will fall as the consumer consumes more of it; the marginal utility of good Y will rise as the consumer consumes less of it. The result is that the value of the left-hand side of Equation 7.1 will fall and the value of the right-hand side will rise as the consumer shifts spending from Y to X. When the two sides are equal, total utility will be maximized. In terms of the marginal decision rule, the consumer will have achieved a solution at which the marginal benefit of the activity (spending more on good X) is equal to the marginal cost: We can extend this result to all goods and services a consumer uses. Utility maximization requires that the ratio of marginal utility to price be equal for all of them, as suggested in Equation 7.3: Equation 7.3 states the utility-maximizing conditionUtility is maximized when total outlays equal the budget available and when the ratios of marginal utilities to prices are equal for all goods and services.: Utility is maximized when total outlays equal the budget available and when the ratios of marginal utilities to prices are equal for all goods and services. Consider, for example, the shopper introduced in the opening of this chapter. In shifting from cookies to ice cream, the shopper must have felt that the marginal utility of spending an additional dollar on ice cream exceeded the marginal utility of spending an additional dollar on cookies. In terms of Equation 7.1, if good X is ice cream and good Y is cookies, the shopper will have lowered the value of the left-hand side of the equation and moved toward the utility-maximizing condition, as expressed by Equation 7.1. If we are to apply the marginal decision rule to utility maximization, goods must be divisible; that is, it must be possible to buy them in any amount. Otherwise we cannot meaningfully speak of spending $1 more or $1 less on them. Strictly speaking, however, few goods are completely divisible. Even a small purchase, such as an ice cream bar, fails the strict test of being divisible; grocers generally frown on requests to purchase one-half of a $2 ice cream bar if the consumer wants to spend an additional dollar on ice cream. Can a consumer buy a little more movie admission, to say nothing of a little more car? In the case of a car, we can think of the quantity as depending on characteristics of the car itself. A car with a compact disc player could be regarded as containing “more car” than one that has only a cassette player. Stretching the concept of quantity in this manner does not entirely solve the problem. It is still difficult to imagine that one could purchase “more car” by spending $1 more. Remember, though, that we are dealing with a model. In the real world, consumers may not be able to satisfy Equation 7.3 precisely. The model predicts, however, that they will come as close to doing so as possible. A college student, Ramón Juárez, often purchases candy bars or bags of potato chips between classes; he tries to limit his spending on these snacks to $8 per week. A bag of chips costs $0.75 and a candy bar costs $0.50 from the vending machines on campus. He has been purchasing an average of 6 bags of chips and 7 candy bars each week. Mr. Juárez is a careful maximizer of utility, and he estimates that the marginal utility of an additional bag of chips during a week is 6. In your answers use B to denote candy bars and C to denote potato chips. © 2010 Jupiterimages Corporation In preparation for sitting in the slow, crowded lanes for single-occupancy-vehicles, T. J. Zane used to stop at his favorite coffee kiosk to buy a $2 cup of coffee as he headed off to work on Interstate 15 in the San Diego area. Since 1996, an experiment in road pricing has caused him and others to change their ways—and to raise their total utility. Before 1996, only car-poolers could use the specially marked high-occupancy-vehicles lanes. With those lanes nearly empty, traffic authorities decided to allow drivers of single-occupancy-vehicles to use those lanes, so long as they paid a price. Now, electronic signs tell drivers how much it will cost them to drive on the special lanes. The price is recalculated every 6 minutes depending on the traffic. On one morning during rush hour, it varied from $1.25 at 7:10 a.m., to $1.50 at 7:16 a.m., to $2.25 at 7:22 a.m., and to $2.50 at 7:28 a.m. The increasing tolls over those few minutes caused some drivers to opt out and the toll fell back to $1.75 and then increased to $2 a few minutes later. Drivers do not have to stop to pay the toll since radio transmitters read their FasTrak transponders and charge them accordingly. When first instituted, these lanes were nicknamed the “Lexus lanes,” on the assumption that only wealthy drivers would use them. Indeed, while the more affluent do tend to use them heavily, surveys have discovered that they are actually used by drivers of all income levels. Mr. Zane, a driver of a 1997 Volkswagen Jetta, is one commuter who chooses to use the new option. He explains his decision by asking, “Isn’t it worth a couple of dollars to spend an extra half-hour with your family?” He continues, “That’s what I used to spend on a cup of coffee at Starbucks. Now I’ve started bringing my own coffee and using the money for the toll.” We can explain his decision using the model of utility-maximizing behavior; Mr. Zane’s out-of-pocket commuting budget constraint is about $2. His comment tells us that he realized that the marginal utility of spending an additional 30 minutes with his family divided by the $2 toll was higher than the marginal utility of the store-bought coffee divided by its $2 price. By reallocating his $2 commuting budget, the gain in utility of having more time at home exceeds the loss in utility from not sipping premium coffee on the way to work. From this one change in behavior, we do not know whether or not he is actually maximizing his utility, but his decision and explanation are certainly consistent with that goal. Source: John Tierney, “The Autonomist Manifesto (Or, How I learned to Stop Worrying and Love the Road),” New York Times Magazine, September 26, 2004, 57–65. In order for the ratios of marginal utility to price to be equal, the marginal utility of a candy bar must be 4. Let the marginal utility and price of candy bars be MUB and PB, respectively, and the marginal utility and price of a bag of potato chips be MUC and PC, respectively. Then we want We know that PC is $0.75 and PB equals $0.50. We are told that MUC is 6. Thus Solving the equation for MUB, we find that it must equal 4. Choices that maximize utility—that is, choices that follow the marginal decision rule—generally produce downward-sloping demand curves. This section shows how an individual’s utility-maximizing choices can lead to a demand curve. Suppose, for simplicity, that Mary Andrews consumes only apples, denoted by the letter A, and oranges, denoted by the letter O. Apples cost $2 per pound and oranges cost $1 per pound, and her budget allows her to spend $20 per month on the two goods. We assume that Ms. Andrews will adjust her consumption so that the utility-maximizing condition holds for the two goods: The ratio of marginal utility to price is the same for apples and oranges. That is, Here MUA and MUO are the marginal utilities of apples and oranges, respectively. Her spending equals her budget of $20 per month; suppose she buys 5 pounds of apples and 10 of oranges. Now suppose that an unusually large harvest of apples lowers their price to $1 per pound. The lower price of apples increases the marginal utility of each $1 Ms. Andrews spends on apples, so that at her current level of consumption of apples and oranges Ms. Andrews will respond by purchasing more apples. As she does so, the marginal utility she receives from apples will decline. If she regards apples and oranges as substitutes, she will also buy fewer oranges. That will cause the marginal utility of oranges to rise. She will continue to adjust her spending until the marginal utility per $1 spent is equal for both goods: Suppose that at this new solution, she purchases 12 pounds of apples and 8 pounds of oranges. She is still spending all of her budget of $20 on the two goods [(12 x $1)+(8 x $1)=$20]. Figure 7.3 Utility Maximization and an Individual’s Demand Curve Mary Andrews’s demand curve for apples, d, can be derived by determining the quantities of apples she will buy at each price. Those quantities are determined by the application of the marginal decision rule to utility maximization. At a price of $2 per pound, Ms. Andrews maximizes utility by purchasing 5 pounds of apples per month. When the price of apples falls to $1 per pound, the quantity of apples at which she maximizes utility increases to 12 pounds per month. It is through a consumer’s reaction to different prices that we trace the consumer’s demand curve for a good. When the price of apples was $2 per pound, Ms. Andrews maximized her utility by purchasing 5 pounds of apples, as illustrated in Figure 7.3 "Utility Maximization and an Individual’s Demand Curve". When the price of apples fell, she increased the quantity of apples she purchased to 12 pounds. Notice that, in this example, Ms. Andrews maximizes utility where not only the ratios of marginal utilities to price are equal, but also the marginal utilities of both goods are equal. But, the equal-marginal-utility outcome is only true here because the prices of the two goods are the same: each good is priced at $1 in this case. If the prices of apples and oranges were different, the marginal utilities at the utility maximizing solution would have been different. The condition for maximizing utility—consume where the ratios of marginal utility to price are equal—holds regardless. The utility-maximizing condition is not that consumers maximize utility by equating marginal utilities. The market demand curves we studied in previous chapters are derived from individual demand curves such as the one depicted in Figure 7.3 "Utility Maximization and an Individual’s Demand Curve". Suppose that in addition to Ms. Andrews, there are two other consumers in the market for apples—Ellen Smith and Koy Keino. The quantities each consumes at various prices are given in Figure 7.5 "Deriving a Market Demand Curve", along with the quantities that Ms. Andrews consumes at each price. The demand curves for each are shown in Panel (a). The market demand curve for all three consumers, shown in Panel (b), is then found by adding the quantities demanded at each price for all three consumers. At a price of $2 per pound, for example, Ms. Andrews demands 5 pounds of apples per month, Ms. Smith demands 3 pounds, and Mr. Keino demands 8 pounds. A total of 16 pounds of apples are demanded per month at this price. Adding the individual quantities demanded at $1 per pound yields market demand of 40 pounds per month. This method of adding amounts along the horizontal axis of a graph is referred to as summing horizontally. The market demand curve is thus the horizontal summation of all the individual demand curves. Figure 7.5 Deriving a Market Demand Curve The demand schedules for Mary Andrews, Ellen Smith, and Koy Keino are given in the table. Their individual demand curves are plotted in Panel (a). The market demand curve for all three is shown in Panel (b). Individual demand curves, then, reflect utility-maximizing adjustment by consumers to various market prices. Once again, we see that as the price falls, consumers tend to buy more of a good. Demand curves are downward-sloping as the law of demand asserts. We saw that when the price of apples fell from $2 to $1 per pound, Mary Andrews increased the quantity of apples she demanded. Behind that adjustment, however, lie two distinct effects: the substitution effect and the income effect. It is important to distinguish these effects, because they can have quite different implications for the elasticity of the demand curve. First, the reduction in the price of apples made them cheaper relative to oranges. Before the price change, it cost the same amount to buy 2 pounds of oranges or 1 pound of apples. After the price change, it cost the same amount to buy 1 pound of either oranges or apples. In effect, 2 pounds of oranges would exchange for 1 pound of apples before the price change, and 1 pound of oranges would exchange for 1 pound of apples after the price change. Second, the price reduction essentially made consumers of apples richer. Before the price change, Ms. Andrews was purchasing 5 pounds of apples and 10 pounds of oranges at a total cost to her of $20. At the new lower price of apples, she could purchase this same combination for $15. In effect, the price reduction for apples was equivalent to handing her a $5 bill, thereby increasing her purchasing power. Purchasing power refers to the quantity of goods and services that can be purchased with a given budget. To distinguish between the substitution and income effects, economists consider first the impact of a price change with no change in the consumer’s ability to purchase goods and services. An income-compensated price changeAn imaginary exercise in which we assume that when the price of a good or service changes, the consumers income is adjusted so that he or she has just enough to purchase the original combination of goods and services at the new set of prices. is an imaginary exercise in which we assume that when the price of a good or service changes, the consumer’s income is adjusted so that he or she has just enough to purchase the original combination of goods and services at the new set of prices. Ms. Andrews was purchasing 5 pounds of apples and 10 pounds of oranges before the price change. Buying that same combination after the price change would cost $15. The income-compensated price change thus requires us to take $5 from Ms. Andrews when the price of apples falls to $1 per pound. She can still buy 5 pounds of apples and 10 pounds of oranges. If, instead, the price of apples increased, we would give Ms. Andrews more money (i.e., we would “compensate” her) so that she could purchase the same combination of goods. With $15 and cheaper apples, Ms. Andrews could buy 5 pounds of apples and 10 pounds of oranges. But would she? The answer lies in comparing the marginal benefit of spending another $1 on apples to the marginal benefit of spending another $1 on oranges, as expressed in Equation 7.5. It shows that the extra utility per $1 she could obtain from apples now exceeds the extra utility per $1 from oranges. She will thus increase her consumption of apples. If she had only $15, any increase in her consumption of apples would require a reduction in her consumption of oranges. In effect, she responds to the income-compensated price change for apples by substituting apples for oranges. The change in a consumer’s consumption of a good in response to an income-compensated price change is called the substitution effectThe change in a consumers consumption of a good in response to an income-compensated price change.. Suppose that with an income-compensated reduction in the price of apples to $1 per pound, Ms. Andrews would increase her consumption of apples to 9 pounds per month and reduce her consumption of oranges to 6 pounds per month. The substitution effect of the price reduction is an increase in apple consumption of 4 pounds per month. The substitution effect always involves a change in consumption in a direction opposite that of the price change. When a consumer is maximizing utility, the ratio of marginal utility to price is the same for all goods. An income-compensated price reduction increases the extra utility per dollar available from the good whose price has fallen; a consumer will thus purchase more of it. An income-compensated price increase reduces the extra utility per dollar from the good; the consumer will purchase less of it. In other words, when the price of a good falls, people react to the lower price by substituting or switching toward that good, buying more of it and less of other goods, if we artificially hold the consumer’s ability to buy goods constant. When the price of a good goes up, people react to the higher price by substituting or switching away from that good, buying less of it and instead buying more of other goods. By examining the impact of consumer purchases of an income-compensated price change, we are looking at just the change in relative prices of goods and eliminating any impact on consumer buying that comes from the effective change in the consumer’s ability to purchase goods and services (that is, we hold the consumer’s purchasing power constant). To complete our analysis of the impact of the price change, we must now consider the $5 that Ms. Andrews effectively gained from it. After the price reduction, it cost her just $15 to buy what cost her $20 before. She has, in effect, $5 more than she did before. Her additional income may also have an effect on the number of apples she consumes. The change in consumption of a good resulting from the implicit change in income because of a price change is called the income effectThe change in consumption of a good resulting from the implicit change in income because of a price change. of a price change. When the price of a good rises, there is an implicit reduction in income. When the price of a good falls, there is an implicit increase. When the price of apples fell, Ms. Andrews (who was consuming 5 pounds of apples per month) received an implicit increase in income of $5. Suppose Ms. Andrews uses her implicit increase in income to purchase 3 more pounds of apples and 2 more pounds of oranges per month. She has already increased her apple consumption to 9 pounds per month because of the substitution effect, so the added 3 pounds brings her consumption level to 12 pounds per month. That is precisely what we observed when we derived her demand curve; it is the change we would observe in the marketplace. We see now, however, that her increase in quantity demanded consists of a substitution effect and an income effect. Figure 7.6 "The Substitution and Income Effects of a Price Change" shows the combined effects of the price change. Figure 7.6 The Substitution and Income Effects of a Price Change This demand curve for Ms. Andrews was presented in Figure 7.5 "Deriving a Market Demand Curve". It shows that a reduction in the price of apples from $2 to $1 per pound increases the quantity Ms. Andrews demands from 5 pounds of apples to 12. This graph shows that this change consists of a substitution effect and an income effect. The substitution effect increases the quantity demanded by 4 pounds, the income effect by 3, for a total increase in quantity demanded of 7 pounds. The size of the substitution effect depends on the rate at which the marginal utilities of goods change as the consumer adjusts consumption to a price change. As Ms. Andrews buys more apples and fewer oranges, the marginal utility of apples will fall and the marginal utility of oranges will rise. If relatively small changes in quantities consumed produce large changes in marginal utilities, the substitution effect that is required to restore the equality of marginal-utility-to-price ratios will be small. If much larger changes in quantities consumed are needed to produce equivalent changes in marginal utilities, then the substitution effect will be large. The magnitude of the income effect of a price change depends on how responsive the demand for a good is to a change in income and on how important the good is in a consumer’s budget. When the price changes for a good that makes up a substantial fraction of a consumer’s budget, the change in the consumer’s ability to buy things is substantial. A change in the price of a good that makes up a trivial fraction of a consumer’s budget, however, has little effect on his or her purchasing power; the income effect of such a price change is small. Because each consumer’s response to a price change depends on the sizes of the substitution and income effects, these effects play a role in determining the price elasticity of demand. All other things unchanged, the larger the substitution effect, the greater the absolute value of the price elasticity of demand. When the income effect moves in the same direction as the substitution effect, a greater income effect contributes to a greater price elasticity of demand as well. There are, however, cases in which the substitution and income effects move in opposite directions. We shall explore these ideas in the next section. The nature of the income effect of a price change depends on whether the good is normal or inferior. The income effect reinforces the substitution effect in the case of normal goods; it works in the opposite direction for inferior goods. A normal good is one whose consumption increases with an increase in income. When the price of a normal good falls, there are two identifying effects: In the case of a normal good, then, the substitution and income effects reinforce each other. Ms. Andrews’s response to a price reduction for apples is a typical response to a lower price for a normal good. An increase in the price of a normal good works in an equivalent fashion. The higher price causes consumers to substitute more of other goods, whose prices are now relatively lower. The substitution effect thus reduces the quantity demanded. The higher price also reduces purchasing power, causing consumers to reduce consumption of the good via the income effect. In the chapter that introduced the model of demand and supply, we saw that an inferior good is one for which demand falls when income rises. It is likely to be a good that people do not really like very much. When incomes are low, people consume the inferior good because it is what they can afford. As their incomes rise and they can afford something they like better, they consume less of the inferior good. When the price of an inferior good falls, two things happen: The case of inferior goods is thus quite different from that of normal goods. The income effect of a price change works in a direction opposite to that of the substitution effect in the case of an inferior good, whereas it reinforces the substitution effect in the case of a normal good. Figure 7.7 Substitution and Income Effects for Inferior Goods The substitution and income effects work against each other in the case of inferior goods. The consumer begins at point A, consuming q1 units of the good at a price P1. When the price falls to P2, the consumer moves to point B, increasing quantity demanded to q2. The substitution effect increases quantity demanded to qs, but the income effect reduces it from qs to q2. Figure 7.7 "Substitution and Income Effects for Inferior Goods" illustrates the substitution and income effects of a price reduction for an inferior good. When the price falls from P1 to P2, the quantity demanded by a consumer increases from q1 to q2. The substitution effect increases quantity demanded from q1 to qs. But the income effect reduces quantity demanded from qs to q2; the substitution effect is stronger than the income effect. The result is consistent with the law of demand: A reduction in price increases the quantity demanded. The quantity demanded is smaller, however, than it would be if the good were normal. Inferior goods are therefore likely to have less elastic demand than normal goods. Ilana Drakulic has an entertainment budget of $200 per semester, which she divides among purchasing CDs, going to concerts, eating in restaurants, and so forth. When the price of CDs fell from $20 to $10, her purchases rose from 5 per semester to 10 per semester. When asked how many she would have bought if her budget constraint were $150 (since with $150 she could continue to buy 5 CDs and as before still have $100 for spending on other items), she said she would have bought 8 CDs. What is the size of her substitution effect? Her income effect? Are CDs normal or inferior for her? Which exhibit, Figure 7.6 "The Substitution and Income Effects of a Price Change" or Figure 7.7 "Substitution and Income Effects for Inferior Goods", depicts more accurately her demand curve for CDs? © 2010 Jupiterimages Corporation The fact that income and substitution effects move in opposite directions in the case of inferior goods raises a tantalizing possibility: What if the income effect were the stronger of the two? Could demand curves be upward sloping? The answer, from a theoretical point of view, is yes. If the income effect in Figure 7.7 "Substitution and Income Effects for Inferior Goods" were larger than the substitution effect, the decrease in price would reduce the quantity demanded below q1. The result would be a reduction in quantity demanded in response to a reduction in price. The demand curve would be upward sloping! The suggestion that a good could have an upward-sloping demand curve is generally attributed to Robert Giffen, a British journalist who wrote widely on economic matters late in the nineteenth century. Such goods are thus called Giffen goods. To qualify as a Giffen good, a good must be inferior and must have an income effect strong enough to overcome the substitution effect. The example often cited of a possible Giffen good is the potato during the Irish famine of 1845–1849. Empirical analysis by economists using available data, however, has refuted the notion of the upward-sloping demand curve for potatoes at that time. The most convincing parts of the refutation were to point out that (a) given the famine, there were not more potatoes available for purchase then and (b) the price of potatoes may not have even increased during the period! A recent study by Robert Jensen and Nolan Miller, though, suggests the possible discovery of a pair of Giffen goods. They began their search by thinking about the type of good that would be likely to exhibit Giffen behavior and argued that, like potatoes for the poor Irish, it would be a main dietary staple of a poor population. In such a situation, purchases of the item are such a large percentage of the diet of the poor that when the item’s price rises, the implicit income of the poor falls drastically. In order to subsist, the poor reduce consumption of other goods so they can buy more of the staple. In so doing, they are able to reach a caloric intake that is higher than what can be achieved by buying more of other preferred foods that unfortunately supply fewer calories. Their preliminary empirical work shows that in southern China rice is a Giffen good for poor consumers while in northern China noodles are a Giffen good. In both cases, the basic good (rice or noodles) provides calories at a relatively low cost and dominates the diet, while meat is considered the tastier but higher cost-per-calorie food. Using detailed household data, they estimate that among the poor in southern China a 10% increase in the price of rice leads to a 10.4% increase in rice consumption. For wealthier households in the region, rice is inferior but not Giffen. For both groups of households, the income effect of a price change moves consumption in the opposite direction of the substitution effect. Only in the poorest households, however, does it swamp the substitution effect, leading to an upward-sloping demand curve for rice for poor households. In northern China, the net effect of a price increase on quantity demanded of noodles is smaller, though it still leads to higher noodle consumption in the poorest households of that region. In a similar study, David McKenzie tested whether tortillas were a Giffen good for poor Mexicans. He found, however, that they were an inferior good but not a Giffen good. He speculated that the different result may stem from poor Mexicans having a wider range of substitutes available to them than do the poor in China. Because the Jensen/Miller study is the first vindication of the existence of a Giffen good despite a very long search, the authors have avoided rushing to publication of their results. Rather, they have made available a preliminary version of the study reported on here while continuing to refine their estimation. Sources: Robert Jensen and Nolan Miller, “Giffen Behavior: Theory and Evidence,” KSG Faculty Research Working Papers Series RWP02-014, 2002 available at ksghome.harvard.edu/~nmiller/giffen.html or http://ssrn.com/abstract=310863. At the authors’ request we include the following note on the preliminary version: “Because we have received numerous requests for this paper, we are making this early draft available. The results presented in this version, while strongly suggestive of Giffen behavior, are preliminary. In the near future we expect to acquire additional data that will allow us to revise our estimation technique. In particular, monthly temperature, precipitation, and other weather data will enable us to use an instrumental variables approach to address the possibility that the observed variation in prices is not exogenous. Once available, the instrumental variables results will be incorporated into future versions of the paper.” ; David McKenzie, “Are Tortillas a Giffen Good in Mexico?” Economics Bulletin 15:1 (2002): 1–7. One hundred fifty dollars is the income that allows Ms. Drakulic to purchase the same items as before, and thus can be used to measure the substitution effect. Looking only at the income-compensated price change (that is, holding her to the same purchasing power as in the original relative price situation), we find that the substitution effect is 3 more CDs (from 5 to 8). The CDs that she buys beyond 8 constitute her income effect; it is 2 CDs. Because the income effect reinforces the substitution effect, CDs are a normal good for her and her demand curve is similar to that shown in Figure 7.6 "The Substitution and Income Effects of a Price Change". Economists typically use a different set of tools than those presented in the chapter up to this point to analyze consumer choices. While somewhat more complex, the tools presented in this section give us a powerful framework for assessing consumer choices. We will begin our analysis with an algebraic and graphical presentation of the budget constraint. We will then examine a new concept that allows us to draw a map of a consumer’s preferences. Then we can draw some conclusions about the choices a utility-maximizing consumer could be expected to make. As we have already seen, a consumer’s choices are limited by the budget available. Total spending for goods and services can fall short of the budget constraint but may not exceed it. Algebraically, we can write the budget constraint for two goods X and Y as: where PX and PY are the prices of goods X and Y and QX and QY are the quantities of goods X and Y chosen. The total income available to spend on the two goods is B, the consumer’s budget. Equation 7.7 states that total expenditures on goods X and Y (the left-hand side of the equation) cannot exceed B. Suppose a college student, Janet Bain, enjoys skiing and horseback riding. A day spent pursuing either activity costs $50. Suppose she has $250 available to spend on these two activities each semester. Ms. Bain’s budget constraint is illustrated in Figure 7.9 "The Budget Line". For a consumer who buys only two goods, the budget constraint can be shown with a budget line. A budget lineGraphically shows the combinations of two goods a consumer can buy with a given budget. shows graphically the combinations of two goods a consumer can buy with a given budget. The budget line shows all the combinations of skiing and horseback riding Ms. Bain can purchase with her budget of $250. She could also spend less than $250, purchasing combinations that lie below and to the left of the budget line in Figure 7.9 "The Budget Line". Combinations above and to the right of the budget line are beyond the reach of her budget. Figure 7.9 The Budget Line The budget line shows combinations of the skiing and horseback riding Janet Bain could consume if the price of each activity is $50 and she has $250 available for them each semester. The slope of this budget line is −1, the negative of the price of horseback riding divided by the price of skiing. The vertical intercept of the budget line (point D) is given by the number of days of skiing per month that Ms. Bain could enjoy, if she devoted all of her budget to skiing and none to horseback riding. She has $250, and the price of a day of skiing is $50. If she spent the entire amount on skiing, she could ski 5 days per semester. She would be meeting her budget constraint, since: The horizontal intercept of the budget line (point E) is the number of days she could spend horseback riding if she devoted her $250 entirely to that sport. She could purchase 5 days of either skiing or horseback riding per semester. Again, this is within her budget constraint, since: Because the budget line is linear, we can compute its slope between any two points. Between points D and E the vertical change is −5 days of skiing; the horizontal change is 5 days of horseback riding. The slope is thus . More generally, we find the slope of the budget line by finding the vertical and horizontal intercepts and then computing the slope between those two points. The vertical intercept of the budget line is found by dividing Ms. Bain’s budget, B, by the price of skiing, the good on the vertical axis (PS). The horizontal intercept is found by dividing B by the price of horseback riding, the good on the horizontal axis (PH). The slope is thus: Simplifying this equation, we obtain After canceling, Equation 7.9 shows that the slope of a budget line is the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis. It is easy to go awry on the issue of the slope of the budget line: It is the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis. But does not slope equal the change in the vertical axis divided by the change in the horizontal axis? The answer, of course, is that the definition of slope has not changed. Notice that Equation 7.8 gives the vertical change divided by the horizontal change between two points. We then manipulated Equation 7.8 a bit to get to Equation 7.9 and found that slope also equaled the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis. Price is not the variable that is shown on the two axes. The axes show the quantities of the two goods. Suppose Ms. Bain spends 2 days skiing and 3 days horseback riding per semester. She will derive some level of total utility from that combination of the two activities. There are other combinations of the two activities that would yield the same level of total utility. Combinations of two goods that yield equal levels of utility are shown on an indifference curveGraph that shows combinations of two goods that yield equal levels of utility..Limiting the situation to two goods allows us to show the problem graphically. By stating the problem of utility maximization with equations, we could extend the analysis to any number of goods and services. Because all points along an indifference curve generate the same level of utility, economists say that a consumer is indifferent between them. Figure 7.10 "An Indifference Curve" shows an indifference curve for combinations of skiing and horseback riding that yield the same level of total utility. Point X marks Ms. Bain’s initial combination of 2 days skiing and 3 days horseback riding per semester. The indifference curve shows that she could obtain the same level of utility by moving to point W, skiing for 7 days and going horseback riding for 1 day. She could also get the same level of utility at point Y, skiing just 1 day and spending 5 days horseback riding. Ms. Bain is indifferent among combinations W, X, and Y. We assume that the two goods are divisible, so she is indifferent between any two points along an indifference curve. Figure 7.10 An Indifference Curve The indifference curve A shown here gives combinations of skiing and horseback riding that produce the same level of utility. Janet Bain is thus indifferent to which point on the curve she selects. Any point below and to the left of the indifference curve would produce a lower level of utility; any point above and to the right of the indifference curve would produce a higher level of utility. Now look at point T in Figure 7.10 "An Indifference Curve". It has the same amount of skiing as point X, but fewer days are spent horseback riding. Ms. Bain would thus prefer point X to point T. Similarly, she prefers X to U. What about a choice between the combinations at point W and point T? Because combinations X and W are equally satisfactory, and because Ms. Bain prefers X to T, she must prefer W to T. In general, any combination of two goods that lies below and to the left of an indifference curve for those goods yields less utility than any combination on the indifference curve. Such combinations are inferior to combinations on the indifference curve. Point Z, with 3 days of skiing and 4 days of horseback riding, provides more of both activities than point X; Z therefore yields a higher level of utility. It is also superior to point W. In general, any combination that lies above and to the right of an indifference curve is preferred to any point on the indifference curve. We can draw an indifference curve through any combination of two goods. Figure 7.11 "Indifference Curves" shows indifference curves drawn through each of the points we have discussed. Indifference curve A from Figure 7.10 "An Indifference Curve" is inferior to indifference curve B. Ms. Bain prefers all the combinations on indifference curve B to those on curve A, and she regards each of the combinations on indifference curve C as inferior to those on curves A and B. Although only three indifference curves are shown in Figure 7.11 "Indifference Curves", in principle an infinite number could be drawn. The collection of indifference curves for a consumer constitutes a kind of map illustrating a consumer’s preferences. Different consumers will have different maps. We have good reason to expect the indifference curves for all consumers to have the same basic shape as those shown here: They slope downward, and they become less steep as we travel down and to the right along them. Figure 7.11 Indifference Curves Each indifference curve suggests combinations among which the consumer is indifferent. Curves that are higher and to the right are preferred to those that are lower and to the left. Here, indifference curve B is preferred to curve A, which is preferred to curve C. The slope of an indifference curve shows the rate at which two goods can be exchanged without affecting the consumer’s utility. Figure 7.12 "The Marginal Rate of Substitution" shows indifference curve C from Figure 7.11 "Indifference Curves". Suppose Ms. Bain is at point S, consuming 4 days of skiing and 1 day of horseback riding per semester. Suppose she spends another day horseback riding. This additional day of horseback riding does not affect her utility if she gives up 2 days of skiing, moving to point T. She is thus willing to give up 2 days of skiing for a second day of horseback riding. The curve shows, however, that she would be willing to give up at most 1 day of skiing to obtain a third day of horseback riding (shown by point U). Figure 7.12 The Marginal Rate of Substitution The marginal rate of substitution is equal to the absolute value of the slope of an indifference curve. It is the maximum amount of one good a consumer is willing to give up to obtain an additional unit of another. Here, it is the number of days of skiing Janet Bain would be willing to give up to obtain an additional day of horseback riding. Notice that the marginal rate of substitution (MRS) declines as she consumes more and more days of horseback riding. The maximum amount of one good a consumer would be willing to give up in order to obtain an additional unit of another is called the marginal rate of substitution (MRS)The maximum amount of one good a consumer would be willing to give up in order to obtain an additional unit of another., which is equal to the absolute value of the slope of the indifference curve between two points. Figure 7.12 "The Marginal Rate of Substitution" shows that as Ms. Bain devotes more and more time to horseback riding, the rate at which she is willing to give up days of skiing for additional days of horseback riding—her marginal rate of substitution—diminishes. We assume that each consumer seeks the highest indifference curve possible. The budget line gives the combinations of two goods that the consumer can purchase with a given budget. Utility maximization is therefore a matter of selecting a combination of two goods that satisfies two conditions: Figure 7.13 "The Utility-Maximizing Solution" combines Janet Bain’s budget line from Figure 7.9 "The Budget Line" with her indifference curves from Figure 7.11 "Indifference Curves". Our two conditions for utility maximization are satisfied at point X, where she skis 2 days per semester and spends 3 days horseback riding. Figure 7.13 The Utility-Maximizing Solution Combining Janet Bain’s budget line and indifference curves from Figure 7.9 "The Budget Line" and Figure 7.11 "Indifference Curves", we find a point that (1) satisfies the budget constraint and (2) is on the highest indifference curve possible. That occurs for Ms. Bain at point X. The highest indifference curve possible for a given budget line is tangent to the line; the indifference curve and budget line have the same slope at that point. The absolute value of the slope of the indifference curve shows the MRS between two goods. The absolute value of the slope of the budget line gives the price ratio between the two goods; it is the rate at which one good exchanges for another in the market. At the point of utility maximization, then, the rate at which the consumer is willing to exchange one good for another equals the rate at which the goods can be exchanged in the market. For any two goods X and Y, with good X on the horizontal axis and good Y on the vertical axis, How does the achievement of The Utility Maximizing Solution in Figure 7.13 "The Utility-Maximizing Solution" correspond to the marginal decision rule? That rule says that additional units of an activity should be pursued, if the marginal benefit of the activity exceeds the marginal cost. The observation of that rule would lead a consumer to the highest indifference curve possible for a given budget. Suppose Ms. Bain has chosen a combination of skiing and horseback riding at point S in Figure 7.14 "Applying the Marginal Decision Rule". She is now on indifference curve C. She is also on her budget line; she is spending all of the budget, $250, available for the purchase of the two goods. Figure 7.14 Applying the Marginal Decision Rule Suppose Ms. Bain is initially at point S. She is spending all of her budget, but she is not maximizing utility. Because her marginal rate of substitution exceeds the rate at which the market asks her to give up skiing for horseback riding, she can increase her satisfaction by moving to point D. Now she is on a higher indifference curve, E. She will continue exchanging skiing for horseback riding until she reaches point X, at which she is on curve A, the highest indifference curve possible. An exchange of two days of skiing for one day of horseback riding would leave her at point T, and she would be as well off as she is at point S. Her marginal rate of substitution between points S and T is 2; her indifference curve is steeper than the budget line at point S. The fact that her indifference curve is steeper than her budget line tells us that the rate at which she is willing to exchange the two goods differs from the rate the market asks. She would be willing to give up as many as 2 days of skiing to gain an extra day of horseback riding; the market demands that she give up only one. The marginal decision rule says that if an additional unit of an activity yields greater benefit than its cost, it should be pursued. If the benefit to Ms. Bain of one more day of horseback riding equals the benefit of 2 days of skiing, yet she can get it by giving up only 1 day of skiing, then the benefit of that extra day of horseback riding is clearly greater than the cost. Because the market asks that she give up less than she is willing to give up for an additional day of horseback riding, she will make the exchange. Beginning at point S, she will exchange a day of skiing for a day of horseback riding. That moves her along her budget line to point D. Recall that we can draw an indifference curve through any point; she is now on indifference curve E. It is above and to the right of indifference curve C, so Ms. Bain is clearly better off. And that should come as no surprise. When she was at point S, she was willing to give up 2 days of skiing to get an extra day of horseback riding. The market asked her to give up only one; she got her extra day of riding at a bargain! Her move along her budget line from point S to point D suggests a very important principle. If a consumer’s indifference curve intersects the budget line, then it will always be possible for the consumer to make exchanges along the budget line that move to a higher indifference curve. Ms. Bain’s new indifference curve at point D also intersects her budget line; she’s still willing to give up more skiing than the market asks for additional riding. She will make another exchange and move along her budget line to point X, at which she attains the highest indifference curve possible with her budget. Point X is on indifference curve A, which is tangent to the budget line. Having reached point X, Ms. Bain clearly would not give up still more days of skiing for additional days of riding. Beyond point X, her indifference curve is flatter than the budget line—her marginal rate of substitution is less than the absolute value of the slope of the budget line. That means that the rate at which she would be willing to exchange skiing for horseback riding is less than the market asks. She cannot make herself better off than she is at point X by further rearranging her consumption. Point X, where the rate at which she is willing to exchange one good for another equals the rate the market asks, gives her the maximum utility possible. Figure 7.14 "Applying the Marginal Decision Rule" showed Janet Bain’s utility-maximizing solution for skiing and horseback riding. She achieved it by selecting a point at which an indifference curve was tangent to her budget line. A change in the price of one of the goods, however, will shift her budget line. By observing what happens to the quantity of the good demanded, we can derive Ms. Bain’s demand curve. Panel (a) of Figure 7.15 "Utility Maximization and Demand" shows the original solution at point X, where Ms. Bain has $250 to spend and the price of a day of either skiing or horseback riding is $50. Now suppose the price of horseback riding falls by half, to $25. That changes the horizontal intercept of the budget line; if she spends all of her money on horseback riding, she can now ride 10 days per semester. Another way to think about the new budget line is to remember that its slope is equal to the negative of the price of the good on the horizontal axis divided by the price of the good on the vertical axis. When the price of horseback riding (the good on the horizontal axis) goes down, the budget line becomes flatter. Ms. Bain picks a new utility-maximizing solution at point Z. Figure 7.15 Utility Maximization and Demand By observing a consumer’s response to a change in price, we can derive the consumer’s demand curve for a good. Panel (a) shows that at a price for horseback riding of $50 per day, Janet Bain chooses to spend 3 days horseback riding per semester. Panel (b) shows that a reduction in the price to $25 increases her quantity demanded to 4 days per semester. Points X and Z, at which Ms. Bain maximizes utility at horseback riding prices of $50 and $25, respectively, become points X′ and Z′ on her demand curve, d, for horseback riding in Panel (b). The solution at Z involves an increase in the number of days Ms. Bain spends horseback riding. Notice that only the price of horseback riding has changed; all other features of the utility-maximizing solution remain the same. Ms. Bain’s budget and the price of skiing are unchanged; this is reflected in the fact that the vertical intercept of the budget line remains fixed. Ms. Bain’s preferences are unchanged; they are reflected by her indifference curves. Because all other factors in the solution are unchanged, we can determine two points on Ms. Bain’s demand curve for horseback riding from her indifference curve diagram. At a price of $50, she maximized utility at point X, spending 3 days horseback riding per semester. When the price falls to $25, she maximizes utility at point Z, riding 4 days per semester. Those points are plotted as points X′ and Z′ on her demand curve for horseback riding in Panel (b) of Figure 7.15 "Utility Maximization and Demand". Suppose the consumer in part (a) is indifferent among the combinations of hamburgers and pizzas shown. In the grid you used to draw the budget lines, draw an indifference curve passing through the combinations shown, and label the corresponding points A, B, and C. Label this curve I. © 2010 Jupiterimages Corporation Economist R. A. Radford spent time in prisoner of war (P.O.W.) camps in Italy and Germany during World War II. He put this unpleasant experience to good use by testing a number of economic theories there. Relevant to this chapter, he consistently observed utility-maximizing behavior. In the P.O.W. camps where he stayed, prisoners received rations, provided by their captors and the Red Cross, including tinned milk, tinned beef, jam, butter, biscuits, chocolate, tea, coffee, cigarettes, and other items. While all prisoners received approximately equal official rations (though some did manage to receive private care packages as well), their marginal rates of substitution between goods in the ration packages varied. To increase utility, prisoners began to engage in trade. Prices of goods tended to be quoted in terms of cigarettes. Some camps had better organized markets than others but, in general, even though prisoners of each nationality were housed separately, so long as they could wander from bungalow to bungalow, the “cigarette” prices of goods were equal across bungalows. Trade allowed the prisoners to maximize their utility. Consider coffee and tea. Panel (a) shows the indifference curves and budget line for typical British prisoners and Panel (b) shows the indifference curves and budget line for typical French prisoners. Suppose the price of an ounce of tea is 2 cigarettes and the price of an ounce of coffee is 1 cigarette. The slopes of the budget lines in each panel are identical; all prisoners faced the same prices. The price ratio is 1/2. Suppose the ration packages given to all prisoners contained the same amounts of both coffee and tea. But notice that for typical British prisoners, given indifference curves which reflect their general preference for tea, the MRS at the initial allocation (point A) is less than the price ratio. For French prisoners, the MRS is greater than the price ratio (point B). By trading, both British and French prisoners can move to higher indifference curves. For the British prisoners, the utility-maximizing solution is at point E, with more tea and little coffee. For the French prisoners the utility-maximizing solution is at point E′, with more coffee and less tea. In equilibrium, both British and French prisoners consumed tea and coffee so that their MRS’s equal 1/2, the price ratio in the market. Source: R. A. Radford, “The Economic Organisation of a P.O.W. Camp,” Economica 12 (November 1945): 189–201; and Jack Hirshleifer, Price Theory and Applications (Englewood Cliffs, NJ: Prentice Hall, 1976): 85–86. The tangency point at B shows the combinations of hamburgers and pizza that maximize the consumer’s utility, given the budget constraint. At the point of tangency, the marginal rate of substitution (MRS) between the two goods is equal to the ratio of prices of the two goods. This means that the rate at which the consumer is willing to exchange one good for another equals the rate at which the goods can be exchanged in the market. In this chapter we have examined the model of utility-maximizing behavior. Economists assume that consumers make choices consistent with the objective of achieving the maximum total utility possible for a given budget constraint. Utility is a conceptual measure of satisfaction; it is not actually measurable. The theory of utility maximization allows us to ask how a utility-maximizing consumer would respond to a particular event. By following the marginal decision rule, consumers will achieve the utility-maximizing condition: Expenditures equal consumers’ budgets, and ratios of marginal utility to price are equal for all pairs of goods and services. Thus, consumption is arranged so that the extra utility per dollar spent is equal for all goods and services. The marginal utility from a particular good or service eventually diminishes as consumers consume more of it during a period of time. Utility maximization underlies consumer demand. The amount by which the quantity demanded changes in response to a change in price consists of a substitution effect and an income effect. The substitution effect always changes quantity demanded in a manner consistent with the law of demand. The income effect of a price change reinforces the substitution effect in the case of normal goods, but it affects consumption in an opposite direction in the case of inferior goods. An alternative approach to utility maximization uses indifference curves. This approach does not rely on the concept of marginal utility, and it gives us a graphical representation of the utility-maximizing condition. The table shows the total utility Joseph derives from eating pizza in the evening while studying. |Pieces of pizza/evening||Total Utility| The table shows the total utility (TU) that Jeremy receives from consuming different amounts of two goods, X and Y, per month. Sid is a commuter-student at his college. During the day, he snacks on cartons of yogurt and the “house special” sandwiches at the Student Center cafeteria. A carton of yogurt costs $1.20; the Student Center often offers specials on the sandwiches, so their price varies a great deal. Sid has a budget of $36 per week for food at the Center. Five of Sid’s indifference curves are given by the schedule below; the points listed in the tables correspond to the points shown in the graph. Consider a consumer who each week purchases two goods, X and Y. The following table shows three different combinations of the two goods that lie on three of her indifference curves—A, B, and C. |Indifference Curve||Quantities of goods X and Y, respectively||Quantitities of goods X and Y, respectively||Quantities of goods X and Y, respectively| |A||1 unit of X and 4 of Y||2 units of X and 2 of Y||3 units of X and 1 of Y| |B||1 unit of X and 7 of Y||3 units of X and 2 of Y||5 units of X and 1 of Y| |C||2 units of X and 5 of Y||4 units of X and 3 of Y||7 units of X and 2 of Y|

You are watching: What is the momentum p of one of these electrons? Measuring Photon Momentum The quantum of EM radiation we speak to a photon has actually properties analogous come those of corpuscle we can see, such as grains the sand. A photon interacts as a unit in collisions or as soon as absorbed, quite than as comprehensive wave. Enormous quanta, favor electrons, additionally act like macroscopic particles—something us expect, since they room the smallest systems of matter. Particles lug momentum as well as energy. Despite photons having actually no mass, there has actually long been evidence that EM radiation tote momentum. (Maxwell and also others who studied EM waves predicted the they would lug momentum.) that is now a well-established truth that photons do have momentum. In fact, photon momentum is said by the photoelectric effect, whereby photons hit electrons the end of a substance. Number 1 shows macroscopic proof of photon momentum. Figure 1. The tails of the Hale-Bopp comet suggest away native the Sun, proof that light has actually momentum. Dust create from the human body of the comet develops this tail. Particles of dust are pushed away from the sunlight by light mirroring from them. The blue ionized gas tail is also produced through photons interacting with atoms in the comet material. (credit: Geoff Chester, U.S. Navy, via Wikimedia Commons) Figure 1 shows a comet v two prominent tails. What most civilization do not know around the tails is that they always suggest away native the Sun rather than trailing behind the comet (like the tail that Bo Peep’s sheep). Comet tails space composed the gases and dust evaporated native the human body of the comet and also ionized gas. The dust corpuscle recoil away from the Sun as soon as photons scatter native them. Evidently, photons bring momentum in the direction the their movement (away native the Sun), and also some the this momentum is transferred to dust particles in collisions. Gas atoms and also molecules in the blue tail room most affected by various other particles of radiation, such together protons and also electrons emanating from the Sun, quite than by the momentum of photons. Making Connections: conservation of Momentum Not just is inert conserved in all realms of physics, however all varieties of corpuscle are found to have actually momentum. We suppose particles with mass to have actually momentum, however now we view that massless particles including photons also carry momentum. Figure 2. The Compton effect is the name offered to the scattering that a photon by an electron. Energy and momentum space conserved, bring about a palliation of both for the scattered photon. Studying this effect, Compton proved that photons have momentum. Momentum is conserved in quantum mechanics just as it is in relativity and classical physics. Few of the earliest straight experimental evidence of this came from scattering that x-ray photons by electron in substances, named Compton scattering after ~ the American physicist Arthur H. Compton (1892–1962). About 1923, Compton observed that x rays scattered from materials had a decreased energy and correctly analyzed this together being because of the scattering that photons from electrons. This phenomenon could be tackled as a collision between two particles—a photon and an electron at rest in the material. Energy and momentum room conserved in the collision. (See figure 2) He won a Nobel prize in 1929 because that the exploration of this scattering, now dubbed the Compton effect, since it aided prove the photon momentum is offered by We have the right to see that photon momentum is small, due to the fact that Example 1. Electron and Photon momentum ComparedCalculate the momentum of a clearly shows photon that has actually a wavelength of 500 nm.Find the velocity of an electron having actually the very same momentum.What is the energy of the electron, and also how does the compare with the power of the photon?Strategy Finding the photon inert is a straightforward application of its definition: Photon momentum is given by the equation: Entering the given photon wavelength yields Solution for Part 2 Since this momentum is without doubt small, we will use the timeless expression p = mv to find the velocity of one electron v this momentum. Solving for v and also using the well-known value for the massive of one electron gives Solution for Part 3 The electron has actually kinetic energy, which is classically provided by Converting this come eV by multiply by The photon power E is which is about five order of magnitude greater.Discussion Photon momentum is undoubtedly small. Even if we have huge numbers of them, the complete momentum they lug is small. An electron through the exact same momentum has actually a 1460 m/s velocity, i m sorry is clearly nonrelativistic. A an ext massive bit with the very same momentum would have actually an also smaller velocity. This is borne the end by the fact that that takes far less energy to give an electron the very same momentum as a photon. Yet on a quantum-mechanical scale, especially for high-energy photons interacting with little masses, photon momentum is significant. Also on a huge scale, photon momentum can have an impact if there are sufficient of them and if over there is naught to prevent the slow-moving recoil the matter. Comet tails are one example, but there are additionally proposals come build space sails the use large low-mass mirrors (made that aluminized Mylar) come reflect sunlight. In the vacuum that space, the mirrors would slowly recoil and also could actually take spacecraft from place to place in the solar system. (See number 3.) Figure 3. (a) space sails have been proposed that usage the momentum of sunlight mirroring from large low-mass sails come propel spacecraft around the solar system. A Russian test design of this (the Cosmos 1) was launched in 2005, however did not make it into orbit due to a rocket failure. (b) A U.S. Variation of this, labeled LightSail-1, is booked for psychological launches in the very first part the this decade. The will have actually a 40-m2 sail. (credit: Kim Newton/NASA) Relativistic Photon Momentum There is a relationship in between photon momentum p and also photon energy E the is continual with the relation offered previously for the relativistic complete energy the a particle as E2 = (pc)2 + (mc)2. We know m is zero for a photon, yet p is not, so that E2 = (pc)2 + (mc)2 becomes E = pc, or To examine the validity that this relation, note that as determined experimentally and also discussed above. Thus, p=E/c is tantamount to Compton’s result p=h/λ. For a additional verification the the relationship between photon energy and also momentum, see example 2. Almost every detection systems talked around thus far—eyes, photographic plates, photomultiplier tube in microscopes, and also CCD cameras—rely top top particle-like nature of photons interacting with a sensitive area. A readjust is caused and either the readjust is cascaded or zillions of points are tape-recorded to type an image we detect. These detectors are used in biomedical imaging systems, and there is continuous research into boosting the effectiveness of receiving photons, an especially by cooling detection systems and reducing heat effects. Example 2. Photon Energy and also Momentum We will certainly take the power E discovered in instance 1, division it through the speed of light, and see if the very same momentum is acquired as before.Solution Given that the energy of the photon is 2.48 eV and converting this come joules, we get This value for momentum is the exact same as found prior to (note that unrounded values are supplied in every calculations to stop even tiny rounding errors), an supposed verification the the connection Note the the forms of the constants h = 4.14 × 10–15 eV ⋅ s and also hc = 1240 eV ⋅ nm might be particularly useful because that this section’s Problems and Exercises. Section SummaryPhotons have actually momentum, provided by Conceptual QuestionsWhich formula might be offered for the inert of every particles, through or there is no mass?Is there any type of measurable difference between the momentum of a photon and also the momentum of matter?Why don’t us feel the momentum of sunlight once we are on the beach? Problems & Exercises(a) uncover the inert of a 4.00-cm-wavelength microwave photon. (b) talk about why you mean the answer come (a) to be an extremely small.(a) What is the inert of a 0.0100-nm-wavelength photon that can detect details of an atom? (b) What is its energy in MeV?(a) What is the wavelength the a photon that has actually a inert of 5.00 × 10−29 kg · m/s? (b) discover its power in eV.(a) A γ-ray photon has a inert of 8.00 × 10−21 kg · m/s. What is that wavelength? (b) calculate its energy in MeV.(a) calculation the momentum of a photon having a wavelength of 2.50 μm. (b) discover the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and also how does that compare with that that the photon?Repeat the previous trouble for a 10.0-nm-wavelength photon.(a) calculate the wavelength that a photon that has actually the same momentum together a proton relocating at 1.00% that the rate of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?(a) discover the momentum of a 100-keV x-ray photon. (b) uncover the tantamount velocity of a neutron through the very same momentum. (c) What is the neutron’s kinetic power in keV?Take the ratio of relativistic remainder energy, E = γmc2, come relativistic momentum, p = γmu, and show the in the limit the mass philosophies zero, you discover photon momentum: the lot of inert a photon has, calculation by Compton effect: the phenomenon by which x light ray scattered from products have lessened energy Selected solutions to Problems & Exercises 1. (a) 1.66 × 10−32 kg ⋅ m/s; (b) The wavelength the microwave photons is large, for this reason the momentum they lug is an extremely small. 3. (a) 13.3 μm; (b) 9.38 × 10−2 eV 5. (a) 2.65 × 10−28 kg · m/s; (b) 291 m/s; (c) electron 3.86 × 10−26 J, photon 7.96 × 10−20 J, ratio 2.06 × 106 7. (a) 1.32 × 10−13 m; (b) 9.39 MeV; (c) 4.70 × 10−2 MeV 9. E = γmc2 and P = γmu, so As the mass of particle approaches zero, that velocity u will strategy which is continuous with the equation because that photon energy. See more: 26 Days From Today ? What Date Is 26 Days From Today 11. (a) 3.00 × 106 W; (b) Headlights are way too bright; (c) force is as well large.

Who is my neighbor? 437 Pages2.21 MB9641 DownloadsFormat: EPUB Foundations Book II 310 Pages0.70 MB9976 DownloadsFormat: PDF/EPUB 711 Pages3.50 MB5519 DownloadsFormat: EPUB Issues and innovations in foreign language education 187 Pages3.77 MB6594 DownloadsFormat: EPUB Morality as self-respect, the categorical imperative 596 Pages0.64 MB2034 DownloadsFormat: EPUB A finite geometry is any geometric system that has only a finite number of familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite. Here, we obtain a finite geometry by restricting the system to one of the planes. Axioms for Fano's Geometry. Undefined Finite geometries without the axiom of parallels book. Description Finite geometries without the axiom of parallels FB2 point, line, and incident. Axiom Finite geometries without the axiom of parallels book. There exists at least one line. Axiom 2. Every line has exactly three points incident to it. Axiom 3. The purpose of this book is to present an introduction to developments which had taken place in finite group theory related to finite geometries. This book is practically self-contained and readers are assumed to have only an elementary knowledge of linear : Paperback. Since finite geometries are analogs of continuous geometries, one may be interested in the the finite analog of curves. A k-arc in a projective plane is a set of k points no three of which are incident with the same line. By axiom 3, every plane contains a 4-arc. Fano initially considered a finite three-dimensional geometry consisting of 15 points, 35 lines, and 15 planes. Here, we obtain a finite geometry by restricting the system to one of the planes. Axioms for Fano's Geometry Undefined Terms. point, line, and incident. Axiom 1. File Size: KB. Axiom of Parallels Given a line and a point outside it there is exactly one line through the given point which lies in the plane of the given line and point so that the two lines do not meet. Note that, while asserting that there is a line through the given point that doesn't meet the given line, it also says there is only one such line. All of this is considered to live in a single plane, in violation of axiom The other axioms talking about planes are all satisfied, although they add little of interest to the picture. If you want an affine three-space instead of only an affine plane, you need more points. A common way. Details Finite geometries without the axiom of parallels EPUB An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'. The term has subtle differences in definition when used in the context of different fields of study. As defined in. The axiom of the parallels in Euclidean geometry asserts that to a given straight line through a given point there exists exactly one parallel; apart from the device used by Bolyai and Lobatschewsky to deny this axiom by assuming the existence of several parallels, there was a third possibility, that of denying the existence of any : An Introduction to Non-Euclidean Geometry covers some introductory topics related to non-Euclidian geometry, including hyperbolic and elliptic geometries. This book is organized into three parts encompassing eight chapters. The first part provides mathematical proofs of Euclid’s fifth postulate concerning the extent of a straight line and the. Foundations of geometry is the study of geometries as axiomatic are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. Graduation date: Among the geometries with n points on every line (with n an integer greater than one), those in which there are no parallels and those in which the axiom of parallels holds. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Wallis' axiom for parallel lines. Ask Question Asked 1 year, 10 months ago. Midline theorem without the axiom of parallels. Geometrical problem to show equal areas. It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomorphic to PG(5,2). Since the GQ(2,4) features only two kinds of geometric hyperplanes, namely point’s perp. Non-Euclidean Geometry is not not Euclidean Geometry. The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of Euclidean Geometry (in a complete system such as Hilbert's). In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. A line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are also. In geometry the parallel postulate is one of the axioms of Euclidean mes it is also called Euclid's fifth postulate, because it is the fifth postulate in Euclid's Elements. The postulate says that: if you cut a line segment with two lines, and the two interior angles the lines form add up to less than two right angles, then the two lines will eventually meet if you extend them. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. You can write a book review and share your experiences. Other readers. A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 continues the synthetic approach as it introduces Euclid's geometry and ideas of non-Euclidean geometry. Affine geometry Last updated Novem In affine geometry, one uses Playfair's axiom to find the line through C1 and parallel to B1B2, and to find the line through B2 and parallel to B1C1: their intersection C2 is the result of the indicated translation. Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary version we also drop the (Cantor's) axiom of continuity, Greenberg calls such geometries Archimedean H-planes in his survey paper. Parallel Lines in Euclidean Geometry an axiom to ensure the existence of parallels, since this can be proved from the other There are other geometries. On the other hand, hyperbolic geometry is an important geometry. We’ll talk about that later. In the previous section, we showed the following is a theorem in neutral geometry. File Size: KB. Eventually these alternate geometries were scholarly acknowledged as geometries, which could stand alone to Euclidean geometry. The two non-Euclidean geometries were known as hyperbolic and elliptic. Hyperbolic geometry was explained by taking the acute angles for C and D on the Saccheri Quadrilateral while elliptic assumed them to be obtuse. Starting right at the beginning in Book 1, Proposition 1, the construction of an equilateral triangle, Euclid assumes without proof that the two circles he created have a point of intersection. Download Finite geometries without the axiom of parallels PDF From the diagram it may seem clear that the circles intersect, but this is not a substitute for proof; there are many examples where what seems obvious. In non-Euclidean geometries, the fifth postulate is replaced with one of its negations: through a point not on a line, either there is none (B) or more than 1 (C) line parallel to the given one. Carl Friedrich Gauss was apparently the first to arrive at the conclusion that no contradiction may be obtained this way. The focus of this chapter is on pointless geometries. The concept of point is assumed as the main primitive term for an axiomatic foundation of geometry. In pointless geometry, regions are considered as individuals, i.e. in the vocabulary of logic, first order objects, while points are represented by classes (or sequences), that is, second Cited by: the relation between Euclidean and non-Euclidean geometries. The Bolyai construc-tion of limiting parallels is shortly discussed from the reconstructed Euclidean point of view. 1 The Standard Interpretation of the Fifth Postulate From Proclus up to our days a. Euclid’s approach to similarity introduces the Archimedean axiom, and the concept of rational approximations to irrational ratios. I.e. two pairs of line segments, both of whose ratios are rational, can be determined by a finite subdivision. § 4 that every finite projective ¿-dimensional geometry satisfying the definition of §1 is a PG(k,p") it ¿>2. § 3. The modulus 2. The method used in § 2 to obtain the PG(k, s) from the G F [ s ] may be described as analytic geometry in a finite field. It may be applied to any field. Full text of "Geometrical researches on the theory of parallels" that which is good," does not mean dem- onstrate everything. "Prom nothing assumed, nothing can be proved. "Geometry without axioms," was a went through several editions, and still has historical value. " In fact this first of the Non-Euclidean geometries. Or do we? Is it an axiom of Euclid that such lines are infinite? Apparently not. So what ultimately followed was the construction by Bernhard Riemann () of a different kind of non-Euclidean geometry, one where there are no parallels and all lines are finite. But what does this mean that all lines are finite?1. Introduction. InDavid Hilbert published the Grundlagen der Geometrie, a book that opened up research in the foundations of fact, the Grundlagen took the axiomatic method both as a culmination of geometry and as the beginning of a new phase of research. In that new phase, the links between the postulates were not seen as the cold expression of their logical relations or Cited by: 6.The first cracks in this inspiring picture appeared in the latter half of the 19th century when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries. Riemann's geometry was modeled on a sphere, Lobachevsky's on a hyperboloid of rotation. 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We propose an all-optical half adder based on two different cross structures in two-dimensional photonic crystals. One cross structure contains nonlinear materials and functions as an “AND” logic gate. The other one only contains linear materials and acts as an “XOR” logic gate. The system is demonstrated numerically by the FDTD method to work as expected. The optimal operating speed without considering the response time of the nonlinear material, the least ON to OFF logic-level contrast ratio, and the minimum power for this half adder obtained were 0.91Tbps, 16dB and 436mW, respectively. The proposed structure has the potential to be used for constructing all-optical integrated digital computing circuits. ©2008 Optical Society of America Photonic Crystals (PhCs), a new class of artificial materials first predicted by Yablonovitch and John , is a promising candidate as a platform to build future photonic integrated devices with dimensions in the order of the operating wavelength , due to their small sizes and unique capability to modify photon interaction with host materials. In the last three decades of the 20th century, optical bistability and optical logic devices were studied extensively . Yet it was blocked by difficulties in optical integrations on a small chip. The creations of PhCs bring new hopes to all-optical logic circuits integration on a small chip. First, optical bistability was found in PhC microcavity with nonlinear materials . Then, optical switches using optical bistability in PhCs and basic optical logic gates using PhC optical switches were reported [6–10]. Besides, basic optical logic gates using PhC splitters were reported [11–13]. Also, we have seen some kind of optical integrated circuits on PhCs [14–15]. It is, however, only the very beginning in developing optical integrated logic circuits and systems on the platforms of PhCs. For example, until now, there is even no answer to what are the most typical and practical structures for the basic all-optical logic gate “ AND”, “OR”, and “NOT” built on the platforms of PhCs. It is known that full adders are the basic parts of a central processing unit (CPU). Since full adders are generally built with half adders, it is important to investigate all-optical half adders (AOHA) for the realization of all-optical CPU. In this paper we show an AOHA on the platform of 2-D PhCs. Its operating speed is calculated to be greater than 0.9Tbps when neglecting the response time of the nonlinear material. This paper is organized as follows. In Sec. 2, we first give the models of the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs, then a model of an all-optical half adder, and last a description of simulation method and parameters. In Sec. 3, the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs are first investigated and optimized, and then the all-optical half adder on the platform of two-dimensional PhCs is investigated. Finally in Sec. 4, we draw a brief conclusion. 2.1 Models of the basic all-optical logic elements “AND gate” and “XOR gate” on the platform of two-dimensional PhCs Figure 1 shows the models of all-optical “AND” (a) and “XOR” (b) logic gates on the platform of 2-D PhCs. In Fig. 1, the hollow circles indicate linear dielectric rods, and the black solid dots indicate Kerr-type nonlinear rods. In Fig. 1(a), a nonlinear diffraction rod is put right in the intersection center of the two waveguides. The radius of the three nonlinear rods is the same as that of the linear rods in the 2-D PhC. In Fig. 1(b), the center of the diffraction rod is at (1.21a, 1.21a) with (0, 0) being the center of the bottom-left rod at the waveguide-intersection region. The radius of the diffraction rod is 0.3a. These parameters are determined through optimization. The so called optimization in this paper means a process of minimizing the operating power, the response time, and the size of the devices, and maximizing the operational bandwidth and the ON (logic 1) to OFF (logic 0) logic-level contrast ratio in the systems by scanning operating parameters. In the following simulations, each of the two structures in Fig. 1 is a 17a×17a 2-D square lattice PhC with a lattice constant a=520.8nm. The dielectric rods consisting of the PhCs are silicon (Si) cylinders, of which the relative permittivity ε and the radius are 11.56 and 0.18 a, respectively. The background material is air. The third-order nonlinear susceptibility of the Kerr-type nonlinear rods in Fig. 1(a) is chosen to be χ (3)=1.0.10-4 µm2/V2, which corresponds approximately to AlGaAs with a Kerr coefficient of n 2=1.5×10-17m2/W [7,8,16]. For a uniform PhC without defects under the above operating parameters, we find out that there exists a large bandgap by a standard plane-wave expanding method: the light with wavelengths between 2.250a and 3.304a cannot pass through the uniform PhC and thus is completely reflected . We also find that, a photonic-crystal waveguide formed by removing a row of rods in the uniform PhC can guide light with wavelength λ 0=2.9762a=1.550µm, which is the operating wavelength we desired for applications in optical communications. The basic idea for constructing the “AND” gate in Fig. 1(a) is as follows. The two black rods bring a uniformity break to the left waveguide in Fig. 1(a), so that reflection will be produced when there is a wave propagating in it. Nonlinear material is introduced to promote the ON to OFF logical-level contrast ratio. Finally, with proper choice of the distance between the uniformity-breaking point and the diffraction rod at the crosspoint of the two waveguides, the structure may function as an “AND” gate through the interference of the reflected waves from the uniformity-breaking part of the waveguide and the waves diffracted by the diffraction rod at the waveguide crosspoint. The principle of the “XOR” gate in Fig. 1(b) is based on the wave-splitting property of the diffraction rod deviated from the waveguide-cross center. This is clearly seen from the field distributions obtained through simulations in section 3 in the following. 2.2 A Model of an all-optical half adder on the platform of two-dimensional PhCs It is known that a half adder adds up two one-bit binary numbers (A and B). The outputs of it are the sum S=A⊕B and the carry C=A·B. Then referring to the all-optical “XOR” and “AND” gates in Fig. 1, we may construct an AOHA, as shown in Fig. 2. The up-right rod and bottom-left rod of the four rods at the waveguide crosspoints in Fig. 2 play the role of splitters . It is obvious that this structure is symmetrical for data A and data B, since the two signals entering into both A and B have to go over the same distance to reach the output port S and C. In simulations of the AOHA, all parameters, including the nonlinear third-order susceptibility, the dielectric constant, the wavelength of the excitation source, the radius, and the position of the nonlinear rods and the diffraction rods are the same as that in the models shown in Fig. 1. 2.3 Simulation method In order to demonstrate the functions and to investigate the properties of the models presented above, we used the FDTD method for simulations. Since the FDTD method is well known and widely used in computational electromagnetism, we omit the description of the FDTD method in this paper. Here we just write out the related expressions concerning nonlinear materials in the models. The Kerr effect of the nonlinear material is modeled by introducing an intensity-dependent increment of the refractive index: where Δn is the increment of the refractive index due to the nonlinearity of the material induced by the electric fields of the waves in the models, I is the local intensity of light and is proportional to |E|2, sat I is the saturation intensity of the nonlinearity, and 2 n is proportional to the third-order nonlinear susceptibility χ (3). Equation (1) can be rewritten in the following form: where n̄2=βn 2 and are the dimensionless nonlinear refractive index and the dimensionless field amplitude, respectively, and α=β/Isat is a dimensionless coefficient. The constant β is determined from the initial condition to be where Pin is the total power per unit length launched into the crystal and the integral is taken along the phase front of the input field. The input field u(x,0) is normalized such that the integral length in Eq. (3) is equal to 1µm. In our computations, for simplicity, we assume that the fields are much less than that required for saturation, i.e., The increment in the refractive index is thus simplified from Eq. (2) to be Denoting the linear refractive index by nL, the overall dielectric constant of the Kerr materials can be written out to be In order to include optical nonlinearity into the FDTD algorithm, a nonlinear polarization term is added to the linear polarization term in Maxwell’s equations . Thus, the electric field is related to the displacement vector by are the linear and nonlinear polarization term, respectively. Here χ (1)(t) and χ (3)(t) are respectively the first and third-order susceptibility, and r=(x,y) is the position vector . We point out that, in simulations the input ports of the structures in Fig. 1 and 2 are excited by continuous wave (CW) sources. The electric-field polarization of the wave is chosen to be parallel to the y-axis, which is the axis of the dielectric rods. The wave propagates in the (x,z) plane. We also point out that, in our simulations the mesh sizes in the x- and z- directions are set to be a/16 and the time step is set to be 8.33.10-2 fs, which meets the requirements of Courant stable condition. And the calculated area is surrounded by a perfect matched layer (PML) boundary. Another point to be mentioned is that we use a 2-D rather than a 3-D geometry in our numerical simulations. According to Ref. , 2-D simulations can be used to estimate the power needed to operate a true 3-D device, reducing the computation time considerably, while still capturing the essential physics of the problem. 3. Numerical results and discussion Since one needs “AND” and “XOR” gates to build the half adder and the performance of the half adder depends on the two gates used, we first study the two basic gates separately, and then investigate the half adder. 3.1 The “AND” logic gate To demonstrate the function of the structure in Fig. 1(a), a numerical experiment is performed. We first apply a CW signal with a power P 0(P 0=2.3×10-11 a/n 2=806 mW) at port A or B separately, and measure the output powers at steady state. Then we apply CW signals with the same power simultaneously both at A and B, and measure the output power at steady state. We found that the output power is 0.014P 0 (logic 0) for separate excitation at A, 0.005P 0 (logic 0) for separate excitation at B, and 1.46P 0 (logic 1) for simultaneous excitations at both input ports. This demonstrates that the structure in Fig. 1(a) does operate as an optical “AND” logic gate. The smallest ON to OFF logic-level contrast ratio for the “AND” logic gate is calculated to be 20.2 dB. For getting better insight into the physics of the structure in Fig. 1(a), the field distributions at steady state operation are illustrated in Fig. 3, from which we see clearly that the structure functions as an “AND” gate. 3.1.1 The optimal logic 0 to 1 turn-over threshold power In this section, we repeat the experiment shown in Fig. 3(c) by varying the radius rc of the nonlinear rod at the waveguide crosspoint. The present numerical experiment shows a very interesting phenomenon that the logic 0 to 1 turn-over threshold power varies with rc. Denoting the input power at a single port by Psin and the steady state output power by Pout, respectively, we obtain the result shown in Fig. 4. When rc is smaller than 0.34a, the minimum power to observe the logic 0 to 1 turn-over is larger than 218mW and the transmittance is low. On the other hand, when rc is larger than 0.34a, the resulting transmission decreases sharply. Thus, we get an optimal rc=0.34a at which the power to realize the logic 0 to 1 turn-over is 218mW with the transmission for separate input power being approximately 95%. We point out that the relative refractive index change of the nonlinear rods at the waveguide sidewalls to realize the OFF to ON transition is Δn/n 0=0.064 (at the rod boundary) ~0.15 (at the rod center), where n 0=2.168 is the refractive index at weak field intensity. It seems that the relative index change is quite large and unrealistic for conventional Kerr materials. In the simulation, however, the third-order susceptibility of the nonlinear rods is taken to be approximately that of AlGaAs, so the simulation result is practical. The large change of refraction index is due to the high field intensity in the resonator and the relatively large input power (218mW). Yet, the relative index change of the nonlinear rod at the waveguide crosspoint is only Δn/n 0=0.06 (at the rod boundary) ~0.003 (at the rod center). We hope: with the development of new nonlinear materials with high third-order susceptibility, the input power will be greatly reduced. The above optimal power level is many orders of magnitude lower than that required by other small all-optical ultra-fast switches. This is explained as follows. First, the transverse area of the modes in the PhC is only 0.67λ 2; consequently, much less power is needed than in other systems with larger transverse mode area. Second, the nonlinear rods in the system introduced cavities which enhance the nonlinearity and reduce the threshold power similar to that described in Ref. . 3.1.2 The Contrast Ratio and the Bandwidth The ON to OFF logic-level contrast ratio is shown in Fig. 5 for different input powers. From Fig. 5 we see that for a fixed input power, the operating bandwidth, which is defined as the region in which the contrast ratio is no less than a given value, is limited. We also find that the operating bandwidth is different with the input power. For an input power of 274mW, the bandwidth for the contrast ratio larger than 15dB is 6nm. Furthermore, from Fig. 5 we can see that the higher the input power is, the longer the operating wavelength gets. This can be explained as follows. Each waveguide-uniformity-breaking region can be regarded as a resonator filled with nonlinear capacitive rods. Only waves with their wavelengths near the resonance wavelength of the resonator can tunnel through the resonator and reach the output port. Noting that the resonance wavelength is proportional to the square root of capacitance in a resonant circuit, the capacitance of a nonlinear rod is proportional to its refractive index, and the refractive index increases with the power, as can be seen from Eqs. (5) and (6), we conclude that the resonant wavelength will increase with the input power, in agreement with that shown in Fig. 5. So, the operating wavelength can be tuned by the input power. 3.2 The “XOR” logic gate Now we perform simulations with the structure shown in Fig. 1(b) in the same way as that in 3.1. When we apply a single CW signal with power P 0 at port A (B), the output power from port Y at steady state is found to be 0.49 P 0 (0.48 0 P), corresponding to logic 1, which is approximately equal to the output power from the idle port. When we excite the two ports A and B simultaneously each with power P 0, the steady state output power at port Y is calculated to be 0.0004 P 0, corresponding to logic 0. This demonstrates that the structure shown in Fig. 1(b) works as an optical “XOR” logic gate. The smallest ON to OFF logic-level contrast ratio for the “XOR” logic gate is found to be 30.8dB. For getting better insight into the physics of the structure in Fig. 1(b), the field distributions at steady state operation are illustrated in Fig. 6, from which we see clearly that the structure functions as an “XOR” gate. As mentioned in Ref. , from Fig. 6 we see that there exists a phase difference of π between the wave at Y port in Fig. 6(a) and that in Fig. 6(b). This explains why the output logic at Y port becomes 0 when port A and B are excited separately and simultaneously by logic-1 signals. To show the effect of the XOR, we have calculated the spectrum of the ON to OFF logic-level contrast ratio, as shown in Fig. 7, from which we may see that the bandwidth for the contrast ratio over 25dB is larger than 32nm. This indicates that XOR logic gate has a large bandwidth in the fiber-optic-communication wavelength band. 3.3 Half Adder Now we move to study the structure indicated in Fig. 2. Necessary parameters for simulations are given in Sec. 2. Applying different signals at port A and B, and calculating the output power from port S and C, we obtain Table 1. For convenience of logic function verification, we transformed Table 1 into Table 2, which demonstrates clearly that the structure, indicated in Fig. 2, functions as a half adder, i.e., S=A⊕B, C=A·B. In the same way as that in 3.1.2 we may get the ON to OFF logic-level contrast ratio of the half adder to be 16dB for an operating bandwidth of 3nm. We have also calculated the operating speed, an important parameter for the half adder, through the time-domain response of the structure. The operating speed is defined as the inverse of the response time, which is the sum of the rising time tr and the recovery time or falling time tf, as indicated in Fig. 8. We find that the structure can have an optimal operating speed of 0.91Tbit/s. This speed, influenced by the quality factor of the resonator introduced by three nonlinear rods, is obtained without considering the response time for action between the wave and the Kerr material as discussed in Ref. . To promote the operating speed, the quality factor of the resonator introduced by the nonlinear rods should be as small as possible . Also, ultra-fast wave-response-time materials should be used. The operating speed may be affected by the operation power, as shown in Fig. 9, from which we can see that greater input power leads to higher operating speed. This may be explained as follows. As mentioned in section 3.2, the refractive index of nonlinear rods increases with the input power. Since increasing refractive index of the nonlinear rods means increasing uniformity break in the waveguides and also increasing reflection of waves by the nonlinear rods, thus the quality factor of the resonators introduced by the nonlinear rods decreases, so that the operating speed of the structure increases. On the basis of the optical “XOR” and “AND” logic gates built with 2-D PhCs, we constructed an all-optical half adder. Numerical simulations demonstrated successfully by the FDTD method that the structure presented does function as an all-optical half adder. The ON to OFF logic-level contrast ratio for this half adder could reach at least 16dB and the optimal operating speed is found to be as high as 0.91Tbits/s when omitting the material-wave-response time of Kerr effect. This structure is useful in designing all-optical signal processing circuits and optical computer systems. We thank the supports from the Chinese Natural Science Foundation (Grant No. 60877034), the Guangdong Natural Science Foundation (Key Project No. 8251806001000004), and the Shenzhen Science Bureau. References and links 2. S. John, “Strong localization of photons in certain disoordered dielectric superlattices,” Phys. Rev. Lett. 58, 2846–2489 (1987). [CrossRef] 3. J. D. Joannopoulos, “Photonics crystals: putting a new twist on light,” Nature (London) 386,143–149 (1997). [CrossRef] 4. H. M. Gibbs, “Optical Bistability,” in Controlling Light with Light, (Academic Press, Orlando, 1985). 5. E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystal doped by a microcavity,” Phys. Rev. B 62, 7683–7686 (2000). [CrossRef] 6. K. M. F. Yanik, S. Fan, and M. Soijacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003). [CrossRef] 7. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. 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Analytical description of quasivacuum oscillations of solar neutrinos We propose a simple prescription to calculate the solar neutrino survival probability in the quasivacuum oscillation (QVO) regime. Such prescription is obtained by matching perturbative and exact analytical results, which effectively take into account the density distribution in the Sun as provided by the standard solar model. The resulting analytical recipe for the calculation of is shown to reach its highest accuracy ( in the whole QVO range) when the familiar prescription of choosing the solar density scale parameter at the Mikheyev-Smirnov-Wolfenstein (MSW) resonance point is replaced by a new one, namely, when is chosen at the point of “maximal violation of adiabaticity” (MVA) along the neutrino trajectory in the Sun. The MVA prescription admits a smooth transition from the QVO regime to the MSW transition one. We discuss in detail the phase acquired by neutrinos in the Sun, and show that it might be of relevance for the studies of relatively short timescale variations of the fluxes of the solar lines in the future real-time solar neutrino experiments. Finally, we elucidate the role of matter effects in the convective zone of the Sun. PACS number(s): 26.65.+t, 14.60.Pq The solar neutrino problem , emerging as a deficit of the observed solar neutrino rates [2, 3, 4, 5, 6, 7, 8] with respect to standard solar model (SSM) predictions [1, 9, 10], can be explained through neutrino flavor oscillations , possibly affected by the presence of matter [12, 13, 14] (see [15, 16] for reviews of oscillation solutions and [17, 18] for recent analyses). The analysis of neutrino oscillations requires a detailed study of the evolution of the flavor states , being any linear combination of and , along the neutrino trajectory. In the simplest case, the active states are assumed to be superpositions, through a mixing angle , of two vacuum mass eigenstates , characterized by a mass squared gap (see, e.g., ). The corresponding neutrino evolution equation in the flavor basis, involves then the following vacuum and matter terms in the Hamiltonian , with the usual definitions for the neutrino wavenumber in vacuum, and for the neutrino potential in matter, being the electron density at the point of neutrino trajectory in the Sun, located at radial distance from the Sun center.111The solar neutrinos reaching the Earth move practically radially in the Sun. The effect of off-center trajectories is negligible for our results, see Appendix A. The distribution is usually taken from SSM calculations [9, 10]. The Hamiltonian is diagonalized in the matter eigenstate basis through the mixing angle in matter defined by being the neutrino wavenumber in matter, In this work, we focus on approximate solutions of Eq. (1) for neutrino flavor transitions at relatively small values of [20, 21, 22, 23, 24], characterizing the so-called quasivacuum oscillation (QVO) [25, 26, 27, 28, 17, 18] regime222The lower part of the QVO range ( eV/MeV) corresponds to the vacuum oscillation (VO) regime . In this work, we do not explicitly distinguish the VO range, but simply treat it as part of the QVO range. Let us note that VO solutions of the solar problem are disfavored by the current data [7, 17, 18]., In order to define our goals more precisely, we recall that, both in the QVO regime and in the “Mikheyev-Smirnov-Wolfenstein” (MSW) [12, 13] regime, which takes place at eV/MeV, the solar survival probability at the Earth surface can be written as where is the average probability, The main ingredients of the above equations are: The mixing angle in matter at the production point , with in the QVO regime; The level crossing (or jump) probability of the transition from at to at the surface of the Sun (); and the phase accumulated on the path from to the Earth surface, A.U.,333In this work we consider only solar matter effects. Earth matter effects are not relevant in the QVO regime, while in the MSW regime in which effectively , they can always be implemented through a modification of the expression for , see the discussion in . where the phases , , have a simple physical interpretation : , being the probability amplitude of the transition between the initial matter-eigenstate and the final vacuum-mass eigenstate . The phase can also be decomposed (see, e.g., Appendix A) as a sum of a “solar” phase acquired on the path in the Sun () and a “vacuum” phase , acquired in vacuum (), As a consequence of Eqs. (10)–(15), solving the neutrino evolution equation (1) basically reduces to calculating and . This task can be accomplished through numerical integration of Eq. (1) for , leading to “exact” solutions.444In this work, high precision numerical results (which we will call “exact”) for SSM density [9, 10] are obtained through the computer codes developed in . However, suitable approximations to the exact results are also useful, both to speed up the calculations and to clarify the inherent physics. The starting point of such approximations is usually the analytical solution of Eq. (1) for the case of exponential density in the Sun [20, 21] (see also [31, 32, 33, 34]). Deviations of the SSM density from the exponential profile can then be incorporated by an appropriate prescription for the choice of the scale height parameter characterizing the realistic change of the electron density along the neutrino trajectory in the Sun. The exponential density analytical results are derived assuming that const. A well-known and widely used prescription for precision calculations of the jump probability in the MSW regime is to use the “running” scale height parameter in Eq. (16), where for given and for , is chosen to coincide with the MSW resonance point, . In the present work, we show how the analytical solution (Sec. II) and the “running resonance” prescription for (Sec. III) can be smoothly extended from the MSW range down to the QVO range, and we give a justification for our procedure. The extension is first achieved by matching the resonance prescription to a perturbative expression for , in the limit of small (Sec. III). A more satisfactory match is then reached (Sec. IV) by replacing the resonance prescription, , which can be implemented only for , with the “maximal violation of adiabaticity” (MVA) prescription, , where for given and any given , is the point of the neutrino trajectory in the Sun at which adiabaticity is maximally violated (or, more precisely, the adiabaticity function has a minimum). We also show how the phase can be easily and accurately calculated in the QVO range through perturbative expressions at , and discuss the conditions under which might play a phenomenological role (Sec. V). In Sec. VI it is shown that the perturbative results essentially probe the low-density, convective zone of the Sun. Our final prescriptions for the calculation of and , summarized in Sec. VII, allow to calculate using Eqs. (10)–(12) with an accuracy in the whole QVO range. All technical details are confined in Appendixes A–E. While this work was being completed, our attention was brought to the interesting work , where the QVO range is also investigated from a viewpoint that, although being generally different from ours, partially overlaps on the topic of adiabaticity violation. We have then inserted appropriate comments at the end of Sec. IV. Ii Analytical forms for and In this section we recall the analytical expressions for and , valid for an exponential density profile, Assuming as in Eq. (17), the neutrino evolution equation (1) can be solved exactly in terms of confluent hypergeometric functions [20, 21, 31, 32, 34]. The associated expression for can be simplified by making a zeroth order expansion of the confluent hypergeometric functions in the small parameter and by using the asymptotic series expansion in the inverse powers of the large parameter where (bulk of the neutrino production zone). Then one obtains the well-known “double exponential” form for , Concerning the phase , to zeroth order in one gets a compact formula (without using the asymptotic series expansion in ), Both Eqs. (22,23) are valid at any , including .555We remark that the restriction made at the beginning of [20, 21], which is equivalent to take for the usual choice , was basically functional to obtain in a certain region of the parameter space (as it was implied by the results of the Homestake experiment). However, such restriction does not play any role in the derivation of the Eqs. (22) and (23) in [20, 21], although this was not emphasized at the time. This has also been recently noticed in . Iii The modified resonance prescription In this section, we show how the resonance prescription for calculation of (and ), valid in the MSW regime and in the first octant of , can be modified to obtain accurate values for also for and for . The resonance prescription in the MSW range is based on the following approximations: oscillations are assumed (and where shown in ) to be averaged out, so that effectively and becomes irrelevant666The same conclusion is valid for all other oscillating terms (and their phases) in [21, 22].; is taken from Eq. (22), but with a variable scale height parameter [Eq. (16)], with “running” with the resonance (RES) point , when applicable. In the absence of resonance crossing, it was customary to take in the MSW regime (see, e.g., ). In particular, in the MSW analysis of , was taken in the second octant of , where and Eq. (27) is never satisfied. The indicated resonance prescription for the calculation of is known to work very well over at least three decades in (– eV/MeV) , with a typical accuracy of a few in . However, in the lowest MSW decade (– eV/MeV), such prescription is not very accurate both in the first octant (where it tends to underestimate the effective value of ) and in the second octant (where is small but not exactly zero). It was found numerically in that, for small , a relatively small and constant value of () provided a better approximation to in the first octant777The numerical results for were obtained in by using the density provided by the Bahcall-Ulrich 1988 standard solar model .; this observation has been also discussed and extended to the second octant in (where is used for ). Here we derive (and improve) such prescriptions by means of perturbation theory. The key result is worked out in Appendix B, where we find a perturbative solution of the neutrino evolution equation (1), in the limit of small values , by treating the vacuum term as a perturbation of the dominant matter term in the Hamiltonian [Eq. (3)]. The solution888The perturbative expansion can be expressed in terms of dimensionless parameters such as or . However, for simplicity, we use the notation , etc., instead of , , etc. can be expressed in terms of the following dimensionless integral, where , and is given by Eq. (5). In particular, the effective value of (in the limit of small ) is given by , namely, independently of (i.e., both in the first and in the second octant). The same value is obtained through exact numerical calculations. The appearance of integrals over the whole density profile indicates that, for small , the behavior of becomes nonlocal, as was also recently noticed in . We further elaborate upon the issue of nonlocality in Sec. VI, where we show that the perturbative results are actually dominated by matter effects in the convective zone of the Sun (), where the function resembles a power law rather than an exponential. In order to match the usual resonance prescription with the value in the regime of small , we observe that, for the SSM density distribution [9, 10] it is at . Thus, we are naturally led to the following “modified resonance prescription,” where “otherwise” includes cases with , for which is not defined. Such a simple recipe provides a description of which is continuous in the mass-mixing parameters, and is reasonably accurate both in the QVO range ( eV/MeV) and in the lowest MSW decade (– eV/MeV). Figure 1 shows isolines of in the bilogarithmic plane charted by the variables999In all the figures of this work, we extend the interval somewhat beyond the QVO range, in order to display the smooth transition to the MSW range. eV/MeV and .101010The variable was introduced in to chart both octants of the solar mixing angle in logarithmic scale. The solid lines refer to the exact numerical calculation of , while the dotted lines are obtained through the analytical formula for [Eq. (22)], supplemented with the modified resonance prescription [Eq. (31)]. Also shown are, in the first octant, isolines of resonance radius for and 0.904. The maximum difference between the exact (numerical) results and those obtained using the analytic expression for , Eq. (22), amounts to , and is typically much smaller. Since is not a directly measurable quantity, we propagate the results of Fig. 1 to probability amplitudes observable at the Earth, namely, to the average probability [Eq. (11)] and to the prefactor of the oscillating term [Eq. (12)].111111 The phase is separately studied in Sec V. Figure 2 shows isolines of for SSM density, derived numerically (solid lines) and by using the analytic expression for and the modified resonance prescription (dotted lines). The maximum difference is . Figure 3 shows, analogously, isolines of . The difference amounts to . From Figs. 1–3, the modified resonance prescription for [Eq. (31)] emerges as a reasonable and remarkably simple approximation to the exact results for , valid in both the MSW and the QVO regimes. However, it does not reproduce the exact behavior of with the requisite high precision of few % for eV/MeV and . This difference can be understood and removed, to a large extent, through the improved prescription discussed in the next Section. Iv The prescription of maximum violation of adiabaticity In this section we generalize and improve Eq. (31), by replacing the point of resonance with the point where adiabaticity is maximally violated (), more precisely, where the adiabaticity function has its absolute minimum on the neutrino trajectory. Let us recall that the validity of the resonance prescription is based on the fact that in a relatively shorth part of the trajectory, where the propagation is locally nonadiabatic. The resonance condition, however, can be fulfilled only in the first octant of . The most general condition for nonadiabaticy, as introduced already in the early papers [13, 35, 41, 42] on the subject, has instead no particular restriction in . Such alternative condition can be expressed, in the basis relevant for the calculation of , in terms of the ratio between the diagonal term and the off-diagonal term () in the (traceless) Hamiltonian. More specifically, a transition is nonadiabatic if the ratio satisfies the inequality at least in one point of the neutrino trajectory in the Sun. If is large along the whole trajectory, , the transition is adiabatic. The minimal value of identifies the point of “maximum violation of adiabaticity,” , We show in Appendix B, that, along the solar trajectory, the MVA point is uniquely defined, for any in both octants. In particular, such point can be unambiguously characterized through the condition121212A handy approximation to the MVA condition, which by-passes the calculation of derivatives with a modest price in accuracy, is discussed at the end of Appendix B. In the first octant, in general, differs from , although one can have in some limiting cases (see Appendix B for a more general discussion). For instance, as Eq. (32) indicates, the two points practically coincide, , in the case of nonadiabtic transitions at small mixing angles (). Indeed, let us consider for illustration the simplified exponential case of . In this (“exp”) case it is easy to find from Eq. (32) that: Obviously, at small mixing angles () we have . However, this is no longer true for the nonadiabatic transitions at large mixing angles in the QVO regime we are interested in. In the latter case, as it follows from Eq. (35), . We discuss in Appendix B the more realistic case of SSM density. As far as the calculation of is concerned, it turns out that in the limits of small (or of large ). Therefore, the MVA condition smoothly extends the more familiar resonance condition in both directions of large mixing and of small , which are relevant to pass from the MSW to the QVO regime. The inequality , derived for exponential density, persists in the QVO range for the realistic case of SSM density, implying that with defined as in Eq. (16). As a conseqence, a difference arises in the value of if is replaced by in the prescription for calculating . Explicitly, our MVA prescription reads Figure 4 shows curves of iso- obtained through Eq. (37). The value of presents weak variations (within a factor of two) in the whole mass-mixing plane and, by construction, smoothly reaches the plateau for eV/MeV. Figures 5, 6, and 7 are analogous to Figs 1, 2 and 3, respectively, modulo the replacement of the resonance condition [Eq. (31)] with the MVA condition [Eq. (37)]. The MVA prescription clearly improves the calculation of , , and , with an accuracy better than a few percent in the whole plane plotted: , , . We conclude that the analytical formula for [Eq. (22)], valid for an exponential density, can be applied with good accuracy to the case of SSM density, provided that the scale height parameter is chosen according to the MVA prescription, Eq. (37). A final remark is in order. We agree with the author of about the fact that, in order to understand better the behavior of in the QVO regime, the concept of adiabaticity violation on the whole neutrino trajectory is to be preferred to the concept of adiabaticity violation at the resonance point. However, we do not share his pessimism about the possibility of using the running value for accurate calculations of : indeed, Figs. 5-7 just demonstrate this possibility. Such pessimism seems to originate from the observation that, as decreases, starts to get nonlocal contributions from points rather far from . In our formalism, this behavior shows up in the limit [Eq. (30)], where, as mentioned in the previous Section, the effective value of gets contributions from an extended portion of the density profile [see also Sec. VI and Appendix C for further discussions]. Our prescription (37), however, effectively takes this fact into account, by matching the “local” behavior of for large with the “nonlocal” behavior of at small . In conclusion, the MVA prescription, appropriately modified [Eq. (37)] to match the limit, allows a description of which is very accurate in the whole QVO range, and which smoothly matches the familiar resonance prescription up in the MSW range. V The phase acquired in the sun In this Section, we discuss the last piece for the calculation of , namely, the solar phase . As we will see, this phase can significantly affect the quasivacuum oscillations of almost “monochromatic” solar neutrinos (such as those associated to the Be and pep spectra). Indeed, there might be favorable conditions in which the phase (often negligible in current practical calculations) could lead to observable effects and should thus be taken into account. First, let us observe that, in the QVO range, the size of the solar phase is of , as indicated by Eq. (23) for exponential density , and also confirmed through numerical calculations for SSM density . Figure 9 shows, in particular, exact results for the ratio , as a function of , using the SSM density. It appears that, neglecting with respect to the vacuum phase , is almost comparable to neglect as compared with . Remarkably, there are cases in which corrections of are nonnegligible, e.g., in the study of time variations of over short time scales [22, 23, 24, 44], induced by the Earth orbit eccentricity (). In fact, the fractional monthly variation of from aphelion to perihelion is . Real-time experiments aiming to detect time variations of the flux in monthly bins might thus test terms of , as also emphasized at the end of Sec. VI in the work . Secondly, let us recall that the oscillating term gets averaged to zero when the total phase is very large . The approximation holds in the MSW regime, but it becomes increasingly inaccurate (and is eventually not applicable) as decreases down to the QVO regime. In order to understand when starts to be observable (at least in principle) one can consider an optimistic situation, namely, an ideal measurement of with a real-time detector having perfect energy resolution, and monitoring the flux from narrowest solar spectra (the Be and pep neutrino lines). In this case, the most important—and basically unavoidable—source of smearing is the energy integration [22, 45, 46, 44, 47] over the lineshape.131313Smearing over the production zone is irrelevant in the QVO regime , see also Appendix A. It has been shown in (see also [22, 24]) that such integrations effectively suppresses the oscillating term at the Earth through a damping factor , calculable in terms of the lineshape. Figure 8 shows the factor for the pep line and the for two Be lines, characterized by average energies , 863.1, and 385.5 keV, respectively. The Be and pep lineshapes [having keV widths] have been taken from and , respectively. Figure 8 proves that is observable, at least in principle, in the whole QV range eV/MeV, as far as the narrowest Be line is considered. Of course, the observability of becomes more critical (or even impossible) by increasing the initial energy spread (e.g., for continuous spectra) or by performing measurements (like in current experiments) with additional and substantial detection smearing in the energy or time domain . With the above caveats in mind, we set out to describe accurately in the whole QVO range. This task is accomplished in Appendixes D and E where, by means of the same perturbative method applied earlier to , we obtain the and expressions for , respectively. The final perturbative result is which is in excellent agreement with the exact result for shown in Fig. 9, in the whole QVO range. The coefficient in Eq. (38) is just the real part of the integral in Eq. (28) (see Appendix D) and is already sufficient for an accurate description of in the QVO range. Remarkably, the right magnitude of this coefficient can also be obtained from the analytical expression (23), which would give (see Appendix D). We also keep the small term in Eq. (38), because it neatly shows that starts to become -dependent for increasing values of , consistently with the exact numerical results of Fig. 9. Figure 10 shows the error one makes on the phase , by using increasingly better approximations, for two representative values of . The error is given as the absolute difference between approximate and exact results for , in units of a period (). The lowest possible approximation is simply to neglect with respect to , so that and , namely, neutrino oscillations are effectively started at the Sun center (“empty Sun”). A better approximation is to start oscillations at the sun edge, and , by assuming that the high sun density damps oscillations up to the surface [12, 49] (“superdense Sun”). Such two approximations (and especially the first) are widely used in phenomenological analyses. Figure 10 shows, however, that they can produce a considerable phase shift (even larger than a full period for the case of “empty” Sun) in the upper QVO range. The fact that the SSM density is neither nor is correctly taken into account through the term in Eq. (38), and even more accurately through the full expression for in Eq. (38), as evident from Fig. 10. The errors estimated in Fig. 10 show that different approximations for can affect calculations for future experiments. In particular, the use of the familiar “empty Sun” or “superdense Sun” approximations can possibly generate fake phase shifts in time variation analyses for real-time detectors sensitive to neutrino lines, such as Borexino or KamLAND . According to the estimates in , such two experiments might be sensitive to monthly-binned seasonal variations for . Figure 10 shows that, in the upper part of such sensitivity range, the empty Sun approximation gives (and thus ) totally out of phase, as compared with exact results. Analogously, the superdense Sun approximation induces a phase shift that, although smaller than in the previous case, can still be as large as in the quoted sensitivity range, and can thus produce a big difference in the calculation of . It is an unfortunate circumstance, however, that the dominant term in is proportional to . This fact implies that, neglecting in the total phase , is basically equivalent to introduce a small bias of the kind . On one hand, such bias is sufficient to produce a substantial difference in when approaches eV/MeV, and is thus observable in principle. On the other hand, is not known a priori, but must be derived from the experiments themselves, and it will be hardly known with a precision of for some time. Therefore, although may produce a big effect at fixed values of , it might be practically obscured by uncertainties in the fitted value of . In conclusion, we have found a simple and accurate expression for the solar phase , to be used in the calculation of the total phase . The solar phase produces effects of , which can be nonnegligible in high-statistics, real-time experiments sensitive to short-time variations associated to neutrino lines, such as Borexino and KamLAND, as was also emphasized in [22, 23] and more recently in . In such context, we recommend the use of Eq. (38) for precise calculations of at fixed value of , although the observability of certainly represents a formidable challenge. Vi Probing the convective zone of the sun In this section we elaborate upon the perturbative results discussed previously for and (and detailed in Appendix C and D). In particular, we show that they are related to the density in the convective zone of the Sun, corresponding to . The results crucially depend upon the quantity , which is defined in Eq. (28) as an integral over an oscillating function, having the density as inner argument. Numerical evaluation of such integral for SSM density gives (see Appendix C). It turns out that the largest contribution to comes from the outer regions of the Sun, where the integrand oscillates slowly, while in the inner regions the integrand oscillates rapidly, with vanishing net contribution to the real and imaginary parts of . Numerical inspection shows that the value of is dominated by the range, which happens to correspond to the convective zone of the Sun . Therefore, it is sufficient to consider such zone to estimate . In the convective zone, the density profile is better described by a power law rather than by an exponential function (see, e.g., ). A good approximation to the SSM density for is: with and mol/cm. By adopting such expression for the density, and using the position , we get the following expression for , where and . Once again, we note that the above integrand gives a very small contribution outside the convective zone , since the exponent becomes large. Therefore, the upper limit can be shifted from to without appreciable numerical changes, and with the advantage that the result can be cast in a compact analytical form, reproducing the exact SSM numerical result with good accuracy: Using the above equation and the results of Appendixes C and D, the small- limit for and can be explicited as Such results show that the effective values of and of at small are connected to the parameters describing the power-law dependence of in the convective zone of the Sun [Eq. (39)]. Therefore, while for relatively large neutrino oscillations in matter probe the inner “exponential bulk” of the solar density profile, for small they mainly probe the outer, “power-law” zone of convection. Vii Summary of recipes for calculation of in the QVO regime We summarize our best recipe for calculation of as follows. In the QVO regime, for any given value of the mixing angle and of the neutrino wavenumber , the survival probability reads The value of can be evaluated with high precision by using the analytical form provided that the density scale parameter is calculated as where is the point where adiabaticity is maximally violated along the neutrino trajectory in the Sun (see Appendix B). Such expression for smoothly joins the more familiar resonance prescription when passing from the QVO to the MSW regime. The value of the Sun phase can be calculated with high precision by using the perturbative result, valid for eV/MeV, We have shown that such phase can play a role in precise calculations of in the QVO range. It is not necessary to extend the calculation of in the MSW range, where effectively , and is not observable even in principle. Notice that the neutrino production point does not appear in the calculation of both and in the QVO range. Finally, we have checked that the above recipe allows the calculation of with an accuracy (and often much better than a percent) in both octants of for the whole QVO range ( eV/MeV). Figure 11 shows, as an example, a graphical comparison with the exact results for at the exit from the Sun ().141414The function at the Earth () can not be usefully plotted in the range of Fig. 11, due to its rapidly oscillating behavior. It appears at glance that our recipe represents an accurate substitute to exact numerical calculations of , in the whole QVO range eV/MeV. We have worked out a simple and accurate prescription to calculate the solar neutrino survival probability in the quasivacuum oscillation regime. Such prescription adapts the known analytical solution for the exponential case to the true density case, as well as to the perturbative solution of neutrino evolution equations in the limit of small . The accuracy of the prescription is significantly improved (up to in in the whole QVO range) by replacing the familiar prescription of choosing the scale height parameter at the MSW resonance point by a new one: is chosen at the point of maximal violation of adiabaticity (MVA) along the neutrino trajectory in the Sun. Such generalization preserves a smooth transition of our prescription from the QVO to the MSW oscillation regime, where the two prescriptions practically coincide. We show that at sufficiently small in the QVO regime, the effective value of is determined by an integral over the electron density distribution in the Sun, : where . The main contribution in the above integral is shown to come from the region corresponding to the convective zone of the Sun, . Thus, if quasivacuum oscillations take place, solar neutrino experiments might provide information about the density distribution in the convective zone of the Sun. We also discuss in detail the phase acquired by neutrinos in the Sun, whose observability, although possible in principle, poses a formidable challenge for future experiments aiming to observe short timescale variations of fluxes from solar lines. Acknowledgements.A.M. acknowledges kind hospitality at SISSA during the initial stage of this work. S.T.P. would like to acknowledge the hospitality of the Aspen Center for Physics where part of this work was done. The work of E.L., A.M., and A.P. was supported in part by the Italian MURST under the program “Fisica Astroparticellare.” The work of S.T.P. was supported in part by the EEC grant ERBFMRXCT960090 and by the Italian MURST under the program “Fisica Teorica delle Interazioni Fondamentali.” Appendix A General form of and its QV limit In this Appendix we recall the derivation of the basic equations for given in the Introduction, together with further details relevant for Appendixes B–E. Given an initial solar state ( being the transpose), its final survival amplitude at the detector can be factorized as where the matrices act as follows (from right to left): (i) rotates (at ) the initial state into the matter mass basis; (ii) is a generic parametrization for the evolution of the matter mass eigenstates from the origin () up to the exit from the Sun where , in terms of the so-called crossing probability and of two generic phases and ; evolves the mass eigenstates in vacuum along one astronomical distance , with defined in Eq. (15); and finally rotates the mass basis back to the flavor basis at the detection point ().151515During nighttime, one should insert a fifth matrix to take into account the evolution within the Earth, which we do not consider in this work (focussed on solar matter effects in the QVO range). Earth matter effects can always be added afterwards as a calculable modification to , since they turn out to be nonnegligible only in the MSW regime, when oscillations are averaged out and , see and references therein. which, together with the identification leads to Eqs. (10)–(12).161616One can make contact with the phase notation of [21, 22, 30] through the following identifications: ; ; ; and . The notation in the present work explicitly factorizes out the contribution of the vacuum phase for . In the QVO regime relevant for our work ( eV/MeV), the negligibility of is evident from the fact that for , so that the prefactor is negligibly small, while the nonvanishing terms can be written as: It was shown in that the above equations are not spoiled by Earth matter effects,171717Equations (56) and (57) in this work coincide with Eq. (28) in , modulo the identification , valid in the QVO regime. as they turn out to be negligible in the QVO range—a fortunate circumstance that considerably simplifies the calculations. We have verified the applicability of the approximations in Eq. (56) and (57) in the QVO regime, by checking that our exact results do not change in any appreciable way by setting from the start. In particular, the results of the numerical integration of the evolution equations vary very little by forcing the initial state to be rather than . We have also verified that none of our figures changes in a graphically perceptible way in the QV range, by moving the point within the production zone not only radially but also for off-center trajectories.181818 Notice that the -independence of implies that no smearing over the production zone is necessary, as also emphasized in . Therefore, in the following appendixes, we will neglect corrections of

This title is out of print. Chapter Openers and Section Openers introduce each chapter and section with interesting and motivational applications, illustrating the real-world nature of the Problem Solving begins in Chapter 1 where students are introduced to problem solving and critical thinking. The problem-solving theme is then continued throughout the text, and special problem-solving exercises are presented in the exercise sets. Critical Thinking Skills are featured in sections on inductive reasoning and the important skills of estimation and dimensional analysis. Profiles in Mathematics present the stories of people who have advanced the discipline of mathematics in brief historical sketches and vignettes. Mathematics Today relates mathematics to everyday life, helping students to recognize the need for math and gain an appreciation for math in their lives. Did You Know? features highlights the connections between mathematics and a variety of other disciplines, including history, the arts and sciences, and technology in colorful and engaging boxed features. Timely Tips, added to assist students, help with concept comprehension or relate the material to other sections of the book. Technology Tips appear as notes that have been added in selected sections to explain how a graphing calculator and/or Microsoft Excel may be used to work certain problems. Exercise Sets include diverse and numerous exercise types such as Concept/Writing, Practice the Skills, Problem Solving, Challenge Problem/Group Activity, Recreational Mathematics, and Internet/Research Activities. Chapter Summaries, Review Exercises, and Chapter Tests comprise end-of-chapter sections that help students review material and prepare for tests. Group Projects appear at the end of each chapter and are suggested projects that can be used to have students work together. These projects can also be assigned to individual students if desired. In this edition, certain topics have been revised or expanded in order to introduce new material and increase understanding: Chapter 14: Graph Theory includes new creative exercises, and a revised method of representing buildings as graphs. Chapter 15: Voting and Apportionment includes more real-life examples and exercises. In addition, several important improvements have been made to the presentation of the material: Chapter 1 Critical Thinking Skills 1.1 Inductive Reasoning 1.3 Problem Solving Chapter 2 Sets 2.1 Set Concepts 2.3 Venn Diagrams and Set Operations 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets 2.5 Application of Sets 2.6 Infinite Sets Chapter 3 Logic 3.1 Statements and Logical Connectives 3.2 Truth Tables for Negation, Conjunction, and Disjunction 3.3 Truth Tables for the Conditional and Biconditional 3.4 Equivalent Statements 3.5 Symbolic Arguments 3.6 Euler Diagrams and Syllogistic Arguments 3.7 Switching Circuits Chapter 4 Systems of Numeration 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration 4.2 Place-Value or Positional-Value Numeration Systems 4.3 Other Bases 4.4 Computation in Other Bases 4.5 Early Computational Methods Chapter 5 Number Theory and the Real Number System 5.1 Number Theory 5.2 The Integers 5.3 The Rational Numbers 5.4 The Irrational Numbers and the Real Number System 5.5 Real Numbers and Their Properties 5.6 Rules of Exponents and Scientific Notation 5.7 Arithmetic and Geometric Sequences 5.8 Fibonacci Sequence Chapter 6 Algebra, Graphs, and Functions 6.1 Order of Operations 6.2 Linear Equations In One Variable 6.4 Applications of Linear Equations In One Variable 6.6 Linear Inequalities 6.7 Graphing Linear Equations 6.8 Linear Inequalities In Two Variables 6.9 Solving Quadratic Equations by Using Factoring and By Using the Quadratic Formula 6.10 Functions and Their Graphs Chapter 7 Systems of Linear Equations and Inequalities 7.1 Systems of Linear Equations 7.2 Solving Systems of Linear Equations By the Substitution and Addition Methods 7.4 Solving Systems of Linear Equations by Using Matrices 7.5 Systems of Linear Inequalities 7.6 Linear Programming Chapter 8 The Metric System 8.1 Basic Terms and Conversions Within The Metric System 8.2 Length, Area, and Volume 8.3 Mass and Temperature 8.4 Dimensional Analysis and Conversions To and From the Metric System Chapter 9 Geometry 9.1 Points, Lines, Planes, and Angles 9.3 Perimeter and Area 9.4 Volume and Surface Area 9.5 Transformational Geometry, Symmetry, and Tessellations 9.7 Non-Euclidean Geometry and Fractal Geometry Chapter 10 Mathematical Systems 10.2 Finite Mathematical Systems 10.3 Modular Arithmetic Chapter 11 Consumer Mathematics 11.2 Personal Loans and Simple Interest 11.3 Compound Interest 11.4 Installment Buying 11.5 Buying a House with a Mortgage 11.6 Ordinary Annuities, Sinking Funds, and Retirement Investments Chapter 12 Probability 12.1 The Nature of Probability 12.2 Theoretical Probability 12.4 Expected Value (Expectation) 12.5 Tree Diagrams 12.6 Or and And Problems 12.7 Conditional Probability 12.8 The Counting Principle and Permutations 12.10 Solving Probability Problems By Using Combinations 12.11 Binomial Probability Chapter 13 Statistics 13.1 Sampling Techniques 13.2 The Misuses of Statistics 13.3 Frequency Distributions 13.4 Statistical Graphs 13.5 Measures of Central Tendency 13.6 Measures of Dispersion 13.7 The Normal Curve 13.8 Linear Correlation and Regression Chapter 14 Graph Theory 14.1 Graphs, Paths, and Circuits 14.2 Euler Paths and Euler Circuits 14.3 Hamilton Paths and Hamilton Circuits Chapter 15 Voting and Apportionment 15.1 Voting Methods 15.2 Flaws of Voting 15.3 Apportionment Methods 15.4 Flaws of the Apportionment Methods Pearson offers affordable and accessible purchase options to meet the needs of your students. Connect with us to learn more. K12 Educators: Contact your Savvas Learning Company Account General Manager for purchase options. Instant Access ISBNs are for individuals purchasing with credit cards or PayPal. Savvas Learning Company is a trademark of Savvas Learning Company LLC. <>Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy. Christine Abbott received her undergraduate degree in mathematics from SUNY Brockport and her graduate degree in mathematics education from Syracuse University. Since then she has taught mathematics at Monroe Community College and has recently chaired the department. In her spare time she enjoys watching sporting events, particularly baseball, college basketball, college football and the NFL. She also enjoys spending time with her family, traveling, and reading Dennis Runde has a BS degree and an MS degree in Mathematics from the University of Wisconsin--Platteville and Milwaukee respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for over fifteen years at Manatee Community College in Florida and for almost ten at Saint Stephen's Episcopal School. 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Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general. Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product? The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . . Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . . Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why? Can you make sense of these three proofs of Pythagoras' Theorem? Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they? Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . . Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . . Can you use the diagram to prove the AM-GM inequality? Euler found four whole numbers such that the sum of any two of the numbers is a perfect square... Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34? If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately. The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article. If you think that mathematical proof is really clearcut and universal then you should read this article. Can you see how this picture illustrates the formula for the sum of the first six cube numbers? Four jewellers share their stock. Can you work out the relative values of their gems? Is the mean of the squares of two numbers greater than, or less than, the square of their means? The sums of the squares of three related numbers is also a perfect square - can you explain why? Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement. In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot. What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle? Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total. Can you find the areas of the trapezia in this sequence? A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base. Here are some examples of 'cons', and see if you can figure out where the trick is. This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . . There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely? Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares. Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Can you discover whether this is a fair game? Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . . The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values. Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens? In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again. This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point. The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

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Updating reference tables mysql Free java femdom chatrooms Happy to oblige - there's a page here in our solutions centre.S152 - SQL Primer as Used in My SQL Which database should I use?[#3412]Comment by Hari K T (published 2009-06-16) Suggested link. [#3379]Comment by Ranjeet (published 2008-11-09)GR8 man... Thanks a Ton [#3378]Comment by Boka (published 2008-11-09) Suggested link.thankx [#3377]Comment by Ehsan (published 2008-11-09)Thank u for this article, it help me to understand joins completly [#3376]Comment by Ehsan (published 2008-11-09)Thank u for this article, it help me to understand joins completly [#3375]Comment by Anon (published 2008-11-09)extremely helpful! [#3374]Comment by Aftvin (published 2008-11-09)Thank you! [#3372]Comment by Swetha (published 2008-11-09)very nicely and simply explained! [#3369]Comment by ME (published 2008-11-09)Really great!!! [#3368]Comment by Vamshi (published 2008-07-09)wonderful example! [#3367]Comment by Santosh Thapliyal (published 2008-11-09)i m thank for your help for me understand this topik [#3366]Comment by erastus (published 2008-07-02)it's until now i have gotten it! [#3334]Comment by Anonymous (published 2007-09-02)great explanation on a subject that's very confusing.Good [#3411]Comment by sahil (published 2009-06-02)great it was an easy explanation [#3410]Comment by Anon (published 2009-06-02)Thank you Laxman Kumar [#3407]Comment by rea (published 2009-06-02)this is a great tutorial. [#3365]Comment by Anon (published 2008-06-25)This should be in the My SQL comments section of the handbook. [#3336]Comment by Sameer (published 2007-09-02)Nobody can clear JOINS as this manual has done. [#3331]Comment by kj (published 2007-09-02)No one can explain with more simplicity than this.- (2005-12-01) Matching in My SQL - (2005-09-24) Getting a list of unique values from a My SQL column - (2005-04-14) My SQL - Optimising Selects - (2004-12-21) A Change is as good as a rest Christmas break Review of the Autumn My SQL - LEFT JOIN and RIGHT JOIN, INNER JOIN and OUTER JOINAutomatic service upgrades Signage Railway train service, Melksham station Linux - where to put swap space Aladdin, or careful what you wish. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96 at 50 posts per page This is a page archived from The Horse's Mouth at - the diary and writings of Graham Ellis. It is a common mistake to think that there is no difference putting the conditions in the ON clause and the WHERE clause. The following post explains it with examples: [#3973]Comment by Rahul (published 2011-08-18)Thanks dear it is very helpful [#3967]Comment by Rahul (published 2011-08-18)Thanks dear it is very helpful [#3966]Comment by Anon (published 2011-03-19)That's really help！ Thanks！ [#3907]Comment by Rajeev Sharma (published 2011-02-18)Yes, Its really a good start to understand joins.Every attempt was made to provide current information at the time the page was written, but things do move forward in our business - new software releases, price changes, new techniques. Good Work [#3883]Comment by v (published 2011-02-18) Suggested link. [#3881]Comment by Juan (published 2011-02-18)It's like Plug and Play...Python, Lua, Tcl, C and C training - public course schedule [here]Private courses on your site - see [here]Please ask about maintenance training for Perl, PHP, Java, Ruby, My SQL and Linux / Tomcat systems In a database such as My SQL, data is divided into a series of tables (the "why" is beyond what I'm writing today) which are then connected together in SELECT commands to generate the output required. I mean we can get better results by using " NOT IN". [#3323]Comment by edw (published 2007-06-26)I'm very pleased with your clear explanation. [#3321]Comment by El Sayed (published 2007-06-13)I'll use this idea explaining Joins 2morow is A Thanks [#3320]Comment by Veejay (published 2007-07-10)Its excellent, to the point and very clear. [#3319]Comment by Priya Saini (published 2007-04-24)This is 'Simply' the best explaination. nw not any more [#3311]Comment by vellaidurai (published 2007-04-03)this is one of best and very simple explanations and keep this.. But only one thing that reduced my rating from 5 to 4 is missing of enough detail regarding INNER & OUTER Joins. Excellent - thanks very much, very very useful indeed. Its easier to understand [#3489]Comment by sagar (published 2010-03-21)good simple but strong [#3485]Comment by Linu Varghese (published 2010-03-09)Nice Explanation, So Simple TO Under Stand,, Thanks [#3481]Comment by Andy (published 2010-03-09) Suggested link. [#3480]Comment by Raj (published 2010-02-08)This is very nice example to understanding the concept of join query. [#3453]Comment by Anon (published 2010-01-18)Great explanation of joins. [#3452]Comment by Yogesh (published 2010-01-18)this is very simple and it cleared all doubts about joins like difference left and left outer join. [#3451]Comment by Zheka82 (published 2009-12-28)Thank you much! [#3784]Comment by rajendra (published 2010-10-08)good example [#3782]Comment by Om Prakash Yadav (published 2010-10-08)Excellent [#3775]Comment by Abhiranjan Jagannath (published 2010-09-11)Simple and clear. [#3772]Comment by Dhawal (published 2010-09-11)thanks a lot sir, for posting this article.... [#3771]Comment by manikandan (published 2010-09-11)Excellent example....... [#3741]Comment by Anon (published 2010-07-15)Thanks really helped [#3653]Comment by Rahul Dhawan (published 2010-07-15)This is the best example that i have ever read, now joins are really clear in my mind,,thank you very much [#3650]Comment by sachin negi (published 2010-05-18)hey!! [#3465]Comment by Anon (published 2010-01-24) Suggested link. [#3459]Comment by Jack (published 2010-01-18)Awesome. :) [#3450]Comment by ARUL, COIMBATORE (published 2009-12-28)REALLY A FANTASTIC ARTICLE [#3449]Comment by Anon (published 2009-12-28)Very good explainaiton. [#3447]Comment by Sunny (published 2009-12-28)I really got helped. [#3444]Comment by Geo (published 2009-12-28)excellent explanation!! [#3443]Comment by Anon (published 2009-12-28)Thanks a lot for the info. [#3441]Comment by Chaitra Yale (published 2009-10-27)Very nice examples and explanation of Mysql joins. Extra link to three way join in solution centre (would have added that link in the comment except that it "spam trapped! Since I wrote this page (which seems to be generating a lot of traffic in its own right), I've also been asked to provide examples of joining more than two tables - both with a regular join, and also with left joins. [#3416]Comment by mike (published 2009-06-16) Suggested i know how to use it :) thanks a lot [#3414]Comment by Anon (published 2009-06-16)i appreciate the way it has been specified the diference between the left join and right join [#3413]Comment by Sev (published 2009-06-16)You covered most of the stuff correctly, but you messed up around the bottom. [#3389]Comment by RAjesh (published 2009-06-02)There are no words also in english to praise. [#3388]Comment by Anon (published 2008-12-31)I'm glad it's not just me that has struggled understanding join vs left vs right join, even more glad you have written such a simple explanation for us all!

ALGEBRA 2/TRIGONOMETRY. SAMPLE ... The following items were gathered from the Regents Examination in. Algebra ... The correct answer choice is (2). 2. SUP P O R T MAT E R I AL S nys regents exam Algebra 2/Trigonometry sample questions The following items were gathered from the Regents Examination in Algebra 2/Trigonometry. The test sampler can be located by visiting http://www.emsc.nysed.gov/osa/mathre/algebra2trigonometrysampler.pdf The expression (3 7i)2 is equivalent to 40 + 0i 40 42i 58 + 0i 58 – 42i This question is designed to asses whether or not a student can multiply two binomials with imaginary numbers. If a student does not remember how to distribute two binomials or is unable to simplify the resulting expression, they can use the fx-9750GII to simplify the imaginary expression. a) To select the RUN mode from the Icon Main Menu, press 1 or !$BNto the desired icon and press l. To input the expression, enter the following: The correct answer choice is (2). If f(x) = 1 x 3 and g(x) = 2x + 5, what is the value of (g ◦ f)(4)? 2 To evaluate this composite function, a student could substitute 4 into the expression 1 x 3 , take the resulting answer and substitute it into 2x + 5. A student can use the 2 Table mode of the fx-9750GII to evaluate this composite function in much the same way. To select the TABLE mode from the Icon Main Menu, press 5. From the initial Table screen, input the f(x) equation for Y1: by entering: To input the g(x) equation for Y2:, enter the following: To view the table for both functions, press u(TABL). With the cursor in the x-column, enter 4lto see the value of When x is 4, 1 x 3 , when x is 4. 2 1 x 3 is equal to 1. With the cursor in the x-column enter 2 n1lto see the value of 2x + 5 when x = 1. The value of the second equation (Y2) is 3, so the answer choice is (3). What are the values of in the interval 0° ≤ < 360° that satisfy the equation tan 3 0 ? 72°, 108°, 252°, 288° 60°, 120°, 240°, 300° Students who do not know how to solve this equation or students who want to check their work can utilize the Equation Editor on the fx-9750GII for problems like this. a) To select the EQUATION mode from the Icon Main Menu, press 8. Before the equation can be solved, pressLp (SET UP)NNNwd to change the angle setting to degrees. From there, input the equation, enter the following: Note: The variable x was used instead of . The answer is 60; answer choice (1) and (4) could be possibilities. Solve the equation again by pressing q(REPT), using x = 240. Any other answer choice substituted for x, would result in an answer > 360°, the answer is (1). The expression log8 64 , is equivalent to Most calculators can perform base 10 logarithms; however, the fx-9750GII can just as easily calculate logarithms with different bases. a) From the RUN mode, enter the following: The correct answer choice is (2). The expression cos 4x cos 3x + sin 4x sin 3x is equivalent to (1) sin x (3) cos x (2) sin 7x (4) cos 7x A student can use the TABLE mode of the fx-9750GII to help evaluate this problem. The given expression, along with each answer choice, can be entered to see if the resulting tables are equal. a) To enter the expression, from the initial TABLE screen, enter the following: To enter the first answer choice, enter the following: To view the tables, select u. If the YI and Y2 columns match up, that answer choice is the correct answer. The two columns do not match up; therefore, answer choice (1) does not work. To return to the initial Table screen, press q(FORM) or d. Highlight Y2: and begin entering answer choice (2). The two columns do not match up; therefore, answer choice (2) does not work. e) Repeat the above steps for answer choice (3). The YI and Y2 columns show equal values; the correct answer choice is (3). The value of the expression 2 (n2 2n ) is The fx-9750GII can evaluate summation expressions in the RUN mode by going to the Option menu and selecting the Sigma Notation tab. a) To input the problem, enter the following from the Run mode: The correct answer choice is (3). 5 For which equation does the sum of the roots equal 3 and the product of the roots 4 equal 2? (1) 4x 2 8x 3 0 4x 2 8x 3 0 4x 2 3x 8 0 4x 2 3x 2 0 To answer this problem, a student would have to factor each polynomial, then multiply and add the roots together. To save time in a testing situation, the student could use the Polynomial mode in the Equation Editor to quickly find the roots of the equations then multiply and add the roots to see if they meet the conditions given in the question. To enter polynomial equations in the fx-9750GII, they must be in standard form, and only the coefficients and constant are entered. a) From the initial Equation Editor screen, selectwfor Polynomial, thenqto enter a second degree polynomial equation. To enter the first equation, input the following: 3 , continue to enter each answer choice. Pressdor 4 q(REPT) to return to the previous screen. Since 1.5 + 0.5 does not equal 4x 2 3x 8 0 has roots that when added, equal 3 and when multiplied equal 2. 4 The correct answer choice is (3). The remaining problems were taken from the free response section of the test sampler. Matt places $1,200 in an investment account earning an annual rate of 6.5%, compounded continuously. Using the formula v Pe rt , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, that Matt will have in the account after 10 years. The student must be able to correctly substitute values into the given formula. Using the fx-9750GII, the student can quickly substitute values into a formula and check their work. From the Icon Main Menu, select8for the Equation Mode. From the initial Equation Editor screen, select efor Solver. From here, formulas can be entered using ato enter the desired letters. To input the formula, enter the following: From here, assign the appropriate values to the given variables, move the cursor to the unknown value and solve by entering the following: The correct answer is $2298.65. Solve the equation 8x 3 4x 2 18x 9 0 algebraically for all values of x. A student could use the fx-9750GII to check their solutions or use the answers to factor the cubic function for a portion of the points. a) From the initial Equation Editor screen, selectwfor Polynomial, thenwfor a third degree polynomial equation. To input this polynomial and solve, enter the following: The answer is x = 1.5, x = 0.5, and x = 1.5. The table below shows the results of an experiment involving the growth of bacteria. Time (x) in minutes Number of Bacteria (y) Write a power regression equation for this set of data, rounding all values to three decimal places. Using this equation, predict the bacteria’s growth, to the nearest integer, after 15 minutes. In the STAT mode of the fx-9750GII, the student can enter lists of data, graph the data, view two-variable statistical calculations or perform ten different types of regressions. a) From the Icon Main Menu, select 2 for the Statistics mode. From the initial List Editor Screen, enter Time in List 1, pressing lafter each entry. After the last entry, press $and begin entering the Number of Bacteria, pressing lafter each entry. To calculate power regression, enterw(CALC), e(REG),u( ), thene(Pwr). To copy this equation to the Table mode, press u(COPY), thenl. Press pfor the Icon Main Menu and 5 for Table mode. The power regression equation is displayed; press qto select it, then u to display the table. To display the number of bacteria after 15 minutes, enter 15l anywhere in the x-column. The correct answer is 1009 bacteria.

Transformation Groups for Beginners S. V. Duzhin B. D. Tchebotarevsky Contents Preface 5 Introduction 6 Chapter 1. Algebra of points 11 x1. Checkered plane 11 x2. Point addition 13 x3. Multiplying points by numbers 17 x4. Centre of gravity 19 x5. Coordinates 21 x6. Point multiplication 24 x7. Complex numbers 28 Chapter 2. Plane Movements 37 x1. Parallel translations 37 x2. Re�ections 39 x3. Rotations 41 x4. Functions of a complex variable 44 x5. Composition of movements 47 x6. Glide re�ections 52 x7. Classi�cation of movements 53 x8. Orientation 56 x9. Calculus of involutions 57 Chapter 3. Transformation Groups 61 x1. A rolling triangle 61 x2. Transformation groups 63 3 4 Contents x3. Classi�cation of �nite groups of movements 64 x4. Conjugate transformations 66 x5. Cyclic groups 70 x6. Generators and relations 73 Chapter 4. Arbitrary groups 79 x1. The general notion of a group 79 x2. Isomorphism 85 x3. The Lagrange theorem 94 Chapter 5. Orbits and Ornaments 101 x1. Homomorphism 101 x2. Quotient group 104 x3. Groups presented by generators and relations 107 x4. Group actions and orbits 108 x5. Enumeration of orbits 111 x6. Invariants 117 x7. Crystallographic groups 118 Chapter 6. Other Types of Transformations 131 x1. A�ne transformations 131 x2. Projective transformations 134 x3. Similitudes 139 x4. Inversions 144 x5. Circular transformations 147 x6. Hyperbolic geometry 150 Chapter 7. Symmetries of Di�erential Equations 155 x1. Ordinary di�erential equations 155 x2. Change of variables 158 x3. The Bernoulli equation 160 x4. Point transformations 163 x5. One-parameter groups 168 x6. Symmetries of di�erential equations 170 x7. Solving equations by symmetries 172 Answers, Hints and Solutions to Exercises 179 Preface 5 Preface The �rst Russian version of this book was written in 1983-1986 by B. D. Tcheb- otarevsky and myself and published in 1988 by \Vysheishaya Shkola" (Minsk) under the title \From ornaments to di�erential equations". The pictures were drawn by Vladimir Tsesler. Years went by, and I was receiving positive opinions about the book from known and unknown people. In 1996 I decided to translate the book into English. In the course of this work I tried to make the book more consistent and self-contained. I deleted some unimportant fragments and added several new sections. Also, I corrected many mistakes (I can only hope I did not introduce new ones). The translation was accomplished by the year 2000. In 2000, the English text was further translated into Japanese and published by Springer Verlag Tokyo under the title \Henkangun Nyu�mon" (\Introduction to Transformation Groups"). The book is intended for high school students and university newcomers. Its aim is to introduce the concept of a transformation group on examples from di�erent areas of mathematics. In particular, the book includes an elementary exposition of the basic ideas of S. Lie related to symmetry analysis of di�erential equations that has not yet appeared in popular literature. The book contains a lot of exercises with hints and solutions. which will allow a diligent reader to master the material. The present version, updated in 2002, incorporates some new changes, including the correction of errors and misprints kindly indicated by the Japanese translators S. Yukita (Hosei University, Tokyo) and M. Nagura (Yokohama National University). S. Duzhin September 1, 2002 St. Petersburg 6 Contents Introduction Probably, the one most famous book in all history of mathematics is Euclid’s \Ele- ments". In Europe it was used as a standard textbook of geometry in all schools during about 2000 years. One of the �rst theorems is the following Proposition I.5, of which we quote only the �rst half. Theorem 1. (Euclid) In isosceles triangles the angles at the base are equal to one another. Proof. Every high school student knows the standard modern proof of this proposi- tion. It is very short. s A �A � A � A � A H � � A � A � A B s� AsC Figure 1. An isosceles triangle Standard proof. Let ABC be the given isosceles triangle (Fig.1). Since AB = AC, there exists a plane movement (re�ection) that takes A to A, B to C and C to B. Under this movement, \ABC goes into \ACB, therefore, these two angles are equal. It seems that there is nothing interesting about this theorem. However, wait a little and look at Euclid’s original proof (Fig.2). r A �A � A � A � A � A � B r ArC F r!��!�a!a!a!a!a!!aAaAaArG � A � A D E Figure 2. Euclid’s proof Introduction 7 Euclid’s original proof. On the prolongations AD and AE of the sides AB and AC choose two points F and G such that AF = AG. Then 4ABG = 4ACF, hence \ABG = \ACF . Also 4CBG = 4BCF, hence \CBG = \BCF . Therefore \ABC = \ABG�\CBG = \ACF �\BCF = \ACB. � In mediaeval England, Proposition I.5 was known under the name of pons asinorum (asses’ bridge). In fact, the part of Figure 2 formed by the points F , B, C, G and the segments that join them, really resembles a bridge. Poor students who could not master Euclid’s proof were compared to asses that could not surmount this bridge. Figure 3. Asses’s Bridge From a modern viewpoint Euclid’s argument looks cumbersome and weird. In- deed, why did he ever need these auxiliary triangles ABG and ACF? Why was not he happy just with the triangle ABC itself? The reason is that Euclid just could not use movements in geometry: this was forbidden by his philosophy stating that \mathematical objects are alien to motion", This example shows that the use of movements can elucidate geometrical facts and greatly facilitate their proof. But movements are important not only if studied separately. It is very interesting to study the social behaviour of movements, i.e. the structure of sets of movements (or more general transformations) interrelated between themselves. In this area, the most important notion is that of a transformation group. The theory of groups, as a mathematical theory, appeared not so long ago, only in XIX century. However, examples of objects that are directly related to transformation groups, were created already in ancient civilizations, both oriental and occidental. This refers to the art of ornament, called \the oldest aspect of higher mathematics expressed in an implicit form" by the famous XX century mathematician Hermann Weyl. 8 Contents The following �gure shows two examples of ornaments found on the walls of the mediaeval Alhambra Palace in Spain. a b Figure 4. Two ornaments from Alhambra Both patterns are highly symmetric in the sense that there are preserved by many plane movements. In fact, the symmetry properties of Figure 4a are very close to those of Figure 4b: each ornament has an in�nite number of translations, rotations by � � 90 and 180 , re�ections and glide re�ections. However, they are not identical. The di�erence between them is in the way these movements are related between themselves for each of the two patterns. The exact meaning of these words can only be explained in terms of group theory which says that symmetry groups of �gures 4a and 4b are not isomorphic (this is the contents of Exercise 129, see page 129). The problem to determine and classify all the possible types of wall pattern symme- try was solved in late XIX century independently by a Russian scientist E. S. Fedorov and a German scientist G. Scho�n�iess. It turned out that there are exactly 17 di�erent types of plane crystallographic groups (see the table on page 126). Of course, signi�cance of group theory goes far beyond the classi�cation of plane ornaments. In fact, it is one of the key notions in the whole of mathematics, widely used in algebra, geometry, topology, calculus, mechanics etc. This book provides an elementary introduction into the theory of groups. We be- gin with some examples from elementary Euclidean geometry where plane movements play an important role and the ideas of group theory naturally arise. Then we ex- plicitly introduce the notion of a transformation group and the more general notion of an abstract group, discuss the algebraic aspects of group theory and its applica- tions in number theory. After this we pass to group actions, orbits, invariants, some classi�cation problems and �nally go as far as the application of continuous groups to the solution of di�erential equations. Our primary aim is to show how the notion of group works in di�erent areas of mathematics thus demonstrating that mathematics is a uni�ed science. Introduction 9 The book is intended for people with high school mathematical education, includ- ing the knowledge of elementary algebra, geometry and calculus. You will �nd many problems given with detailed solutions and lots of exercises for self-study supplied with hints and answers at the end of the book. It goes without saying that the reader who wants to really understand what’s going on, must try to solve as many problems as possible.

Financial markets are chaotic. So chaotic, even, that many economists and investors believe market trends to be the product of ‘random walks’ and that prices cannot be predicted (see generally Malkiel). But randomness shouldn't be worrisome. In fact, random price movements can be good. Gaussian random walk, an assumption used by an options pricing model called Black-Scholes, treats intervals of an asset’s price over time as independent variables. By doing so, the changes in price over time, or the returns of an asset, are assumed to be normally distributed. Otherwise stated, “If transactions are fairly uniformly spread across time, and if the number of transactions per day, week, or month is very large, then the Central Limit Theorem leads us to expect that these price changes will have normal or Gaussian distributions” (Fama, 399). When an asset's returns are normally distributed, the probabilities of those returns are known. Knowing these probabilities can give investors a reliable framework accounting for the risk of holding said asset. When it comes to bitcoin, much has been said about how risky it is. The purpose of this article is to explore how to frame risk and to test how well traditional assumptions, implicit in derivatives pricing, apply to bitcoin. This article will proceed by introducing the derivatives market and giving an overview of the Black-Scholes model. After a brief discussion of why the Black-Scholes model is important and what it does, I will highlight the weaknesses of the model and the (sometimes) unrealistic assumptions it makes. I will then turn to how well the model fits into the bitcoin derivatives market. Particularly, I will show that historical data on daily bitcoin returns from January 2016 till August 2019 exhibit excess kurtosis. Following a discussion of the findings, I will compare the efficacy of Black-Scholes applied to bitcoin with the S&P 500. Finally, the article will end with closing thoughts on why the Black-Scholes model may poorly fit the crypto market and the implications this presents for the future of the quickly growing crypto derivatives market. Imagine that you’re a farmer and you expect to produce 5,000 bushels of corn. Naturally, you want to sell the corn for as much as you can. However, because prices are subject to the demands of an unrelenting market, there’s a possibility that corn may be selling below your production costs after the harvest. Financial derivatives can be used to minimize the total loss suffered by an unexpected crash. If the price of corn is trading for around $3.50/bushel and you would like to “lock-in” a price floor of $3, you could buy a put option at $3 and be safe from a possibility where the price went below $3. Since put options increase in value as the underlying asset trades below its strike price, the price at which the option is purchased, the total loss is the cost of the option. If the cost of the put option at $3 is 10 cents ($500=5000*.10) and if the production costs of corn is $1/bushel, then the minimum profit is $9500. The purpose of the above example is to highlight the utility of derivatives. Of course, the picture of a farmer’s derivatives portfolio can become increasingly more complicated when considering how to use a full suite of futures, options, swaps, etc. Yet, a basic intuition can be gleamed: markets and prices reflect uncertainty/risk and derivatives exist as products to minimize such uncertainty. With this understanding of derivatives, the price of any derivative takes on a special consideration. For, the derivative’s instrumentality is premised upon its ability to represent an actual hedge against the uncertainty of the underlying asset. In the above example, if the price of the put option was $2, instead of 10 cents, and corn was trading at $3.50, the state of the options market can tell you a number of things about the broader corn market. First, if the option is accurately priced and not the result of some egregious error, then the Black-Scholes model would calculate that the volatility of corn prices is a little over 200% (see notes). Such volatility would be extraordinary for the agricultural market and may change your assumptions about what price you would be willing to sell your corn. Second, if your assumptions remain unchanged, buying a put option for $2 would substantially reduce your profit potential and may result in a total loss if the price of corn fell to $3 because of fees. Third, if your assumptions do change and the implied volatility of corn prices is to be believed, producing 1 bushel of corn for $1 becomes risky because there’s a substantially higher probability of corn trading below your costs. Thus, determining the validity of a given option’s price is critically important because the option’s price implies crucial aspects of the underlying market. When it comes to determining the value of an options contract, the process is fairly mechanical. The pervasive Black-Scholes model for options pricing is used to give investors an analytical framework for determining the value of an options contract. The Black-Scholes model is also used by investors and exchanges alike to determine “the greeks,” or partial derivatives such as an option’s or portfolio’s delta, vega, theta, gamma, etc. These partial derivatives have been very useful for risk management as well as prescribed risk-limits created by exchanges/brokerages because they determine how sensitive an option’s or portfolio’s (or another partial derivative) value is to certain parameters. An example of this is when Deribit, a large crypto derivatives exchange, liquidates risky positions; the risk engine tries to create a “delta-neutral” position. In the broader context of the derivatives market, when the Chicago Board Options Exchange launched its first “Autoquote” system in 1986, a program that gave traders updated prices of options being traded, it used Black-Scholes (MacKenzie , 169). It’s not an overstatement to propose that the Black-Scholes model was a “vital contribution” in the context of modern finance’s growth and the accelerating influence derivatives have in financial markets. Such accelerating influence has also been seen in crypto markets. With the recent announcements from bitcoin derivatives exchange LedgerX and Seed CX testing physically-settled bitcoin derivatives to the ongoing development of Bakkt and ErisX, a lot of interest has been generated in offering US customers exposure to crypto derivatives. As for options, only LedgerX and Seed CX currently offer trading for US customers. The news of so many new derivatives exchanges possibly going live in the near future prompts the question: how well does the Black-Scholes model lend itself to the risk management of bitcoin investments? Certainly, the Black-Scholes model is not without its flaws, which will be taken up below. Yet, the model remains as standard curriculum for future investment bankers and is attractive because it provides an easy to implement system to track risk. A crucial feature of Black-Scholes is the implicit assumption that asset returns are normally distributed. By assuming a normal distribution of returns, Black-Scholes offers a framework for predicting the probabilities of certain returns that investors can factor into their hedging strategies. In their original paper, “The Pricing of Options and Corporate Liabilities,” Black and Scholes state this assumption as “[t]he stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is log-normal. The variance rate of the return on the stock is constant” (Black and Scholes, 640). This assumption can be illuminated by looking at the formula. The Black-Scholes formula is: The inputs for the formula are: C = Call option price So = Stock/underlying asset price X = Strike price σ = volatility r = continuously compounded risk-free interest rate q = continuously compounded dividend yield t = time to expiration (% of year). e = exponential term The N(x) function is the standard normal cumulative distribution function. The N(x) function represents the probability “weighting” for the “value” part of the formula (So e-qt ) and the “cost” part (X e-rt). In the original conception of the formula, the “value” part was denoted as the stock price times the N(x) function. This was changed later by Robert Merton, who greatly expanded the Black-Scholes model, to account for dividends. Roughly speaking, the Black-Scholes formula represents an investor’s return (So e-qt) minus the cost of the option. The ‘e-rt’ accounts for the risk-free interest rate, compounded continuously, from the time of purchase to the expiration of the option. Essentially, the ‘e-rt’ represents the “time value of money” and it discounts the strike price (X) to present value. This is done because, ideally, the option’s value should be greater than the current risk-free rate of a Treasury bill (T-bill) or government bond. If an investor could achieve a higher return by buying a T-bill, buying the option would make little sense. Importantly, Black-Scholes uses a log-normal distribution for options prices. However, the returns at expiration ((ln(So/X)+t) are normally distributed. This means that the distribution of prices is skewed so that the mean, median, and mode are different. Since a log-normal distribution has a lower bound of 0, it intuitively makes sense that prices are log-normally distributed because asset prices cannot be negative. The term ‘σ’ represents the asset’s daily volatility. When the growth term (ln(So/X)+t) is divided by the standard deviation of the asset’s daily volatility (σ√t) the distribution becomes a normal distribution. With returns being normally distributed, the volatility of an asset (σ√t) will determine the curve of the distribution when weighted by the N(x) function. Because volatility is weighted by the N(x), the higher the value is for (σ √t), the “flatter” the curve will be. When the N(x) function is N(d1), the function represents the probability of how far into the money the option will be if, and only if, the asset price is above the strike price at expiration. In other words, N(d1) gives the expected value, at time t, of the asset price (So) and counts asset prices less than the strike price as 0. When looking at Figure 2, if 'a' represents a strike price, N(d1) gives the expected value of the option when the asset price is to the right of the ‘a.’ When the asset price is to the left of ‘a,’ N(d1) treats the price as 0. This represents how an option works. In the case of a call option, assume that ‘a’ in Figure 2 denotes a strike price. A call option is a bet that the underlying asset’s price will be above the strike price at the time of expiration. If the price, at expiration, is below the strike price, the call option’s value is 0. Alternatively, if ‘a’ denotes the strike price for a put option and the underlying asset’s price expires to the right of ‘a,’ the put option’s value is 0. N(d2), on the other hand, is “the probability that a call option will be exercised in a risk-neutral world” (Hull, 335). Assuming again that the ‘a’ in Figure 2 denotes a strike price, N(d2) represents the probability of the asset’s price being above (for a call option) or below (for a put) ‘a’ at expiration. Because the total area under a normal distribution curve, e.g. Figure 2, represents all probabilities of an event occurring, and returns are modeled as a normal distribution, Black-Scholes models the total probability of what the future rate of return for an asset will be. N(d2) is the means of determining the probability of whether the price of an asset will be above or below a given strike by modeling the probabilities of an asset’s rate of growth. These probabilities are calculated by determining how many standard deviations away the rate of growth, from the stock price to the strike price, is from the expected rate of growth (r-(σ²/2)). Putting it all together, because the option is only paid if the asset’s price is greater than the strike (for a call) and the probability of this happening is N(d2), the expected payoff for that option in a risk-neutral world is: Volatility ‘σ’ is the most deterministic input for Black-Scholes because higher volatility means that the area of the normal distribution curve will be greater. This also means that the option will be priced higher because (So e-qt) is multiplied by the N(x) function. Thus, option prices can be conceived of as merely probability distributions. If volatility is very stable and there’s a 100% chance that a stock will expire above or below a call or put option’s strike, respectively, then that option is not very valuable. Indeed, the option is useless from a hedging perspective because there’s no risk. Alternatively, if there’s a 50% chance of the stock expiring above or below an option’s strike, that option has value because it’s attractive to investors seeking to reduce the risk of holding the underlying stock. Black-Scholes is by no means perfect. In part, the utility of the Black-Scholes model is hampered by its assumptions about the market. Namely, the model assumes that volatility is not only constant, but also knowable in advance. This assumption is problematic because volatility, itself, can be volatile. The Chicago Board Options Exchange created the Volatility Index (VIX) to tract the 30-day implied volatility of the S&P 500 index options. In 2018 the VIX reached a low of almost 8.5% and a high of over 46%. Volatility is by no means consistent. Moreover, finding volatility is not as straightforward as simply looking up a stock price. Whereas the stock price, the strike price, the risk-free interest rate, the dividend yield, and the time till expiration are all observable, volatility is implied. Volatility must be calculated by looking backwards and projecting that at time t it can be known, or at least cautiously relied on. Black-Scholes also suffers because the market has changed. When the market flash crashed in 1987, an important aspect of the derivatives market was dramatically affected. This was the “volatility smile.” Prior to 1987, implied volatility (IV) for out-of-the-money puts and out-of-the-money calls were almost similar in value. The market priced in unbiased IVs for both calls and puts. However, as shown in Figure 3, this changed after 1987 and the market currently tends to give higher IVs to put options over calls. The volatility smile now demonstrates “skewness.” Skew can represent the fear in the market. If put options are pricing in much higher IVs than calls, it can be interpreted that traders are disproportionately hedging for downside risk. In the case of Figure 3, the graph suggests that there is negative skewness for the S&P 500. Negative skewness indicates that there’s higher probability for values to the left of the mean. Ever since 1987, the market has priced in such skew by valuing the IV for puts higher than calls. Simply put, traders fear a future crash and have a higher demand for this type of hedging. While Black-Scholes, through a normal distribution curve, gives equal probabilities at both ends of the curve, actual markets tend to betray a more pessimistic outlook. Interestingly though, bitcoin traders are much more optimistic. Figure 4 shows the options chain on Deribit (the market of listed options) for bitcoin with a December 27th, 2019 expiration. It can be seen that similarly distanced strikes (7000 and 15000) from the current price of bitcoin ( ~10000) show very different IVs. Whereas the put options (on the right side) being bought at 7000 have an IV of 86.6%, the call options (on the left side) at 15000 have a slightly higher IV of 91.8%. As a result, the out-of-the-money puts are valued far less than out-of-the-money calls. Although this one option chain isn’t indicative of the entire bitcoin options market, it shows that there’s a sizable amount of speculators/investors who are undervaluing downside risk. In these respects, the Black-Scholes model should not be held as sacred. Rather, a good bit of skepticism should be applied when using Black-Scholes to price options. However, Black-Scholes, as noted above, is still widely used and certainly has its uses. Particularly, it’s useful when trying to get an idea of option prices for many assets quickly. In many respects, this easy-to-apply model may have led to some over-reliance for many firms, but fantasy can still serve a purpose. Kurtosis is a measure of “tailedness,” or how well the tails of a sample’s distribution fit into the bell-curve of a normal distribution. Since January 2016 bitcoin has had excessive kurtosis. The formula for sample excess kurtosis is: The inputs are: X = random variable n = sample size s = sample standard deviation Kurtosis is defined as the fourth standardized central moment, and is represented as: For calculating the kurtosis of a distribution of an asset’s returns, the deviation from the mean (the difference between each random variable X and the average of all values) for each daily return is needed. This deviation can be represented as: Statistical moments describe the shape of distributions. Generally speaking, the first and second moments represent the mean and variance, respectively, of a distribution. The third moment represents skewness. As introduced above, skewness is the shift in the distribution away from the mean of a normal distribution. The fourth moment, when standardized, is kurtosis and changes the curve of a normal distribution in different ways. The fourth moment can be represented as: Since Kurtosis is the fourth standardized central moment, the fourth moment must be normalized. Normalization can be achieved by dividing by the sample standard deviation. Thus, the fourth moment is divided by ‘s⁴’ in the formula above. A normal distribution has a kurtosis of 3. If the kurtosis for a distribution is higher than 3, it is called “leptokurtic.” When kurtosis is less than 3, it is called “playkurtic.” When calculating for excess kurtosis, the formula is adjusted to subtract 3 for a sample. Whereas playkurtic distributions have more uniform distributions and can have a flatter curve, “leptokurtic distributions have the property that small and large values around the mean are more likely than for a normal distribution, while intermediate changes are less likely; that is, the probability from the shoulders is moved to the centre and tails” (McAlevey and Stent, 4). As can be seen in Figure 5, a leptokurtic distribution shows very high probabilities around the mean and much higher probabilities at the tails when compared to a normal distribution. This means that, for assets, they are generally less predictable because the probabilities are skewed by giving higher probabilities to very dramatic swings in price. Thus, when an asset shows excess kurtosis, the inherent risk of holding the asset is greater. When thinking about risk, kurtosis can be especially helpful. Disregarding fundamental assumptions like 'random walk' for the moment, finding the kurtosis of returns for any given time-frame can give investors a picture of how volatility is distributed. Finding out whether returns are actually normally distributed or not then adds nuance to how and when assets are considered risky. A common interpretation of risk simply focuses on volatility. The more volatile an asset is, the more risk. Conversely the more stable an asset is, the safer. However, this dualism of volatile/risky and stable/safe brushes over the nature of volatility and lumps even normally distributed returns that have a very wide curve into the "risky" category. Yet, if returns are normally distributed, then the probabilities for those returns are knowable - no matter how wide the curve is. For example, imagine an asset where the edges of the normally distributed returns reach -50% and 50%. Such an asset would be considered very volatile. But if the asset follows a normal distribution, it can be known that the tails and edges of the curve represent 2 and 3 standard deviations from the mean, while somewhere in the shoulders of the curve represents 1 standard deviation. When that information is known, an investment strategy can be tailored around such probabilities and even very volatile assets can be traded just like less volatile ones. Therefore, instead of treating risk and volatility as parallel to each other, they should be related orthogonally. This orthogonal relation can produce a Volatility-Risk compass. In Figure 6, imagine that the Unpredictable-Predictable axis relates to price and the Knowable-Unknowable axis relates to probability distributions of returns. In the two bottom quadrants the prices of an asset are predictable, while in the two top quadrants prices are unpredictable. Likewise, in the two left-side quadrants the probability distributions of returns are knowable, while in the two right-side quadrants the probabilities are unknowable (or at least unreliable). In this picture, the top quadrants represent 'random walk' and the left-side quadrants represent normally distributed returns. So then, in Figure 7, the "ideal" asset in the Black-Scholes model is represented in the top left quadrant. Such an asset follows a 'random walk' and its prices are unpredictable. However, the returns are normally distributed and thus the probabilities are knowable. Alternatively, the bottom right quadrant represents the antitheses of the Black-Scholes model. The price of the asset in the bottom right quadrant is predictable but the probability distributions are essentially unknowable. This asset can be thought of as being manipulated. Either through insider trading or universal acceptance of a prescribed technical analysis, the price is perfectly predictable but the asset moves erratically. There's no uniform logic to how the returns are distributed. Instead, price action, the asset's price plotted over time, is perfectly determinative and trying to divine probabilities of returns is irrelevant. For the bottom left quadrant, the price is predictable and the probabilities are knowable. Such an asset could be thought of as a "stablecoin" where there shouldn't be any deviations in price and thus the probabilities of future returns are knowable because there won't be daily changes. Finally, the top right quadrant shows an asset that presumably follows a 'random walk' but the distribution of returns are not normal. With the Volatility-Risk compass, a clearer picture can be drawn to determine when an asset is risky. In this sense, kurtosis can be used to indicate which quadrant an asset falls into. The kurtosis of an asset quantifies its options' risk because excess kurtosis means that the probabilities priced in by the Black-Scholes model are less reliable. Furthermore, finding the kurtosis of assets held in a portfolio can provide a more accurate measurement of that portfolio's value when looking at 'risk-adjusted returns.' This is because kurtosis will indicate the 'tail risk' of a portfolio. By identifying 'tail risk' and not just volatility, investors can gain a better understanding of the risk present in their portfolios. Therefore, when trying to account for the risk present in bitcoin, focusing on bitcoin's volatility is not sufficient. It's important to test how the volatility is distributed. If returns are normally distributed, then accurate prices for bitcoin options can be given and risk can be quantified. If returns are not normally distributed, then the probabilities of returns, reflected in option prices, are much less reliable. By looking at the daily returns for bitcoin from 2016 to August, 2019, taken from coinmarketcap, excess kurtosis can be found. Figure 8 shows the distribution of 2016, 2017, and 2018 daily returns for bitcoin overlaid with a normal distribution curve for those same returns. As is clear from the charts, bitcoin does not follow a normal distribution. Rather, comparing Figure 8 with Figure 5, it can be seen that bitcoin looks more similar to a leptokurtic distribution. This observation is confirmed when calculating the excess kurtosis for 2016, 2017, and 2018, with the results being 10.03, 3.29, and 2.05. The range in each chart is from -20.00% to 20.00% daily returns. For 2017, there were two observations that fell outside of this range. These values were actually 22% in daily returns. 2017 was a particularly volatile year for bitcoin and there were observations at both of the edges of the range. Volatility slightly dropped in 2018, but significant ‘tail risk’ was still present. Notably, 2018 saw more extreme volatility to the downside. So far, in 2019, bitcoin’s kurtosis has not subsided. Rather, there is a slight uptick in 2019 compared to 2018 with excess kurtosis being 3.92. While there is far greater probability of daily returns being close to the mean for 2019, the probabilities for the tails are relatively evenly distributed. This shows a classic leptokurtic distribution, where there is a sharp drop off from the mean and “fatter” or more even distributions in the tails. Overall the excess kurtosis shows that probabilities for bitcoin’s daily returns are skewed to overrepresent the mean and tails from what should be expected when using the Black-Scholes model. This can be troublesome when pricing options because implied volatility becomes less reliable. Instead of volatility being normally distributed, and thus giving outsized moves greater than 2 standard deviations very low probabilities, excess kurtosis means that very large changes in price are less predictable. As a result, an asset may show relatively low volatility, and subsequently give very low probabilities to large changes in price, but such volatility is not truly indicative of how volatile an asset can be. Looking back to the Volatility-Risk compass in Figure 7, bitcoin is likely represented in the top right quadrant. Presumably, bitcoin's price is unpredictable and random. However, the probabilities of returns do not follow a normal distribution. Thus far, it has been shown that bitcoin exhibits excess kurtosis. So is bitcoin an outlier when compared to the broader stock market? Yes, and no. Excess kurtosis is by no means only germane to bitcoin. Plenty of assets exhibit excess kurtosis, even indexes like the S&P 500. In 2018, the S&P 500 had an excess kurtosis equal to 3.09. While this is lower than bitcoin’s 2016, 2017, and 2019 excess kurtosis, it’s actually higher than bitcoin’s 2018’s kurtosis. Of course, 2018 was a very volatile year for the S&P 500. At the beginning and end of 2018, there were sharp declines in price and very aggressive price rallies. In fact, for 2018, the S&P 500 saw the largest spike in the one-year rolling kurtosis that has been observed in almost 30 years. As Figure 11 shows, the beginning of 2018 represented relatively extreme kurtosis when the index suddenly fell >10%. However, Figure 11 also shows that, give or take a few spikes, excess kurtosis has been low since 1991. A histogram of the S&P 500’s daily returns for 2018 show a distribution much closer to a normal distribution when compared to bitcoin. In Figure 12, excess kurtosis is clearly present because there are many days where the returns are well outside a normal distribution curve. Yet, for the 250 observations of daily returns in 2018 for the S&P 500, only 6 fall outside the normal distribution curve. In 2017, for the 364 observations of daily returns for bitcoin (bitcoin trades everyday of the year and observations of returns begin Jan, 02), 28 fell outside of the normal distribution curve. In this respect, when only focusing on instances that fall outside the tails of the normal distribution curve, S&P 500 presented a lower chance of experiencing extreme, and hence unpredictable, price swings. What can be taken away from these findings? While it’s certainly true that the S&P 500 does not fall into a normal distribution curve, it is also true that it fits a normal distribution better than bitcoin. Why this is can only be left to speculation at this point. I believe that there can be at least three possible reasons posited. First, bitcoin represents a different asset class and follows different assumptions from the broader market. Second, bitcoin, at this time, is an immature market and will be tamed by institutional investment. Or third, the Black-Scholes model is becoming less reliable overall and volatility, as well as unpredictable distributions of such volatility, will become the ‘new normal.’ Regardless of which one of these reasons is true, it must be concluded that bitcoin options are likely mis-priced and ‘delta-hedged’ bitcoin portfolios are not valued accurately. This can be problematic, especially in the context of many new crypto derivatives exchanges coming to the market, because the relatively high implied volatility of bitcoin does not tell the whole story. Lurking behind such volatility is excess kurtosis, and as a result an investor’s ability to hedge away risk is greatly reduced. From Brenner’s and Subrahmanyan’s “A Simple Formula To Compute Implied Standard Deviation” we can derive from Black-Scholes: C = Option price S = Current stock price T = Time until expiration σ = volatility In the corn example, the option price was $2 and the current price of corn was $3.50 and assume that option contract is for 6 months, or .5 years. It should be noted that this formula is for deriving volatility from a call option’s price. The example used a put option. As a result, the volatility given will be slightly inaccurate. Yet, for simplicity, and because this is a tangent anyways, I’m going to treat the option as a call option. Using the equation above: σ = √(2 π/.5)*(2/3.5) σ = 202.73% Brenner, Menachem and Marti Subrahmanyan, “A Simple Formula to Compute the Implied Standard Deviation.” Financial Analysts Journal 44(5) (1988). 81 Black, Fischer & Myron Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81(3) (1973). Fama, Eugene F. “Efficient Capital Markets: A review of Theory and Empirical Work.” The Journal of Finance 25(2) (1970). Hull, John C. Options, Futures, and Other Derivatives. Tenth Edition. New York: Pearson Education, (2018). MacKenzie, Donald. An Engine, Not a Camera. Cambridge, MA: MIT Press, 2006. McAlevey, Lynn G. and Alan F. Stent. “Kurtosis: a Forgotten Moment.” International Journal of Mathematical Education in Science and Technology 49(1) (2017). Malkiel, Burton G. “The Efficient Market Hypothesis and Its Critics.” Journal of Economic Perspectives 17(1) (2003). Westfall, Peter H. “Kurtosis as Peakedness, 1905–2014. R.I.P.” Am Stat 68(3) (2014).

On geometry influence on the behavior of a quantum mechanical scalar particle with intrinsic structure in external magnetic and electric fields Relativistic theory of the Cox’s scalar not point-like particle with intrinsic structure is developed on the background of arbitrary curved space-time. It is shown that in the most general form, the extended Proca-like tensor first order system of equations contains non minimal interaction terms through electromagnetic tensor and Ricci tensor . In relativistic Cox’s theory, the limiting procedure to non-relativistic approximation is performed in a special class of curved space-time models. This theory is specified in simple geometrical backgrounds: Euclid’s, Lobachevsky’s, and Riemann’s. Wave equation for the Cox’s particle is solved exactly in presence of external uniform magnetic and electric fields in the case of Minkowski space. Non-trivial additional structure of the particle modifies the frequency of a quantum oscillator arising effectively in presence if external magnetic field. Extension of these problems to the case of the hyperbolic Lobachevsky space is examined. In presence of the magnetic field, the quantum problem in radial variable has been solved exactly; the quantum motion in z-direction is described by 1-dimensional Schrödinger-like equation in an effective potential which turns out to be too difficult for analytical treatment. In the presence of electric field, the situation is similar. The same analysis has been performed for spherical Riemann space model. O.V. Veko111Kalinkovichi Gymnasium, Belarus,firstname.lastname@example.org, K.V. Kazmerchuk222Mosyr State Pedagogical University, Belarus, email@example.com, E.M. Ovsiyuk333Mosyr State Pedagogical University, Belarus, firstname.lastname@example.org, V.V. Kisel444Belarusian State University of Informatics and Radioelectronics, V.M. Red’kov555B.I. Stepanov Institute of Physics, NAS of Belarus, email@example.com PACS numbers: 02.30.Gp, 02.40.Ky, 03.65Ge, 04.62.+v MSC 2010: 33E30, 34B30 KeywordsIntrinsic structure, scalar particle, curved space-time, generalized Schrödinger equation, magnetic field, electric field, Minkowski, Lobachevsky, Riemann space models 1 Scalar Cox’s particle with intrinsic structure In 1982 W. Cox proposed a special wave equation for a scalar particle with a larger set of tensor components than the usual Proca’s approach includes: he used the set of a scalar, 4-vector, antisymmetric and (irreducible) symmetric tensor, thus starting with the 20-component wave function (see Section 13). First, let us consider the system of Cox’s equations in the Minkowski space. We will use a Proca-like generalized system obtained after elimination from the initial system of Cox’s equations two second-rank tensors (see Section 13): is a tensor inverse to ( stands for additional parameter responsible for non-trivia intrinsic structure of a scalar particle in Cox’s approach): are expressed through electromagnetic invariants (for more technical details see Section 13). In geometrical models with metrics of special type one cap perform non-relativistic approximation and derive extended Schrödinger type equation (see Section 13): where the notation is used It is a generalized Schrödinger equation for the particle with intrinsic structure. In presence of a pure magnetic field, the above equation (1.3) takes a more simple form where the notation is used (let ) In presence of a pure electric field, the above equation (1.3) takes the form 2 Cox’s particle in the magnetic field, Minkowski space Let the homogeneous magnetic field be directed along the axis : Recalculating the potential to cylindrical coordinates by the formulas The metric tensor in these coordinates and field variables are determined by The Schrödinger equation for this case reads below we will use the notation After using the substitution for the wave function we get the radial Schrödinger equation By physical reasons parameter must be purely imaginary (see Section 13): ; so the radial equation reads With the use of notation equation (2.5) can be written as which coincides with the equation arising in the problem of the usual particle in the magnetic field. Its solutions are known. We present here only an expression for the energy spectrum from this after translating to ordinary units we obtain With the use of notation the formula for the energy levels can be written as Thus, the intrinsic structure of the Cox’s particle modifies the frequency of the quantum oscillator (in fact, this result was firstly produced in different formalism by Kisel ). 3 Cox’s particle in the magnetic field in the Lobachevsky space In a special (cylindrical) coordinate system in the Lobachevsky space, analogue of the uniform magnetic field is determined by the relations (we use dimensionless coordinate obtained by dividing on the curvature radius ): The wave equation in this case reads Below the notation is used: After using the substitution for the wave function: the Schrödinger equation (3.2) gives (the function must be imaginary, ) In this equation, the variables are separated: The radial equation for the function reads the equation for is (remember that ) 4 Analysis of the equation in the variable In equation (3), let us eliminate the first derivative term: Eq. (4.1) can be viewed as the Schrödinger equation in the effective potential field . The corresponding effective force is We find the points of local extremum: and the roots of a quadratic equation When considering the bound states (for motion in the variable ) we have . This means that the square root in (4.3) is an imaginary number. Consequently, the point of zero force (equilibrium points) except cannot exist. The situation is illustrated in the Fig. 1. After the change of variables , the differential equation (4.1) reads Note that singular points are located outside the physical range of the variable. Further progress in analytical treatment of eq. (4.4) (with 5 singular points) is hardly possible. 5 Solution of the radial equation Let us turn to the radial equation (3.7) for the function . It is solvable in hypergeometric functions – see more detail . Below we will write done only final results on energy spectrum. There exist only finite series of bound states, defined by relations obeys the restriction In usual units the last relation can be written as: In the limit of vanishing curvature, we obtain the known result in the flat space 6 Cox’s particle in the electric field, Minkowski space Schrödinger equation for Cox’s particle in the electric field has the form (see Section 13) the notation is used: Let us use cylindric coordinates First, we get (let it be ) Next, we consider the Hamiltonian In explicit form, the extended Schrödinger equation looks as follows (to allow for the imaginary character of , we make formal change ) With the substitution and the notation After separation of the variables ( stands for the separation constant) we derive In fact, (6.6) coincide with the well known equations for an ordinary particle in the uniform electric field. Equation in the variable looks as a one-dimensional Schrödinger equation in the potential of the form : The form of the curve says that any particle moving from the right must be reflected by this barrier in vicinity of the point (we assume that electric force acts in positive direction of the axis ). Mathematic solutions of the equation (6.7) can be expressed in Airy function. Indeed, let us change the variable let it be (for definiteness ) then we arrive at the Airy equation to the turning point there corresponds the value . Eq. (6.9) can be related to the Bessel equation. Indeed, let us introduce the variable then Airy equation gives Applying the substitution , we arrive at the Bessel equation with two linearly independent solutions Thus, general solutions of Airy equation can be constructed as linear combinations of and with the notation , one expresses two independent solutions of the Schrödinger equation as follows

Poker Combinations Inhaltsverzeichnis The poker hand ranking charts are based on the probability for each distinct hand rank. More unlikely combinations are ranked higher. Suchen Sie nach poker combinations-Stockbildern in HD und Millionen weiteren lizenzfreien Stockfotos, Illustrationen und Vektorgrafiken in der. nach poker combinations-Stockbildern in HD und Millionen weiteren lizenzfreien Stockfotos, Illustrationen und Vektorgrafiken in der Shutterstock-Kollektion. Keep forgetting poker combinations? Have this app on your phone and use it as a cheat sheet. Features: Simple, clean material design -Illustrations and. Poker Hand Rankings. () poker. Royal Flush. Straight Flush. Four of a Kind. Full House. A A A*J*: *5*7. Flush. Straight. Three of a kind. *7*6. Two Pair. Suchen Sie nach poker combinations-Stockbildern in HD und Millionen weiteren lizenzfreien Stockfotos, Illustrationen und Vektorgrafiken in der. Im Kartenspiel Poker beschreibt der Begriff Hand die besten fünf Karten, die ein Spieler nutzen kann. Die Rangfolge der einzelnen Kartenkombinationen ist bei. Bluffing in optimal poker play is often not a profitable play in and on itself. Instead, the combination of bluffing and value betting is designed to ensure the optimal. Example: We are dealt AKo, what are our chances of flopping a straight? Back to top. In high-low split games, both the highest-ranking and lowest-ranking hands win, though different rules are used to rank the high and low hands. Top Menu. Poker con Dinero Real. Each full house is ranked first by the rank of its triplet, and then by the final, Paysavecard Guthaben are of its pair. How many possible combinations of AK and TT are out there that our opponent could hold? An understanding of basic probabilities will give your poker game a stronger foundation, Combinatorics (card combinations), statistics (sample size) and other. Find poker hand rankings in order from strongest to weakest and learn what poker hands beats A full house is a combination of a three-of-a-kind and a pair. Im Kartenspiel Poker beschreibt der Begriff Hand die besten fünf Karten, die ein Spieler nutzen kann. Die Rangfolge der einzelnen Kartenkombinationen ist bei. Bluffing in optimal poker play is often not a profitable play in and on itself. Instead, the combination of bluffing and value betting is designed to ensure the optimal. Poker Combinations VideoThere are no cards left for a kicker. Kategorie : Spielbegriff Poker. Straight Flushes are almost as rare as Royal Flushes. Die anderen beiden Karten müssen zwei der zwölf verbliebenen Werte haben und können in vier verschiedenen Farben sein:. Pair 2 cards of the same rank. Namensräume Artikel Click here. What is the click here royal flush in poker? To maximize your chances you should always keep all suited cards 10 or above if you have at least 2 and Poker Combinations the rest. A three-of-a-kind is composed of three cards of the same rank. However, at PalaPoker. Es gibt dann. A player Tipico Wettschein Scannen one pair has 3 kickers, a player with trips has 2 kickers, and a player with 2 pair or quads has 1 kicker. Es gibt ohne Ass als höchste Karte neun verschiedene mögliche höchste Karten und vier verschiedene Farben:. A Royal Flush is the best possible poker see more and of course always beats any other flush. However, at PalaPoker. Diese Hand ist auch als steel wheel bekannt. If they are identical, the player with the higher pair wins. Anzahl möglicher Kombinationen. Two Pair 2 cards of the same rank twice. Bei zwei konkurrierenden High Cards zählt der Kicker, bei Gleichheit der zweite Kicker und so weiter. Jedes der zwei Paare kann einen der dreizehn Werte und zwei der vier Farben haben. Es entscheidet die Höhe des Vierlings. Die Angaben zu den Wahrscheinlichkeiten der unterschiedlichen Hände sind abhängig this web page der Spielvariante; sind https://toplistsiteleri.co/how-to-play-online-casino/beste-spielothek-in-zachow-finden.php davon https://toplistsiteleri.co/australian-online-casino-paypal/spiele-six-shooter-video-slots-online.php, ob es Gemeinschaftskarten gibt z. Poker Combinations - Comments (43)Diese Hand ist eigentlich ein Straight Flush, wird durch ihre Rolle als beste Hand im Poker und ihre Seltenheit jedoch gesondert betrachtet. Eine unabhängig von der Spielerstrategie gültige Berechnung ist somit nicht möglich, und auf die Bestimmung einer möglicherweise optimalen Tauschstrategie kann hier nicht eingegangen werden. We also use third-party cookies that help us analyze and understand how you use this website. A higher rank is only possible when playing with a Joker. What is a flush in poker? Die restlichen Karten werden jetzt noch auf die 39 Karten verteilt, die keinen Flush mit mehr gleichfarbigen Karten bilden würden. Poker Combinations Strategy SectionsIf two players have four-of-a-kind, then the one with the highest four-of-a-kind wins. Er besteht aus einer der zehn möglichen höchsten Karten. Straight 5 cards in a row. Ein Straight besteht aus fünf Karten. Dann wird die Anzahl der Kombinationen, die einen Straight Flush Spielothek in Aschach an Steyr finden würden, wodurch eine ranghöhere Hand entstehen würde abgezogen. Es gibt. Full House 3 and 2 cards of the same rank. For example, it may often surprise players to learn that on boards with a possible straight, there are more combinations of straights than flushes. Click the following article is not enough to. Wrap Around Straight - Poker Terms. Retrieved 13 July Invite your friends and reap rewards! Bei zwei konkurrierenden High Cards source der Kicker, bei Gleichheit der zweite Kicker und so weiter. Deposit Methods. Die wichtigste Änderung stellt ein Deck mit einem Joker dar. Die unterschiedlichen Spielvarianten zeichnen sich dadurch aus, dass es jeweils unterschiedliche Möglichkeiten gibt, um zu einer Hand aus fünf Karten zu gelangen. In case two players have the same pair, then the one with the highest kicker wins. So what is combinatorics? It may sound like rocket science and it is definitely a bit more complex than some other poker concepts, but once you get the hang of combinatorics it will take your game to the next level. Combinatorics is essentially understanding how many combos each of your opponent's potential holdings are and deducing their potential holdings utilizing concepts such as removal and blockers. There are 52 cards in a deck, 13 of each suit, and 4 of each rank with poker hands in total. To simplify things just focus on memorizing all of the potential combos to start:. Here is a short video example of using combinatorics to count the number of ways a non-paired hand AK can be arranged i. So now that we have this memorized, let's look at a hand example and how we can apply combinatorics in game. He flats and we go heads up to a flop of. Our opponent is representing a polarized range here. He is either nutted or representing missed draws so we find ourself in a tough spot. This is where utilizing combinatorics to deduce his value hands vs bluffs come into play. Now we need to narrow down his range given our line and his line. Let's take a look at how we do this So there is exactly 1 combo of AA. We checked flop to add strength to our check call range although a bet with a plan to triple barrel is equally valid in this situation SB vs BTN and because of this our opponent may not put us on an A here. The problem in giving him significant credit for this part of his bluffing range is the question of would he really shove here with good SDV Showdown Value? These are the types of questions we must ask ourselves to further deduce his range along with applying the combinatoric information we now have. Now, this is the range we assigned him in game based on the action and what we perceived our opponents range to be. We are not always correct in applying the exact range of his potential holdings, but so long as you are in the ballpark of that range you can still make quite a few deductions to put yourself in the position to make the correct final decision. This is not enough to call. We ultimately made our decision based on the fact that we felt our opponent was much less likely to jam with his bluffs in this spot. Given that it was already a close decision to begin with, we managed to find what ended up being the correct fold. A good starting point is to simply memorize all of the possible hand combinations listed above near the beginning of the article. Get access to our minute lesson on Combinatorics and PokerStove by clicking on one of the buttons below:. Playing Cash or Tournament Poker Games. Learn to Play Poker Today. How many different combinations of AK are there? If we were so inclined, we could list every possible way of making AK. But most players simply remember that there are 16 combinations of every unpaired hand - 12 are off-suit, and 4 are suited. How many different combinations of 66 are there? Various board cards are known, reducing the available combinations of different types of starting hand. How many combinations of AK are there? The key to this question is understanding that there are now only 3 Aces left in the deck since one of them is clearly on the flop. As such, we have 3 available Aces and 4 available Kings. Returning to our preflop question with the pocket 6s, there are four available Sixes in the deck. How many combinations of 66 are there? We should now be equipped to answer slightly more advanced questions regarding combinations. Example: We hold pocket Aces preflop. However, our opponent holds two of the Aces meaning that the number of combos will be affected due to card removal effects. How many different ways of making a set are there? How many different ways of making top pair are there? This situation is where logic and common sense come in. There are 13 card ranks in a deck, so if we exclude AA, A8 and A9, we must be left with 10 different types of Ax hand that make top pair, each with 12 combinations. So far, we have seen ways of calculating hole card combinations. This second definition will help us calculate the probability of various board runouts. Demonstrate this mathematically using combinations. The result is 22, possible flop combinations. So why is it that a quick online search reveals 19, possible flop combinations? Could it be that those calculations are accounting for the fact that 2 cards in the deck are often known i. So, when players say there are 19, flops, they mean assuming 2 cards from the deck are known. What are our chances of flopping a flush? We already know how many different flops there are 19, so our next goal should be to establish how many ways there are of making a heart flush assuming there are 11 hearts left in the deck. How can we establish the number of different 3 card combinations that can be dealt from a selection of 11 cards? However, when we check with equity calculation software, it tells us that the chance of flopping a flush is 0. Can you see what the discrepancy is? Some of those three heart flops give us a straight flush. We need to know how many so that we can discount it from our total number of flush flops. There we go. We have arrived at 0. Perhaps we are starting to see a pattern for the simplified version of the formula when using combinations to arrive at the likelihood of various flops. There are 8 ways we can hit either a J or Q assuming we hit a T on the flop. This solution is the number of different ways of flopping the straight if we assume the order of the cards is relevant which is not. The mathematical name for a selection where the order matters is permutation rather than combination and involves a slightly different formula. The second part of the formula divide by 3! Or 6 is used to account for duplicate flops where just the ordering is different. Using combinations to answer this question is not strictly necessary; we can confirm the answer by using the basic probability formula for the probability of successive events.

1. Which of the following portions of a receiver can be effective in eliminating image signal interference? 2. What is the term for the time required for the capacitor in an RC circuit to be charged to 63.2% of the applied voltage? 3. How many elevation lobes appear in the forward direction of the antenna radiation pattern shown in Figure E9-2? 4. How many horizontal lines make up a fast-scan (NTSC) television frame? 5. What is the direction of an ascending pass for an amateur satellite? 6. What type of signal is picked up by electrical wiring near a radio antenna? 7. What is the permitted mean power of any spurious emission relative to the mean power of the fundamental emission from a station transmitter or external RF amplifier installed after January 1, 2003, and transmitting on a frequency below 30 MHZ? 8. What is the effect of ringing in a filter? 9. What types of amateur stations may automatically retransmit the radio signals of other amateur stations? 10. Under clear communications conditions, which of these digital communications modes has the fastest data throughput? 11. What is a common use for point contact diodes? 12. For which types of out-of-pocket expenses do the Part 97 rules state that VEs and VECs may be reimbursed? 13. In Figure E6-1, what is the schematic symbol for a PNP transistor? 14. Which of the following tests establishes that a silicon NPN junction transistor is biased on? 15. What happens to the conductivity of a photoconductive material when light shines on it? 16. In polar coordinates, what is the impedance of a series circuit consisting of a resistance of 4 ohms, an inductive reactance of 4 ohms, and a capacitive reactance of 1 ohm? 17. Which of the following is a good technique for making meteor-scatter contacts? 18. What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 14.25 MHz and a Q of 187? 19. How does an impedance-matching circuit transform a complex impedance to a resistive impedance? 20. Which of the following may reduce or eliminate intermodulation interference in a repeater caused by another transmitter operating in close proximity? 21. How is the compensation of an oscilloscope probe typically adjusted? 22. Which of the following would be used to reduce a signal's frequency by a factor of ten? 23. In Figure E6-5, what is the schematic symbol for the NOT operation (inverter)? 24. What is the approximate feed point impedance at the center of a two-wire folded dipole antenna? 25. What term describes station output, including the transmitter, antenna and everything in between, when considering transmitter power and system gains and losses? 26. What is the function of a reactance modulator? 27. What core material property determines the inductance of a toroidal inductor? 28. What is the name of the high-angle wave in HF propagation that travels for some distance within the F2 region? 29. How can an RF power amplifier be neutralized? 30. Which of the following factors may affect the feed point impedance of an antenna? 31. What is one use for a pulse modulated signal? 32. What is the only amateur band where transmission on specific channels rather than a range of frequencies is permitted? 33. What is the purpose of C1 in the circuit shown in Figure E7-3? 34. Which of the following types of amateur station communications are prohibited? 35. How is power-supply voltage normally furnished to the most common type of monolithic microwave integrated circuit (MMIC)? 36. How does the total amount of radiation emitted by a directional gain antenna compare with the total amount of radiation emitted from an isotropic antenna, assuming each is driven by the same amount of power? 37. What is the purpose of a Wilkinson divider? 38. What is an electromagnetic wave? 39. Which of the following can divide the frequency of a pulse train by 2? 40. What do the arcs on a Smith chart represent? 41. Which of the following could account for hearing an echo on the received signal of a distant station? 42. Which of the following special provisions must a space station incorporate in order to comply with space station requirements? 43. Which insulating material commonly used as a thermal conductor for some types of electronic devices is extremely toxic if broken or crushed and the particles are accidentally inhaled? 44. What is the velocity factor of a transmission line? 45. During a VHF/UHF contest, in which band segment would you expect to find the highest level of activity? 46. What is the advantage of including a parity bit with an ASCII character stream? 47. What condition must exist for a circuit to oscillate? 48. What is the modulation index of an FM-phone signal having a maximum frequency deviation of 3000 Hz either side of the carrier frequency, when the modulating frequency is 1000 Hz? 49. What is the typical bandwidth of a properly modulated MFSK16 signal? 50. What is the power factor of an RL circuit having a 45 degree phase angle between the voltage and the current? Warning! You skipped 0 question(s). Log in to set preferences.

INLO–PUB–14/94 P.J. Rijken and W.L. van Neerven August 1994 Massive lepton pair production in hadronic interactions is besides deep inelastic lepton-hadron scattering one of the most important probes of the structure of hadrons. It is well established that one of the dominant production mechanisms is the Drell-Yan (DY) process . Here the lepton pair is the decay product of one of the electroweak vector bosons of the standard model ( and ) which in the Born approximation are produced by the annihilation of quarks and anti-quarks coming from the colliding hadrons. This process is of experimental interest because it provides us with an alternative way to measure the parton densities of the proton and neutron which have been very accurately determined by the deep inelastic lepton hadron experiments. Moreover it enables us to measure the parton densities of unstable hadrons like pions and kaons which is impossible in deep inelastic lepton-hadron scattering. Besides the measurement of the parton densities there are other important tests of perturbative quantum chromo dynamics (QCD) which can be carried out by studying the DY Here we want to mention the scale evolution of the parton densities, although not observed in this process because of the low statistics, and the measurement of the running coupling constant which includes the QCD scale . Finally this process constitutes an important background for other production mechanisms of lepton pairs. Examples are and decays or thermal emission of lepton pairs in heavy-ion collisions . The DY process is also of theoretical interest. Since it is one of the few reactions which can be calculated up to second order in perturbation theory it enables us to study the origin of large QCD corrections which are mostly due to soft gluon bremsstrahlung and virtual gluon contributions. In order to control these corrections in the perturbation series one has constructed various kinds of resummation techniques mostly leading to the exponentiation of the dominant terms -. Another issue is the dependence of the physical quantities on the chosen scheme and the choice of scales. Since the perturbation series is truncated the theoretical cross section will depend on the scheme and the renormalization/factorization scale . These dependences can be reduced by including higher order terms in the perturbation series. An alternative way is to determine itself (optimum scale) by using so called improved perturbation theory like the principle of minimal sensitivity (PMS) , fastest apparent convergence (FAC) or the Brodsky-Lepage-Mackenzie (BLM) procedure . The first fixed target experiment on massive lepton pair production was carried out by the Columbia-BNL group . Later on this process was studied in many other experiments which were carried out at increasing energies (for reviews see ). When the statistics of the data was improving one discovered that the cross section could not be described by the simple parton model given by S.D. Drell and T.M. Yan in . This was revealed for the first time by the NA3 experiment (see also ) where the data show a discrepancy in the normalization between the experimental and theoretical cross section. This discrepancy is expressed by a so called -factor which is defined by the ratio between the experimentally observed cross section and its theoretical prediction. The above group and the experiments carried out later on show that this -factor ranges between 1.5 and 2.5 and is roughly independent of the type of incoming hadrons. The most generally accepted explanation of this -factor was provided by perturbative QCD. The calculation of the order corrections - to the DY cross section in show that a considerable part of the -factor can be attributed to next-to-leading order effects. However the order corrections do not account for the whole -factor. More recent experiments - still indicate that the ratio between the experimental cross section and the order corrected theoretical prediction is about 1.4, a number which might be explained by including QCD corrections beyond order as we will show in this paper. As has been mentioned at the beginning the DY process is one of the few processes where the order corrections to the coefficient function are completely known. The latter refers to the cross section only where denotes the lepton pair invariant mass. This coefficient function has been calculated in the as well as in the DIS scheme. However in the case of the double differential cross section () one has only calculated the order part of the coefficient function which is due to soft and virtual gluon contributions because the remaining part is very complicate to compute. Fortunately as is shown in the literature - the soft plus virtual gluon corrections dominate the total and differential DY cross sections in particular at fixed target energies so that we can restrict to them to make reliable predictions. An analysis of the higher order corrections to the total DY cross section for - and -production at large hadron collider energies has been performed in [24, 25]. Such an analysis is still missing for the DY process at fixed target energies and therefore we present it here. In particular we want to show that the discrepancy in the normalization between the order corrected DY cross section and the one measured at the fixed target experiments can be partially explained by including the order contributions due to soft plus virtual gluon effects. This paper is organized as follows. In section 2 we present the expressions for the various DY cross sections and give a review of the partonic subprocesses included in our analysis. In section 3 the validity of the soft plus virtual gluon approximation will be discussed and we make a comparison between the order corrected cross section and the most recent fixed target DY data. In appendices A and B we give the coefficient functions for () corrected up to order and order respectively. They are presented for arbitrary renormalization and mass factorization scale in the - as well as in the DIS-scheme. 2 Higher order QCD corrections to Massive lepton pair production in hadron-hadron collisions proceeds through the following reaction Here and denote the incoming hadrons and is one of the vector bosons of the standard model (, or ) which subsequently decays into a lepton pair (,). The symbol denotes any inclusive hadronic final state which is allowed by conservation of quantum numbers. Following the QCD improved parton model as originally developed in the double differential DY cross section can be written as Here where denotes the lepton pair invariant mass. The longitudinal momentum fraction of the lepton pair and the Bjørken scaling variable are defined by where stands for the center of mass energy of the incoming hadrons and . The quantity is the pointlike DY cross section which describes the process where and denote the incoming quark and anti-quark respectively. If we limit ourselves to , then gets the form Here the width of the -boson is taken to be energy independent and all fermion masses are neglected since they are much smaller than . The charges of the leptons and quarks are given by The vector- and axial-vector coupling constants of the -boson to the leptons and quarks are equal to The function in (LABEL:diff_cross) stands for the combination of parton densities corresponding to the incoming partons and (). Finally denotes the DY coefficient function which is determined by the partonic subprocess where now represents any multi partonic final state. Both functions and depend in addition to the scaling variables and also on the renormalization and mass factorization scales which are usually put to be equal to . Besides the cross section in (LABEL:diff_cross) one is sometimes also interested in the rapidity distribution of the lepton pair. In this case the left hand side in (LABEL:diff_cross) is replaced by where denotes the rapidity defined by (see (2.3)) Furthermore on the right hand side the coefficient function is replaced by its analogue corresponding to the cross section The coefficient function (LABEL:diff_cross) can be expanded as a power series in the running coupling constant as follows In lowest order the coefficient function of the differential cross section (LABEL:diff_cross) is determined by the subprocess Here either stands for the virtual photon or the -boson and the coefficient function is given by In addition to the process above we have another reaction which instead of a quark or anti-quark has a gluon in the initial state This reaction contributes to . Both contributions and have been calculated in [17, 18, 27] (DIS-scheme) and in (-scheme) and are presented in (A.1) and (A.7), (A.8) respectively. A part of the order corrections to the coefficient function corresponding to has also been calculated in . These corrections originate from the soft plus virtual gluon contributions. They consist of the two-loop corrections to process (2.13) and the one-loop correction to process (2.15) where the gluon is taken to be soft. Furthermore one has also included the bremsstrahlungs process and fermion pair production where the gluons were taken to be soft and the quark–anti-quark pair in the final state of (2.18) has a low invariant mass. All above corrections contribute to and can be found in appendix B for arbitrary factorization and renormalization scale where they are presented in the - as well as in the DIS-scheme. The hard gluon corrections (2.17) and the other two-to-three body processes (see below) are very hard to compute at least for the double differential cross sections. Fortunately as has been shown in - the bulk of the order radiative corrections to the cross sections and is constituted by the soft plus virtual gluon contributions to . Therefore within the experimental and theoretical uncertainties one can assume that the order part of the coefficient function which is only due to soft plus virtual gluon contributions is sufficient to describe the next-to-next-to-leading order DY cross section at fixed target energies. This can be tested for the quantity which is defined by which can also be written as where now stands for the coefficient function corresponding to the integrated cross section . Since the exact order corrections to this coefficient function are completely known see (-scheme) and (DIS-scheme) one can now make a comparison between the exact DY cross section coming from the complete coefficient function and the approximate cross section due to the soft plus virtual gluon part. The full order contribution to the DY coefficient function requires besides the calculation of the subprocesses mentioned above the computation of the following two-to-three body partonic subprocesses. First we have the bremsstrahlungs correction to (2.16) which entails the computation of the one-loop corrections to (2.16). In addition one has to add the subprocesses Reactions (2.21),(2.22),(2.23) and (2.24) contribute to the coefficient functions and respectively. The exact result of the coefficient function calculated up to order for gives an indication about the validity of the soft plus virtual gluon approximation of (or ) for which a complete order calculation is still missing. In one has made a detailed analysis of this approximation for the total cross section of - and - production which is derived from (2.20) by integrating over . From this analysis one infers that the approximation works quite well in order as well as in order when provided the DY coefficient function is computed in the DIS-scheme. that in practice one can only apply it to the cross section measured at the (). The reason that this happens in the DIS-scheme is purely accidental. It originates from the large coefficient of the delta-function appearing in which is small in the -scheme. Apparently the combination of the anomalous dimension (Altarelli-Parisi splitting function) and the remaining part of the coefficient function is very small in the DIS-scheme. It is expected that the approximation will even work better when , a condition which is satisfied by fixed target experiments. In this case the phase space of the multi partonic final state in the above reactions will be reduced so that only soft gluons or fermion pairs with low invariant mass can be radiated off. Their contributions manifest themselves by large logarithms of the type which appear in the coefficient function in the DIS- as well as in the Notice that the above analysis holds if the mass factorization scale is chosen to be . Therefore it is not impossible that the above conclusions have to be altered when a scale completely different from is adopted. Finally one has to bear in mind that a complete next-to-next-to-leading order analysis cannot be carried out yet because the appropriate parton densities are not available. The latter can be attributed to the fact that the three-loop contributions to the Altarelli-Parisi splitting functions or the anomalous dimensions have not been calculated up to so far. Therefore the analysis of the order corrected result for has to be considered with caution. This holds even more for the order corrected differential distribution or . In this section we start with a discussion of the validity of the soft plus virtual gluon () approximation of the order (LABEL:diff_cross). This is done by making a comparison with the integrated section (2.19) for which the coefficient function is completely known up to order . Then we include this approximation in our analysis of the fixed target muon pair data published in -. In particular we show that this correction partially accounts for the difference in the normalization between the data in - and the order corrected cross section calculated in [17, 18, 27, 28]. The calculation of the cross sections (2.19) and (LABEL:diff_cross) will be performed in the DIS- as well as in the -scheme chosen for the coefficient functions as well as for the parton densities. The coefficient functions for up to order can be found in (-scheme) and (DIS-scheme). The coefficient functions for corrected up to order are obtained from [17, 18, 27] (DIS-scheme) and (-scheme). In order to make this paper self-contained we have also presented them in appendix A. The order contribution as far as the soft plus virtual gluon part is concerned has been calculated in and is presented in both schemes in a more amenable form in appendix B. For the next-to-leading order nucleon parton densities we have chosen the MRS(D-) set for which a DIS- () and an -version () exist. Further we use the two-loop (-scheme) corrected running coupling constant with the number of light flavors and the QCD scale is the same as chosen for the MRS(D-) set. For the pion densities we take the leading log parametrization (DO1) in . Using this set one could only fit the old lepton-pair data (for references see ) by allowing an arbitrary normalization (or -factor) with respect to the leading order theoretical DY cross section. In this section it is shown that this factor can be partially explained by including higher order QCD corrections. Next-to-leading (NLO) order parton densities for the pion exist in and but they are only presented in the -scheme. Also here one has to use an arbitrary -factor to fit the data which is smaller than found for the leading order process since a part of the normalization is accounted for by the order corrections. Because of the missing (NLO) parton densities of the pion in the DIS-scheme we prefer to use the leading log parametrization in . Finally we choose the factorization scale to be equal to the renormalization scale where . All numerical results in this paper are produced by our Fortran program DIFDY which can be obtained on request. The plots will be presented at three different fixed target energies given by . At the first energy i.e. one has observed muon pairs produced in the reactions and measured by the E537 group . The second experiment is carried out at by the E615 group where the same lepton pair is measured in the reaction . Finally we discuss the E772 experiment [22, 23] at where the reaction is studied where is either represented by the isoscalar targets and or by (tungsten) which has a large neutron excess. Here we will only make a comparison with the -data. In the case of the E537, E615 experiments is given by whereas E772 used tungsten with . Here and denote the charge and atomic number of the nucleus respectively. Finally notice that at the above energies we can safely neglect the contributions coming from the Z-boson in (2.5) since the virtual photon dominates the cross section. Let us first start with the discussion of the approximation to the coefficient function corresponding to . The soft plus virtual gluon part of the coefficient function, which only appears in , can be written as where the logarithms have to be interpreted in the distributional sense (see ). The coefficients and depend on and the factorization scale . The above coefficients can be read off the explicit form of (3.1) given by eqs. (B.3), (B.8) in and (A.3), (A.8) in . In order to test the approximation to the DY cross section we study the following ratios In the above expressions () denotes the contribution to the DY cross section containing the exact part of the coefficient function where all partonic subprocesses are included. The quantities stand for the contribution to the cross sections where only the soft plus virtual gluon part of the coefficient function according to (3.1) is taken into account. In fig. 1 we have plotted and in the DIS-scheme for the -ranges explored by the three experiments mentioned above. From the figure we infer that the approximation overestimates the exact cross section by less than at small -values. At large -values this becomes better which is to be expected since in the limit the approximation becomes equal to the exact correction. In this limit hard gluon radiation and all other partonic subprocesses like quark-gluon scattering are suppressed because of the reduction in phase space. By comparing with we observe a slight improvement when higher order corrections are included in the denominator as well as in the numerator. In fig. 2 we did the same as in fig. 1 but now for the -scheme. Here we observe that the approximation underestimates the exact DY cross section by more than in particular when the C.M. energy is small like in the case of E537 () or E615 (). Furthermore (3.3) becomes worse than (3.2) in particular in the low -region. Hence we can conclude that for the approximation works better in the DIS-scheme than in the -scheme. In the case of the double differential cross section () the exact order contribution to the coefficient function is not known so that one can only make a comparison on the order level. The part of the coefficient function, of which the explicit form is given up to order in appendices A and B, becomes where the definitions for the distributions indicated by a plus sign can be found in appendix A. To study the approximation we define an analogous quantity as given for in (3.2). In the subsequent figures we plot the ratio where the meaning of and is the same as for and defined below (3.3). Notice that here we cannot present because the exact cross section is still unknown. Starting with the DIS-scheme we have plotted at (E537) for three representative -values as a function of in fig. 3. From this figure one infers that at small around the approximate cross section overestimates the exact one by about . This value is much larger than in the case of the integrated cross section where it was at maximum . The approximation becomes better when either or gets larger. The overestimation is even bigger when the energy increases. This can be observed in fig. 4 (, E615) or fig. 5 (, E772). Here one overestimates the exact cross section at small -values even by . If we repeat our calculations in the -scheme we observe a considerable improvement of the approximation to the double differential cross section (see figs. 6-8). Although like in the case of the approximation underestimates the cross section at high -values the difference with the exact one is less than . Summarizing our findings we conclude that in the case of the DIS-scheme the approximation works better for than for whereas for the -scheme just the opposite is happening, except for where and become close to 1 independent of the chosen scheme. Further from figs. 3-8 it appears that when is integrated over according to (2.19) we get a result which differs from the one obtained from in (2.20) in particular at small . On the first sight this is surprising because one expects the same cross section independent of the order of integration. However both procedures only lead to the same answer for when the full coefficient functions are inserted in the equations for (LABEL:diff_cross) and (2.20). If we limit ourselves to the part of the coefficient functions as given in (3.1) and (3.4) then the two procedures to compute only provides us with the same answer when . This we have also checked for the order contribution. Therefore the expression in (3.1) is not the integrated form of equation (3.4) except if . This explains why at large (3.2) and (3.5) are roughly the same and equal to 1 irrespective of the chosen scheme. The above properties of the approximation also reveal that if becomes much smaller than 1 one has to be cautious in predicting the still unknown from the values obtained for the known (3.3) and (3.5). In the subsequent part of this work we will use as a guiding principle that as long as we expect that the approximation of the second order contribution to will be very close to the exact result. If then one should not trust this approximation and one has to rely on the predictions obtained from the first order corrected cross section. This implies that for the experiments discussed in this paper one can make a reasonable prediction for the second order correction as long as . After having discussed the validity of the above approach at fixed target energies we will now make a comparison with the data of the E537 , E615 and E772 [22, 23] experiments. For that purpose we compute the Born cross section , the order corrected exact cross section and the order corrected cross section . Notice that in the latter only the contribution due to the coefficient function (3.4) (see appendix B) has been included because the other contributions are still missing. The computations have been carried out in the DIS-scheme. The results for the -scheme will be shortly commented upon at the end of this section. Starting with the experiment E537 () we have plotted the quantity in figs. 9 and 10 for the reactions and respectively. Notice that in is defined as which differs from the usual definition in [17, 18, 27, 28]. Since the higher order QCD corrections are calculated for with defined in (2.3) and the cross section is not a Lorentz invariant we had to change the -bins in table III of according to our definition above. Figs. 9 and 10 reveal that the data are in agreement with the order as well as with the order corrected cross section but lie above the result given by the Born approximation. The difference between the latter and the data is observed when we consider the quantity which is presented in figs. 11 and 12 for the above Even the order corrected cross section lies below the data for as can be seen in fig. 12. On the other hand the order corrected cross section is in agreement with experiment over the whole range. The second experiment, E615 also studies the reaction but now for . In fig. 13 we have compared the quantity with the data where is defined in the same way as in (3.6). Apart from the bump, which is due to the resonance at about , the order corrected cross section reasonably describes the experimental results whereas the Born and the order prediction fall below the data. The importance of the order contribution is also revealed when we study the double differential cross section for various regions, see figs. 14-19. The curves predicted by the Born and the order corrections all lie below the data. For even the order contribution is not sufficient to close the gap between theory and experiment. This is due to the presence of the in the region which has not been subtracted from the data. The discrepancy between the order corrected cross section and the data becomes even more clear when we plot the -factor (fig. 20) defined by in fig. 20 and compare the above expression with the experimental -factor which is given by where denotes the order corrected cross section. Fig. 20 shows that neither fit the data. The second order corrected -factor is closer to the data in the small -region. It is a pity that due to the presence of the in the data it is difficult to compare theory with experiment in particular in those regions of where the approximation is supposed to work. Finally we also made a comparison with the data obtained by the E772 experiment for the reaction carried out at . The main goal of this experiment was to find a charge asymmetry in the sea-quark densities of the nucleon i.e. . Here we are also interested whether the data obtained for are in agreement with the order corrected DY cross section. In fig. 21 we have plotted the data for and compared them with the predictions given by the Born, the order corrected and the order corrected cross section. The figure shows that the order corrections are needed to bring theory into agreement with the data. Notice that at this -value one obtains which is quite small for the approximation so that the result has to be interpreted with care. In the next figure (fig. 22) we study the effect of the higher order QCD corrections on the suppression of the cross section near which is caused by the difference between the up-sea and down-sea quark densities. Notice that the reaction is symmetric whereas the reaction is asymmetric around irrespective whether there is charge asymmetry or not. Therefore the reaction leads to an asymmetry even for isoscalar targets like . In fig. 22 we have presented the order corrected cross section for three different parton density sets for the nucleon. They are given by MRS(S0) and MRS(D0) where the former has a symmetric sea () whereas the latter contains an asymmetric sea () parametrization. For comparison we have also shown MRS(D-) which only differs from MRS(D0) that the gluon and sea densities have a much steeper small -behavior (lipatov-pomeron) than the ones given by MRS(D0) and MRS(S0) (non perturbative pomeron). Fig. 22 reveals that there is hardly any suppression of the cross section for while going from the symmetric sea (MRS(S0)) to the asymmetric sea (MRS(D0)) parametrization so that both parton density sets are in agreement with the data. If other parton densities are used like those discussed in the suppression for can be much larger. For the MRS-set it appears that a change in the small -behavior of the parton densities leads to a larger suppression of the cross section (compare MRS(D0) with MRS(D-)) than the introduction of a charge asymmetry in the sea-quarks (MRS(S0) versus MRS(D0)). In addition to the calculations performed in the DIS-scheme we have also presented in figs. 9-21 the order corrected cross section computed in the -scheme. Although the latter is an improvement with respect to the order corrected result it is smaller than the cross section computed in the DIS-scheme except when is large. This is not surprising because figs. 6-8 already indicate that the approximation underestimates the exact cross section in the case of the -scheme. Summarizing the content of this work we can conclude that up to the order level the soft plus virtual gluon contribution gives a fairly good approximation of the exact DY cross section . Therefore we expect that this approximation will also work for the correction as long as the cross section is computed at fixed target energies and for . In this -region we expect that all other partonic subprocesses are suppressed due to the reduction in phase space. This expectation is corroborated by a thorough analysis of the second order contribution to for which the exact coefficient function is known. Because of the missing pieces in the order contribution to the coefficient function corresponding to the cross section and the absence of the next-to-next-to-leading order parton densities we have to rely on the order soft plus virtual gluon approximation to make a comparison with the data. Using this approach we can show that a part of the discrepancy between the data and the order corrected cross section can be attributed to the higher order soft plus virtual gluon contributions. In this appendix we will present the order contributions to the coefficient functions corresponding to coming from the partonic subprocesses in (2.15) and (2.16). Although these processes have been calculated in the DIS-scheme in [17, 18] (see also ) and the -scheme we have some different definitions for the distributions and we have a small disagreement with the coefficient function for the subprocess in . Moreover we want to give a clear definition for the soft plus virtual () gluon part of the coefficient function corresponding to the subprocess. We have recalculated the double differential cross section for the partonic subprocesses (2.15) and (2.16). After performing the mass factorization in the -scheme the coefficients (see the definition in (2.12)) read as follows where the color factor is given by (QCD :

Symmetry boundary conditions inherently provide partial restraints against rigid body deformation and do not introduce any artificial singularity, as can often occur with fixed support boundary conditions. In this simply supported beam analysis, the plasticity model of concrete damage in ABAQUS has been introduced. Deflection of edge protected beams 19 6. Determine the peak stress in the component and at position A. This is because, there exists an axial force which reduces the vertical deformation in order to maintain the equilibrium. 28 N /mm2 for fixed and simply supported boundary. 1 Torsional-lateral buckling - bifurcation analysis with a simple beam using Abaqus 6. 000667 and -0. Should i use any other method? Plz do give ur feed backs. four boundary conditions: freely suspended (FF), simply supported near the four corners (SS), free end conditions and supported on top of an oil film (FFO), and simply supported near the four corners and supported on top of an oil film (SSO). Various boundary value problems arise from the di erent ways that such a beam can be supported. The fixed/simply-supported ends of the beam prescribe these boundary conditions. Behaviour of steel framed structures under fire conditions British Steel Fire Test3: Reference ABAQUS model using grillage representation for slab Research Report Report R00-MD10 Abdel Moniem Sanad The University of Edinburgh School of Civil & Environmental Engineering Edinburgh, UK June, 2000. For simply supported boundary condition, the stiffness of the translational and rotational springs in Eq. , particles). I defined rigid links and elastic lin. boundary condition synonyms, boundary condition pronunciation, boundary condition translation, English dictionary definition of boundary condition. 2 Differential Equations of the Deflection Curve consider a cantilever beam with a concentrated load acting upward at the free end the deflection v is the displacement in the y direction the angle of rotation of the axis. For example, at the ends x=0, L of a bent beam free, simply supported, or built-in conditions may be prescribed. Deflection of secondary beams 18 6. The consequence is that the axial and transverse displacements of the beam are coupled by the boundary conditions. material overlap in Abaqus shell model. Instead, apply the following weaker conditions: (i) the displacement at the built-in end at 0y is zero ( 0ux uy ), (ii) the slope there, uy / , is zero. I am trying to model internal flows using SPH in Abaqus. The easiest one I could think of is fixed end. The electron beam scanning occurs at the top surface of a powder layer and traverses along the X-axis with a constant speed. It is possible to create boundary conditions in ABAQUS CAE and control how they take effect through different steps where we may be applying different load cases to a model. Conditions for the uniqueness and continuous dependence on the input data of the solution of the inverse problem for simply supported beams are established and, in particular, it is shown that the operator which. Compute the largest spring force and largest bending moment in the beam. Stability of beams under various boundary conditions De Breuker 26/09/13 Delft. Select a beam and enter dimensions to get started. with Derichlet boundary conditions , and and. The plate is simply supported on all sides. As it is known from the study of&²%³ deformable bodies (or strength of materials) the equation governing the beam deflection is&²%³. Because the beam is pinned to its support, the beam cannot experience deflection at the left-hand support. free-free beam except that in this case we do not have the two rigid body modes (translation and rotation at ω= 0) since it is not allowed by the boundary conditions. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Positive (+) is an upward reaction. Seed→Edges. The environment immerses engi-neers with familiar Abaqus language for element definitions, loads and boundary conditions, solution parameters and other common Abaqus nomenclature. In this case the boundary conditions which. Boundary conditions in Abaqus/Standard and Abaqus/Explicit. In order to study the free vibration of simply supported circular cylindrical shells, a semi-analytical procedure is discussed in detail. The example below includes a point load, a distributed load, and an applied moment. I defined rigid links and elastic lin. Benchmarks for Structural Fire Modelling Dr M Gillie (with thanks to Dr Z Huang for the Vulcan predictions) Gillie M. The simply supported NSC column described is subjected to each of five b lasts ( labeled trials NS1 through NS5) and the DSAS max imum response results are listed in Table 42 T he column deflection s range between 2. When the model is being analyzed, an equation is generated for each degree of freedom of each node. If defined for step 0, only fixed boundary conditions (ex. (3) Other loadings and boundary conditions FIGURE 4. Simply supported in plane: centroids of both ends were restrained against in-plane y-axis deflection U2 0 but unrestrained against in-plane rotation 01 z 0,UR 2 z. The beam parameters are as shown in Fig. Simply-supported Beam - Surface Load (Again!!) A. Exporting the Finite Element Model. Alternatively however, I used the prediction for a simply supported beam and the results were correct. In this example, the shaped piezoelectric modal sensor of the two-step simply supported beam is studied using the DTM. I have tried every possible configuration of boundary conditions for this. Click on description below to see example. Finite Element Analysis Using ABAQUS EGM 6352 (Spring 2017) • Loads and boundary conditions (nodal force, pressure, Beam 2D Planar Deformable Shell. Let's say I want to reproduce the support conditions for a beam. beam, so you lose the information throughout the beam. The geometry, loading and boundary conditions of the cantilever beam are shown in Figure 1. Today's learning outcomes will be to review where we derive the relationship between load, shear, and moment, and then extend that to the slope and deflection of a beam, and then to determine the maximum deflection of a simply supported beam with a concentrated load at the center, and where that max deflection occurs. Mmax = maximum moment in the beam. Having options of boundary conditions as like this in image. 9: Control Topology for Simply Supported Beam 53 Figure 3. 7 Normalized shear modulus reduction as a function of compressive stress for epoxy Epikote 828,. Benchmarks for Structural Fire Modelling Dr M Gillie (with thanks to Dr Z Huang for the Vulcan predictions) Gillie M. As the stresses are not uniform along the member due to localised loading, the pre-buckling analysis also requires multiple series terms with orthogonal functions. The graphical results were obtained for the same parameters as described for a clamped beam. Mehri1;, A. The beam can represented in two different orientations. Forced Vibration. For clamped boundary, I can use some bolts to fix the beam or plate on a base. However, there are many situations in which a bottom panel is far from being clamped. In order to simulate simply supported plate you should fix all 3 translations and leave rotations unconstrained. Analysis of a Beam with a Distributed Load: In this tutorial, you will model and analyze a simply supported beam with both a distributed load and a concentrated load. This example uses two simple beam structures: a cantilever with various supports at the tip, and a beam with both ends simply supported. 14 DATA SHEET ANALYSIS TYPES • Nonlinear dynamic stress/ displacement • Acoustics • Adiabatic stress • Coupled Eulerian-Lagrangian • Coupled field - Thermo-mechanical - Shock and acoustic- structural ANALYSIS AND MODELING TECHNIQUES • Import • Restart • Recover • Automated mass scaling • Nonstructural mass. Figure 3 shows the ABAQUS numerical model having the same cross-section and reinforcements of the control beam. Design of dog legged and open well staircases. A simply supported beam has 2 supports: hinge and roll. , with mutually independent, identicallydistributedforce amplitudesarrivingat the beam at independentrandomtimes. , Step 3 of the proposed procedure) can be found by considering that the boundary part of the slab is similar to the case of Fig. Depending on the load applied, it undergoes shearing and bending. Bernoulli and Euler developed bending theory further and Coulomb put it all together. The vertical connectors, when located in the middle of the. The beam is shown in the following figure 1, subjected to a distributed load, q(x) and an axial load, P, with the following displacement and strain fields. The critical buckling parameters are obtained for clamped-free and clamped-clamped hybrid beams. Review results. Nielsen Jacobs Babtie, 95 Bothwell St, Glasgow, UK Abstract: Absorbing boundary conditions are required to simulate seismic wave propagation in elastic media. Project Brief. Use the ANSYS four node element shell63 for the model. 1 Simpl y Support ed NSC Column For this study, the simplesupport condition is defined by pin and roller boundary conditions. guided boundary conditions. The numerical simulations were performed with Finite Elements Method (FEM) in Abaqus and Ansys environment and an analytical-numerical method. The graphical results were obtained for the same parameters as described for a clamped beam. variable pinned boundary conditions. The boundary conditions in the upper beam actually depend on how the beam is physically supported (whether the restraint is underneath the beam or in the middle or at the top). Zibdehand Rackwitz[3,4] studiedthe response of beams simply supported and with general boundary conditions subjected to a stream of random moving loading systems of Poissonian pulse type, i. The model corresponding to the case with a distributed load (i. 5 Are there any default boundary conditions representing "pinned" and "encastered" nodes. 2000-11-02 00:00:00 A simple and unified approach is presented for the vibration analysis of a generally supported beam. Edge boundary conditions can only be applied to parts that originated from CAD solid models or the 2D Mesh Generation. We shall use clamped-pinned to make things a little more interesting. Dynamics of Nonlinear Beam on Elastic. The second boundary conditions yields. The influence of buckling length for different boundary conditions proposed by Rhodes was considered to calculate critical flexural-torsional buckling moment. We must be aware with the boundary conditions applicable in such a problem where beam will be simply supported and loaded with multiple point loads. The beam properties and proportions are given below. Double click BCs in the model tree and the Create Boundary Condition dialog box will appear (Figure 22a). A uniform distributed load of 1000 N/m is applied to the lower horizontal members in the vertical downward direction. Ansys Tutorial - Rigid Body Dynamics Beam Engine A beam engine is a type of steam engine where a pivoted overhead beam is used to apply the force from a vertical piston to a vertical connecting rod. along x,y and z directions (both displacement and rotation) 2. As for the boundary conditions, simply fastened is usually taken to mean that the ends of the beam are held stationary, but the slopes at the end points can move. Putra and Thompson also considered the difference between simply supported and guided boundary conditions for both baffled and unbaffled plates. The Edit Boundary Condition dialogue box will open as seen in Fig. The two constants of integration for each region of the beam are evaluated from known conditions pertaining to the slopes and deflections. Methodology. Results obtained are discussed critically with those of other theories. But how can I simulate the simply supported boundary conditions?. and inflatable composite tubes using ABAQUS Finite Element Analysis (FEA) software. Typical boundary conditions are: Simply supported beams: The displacement is zero at the locations of the two supports. Simply-Supported Beams. Both are equivalent if the plate thickness is the same as the length in this case. In this study, the interactive behaviors among transverse magnetic fields, axial loads and external force of a magneto-elastic beam with general boundary conditions are investigated. Keywords: Lateral torsional buckling, finite element method, simply supported steel I-beam 1. But how to give this three dimensional beam the required boundary conditions to make it a simply supported beam? The beam cross section is 50mm x 50mm and length of beam is 500mm. As long as the beam is uniform and the distributed mass along the span is much smaller that than the concentrated mass at mid-span, eqn. Select the Nodal Boundary Condition, Edge Boundary Condition or Surface Boundary Condition command. (b) Click. Flux boundary conditions are also called Neumann boundary conditions. For information on exporting an Exodus File, see Exporting Exodus II Files. In laterally supported beams full lateral support is provided by RC slab. Putra and Thompson also considered the difference between simply supported and guided boundary conditions for both baffled and unbaffled plates. The boundary conditions for a clamped plate generally indicate that the edge deflection and edge slopes are both equal to zero (similar boundary conditions are used for beams with fixed ends). Defining the boundary conditions Boundary conditions in Abaqus are defined in the Load module. and inflatable composite tubes using ABAQUS Finite Element Analysis (FEA) software. In its manual there is a cantilever beam, which is fine. But how can I simulate the simply supported boundary conditions?. The simply sup-ported boundary conditions are realized by clamping small lengths of the spring steel beam ends into rotational guides which revolve on axles in low-friction bearings (McMaster-Carr, 8600N3). 80 Equation B. How to apply. 3 Boundary conditions of both sides cantilevered beam 26 3. Rahmani2 Abstract. 0 2 4 6 8 10 12 14 16-0. The Abaqus environment for NX™ CAE. September 26, 2018. Therefore, the solutions of the differential equation of equilibrium will depend on certain integration constants that are determined by imposing two boundary conditions on each edge. boundary condition synonyms, boundary condition pronunciation, boundary condition translation, English dictionary definition of boundary condition. 3, Version 4. However, there are many situations in which a bottom panel is far from being clamped. 2 Boundary Conditions and Load Application To simulate the idealized simply support boundary conditions given by Trahair (1993), into FE model following criterion are considered: 1. great influence on adjacent span and the bridge failure is a whole progressive collapse process. Common boundary conditions are shown at right. Graphs were developed for various wall boundary conditions based on an element size of 6 inches x 6 inches to facilitate the design of the walls subjected to lateral soil pressure. In other words, VA is equal to 0, and VB is equal to 0. At the built-in end of the beam there cannot be any displacement or rotation of the beam. From this analysis you will learn: 1. We must be aware with the boundary conditions applicable in such a problem where beam will be simply supported and loaded with multiple point loads. 4 Yield load and collapse load for plate having different boundary conditions 40. In order to study the free vibration of simply supported circular cylindrical shells, a semi-analytical procedure is discussed in detail. For simply supported boundary condition, the nan- the effective thickness for the nanotube is 0. Figure3: Example for boundary conditions that change during the analysis (Source – Abaqus Manual). Abaqus Tutorial for Adhesive and Composite Joints. Additionally, the function must satisfy some boundary conditions at the ends of the interval (or possibly within). First, the equations of motion are provided. COMPARISON WITH TEST DATA 16 6. Learn how to create a model of a bending beam and subsequently create a macro and a python script to change the mesh size in the model and rerun it. Dynamics of Nonlinear Beam on Elastic. International Journal of Solids and Structures 7:11, 1555-1571. The stiffness is zero in this case. We have already seen terminologies and various terms used in deflection of beam with the help of recent posts and now we will be interested here to calculate the deflection and slope of a simply supported beam carrying uniformly distributed load throughout length of the beam with the help of this post. Because the beam is pinned to its support, the beam cannot experience deflection at the left-hand support. 5) In order to obtain the nonlinear natural frequency, we start to apply the Galerkin method on which the deflection for the beam with simply supported boundary conditions is expressed as 𝑤 (𝑥, 𝑡) = 𝑊 (𝑡) s i n (𝜋 𝑥) (3. One of the boundary conditions is applied to the boundary grid points, whereas the other can be applied to the -points. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment. four boundary conditions: freely suspended (FF), simply supported near the four corners (SS), free end conditions and supported on top of an oil film (FFO), and simply supported near the four corners and supported on top of an oil film (SSO). The action with the initial value of the concentrated transverse bending force P is specified in the node in the middle of the beam span. LOADING 13 5. Hinton has been analyzed by using Abaqus software. 1 : How do I change the boundary conditions at some of the nodes? Re-specify the boundary condition for only the nodes for which the boundary condition has changed with the OP=NEW option. The Timoshenko beam subjected to uniform load distribution with different boundary conditions has been already solved analytically. concrete to apply load and also for applying boundary condition on nodes. This section discusses the time and frequency responses of the sandwich beam with various boundary conditions. From this analysis you will learn: 1. 53 Figure 3. Chandrashekhara did the modeling of laminated beams by a systematic reduction of the constitutive relations of the three-dimensional anisotropic body. Often you have a beam Pinned on one end and resting on the other end to allow for thermal expansion, or for example to act as a gangway to get from a dock to a ship which moves up and down. Load collectors may be created using the right click context menu in the Model Browser (Create > Load Collector). For simply supported boundary condition, the nan- the effective thickness for the nanotube is 0. ABAQUS: Boundary conditions for CEL interaction (e. 3 Boundary conditions of both sides cantilevered beam 26 3. In this blog I will explain how to set up a simulation to determine the stress intensity factor or J-integral. Deflection of primary beams 16 6. w = distributed load on the beam. If this happens in a single element,. As the vertical deflection near the end of the beam increases and hits the stop, there is a change in the boundary condition. In the other boundary, the degrees of freedoms w 0, w u and w l are restrained while u 0, θ x, u u and u l are unrestrained. They cause stress inside the beam and deflection of the beam. four boundary conditions: freely suspended (FF), simply supported near the four corners (SS), free end conditions and supported on top of an oil film (FFO), and simply supported near the four corners and supported on top of an oil film (SSO). It is possible to create boundary conditions in ABAQUS CAE and control how they take effect through different steps where we may be applying different load cases to a model. Simply supported in plane: centroids of both ends were restrained against in-plane y-axis deflection U2 0 but unrestrained against in-plane rotation 01 z 0,UR 2 z. Boundary conditions: Pertain to the deflections and slopes at the supports of a beam. Shi-rong Li et al. 7: Abaqus Topology of Simply Supported Beam using Elasticity Stresses. I need to analyse its behavior (find buckling critical load). The default is n =1. Determine the maximum deflection. Graphs for the beam equation using homogenous boundary conditions The boundary conditions were then changed to obtain results for a simply-supported beam. Having options of boundary conditions as like this in image. When a simply supported (pin-pin) frame object is modeled with a top-center or bottom-center insertion point, internal constraints are generated which have the effect of a vertical offset. The overall stability of castellated beams is mainly affected by porosity, shape of openings, span depth ratio and other factors. Abaqus Tutorial 4: (Workshop) I Beam. Seed→Edges. Equations (21) and (22) show that, in the case of a so-called simply supported beam, the supports must prevent both lateral deflection and twist but the section is free to warp at the ends. So in case of ABAQUS keywords where boundary conditions are applied on node sets or nodes, following steps are taken by the filter. f, which must be provided by the user. A simply supported beam of 12 m effective span having a cross section as shown in Figure 4. The geometry of the beam is the same as the structure in Chapter 3. A simply supported beam has 2 supports: hinge and roll. pdf), Text File (. However, there are many situations in which a bottom panel is far from being clamped. The structure is a simply supported beam modelled with two dimensional solid elements (quadrilateral elements). 14 DATA SHEET • Meshed beam cross sections • Rigid, display, and • Initial conditions • Boundary conditions • Loads. Putra and Thompson also considered the difference between simply supported and guided boundary conditions for both baffled and unbaffled plates. boundary conditions. Boundary conditions relevant to the problem are as follows: 1. which corresponds to the simply supported beam boundary conditions. Simply supported plate with a central hole of radius b, uniformly load by pressure q o over region c < r < a. At the built-in end of the beam there cannot be any displacement or rotation of the beam. • Constants are determined from boundary conditions x x EI y = ∫ dx ∫ M ( x ) dx + C1x + C2 0 0 • Three cases for statically determinant beams, – Simply supported beam y A = 0, yB = 0 – Overhanging beam y A = 0, yB = 0 – Cantilever beam y A = 0, θ A = 0 • More complicated loadings require multiple. But how can I simulate the simply supported boundary conditions?. In this case the boundary conditions which. The overall stability of castellated beams is mainly affected by porosity, shape of openings, span depth ratio and other factors. at support A and at support B will be zero, while slope will be maximum. Exporting the Finite Element Model. Real Boundary Conditions in FEA. org 29 | P a g e The load q on the plate is supported by the combined effort of the longitudinal x- strips, the transverse y-strips and the diagonal xy-strips (Figs. Simply Supported Beam with Point Load Example. Weinberger Department of Mechanical Engineering, Stanford University, CA 94305-4040 Abstract We present a unifled approach for atomistic modeling of torsion and bending of. Torsion and Bending Periodic Boundary Conditions for Modeling the Intrinsic Strength of Nanowires Wei Cai, William Fong, Erich Elsen, Christopher R. No displacement from Axis: y(0) = 0, y(L) = 0 Bending Moment is Zero: y00(0) = 0, y00(L) = 0. simply-supported beam and 2D-FE results obtained using ABAQUS. e frequency response. ABAQUS - Boundary Conditions Q6. Submit it for a linear static analysis. Determine the deflection at the load point as a function of the number of elements used per side to model the plate. Because the beam is pinned to its support, the beam cannot experience deflection at the left-hand support. The fourth order partial differential equation with variable coefficients governing vibrations of the non-prismatic prestressed Rayleigh beam is solved using the generalised Galerkin method and a modification of the Struble's asymptotic method. Simply supported boundary condition in FEA means that all translations (3 in 3D) are fixed but rotations are uncostrained. By analysis, a mesh density of one inch was ultimately deemed computationa l l y efficient. Free Online Library: Vibration Analysis of Reinforced Concrete Simply Supported Beam versus Variation Temperature. free-free beam except that in this case we do not have the two rigid body modes (translation and rotation at ω= 0) since it is not allowed by the boundary conditions. Often the loads are uniform loads, also called continuous loads, this can be dead loads as well as temporary loads. Index Terms- Equilibrium equations flexure, principle of virtual work, trigonometric shear deformation, thick beam. Shells may be modeled as simply supported shells by inserting short links between the slab and its supporting members. Dynamics of Nonlinear Beam on Elastic. (b) Click. (2009) Analysis of heated structures: Nature and modelling benchmarks. Only Abaqus and ANSYS® solvers are supported. 7 Normalized shear modulus reduction as a function of compressive stress for epoxy Epikote 828,. channel–section simply supported beam under axial compression. ABAQUS/EXPLICIT 6. Simply Supported. Material Properties and Section Assignment. APPLICATION TO SIMPLY-SUPPORTED BEAMS To check its applicability, the method was applied to a simply supported Euler}Bernouilli beam. Only Abaqus and ANSYS® solvers are supported. He did not solve the problem in its most general form, but suggested a solution for a simply supported cylindrical shell, in the form of trigonometric functions which satisfied the boundary conditions. 69 N/mm2 and 28. Antoinea, E. 7 inches, well into the large def ormation regime. The Abaqus environment for NX™ CAE. In this tutorial video the static analysis of a simply-supported beam is performed using ABAQUS 6. Where each of the redundants act, a condition of compatibility must be written. That came with Hooke in 1660. The beam was made of carbon-epoxy symmetrical composite prepared with a pre-preg technology using 8 layers of unidirectional band. Distributed load 13 5. EI of the beam is 441x109 Nmm2, K = 275 N/mm for each spring. Preparing a Part Model: The pictorial view of part model of castellated beam along with loading and boundary conditions is shown in Fig. ABAQUS/EXPLICIT 6. The developed model is used to investigate the shape control of composite beam with three sets of boundary condition clamped-free (C-F), clamped-clamped (C-C) and simply supported (S-S). A mechanism map for initial collapse is generated from these formulas in order to relate the governing collapse mechanism of clamped beams to their geom-etry and material properties. The elty all he8 command indicates that we will be meshing the set all with he8 elements. LARGE SPAN STEEL TRUSS BRIDGE FINITE ELEMENT SIMULATION TO INVESTIGATE THE BOUNDARY CONDITIONS. The information entered here appears in the ABAQUS input deck under the "*BOUNDARY" card, either in the general section if an ABAQUS step of 0 is defined, or under the relevant step card. – The live load pattern is usually symmetric, so symmetric boundary conditions are needed for the calculation of the perturbation stresses used in the formation of the initial stress stiffness matrix. The four boundary conditions give: A 1 = B 1 = B 2 = 0 (23) and A 2 sin k 2 l = 0 (24) which requires either A 2 = 0 (in this case there is no twist), or. TxDOT 0-6816: Applications of Partial Depth Precast Concrete Deck Panels on Horizontally Curved Bridges. 1 Torsional-lateral buckling - bifurcation analysis with a simple beam using Abaqus 6. One describes the remaining boundary conditions in terms of the bending moment of the beam. approach based on shear deformable beam theory for the case of laminated beam with bonded piezoelectric actua-tors. 00111 in rad CIVL 7/8117 Chapter 4. Beams are made of SPSR400. The efficiency of the CLD and the DSLJ damper is compared in beam and plate structures with simply supported and cantilever boundary conditions. The beam is also pinned at the right-hand support. Select the Nodal Boundary Condition, Edge Boundary Condition or Surface Boundary Condition command. From the loading, one would expect the beam to deflect something like as indicated by the deflection curve drawn. Slender beam theory is one of the triumphs of mechanical. 2 Boundary Conditions and Load Application To simulate the idealized simply support boundary conditions given by Trahair (1993), into FE model following criterion are considered: 1. 192) can be set to 1 × 10 9 and 0, respectively. CONCLUSION 23 8. 1 Boundary Conditions The boundary condition is the application of a force and/or constraint. 5 Boundary conditions of one side cantilevered beam with distributed load 39 4. From the loading, one would expect the beam to deflect something like as indicated by the deflection curve drawn. Edge 1-201 is modelled with three different boundary conditions, namely simply supported, encastre. A edge or surface boundary condition will apply nodal boundary conditions to each node on the edge or surface. boundary conditions on the parallel ends of the plate are shown to increase the buckling load compared to simply supported boundary conditions. Hi, this is module 3 of Mechanics and Materials part 4. You can achieve such effect in Abaqus using Pinned boundary condition or selecting Displacement/Rotation and constraining aforementioned degrees of freedom manually. Adhesive joints attract more attention due to their advantage of enabling the development of lightweight, cost-effective and highly integrated structures with a more uniform load distribution and improved damage tolerance. Do the same analysis as above, but use 4 nodes and three elements. Abstract In this investigation, a theoretical model for adaptive beam structures with various boundary conditions was developed, and its potential applicability to electrorheological (ER) material based adaptive beams was introduced. Free Online Library: Vibration Analysis of Reinforced Concrete Simply Supported Beam versus Variation Temperature. The following different boundary conditions are used: (1) Simply supported boundary condition: The degrees of freedoms u 0, w 0, w u and w l are restrained while θ x, u u and u l are unrestrained in one boundary. 10 This document contains an Abaqus tutorial for performing a buckling analysis using the finite element program Abaqus/Standard, version 6. Select a calculator below to get started. The size of the beam is 1x1x8 , the loading consists of a point force of N and the beam is completely fixed (in all directions) on the left end. Other mechanisms, for example twisting of the beam, are not allowed for in this theory. at support A and at support B will be zero, while slope will be maximum. Rahmani2 Abstract. Fig -3: Schematic representation of the control beam using ABAQUS Tables 2, 3 and 4 present the properties of steel and concrete used in the current finite element model. EI of the beam is 441x109 Nmm2, K = 275 N/mm for each spring. In order to study the free vibration of simply supported circular cylindrical shells, a semi-analytical procedure is discussed in detail. Lorinb,c and Q. ( Simply supported beam with round cross section ) (Sorry for saying as fixed support I thought both are same) Having a force at the top surface at a point. Mmax = maximum moment in the beam. In this case, the maximum. Owen and E. (d) Determine whether a function is an eigenfunction of a differential oper-ator. ANSYS Solution. Other designs for brace installations will require different boundary conditions (e. International Journal of Solids and Structures 7:11, 1555-1571.

Table of contents: - What is a rational algebraic expression? - How do you write a rational algebraic expression? - How do you identify a rational expression? - What are the steps in simplifying rational algebraic expression? - What is the first step in simplifying rational expression? - What is the first step in multiplying rational algebraic expression? - What is the final step in simplifying rational algebraic expression? - What are the steps in multiplying? - How do you multiply a rational expression? - How do we multiply rational algebraic expression? - What are the steps in adding and subtracting rational algebraic expressions? - What is the multiplication expression? - How do you solve algebraic expressions? - How do you find expressions? - What are the type of expression? - How many terms are in expression? - Is 2x an algebraic expression? - What does expression mean in algebra? - How do you write an expression? - What is the name of a term without variable in an algebraic expression? - What is algebraic expression and equation? - What is constant and variable? - What are constants in algebra? What is a rational algebraic expression? A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions. The last one may look a little strange since it is more commonly written 4x2+6x−10 4 x 2 + 6 x − 10 . ... Well the same is true for rational expressions. How do you write a rational algebraic expression? Steps for Simplifying a Rational ExpressionDetermine the domain. The excluded values are those values for the variable that result in the expression having a denominator of 0.Factor the numerator and denominator.Find common factors for the numerator and denominator and simplify. How do you identify a rational expression? A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials. Notice that the numerator can be a constant and that the polynomials can be of varying degrees and in multiple forms. What are the steps in simplifying rational algebraic expression? Q and S do not equal 0.Step 1: Factor both the numerator and the denominator. ... Step 2: Write as one fraction. ... Step 3: Simplify the rational expression. ... Step 4: Multiply any remaining factors in the numerator and/or denominator. ... Step 1: Factor both the numerator and the denominator.Step 2: Write as one fraction.더보기•2009. 12. 14. What is the first step in simplifying rational expression? The first step in simplifying a rational expression is to determine the domain, the set of all possible values of the variables. The denominator in a fraction cannot be zero because division by zero is undefined. What is the first step in multiplying rational algebraic expression? Step 1: Completely factor both the numerators and denominators of all fractions. Step 2: Cancel or reduce the fractions. What is the final step in simplifying rational algebraic expression? Step 1: Factor the numerator and the denominator. Step 2: List restricted values. Step 3: Cancel common factors. Step 4: Simplify and note any restricted values not implied by the expression. What are the steps in multiplying? Steps to multiply using Long MultiplicationWrite the two numbers one below the other as per the places of their digits. ... Multiply ones digit of the top number by the ones digit of the bottom number. ... Multiply the tens digit of the top number by the ones digit of the bottom number. ... Write a 0 below the ones digit as shown.더보기 How do you multiply a rational expression? Multiplying Rational ExpressionsMultiply the numerators.Multiply the denominators.Simplify the “new” fraction by canceling common factors. Most of the time, you will need to expand a number as a product of its factors to identify common factors in the numerator and denominator which can be canceled. How do we multiply rational algebraic expression? To multiply rational expressions:Completely factor all numerators and denominators.Reduce all common factors.Either multiply the denominators and numerators or leave the answer in factored form. What are the steps in adding and subtracting rational algebraic expressions? Adding and Subtracting Rational ExpressionsFactor the denominators to find the least common denominator(LCD)Multiply each fraction by the LCD and write the resultant expression over the LCD.By keeping the LCD, add or subtract the numerators. ... Factor the LCD and simplify your rational expression to the lowest terms. What is the multiplication expression? A multiplication expression is a mathematical expression with multiplication. To simplify such an expression, we combine the numbers together and then the like variables together. ... To write our answer in standard form, remember that the number always comes first and to put the variables in alphabetical order. How do you solve algebraic expressions? If you're solving an algebraic equation, then your goal is to get the variable, often known as x, on one side of the equation, while placing the constant terms on the other side of the equation. You can isolate x by division, multiplication, addition, subtraction, finding the square root, or other operations. How do you find expressions? To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations. What are the type of expression? There are four types of expressions exist in C: Arithmetic expressions. Relational expressions. Logical expressions. Conditional expressions. How many terms are in expression? A Term is either a single number or a variable, or numbers and variables multiplied together. So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?" Is 2x an algebraic expression? This is a type of expression having only one term for example, 2x, 5x 2 ,3xy, etc. An algebraic expression having two unlike terms, for example, 5y + 8, y+5, 6y3 + 4, etc. This is an algebraic expression with more than one term and with non -zero exponents of variables. What does expression mean in algebra? In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. How do you write an expression? To write an expression, we often have to interpret a written phrase. For example, the phrase “ 6 added to some number” can be written as the expression x + 6, where the variable x represents the unknown number. How Do You Write Mathematical Expressions from Word Problems? What is the name of a term without variable in an algebraic expression? Constants. Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. What is algebraic expression and equation? An expression is a number, a variable, or a combination of numbers and variables and operation symbols. An equation is made up of two expressions connected by an equal sign. What is constant and variable? A constant is a value that cannot be altered by the program during normal execution, i.e., the value is constant. ... This is contrasted with a variable, which is an identifier with a value that can be changed during normal execution, i.e., the value is variable. What are constants in algebra? In Algebra, a constant is a number on its own, or sometimes a letter such as a, b or c to stand for a fixed number. ... Example: in "x + 5 = 9", 5 and 9 are constants. - Why is self control so important? - How do you find the rotational symmetry? - What is it called when you think before you act? - What happens when you don't think before you act? - What is meant by self-control? - What is lack of self control? - What is the difference between self-control and self-regulation? - Why is self-control so important? - What is an example of intellect? - How do you do the mushroom puzzle in Valhalla? 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Probing nanoscale deformations of a fluctuating interface We consider the contribution of thermal capillary waves to the interaction between a fluid-fluid interface and a nearby nanoparticle. Fluctuations are described thanks to an effective interaction potential which is derived using the renormalization group. The general theory is then applied to a spherical particle interacting with the interface through van der Waals forces. Surprisingly enough, we find that fluctuations contribute significantly to the deformation profile. Our study therefore reveals that thermal fluctuations cannot be ignored when probing nanoscale deformations of a soft interface. Liquid-liquid interfaces Surface tension and related phenomena Interface and surface thermodynamics With the miniaturization of fluidic devices, it is now possible to study simple and complex fluids at the scale of the nanometer. As the size of the system decreases, confinement as well as the importance taken by surface effects are expected to lead to novel transport properties [1, 2]. Accordingly, exploiting the possibilities of nanofluidics requires a fine knowledge of liquids and liquid interfaces at very small scales. In this context, nanoscale measurements of liquid-surface properties have become increasingly popular. For instance, the contribution of individual surface defects to contact angle hysteresis has been evidenced using atomic force microscopy (AFM) with a carbon nanotube probe . Local rheology measurements have been performed using a hanging-fiber AFM , and nanoscale deformations of an interface in response to the interactions with an AFM tip have been characterized recently . Non-contact manipulation of liquid interfaces has also been achieved using magnetic beads or knife-edge electric field tweezers techniques [7, 8], whereas other experiments with near-critical fluids have explored interfacial deformations by a laser beam . Despite continuous progress, some fundamental issues regarding the properties of a liquid interface at the nanoscale remains unsolved. The surface of a liquid is rather difficult to probe since the interaction with the measuring device is expected to induce strong perturbations of the interface . A thorough description of the probe-interface interaction is therefore required in order to get conclusive information regarding liquid parameters such as surface tension or viscosity at very small scales. In recent years, several groups have developed theoretical approaches to describe the interaction of an interface with a nanoscopic probe [11, 12, 13, 14]. The resulting deformation is obtained as the minimum of some elastic energy, but thermally activated fluctuations have been systematically overlooked so far. However, interfacial fluctuations range from a few angstroms to a few nanometers and are therefore expected to become relevant when the size of the probe reaches the nanometer scale. This assertion is also valid for larger probes if the fluids are close to a critical point, in which case fluctuations can lie in the micrometer range . The aim of this Letter is thus to study theoretically the effect of thermal capillary waves on interfacial deformation. We follow here a linear renormalization group (RG) scheme that is commonly used in the context of wetting. Indeed, the critical exponents that characterize the adsorption transition of a liquid film on a solid substrate are affected by thermal fluctuations [17, 18]. To account for the latter, an effective potential can be derived by tracing out small wavelength fluctuations [19, 20]. This renormalized potential is then expressed as a convolution of the bare potential with the fluctuation probability distribution function . The very same idea is applied in this work in order to obtain an effective probe-interface potential. The paper is organized as follows. We first derive the shape equation for a given interaction potential. The effect of interfacial fluctuations is discussed next, and the general theory is then applied to van der Waals forces. The issue of fluctuations of the probe itself is commented in the last section. 2 Euler-Lagrange equation We consider a fluid-fluid interface which is interacting with a nanoprobe, as schematically drawn in fig. 1. We limit the discussion to the non-wetting case of a probe that lies at a given height above the interface, in order to avoid possible issues related to capillary contact . If the probe were absent, the interface would be flat and coincide with the horizontal plane . Interaction with the probe provokes a local deformation of the interface described by a smooth function , with . The equilibrium profile is then obtained as the minimum of the total Hamiltonian , the latter being the sum of two contributions: . In the small-gradient approximation , the capillary-wave Hamiltonian reads with the surface tension and the capillary length, being the mass density difference between the two fluids and the gravitational acceleration. The second term describes the interaction between the probe and the interface, and can be written as The potential , whose explicit form will be specified later, is a function of both the position and the local shape of the interface. It is assumed to be radially symmetric and to vanish beyond a typical distance — for instance, the radius of an AFM tip or the waist of a laser beam . Minimizing the total energy functional =0 then leads to the generalized Young-Laplace equation with the disjoining pressure . This equation has to be solved with the condition that the profile is flat far away from the probe. We make the further assumption that the capillary length is much larger than the size of the probe ( typically lies in the millimeter range). Still, we need to keep it finite in order to enforce the condition when . Eq. (3) can then be solved using the method of matched asymptotic [11, 14]. For , eq. (3) reduces to and the outer solution reads . Here, is the modified Bessel function of second kind and an unknown constant. In the opposite limit , gravitational effects can be neglected and the inner solution follows the equation . The matched asymptotic method then requires that The approximate solution is finally obtained by adding the inner and outer approximations and subtracting their overlapping value, which would otherwise be counted twice. 3 Thermal fluctuations Due to the random motion of the molecules, a liquid interface is by essence a fluctuating object. In the absence of interaction, the statistical properties of the free interface are set by the capillary-wave Hamiltonian . In particular, the probability distribution of height fluctuations reads The width of the distribution corresponds to the mean-square displacement (MSD) of the free interface When considering the interaction with an external probe, one has to account for the roughness of the interface that appears “fuzzy” at the scale of the probe — see fig. 1. In order to anticipate whether fluctuations significantly affect the shape of the interface, we define the dimensionless parameter (with is the size of the probe). Consider for instance the AFM experiment described in : given the tip radius nm and nm, one gets so that fluctuations are expected to be relevant. On the other hand in the experiment with millimeter-size magnetic beads , thermal fluctuations can safely be neglected since . From a theoretical viewpoint, the appropriate formalism to describe thermal fluctuations at a given length scale is the renormalization group (RG). We follow here a linear functional RG scheme that has been developed to describe the wetting transition [19, 20]. This approach can easily adapted to our geometry even though the potential depends explicitly on the position. Starting form the bare interaction potential , thermal fluctuations are traced out through momentum-shell integration (see for instance ref. for technical details). The resulting RG flow equation can be integrated explicitly, yielding to the renormalized potential [20, 21] The renormalized potential is thus obtained as a convolution of the original potential with the probability distribution of height fluctuations . The point is that now includes information regarding the roughness of the fluctuating interface. The consequences on the deformation profile can deduced in a straightforward way. Let us denote the solution of the shape eq. (3) in the absence of fluctuations, i.e. with the bare disjoining pressure . The renormalized profile is then solution of the same eq. (3) but with the renormalized disjoining pressure . Considering the contribution of the fluctuations as a perturbation, we evaluate the correction at lowest order in . We first note that, when , the probability distribution is narrowly centered around . The disjoining pressure is then given by so that the correction to the deformation profile can be written as . Since , we find that the lowest-order correction scales linearly with temperature. 4 Van der Waals forces In the theory developed so far, no specific assumption has been made regarding the interaction potential. The situation that we now discuss is that of a nanoscopic probe interacting with an interface through van der Waals forces. It is assumed for the sake of simplicity that the probe is a sphere of radius . The center of the sphere is held at a fixed height above the reference plane — see fig. 1. The attractive force exerted by the sphere over the interface can be obtained from Hamaker theory with the Hamaker constant. The renormalized disjoining pressure is then deduced from eq. (8) and we get We plot in fig. 2 the relative increase evaluated at and (i.e., where the pressure is maximum), as a function of the particle-interface distance . It vanishes as in the limit , whereas it grows rapidly when the particle gets closer and closer to the interface. For particle-interface distances that are a few times the particle radius, the correction can easily reach 5 or 10 % of the total pressure. Note also that, for a given particle-interface distance, the force acting on the interface depends strongly on . Before solving the shape equation, it is convenient to express the relevant energies in terms of dimensionless constants. The ratio of van der Waals to surface forces defines the Hamaker number , whereas the balance of gravitational to surfaces forces defines the Bond number . Taking J and N/m, we find that in a typical AFM experiment with nm . We can therefore assume that . This amounts to evaluate the renormalized disjoining pressure (8) at , and, consequently, the differential eq. (3) becomes linear . On the other hand the Bond number is also very small for usual interfaces, at least far from a critical point: one gets for instance for nm and mm. Still this parameter has to be kept finite for technical reasons in order to enforce the relaxation to the flat shape far away from the particle. where we define and . This result was previously derived in . The new contribution is the fluctuation-induced correction to the bare profile. It is given by with . The full solution is plotted in fig. 3 for different values of , and for . To illustrate the discussion, we consider for instance the deformation at the induced by a particle located at . Given the fact the , we then have For nm and nm, we find that the correction contributes to of the total deformation at the origin. Note that this proportion only depends of the distance between the probe and the interface. Eq. (13) reveals that fluctuations cannot be ignored when probing nanoscale deformations of a liquid interface. Such a large contribution was a priori quite unexpected, but it also decreases very rapidly with : it drops to when , and is completely negligible when . A fine knowledge of the interface MSD is therefore required in order to interpret experimental data. In the model, it is implicitly assumed that the position of the probe is fixed. But in a real AFM experiment, the tip behaves as an harmonic oscillator and is itself subject to Brownian motion. Consider for instance ref. : in this experiment, the tip oscillates at frequency kHz with a MSD nm. The question is then to know whether fluctuations of the tip are dynamically coupled to interfacial fluctuations [24, 25]. At the nanometer scale, interfacial modes are overdamped with relaxation rate , with the wave number and the mean viscosity of the two fluids. Taking N/m and Pa.s, we find that s for a wavelength of the order of the tip radius nm. Since the relaxation rate of interfacial fluctuations is several orders of magnitude higher than the frequency of the tip, the stationary approach is then fully justified. Still, dynamical issues are expected to be relevant in systems with a very low surface tension, for instance in near-critical fluids . More generally, the viewpoint of dynamical coupling between tip and interface fluctuations is compelling since it would allow to directly relate liquid properties (surface tension, viscosity) to the statistical properties of the probe, with no particular assumption regarding interfacial deformation. Work on this issue is currently under progress. Acknowledgements.The author wishes to thank D. Dean and J. Indekeu for insightful discussions. - \NameBocquet L. Charlaix E. \REVIEWChem. Soc. Rev.3920101073. - \NameSiria A., Poncharal P., Biance A.-L., Fulcrand R., Blase X., Purcell S.T. Bocquet L. \REVIEWNature4942013455. - \NameDelmas M., Monthioux M. and Ondarçuhu T. \REVIEWPhys. Rev. Lett.1062011136102. - \NameDevailly C., Laurent J., Steinberger A., Bellon L. Ciliberto S. arXiv:1311.2217. - \NameLedesma-Alonso R., Legendre D. Tordjeman P. \REVIEWPhys. Rev. Lett.1082012106104. - \NameTsai S. S. H., Griffiths I. M., Li Z., Kim P. Stone H. A. \REVIEWSoft Matter920138600. - \NameShimokawa Y., Kajiya T., Sakai K. Doi M. \REVIEWPhys. Rev. E842011051803. - \NameShimokawa Y. Sakai K. \REVIEWPhys. Rev. E872013063909. - \NameCasner A. Delville J.-P. \REVIEWPhys. Rev. Lett.872001054503. - \NameRaphaël E. de Gennes P.-G. \REVIEWPhys. Rev. E5319963448. - \NameLiu N., Bai Y.-L., Xia M.-F. Ke F.-J. \REVIEWChin. Phys. Lett.2220052012. - \NameLedesma-Alonso R., Tordjeman P. Legendre D. \REVIEWPhys. Rev. E852012061602. - \NameLedesma-Alonso R., Legendre D. Tordjeman P. \REVIEWLangmuir2920137749. - \NameQuinn D. B., Feng J. Stone H. A. \REVIEWLangmuir2920131427. - \EditorLangevin D. \BookLight scattering by liquid surfaces and complementary techniques \PublMarcel Dekker, New York \Year1992. - \NameAarts D. G. A. L., Schmidt M. Lekkerkerker H. N. W. \REVIEWScience3042004847. - \NameFisher M. E. \BookStatistical mechanics of membranes and surfaces \EditorNelson D., Piran T. Weinberg S. \PublWorld Scientific, Singapore \Year2004. - The general theory of the wetting transition, and in particular the way that fluctuations affects the critical exponents, has long been a subject of debate. Recent works have emphasized the major role played by non-local contributions in the interfacial Hamiltonian. See for instance : \NameParry A. O., Romero-Enrique J. M., Bernardino N. R. Rascón C. \REVIEWJ. Phys.: Condens. Matter202008505102. - \NameBrézin E., Halperin B. I. Leibler S. \REVIEWPhys. Rev. Lett.5019831387. - \NameFisher D. S. Huse D. A. \REVIEWPhys. Rev. B321985247. - \NameIndekeu J. O., Koga K., Hooyberghs H. Parry A. O. \REVIEWPhys. Rev. E882013022122. - \NameBarrat J.-L. Hansen J.-P. \BookBasic concepts for simple and complex liquids \PublCambridge University Press, Cambridge \Year2003. - \NameIsraelachvil J. 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Bi-Convex (Double-Convex) lenses have the same radius of curvature on both sides of the lens and function similarly to plano-convex lenses by focusing parallel rays of light to a single point. As a guideline, bi-convex lenses perform with minimum aberration at conjugate ratios between 5:1 and 1:5. What is a double convex lens called? Biconvex Lenses- Definition and Meaning Biconvex lens is the simple lenses which comprise two convex surfaces in spherical form, generally having the same kind of radius of curvature. These are also called a convex-convex lens. A Biconvex lens is as shown in the figure below- What is a double lens? Answer: If your prescription glasses have two areas in the lenses, then it is most likely you have what is known as bifocals. Some people might call this a double lens. What is a convex lens for kids? A convex lens curves outward. It is thicker in the middle than at the edges. When light passes through a convex lens, the light rays bend toward each other. The rays meet at a single point on the other side of the lens. What is double convex lens Class 10? The fact that a double convex lens is thicker across its middle is an indicator that it will converge rays of light that travel parallel to its principal axis. A double convex lens is a converging lens. A double concave lens is also symmetrical across both its horizontal and vertical axis. What is a double concave lens? Double-Concave Lenses (DCV or Biconcave Lenses) are employed in beam expansion, image reduction, and light projection applications. These lenses are also great for extending an optical system’s focal length. Optical lenses with negative focal lengths are double-concave lenses, which feature two concave surfaces. What is the difference between convex lens and double convex lens? DIFFERENCE BETWEEN THE CONVEX AND DOUBLE CONVEX LENS A plane convex lens is a plane on one side and convex on the other, whereas a biconvex or double convex lens, as used in magnifying glasses, is curved in a convex fashion on both sides of the glass. What is double concave lens class 10th? Bi-Concave (Double-Concave) lenses have equal radius of curvature on both sides of the lens and function similarly to plano-concave lenses by causing collimated incident light to diverge. Is magnifying glass double convex? Let’s first figure out the magnifying glass. This is what physicists call a convex lens, usually double-convex, the same shape on both sides. Such a lens, due to the refraction of the light entering it from air, and leaving it back into air, can bring parallel rays of light to a sharp focus. Actual photo! Can a double convex lens form a virtual image? If the object is placed inside F (between F and the lens), the image will be on the same side of the lens as the object and it will be virtual, upright, and enlarged. How do you find the focal length of a double convex lens? The focal length of a double convex lens is equal to radius of curvature of either surface The refractive index of its material is. How does a convex lens work? Convex lenses refract light inward toward a focal point. Light rays passing through the edges of a convex lens are bent most, whereas light passing through the lens’s center remain straight. Convex lenses are used to correct farsighted vision. What is an example of a convex lens? Magnifying Glasses The most common use of convex lens is that it is used in magnifying glasses. Magnifying glasses trick our eyes by creating the illusion of a bigger image behind the lens. This illusion is actually the virtual image formed by the convex lens. Magnifying glasses converge the light at one point. What is the meaning of convex lens? Definition of Convex Lens. This type of lens is thicker at the centre and thinner at the edges. An optical lens is generally made up of two spherical surfaces. If those surfaces are bent outwards, the lens is called a biconvex lens or simply a convex lens. What is a lens definition for kids? Kids Definition of lens 1 : a clear curved piece of material (as glass) used to bend the rays of light to form an image. 2 : a clear part of the eye behind the pupil and iris that focuses rays of light on the retina to form clear images. What is a concave lens Class 10? What is a Concave Lens? A concave lens is a lens that diverges a straight light beam from the source to a diminished, upright, virtual image. It can form both real and virtual images. Concave lenses have at least one surface curved inside. What is concave lens and convex lens Class 10? A concave lens is thinner in the middle and thicker at the edges. A convex lens is thicker in the middle and thinner at the edges. Also known as. It is also known as Diverging Lens. It is also known as Converging Lens. What is a convex mirror Class 7? What is Convex Mirror? Convex Mirror is a curved mirror where the reflective surface bulges out toward the light source. This bulging-out surface reflects light outwards and is not used to focus light. How is a double concave lens formed? A double concave lens has two concave surfaces on both the ends. When light from an object is made to pass through a double concave lens from one of the sides of the lens, a virtual image is formed on the same side of the lens. A double concave lens acts as a diverging lens. Why is a double concave lens called a diverging lens? Concave lenses refract parallel light waves, causing them to disperse. The three light rays traveling into the concave lens shown above travel away from each other. For this reason, concave lens are also called diverging lenses. As a result of this light divergence, concave lenses create only virtual images. What image is formed by a double concave lens? Images in Double Concave Lenses Since none of these rays will intersect, a real image cannot exist. Instead, all images created by a double concave lens are virtual images. What are the three types of convex lenses? Types of Convex Lens There are basically three types of these lenses, they are the plano-convex lens, the double convex lens, and finally, concave-convex lens. What are the types of convex? - Plano-convex lens: This lens has a plano-convex shape. It is curled outwards on one side and flat on the other. - Double Convex lens: From both sides, it curves outwards. - Concave-Convex lenses: From one side, it curves inwards, while from the other, it curves outwards. What is the difference between convex lens and convex mirror? A convex mirror forms only virtual,diminished and erect image for all postion of object ,where as convex lense can form both real,inverted images of various size and virtual,erect and large iamges depending on the position of object. Which lens is used in myopia? Statement1: Myopia is corrected by the use of the concave lens. statement 2: In myopia, the converging power of eye lens increases.

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I’m not sure how, but many of my Mathematics A To Z essays seem to circle around algebra. I mean abstract algebra, not the kind that involves petty concerns like ‘x’ and ‘y’. In abstract algebra we worry about letters like ‘g’ and ‘h’. For special purposes we might even have ‘e’. Maybe it’s that the subject has a lot of familiar-looking words. For today’s term, I’m doing an algebra term, and one that wasn’t requested. But it’ll make my life a little easier when I get to a word that was requested. Also, I lied when I said this was an abstract algebra word. At least I was imprecise. The word appears in a fairly wide swath of mathematics. But abstract algebra is where most mathematics majors first encounter it. And the other uses hearken back to this. If you understand what an algebraist means by “homomorphism” then you understand the essence of what someone else means by it. One of the things mathematicians study a lot is mapping. This is matching the things in one set to things in another set. Most often we want this to be done by some easy-to-understand rule. Why? Well, we often want to understand how one group of things relates to another group. So we set up maps between them. These describe how to match the things in one set to the things in another set. You may think this sounds like it’s just a function. You’re right. I suppose the name “mapping” carries connotations of transforming things into other things that a “function” might not have. And “functions”, I think, suggest we’re working with numbers. “Mappings” sound more abstract, at least to my ear. But it’s just a difference in dialect, not substance. A homomorphism is a mapping that obeys a couple of rules. What they are depends on the kind of things the homomorphism maps between. I want a simple example, so I’m going to use groups. A group is made up of two things. One is a set, a collection of elements. For example, take the whole numbers 0, 1, 2, and 3. That’s a good enough set. The second thing in the group is an operation, something to work like addition. For example, we might use “addition modulo 4”. In this scheme, addition (and subtraction) work like they do with ordinary whole numbers. But if the result would be more than 3, we subtract 4 from the result, until we get something that’s 0, 1, 2, or 3. Similarly if the result would be less than 0, we add 4, until we get something that’s 0, 1, 2, or 3. The result is an addition table that looks like this: So let me call G the group that has as its elements 0, 1, 2, and 3, and that has addition be this modulo-4 addition. Now I want another group. I’m going to name it H, because the alternative is calling it G2 and subscripts are tedious to put on web pages. H will have a set with the elements 0, 1, 2, 3, 4, 5, 6, and 7. Its addition will be modulo-8 addition, which works the way you might have guessed after looking at the above. But here’s the addition table: G and H look a fair bit like each other. Their sets are made up of familiar numbers, anyway. And the addition rules look a lot like what we’re used to. We can imagine mapping from one to the other pretty easily. At least it’s easy to imagine mapping from G to H. Just match a number in G’s set — say, ‘1’ — to a number in H’s set — say, ‘2’. Easy enough. We’ll do something just as daring in matching ‘0’ to ‘1’, and we’ll map ‘2’ to ‘3’. And ‘3’? Let’s match that to ‘4’. Let me call that mapping f. But f is not a homomorphism. What makes a homomorphism an interesting map is that the group’s original addition rule carries through. This is easier to show than to explain. In the original group G, what’s 1 + 2? … 3. That’s easy to work out. But in H, what’s f(1) + f(2)? f(1) is 2, and f(2) is 3. So f(1) + f(2) is 5. But what is f(3)? We set that to be 4. So in this mapping, f(1) + f(2) is not equal to f(3). And so f is not a homomorphism. Could anything be? After all, G and H have different sets, sets that aren’t even the same size. And they have different addition rules, even if the addition rules look like they should be related. Why should we expect it’s possible to match the things in group G to the things in group H? Let me show you how they could be. I’m going to define a mapping φ. The letter’s often used for homomorphisms. φ matches things in G’s set to things in H’s set. φ(0) I choose to be 0. φ(1) I choose to be 2. φ(2) I choose to be 4. φ(3) I choose to be 6. And now look at this … φ(1) + φ(2) is equal to 2 + 4, which is 6 … which is φ(3). Was I lucky? Try some more. φ(2) + φ(2) is 4 + 4, which in the group H is 0. In the group G, 2 + 2 is 0, and φ(0) is … 0. We’re all right so far. One more. φ(3) + φ(3) is 6 + 6, which in group H is 4. In group G, 3 + 3 is 2. φ(2) is 4. If you want to test the other thirteen possibilities go ahead. If you want to argue there’s actually only seven other possibilities do that, too. What makes φ a homomorphism is that if x and y are things from the set of G, then φ(x) + φ(y) equals φ(x + y). φ(x) + φ(y) uses the addition rule for group H. φ(x + y) uses the addition rule for group G. Some mappings keep the addition of things from breaking. We call this “preserving” addition. This particular example is called a group homomorphism. That’s because it’s a homomorphism that starts with one group and ends with a group. There are other kinds of homomorphism. For example, a ring homomorphism is a homomorphism that maps a ring to a ring. A ring is like a group, but it has two operations. One works like addition and the other works like multiplication. A ring homomorphism preserves both the addition and the multiplication simultaneously. And there are homomorphisms for other structures. What makes them homomorphisms is that they preserve whatever the important operations on the strutures are. That’s typically what you might expect when you are introduced to a homomorphism, whatever the field. 9 thoughts on “A Leap Day 2016 Mathematics A To Z: Homomorphism”

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- Does density depend on the amount of substance? - Does the amount of the substance affect how dense it is? - What affects the density of a substance? - Does changing the amount of a substance change its density? - Does density affect weight? - How is density affected by pressure? - Is density equal to pressure? - Does pressure depend on density? - Does water density increase with pressure? - Does density of gas change with pressure? - What is the relationship between mass volume and density? - What is the difference between mass and volume? - Are mass and volume directly related? - Does mass increase as volume increases? - What is an example of a real gas? - What are the 5 assumptions of an ideal gas? - How do you determine which gas behaves most ideally? - What are the two most ideal gases? - What happens to gases at high pressure? - What is the difference between a real and an ideal gas? - Why are real gases not ideal? - What are three major differences between a real gas and an ideal gas? - Which conditions of P and T are most ideal for a gas? - What is a real life example of ideal gas law? - Which plot will give a straight line? - What do you mean by compressibility factor? Does density depend on the amount of substance? Density is the ratio of the mass of an object to its volume. Density is an intensive property, meaning that it does not depend on the amount of material present in the sample. Does the amount of the substance affect how dense it is? Density is an intensive property. This means that regardless of the object’s shape, size, or quantity, the density of that substance will always be the same. Even if you cut the object into a million pieces, they would still each have the same density. What affects the density of a substance? If volume increases without an increase in mass, then the density decreases (Fig. 2.2 A to 2.2 C). Adding additional matter to the same volume also increases density, even if the matter added is a different type of matter (Fig. 2.2 A to 2.2 D). Does changing the amount of a substance change its density? Density (ρ) is the amount of mass (m) per unit volume (V) of a substance. Density is an intensive property, which means the density does not change as the amount of the substance present changes. Does density affect weight? The components of Density are mass and volume whereas, on the other hand, the components of weight are mass and gravity. Gravity does not affect density whereas on the other hand gravity directly affects weight. To calculate density mass is divided by volume, whereas to calculate weight mass is multiplied with gravity. How is density affected by pressure? Density and pressure/temperature Density is directly proportional to pressure and indirectly proportional to temperature. As pressure increases, with temperature constant, density increases. Conversely when temperature increases, with pressure constant, density decreases. Is density equal to pressure? Density is the mass per unit volume of a substance or object, defined as ρ=m/V. Pressure is the force per unit perpendicular area over which the force is applied, p=F/A. Does pressure depend on density? Pressure within a liquid depends only on the density of the liquid, the acceleration due to gravity, and the depth within the liquid. Pressure within a gas depends on the temperature of the gas, the mass of a single molecule of the gas, the acceleration due to gravity, and the height (or depth) within the gas. Does water density increase with pressure? As pressure increases, so does water density. Does density of gas change with pressure? Gas density is a function of the pressure and temperature conditions for the gas. Due to its high compressibility, gas can change its volume significantly with change in pressure. Therefore, density changes (at low pressure) can be significant. What is the relationship between mass volume and density? Density is directly related to the mass and the volume. To find an object’s density, we take its mass and divide it by its volume. If the mass has a large volume, but a small mass it would be said to have a low density. What is the difference between mass and volume? Mass is how much stuff something is made of. Volume is how much space an object takes up. Find two objects with similar MASS. Are mass and volume directly related? We can say that the volume of the object is directly proportional to its mass. As the volume increases the mass of the object increases in direct proportion. Does mass increase as volume increases? How is mass found? A measure of mass per unit of volume. If the volume of the object stays the same but the mass of the object increases then its density becomes greater. What is an example of a real gas? Any gas that exists is a real gas. Nitrogen, oxygen, carbon dioxide, carbon monoxide, helium etc. Real gases have small attractive and repulsive forces between particles and ideal gases do not. Real gas particles have a volume and ideal gas particles do not. What are the 5 assumptions of an ideal gas? The kinetic-molecular theory of gases assumes that ideal gas molecules (1) are constantly moving; (2) have negligible volume; (3) have negligible intermolecular forces; (4) undergo perfectly elastic collisions; and (5) have an average kinetic energy proportional to the ideal gas’s absolute temperature. How do you determine which gas behaves most ideally? Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure, as the potential energy due to intermolecular forces becomes less significant compared with the particles’ kinetic energy, and the size of the molecules becomes less significant compared to the empty space between them. What are the two most ideal gases? The real gas that acts most like an ideal gas is helium. This is because helium, unlike most gases, exists as a single atom, which makes the van der Waals dispersion forces as low as possible. Another factor is that helium, like other noble gases, has a completely filled outer electron shell. What happens to gases at high pressure? High pressures: When gas molecules take up too much space At high pressures, the gas molecules get more crowded and the amount of empty space between the molecules is reduced. Initially the gas molecules move around to take up the entire volume of the container. What is the difference between a real and an ideal gas? It simply means that the particle is extremely small where its mass is almost zero. Ideal gas particle, therefore, does not have volume while a real gas particle does have real volume since real gases are made up of molecules or atoms that typically take up some space even though they are extremely small. Why are real gases not ideal? At relatively low pressures, gas molecules have practically no attraction for one another because they are (on average) so far apart, and they behave almost like particles of an ideal gas. At higher pressures, however, the force of attraction is also no longer insignificant. What are three major differences between a real gas and an ideal gas? |Difference between Ideal gas and Real gas |No definite volume |Elastic collision of particles |Non-elastic collisions between particles |No intermolecular attraction force |Intermolecular attraction force Which conditions of P and T are most ideal for a gas? Low P and High T are most ideal for a gas. This is because ideal gas particles experience no intermolecular forces and these conditions are least… What is a real life example of ideal gas law? Ideal gas laws are used for the working of airbags in vehicles. When airbags are deployed, they are quickly filled with different gases that inflate them. The airbags are filled with nitrogen gases as they inflate. Through a reaction with a substance known as sodium azide, the nitrogen gas is produced. Which plot will give a straight line? If all curves are hyperbolas the gas obeys Boyle’s law at the given temperatures. By plotting V versus 1/P (or P versus 1/V), we obtain a straight line with slope = const. Therefore, a gas is ideal when the plot of V versus 1/P (or P versus 1/V) yields a straight line. What do you mean by compressibility factor? It is a measure of how much the thermodynamic properties of a real gas deviate from those expected of an ideal gas. It may be thought of as the ratio of the actual volume of a real gas to the volume predicted by the ideal gas at the same temperature and pressure as the actual volume.

- Research Article - Open Access Regularization and Iterative Methods for Monotone Variational Inequalities © X. Xu and H.-K. Xu. 2010 - Received: 16 September 2009 - Accepted: 23 November 2009 - Published: 7 December 2009 We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discretization of the regularization method. We prove that both regularization and iterative methods converge in norm. - Variational Inequality - Iterative Method - Nonexpansive Mapping - Monotone Operator - Variational Inequality Problem A typical example of monotone operators is the subdifferential of a proper convex lower semicontinuous function. The dual VIP of (1.1) is the following VIP: The following equivalence between the dual VIP (1.6) and the primal VIP (1.1) plays a useful role in our regularization in Section 2. Lemma 1.1 (cf. ). However, if fails to be Lipschitz continuous or strongly monotone, then the result of the above theorem is false in general. We will assume that is Lipschitz continuous, but do not assume strong monotonicity of . Thus, VIP (1.1) is ill-posed and regularization is needed; moreover, a solution is often sought through iteration methods. In the special case where is of the form , with being a nonexpansive mapping, regularization and iterative methods for VIP (1.1) have been investigated in literature; see, for example, [5–19]; work related to variational inequalities of monotone operators can be found in [20–25], and work related to iterative methods for nonexpansive mappings can be found in [26–33]. The aim of this paper is to provide a regularization and its induced iteration method for VIP (1.1) in the general case. The paper is structured as follows. In the next section we present a general regularization method for VI (1.1) with the regularizer being a Lipschitz continuous and strongly monotone operator. In Section 3, by discretizing the implicit method of the regularization obtained in Section 2, we introduce an iteration process and prove its strong convergence. In the final section, Section 4, we apply the results obtained in Sections 2 and 3 to a convex minimization problem. Since VIP (1.1) is usually ill-posed, regularization is necessary, towards which we let be a Lipschitz continuous, everywhere defined, strongly monotone, and single-valued operator. Consider the following regularized variational inequality problem: To analyze more details of VI (2.1) (or its equivalent fixed point equation (2.2)), we need to impose more assumptions on the operators and . Assume that and are Lipschitz continuous with Lipschiz constants , respectively. We also assume that is -strongly monotone; namely, there is a constant satisfying the property Below we always assume that satisfies (2.6) so that is a -contraction from into itself. Therefore, for such a choice of , has a unique fixed point in which is denoted as whose asymptotic behavior when is given in the following result. Now (2.10) follows immediately from (2.15). Proof of Theorem 2.2. In Theorem 2.2, we have proved that if the solution set of VIP (1.1) is nonempty, then the net of the solutions of the regularized VIPs (2.1) is bounded (and hence converges in norm). The converse is indeed also true; that is, the boundedness of the net implies that the solution set of VIP (1.1) is nonempty. As a matter of fact, suppose that is bounded and is a constant such that for all . By Lemma 1.1, we have From the fixed point equation (2.2), it is natural to consider the following iteration method that generates a sequence according to the recursion: Assume in addition that To prove Theorem 3.1, we need a lemma below. Lemma 3.2 (cf. ). Proof of Theorem 3.1. Let , where . By assumption (i) and Lemma 2.1, is a contraction and has a unique fixed point which is denoted by . Moreover, by Theorem 2.2, converges in norm to the unique solution of VI (2.9). Therefore, it suffices to prove that as . satisfy the assumptions (i)–(iv) of Theorem 3.1. Consider the constrained convex minimization problem: It is known that the minimization (4.1) is equivalent to the variational inequality problem: Therefore, applying Theorems 2.2 and 3.1, we get the following result. Assume that (4.1) has a solution. Assume in addition that The authors are grateful to the anonymous referees for their comments and suggestions which improved the presentation of this paper. This paper is dedicated to Professor William Art Kirk for his significant contributions to fixed point theory. The first author was supported in part by a fund (Grant no. 2008ZG052) from Zhejiang Administration of Foreign Experts Affairs. The second author was supported in part by NSC 97-2628-M-110-003-MY3, and by DGES MTM2006-13997-C02-01. - Baiocchi C, Capelo A: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1984:ix+452.Google Scholar - Giannessi F, Maugeri A: Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York, NY, USA; 1995.View ArticleMATHGoogle Scholar - Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. 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Who can be heirs? Men: 1. son 2. son's son (grandson) 3. father 4. paternal grandfather 5. brother 6. father's brother (uncle) 7. mother's brother 8. brother's son 9. paternal uncle's son 10. brother of paternal uncle 11. father's uncle 12. son of father's uncle's brother 13. father's cousin 14. husband 15. guardian. Women: 1. daughter 2. daughter's daughter (granddaughter) 3. mother 4. maternal grandmother 5. paternal grandmother 6. sister 7. father's sister (aunt) 8. mother's sister 9. wife 10. female guardian. The above is extended, if necessary, with family members, uncles and aunts in the broadest sense, both on maternal and paternal branches, and their relatives. Calculation of inheritance If we are aware of the ratios of the distribution according to the Quran, the calculation of the inheritance is as follows. Consider the legacy as shares. In the first step, we define the number of shares. The second step is to calculate the value of a share. In the simplest case, the number of shares is calculated by looking for the smallest fraction regarding legacy portions, so where denominator is the largest. These fractions, i.e., inheritance rates, are found in the Quran. Use the highest denominator. If this number can be divided with all other numbers in the denominators of the other fractions, it will be the number of shares. Then I define the number of the shares and their owners. I get this by multiplying the total number of shares by the ratios of the heirs. The next step is to calculate the value of one share. To do that, I have to divide the value of the legacy by the number of shares. Multiplying that by the ratios, I get the allowance of all heirs. A person dies and his father, his mother, his daughter's son are left behind. The legacy is 120 hectares of land. The largest number in the denominators of the fractions concerning the legacy is 6. So, that's the number of shares. Multiplying the number of shares by the fractions related the legacy, we get the number of shares per capita, so for example, the mother is 6 x 1/6 = 1 Legacy totals 120 hectares, so 120 : 6 = 20, i.e., one share is worth 20 hectares. From that it can be calculated that: the part of the mother is 1 x 20 = 20 hectares the father's part is 1 x 20 = 20 hectares the part of the daughter is 3 x 20 = 60 hectares the daughter's son 1 x 20 = 60 hectares A person dies and two sisters of his father and two sisters of his mother are left behind. The legacy is 90 hectares of land. The largest number in the denominators is 3, so we have 3 shares. The value of one share 90 : 3 = 30 hectares The two sisters of father get 2 x 30 = 60 hectares The two sisters of mother get 1 x 30 = 30 acres A person dies and his mother, two daughters, wife and paternal uncle are left behind. The legacy is 360 hectares of land. In this example, the largest number in the denominator does not provide a solution because the other numbers cannot be divided with it. Therefore, we need to find the smallest common multiple, which in this case is 3 x 8 = 24. The 24 is divisible by 6, 3, and 8 alike and there is no smaller common multiple. So, the number of shares is 24. Value of one share: 360 : 24 = 15 hectares the part of the mother is 4 x 15 = 60 hectares the part of the two daughters 16 x 15 = 240 hectares, 120 hectares each t he part of wife is 3 x 15 = 45 hectares the part of the paternal uncle is 1 x 15 = 15 hectares A person dies and his wife, mother and paternal uncle are left behind. The legacy is 96 hectares of land. The smallest common multiple of the numbers in the denominators is 12, this will be the number of shares. The value of one share is 96 : 12 = 8 hectares the part of wife 3 x 8 = 24 hectares the part of the mother is 4 x 8 = 32 hectares the part of the paternal uncle is 5 x 8 = 40 hectares. Even though the paternal uncle is in the side branch, the rules in this case have given him most of the land. A person dies and his wife, daughter and two sisters are left behind. The number of shares will be 8. In this case, we see that the two sisters get a total of 3 shares, the division of which can lead to disputes. In order to alleviate the problem, the solution is to multiply the number of shares by two and in that case each participant will get an integer number of shares. Accordingly, the number of shares of the mother is changed to 2, of the daughter to 8, and of the two sisters to 6, (i.e., to 3 each). The total number of shares will be 16. Thus, the value of the legacy must be divided by 16 in order to obtain the value of one share and from this the value of legacy per capita is calculated. A person leaves behind two wives, a mother, a daughter, and a brother The smallest common multiple in the example is 24. In this case, however, this is not the number we can use. According to the rule, each wife's share is 1/8. We need to correct. The way to do that is to multiply the number of our calculated shares by two, i.e., the number of wives. It is always considered as multiplier that occurs as plus in the number of heirs. So, 24 x 2 = 48. That will be the number of shares that we have to distribute in such a way that the number of shares of other heirs should also be corrected. Their shares also need to be doubled. the number of shares of the two wives is 2 x 3 = 6 the number of shares of the mother is 2 x 4 = 8 the number of shares of the daughter is 2 x 12 = 24 the number of shares of the brother is 2 x 5 = 10 Father, mother and ten daughters are left behind. In this example, 6 would be the number of shares, but it cannot be divided according to the above headcount and ratios. If 10 girls get 2/3, then 5 girls get 1/3. As in the previous example, 5 will be the adjustment number and multiplying with it the calculated number of shares, we get the corrected number of shares. This is 6 x 5 = 30. Then the number of shares of each heir is multiplied by 5 and we get what they are entitled to. the part of father is 5 x 1 = 5 shares the part of mother 5 x 1 = 5 shares the part of daughters 5 x 4 = 20 shares. And from here, it’s easy to calculate that every daughter gets 2 shares. The husband and five sisters are left behind. The husband is entitled for half of the legacy and the sisters for two-thirds. The smallest common multiple of the denominators is 6, but this cannot be divided according to the above headcounts and proportions. Considering the above proportions, the legacy cannot be divided, as they make up more than the legacy as a whole. It therefore needs to be corrected. The common denominator of the ratios is 6, so the husband should receive 3/6 and the five sisters 4/6. This is a total of 7/6, which is not possible. On the other hand, 7 is good to be a correction point, so the shares need to be distributed in a 3 : 4 ratio, which is 7. So, the inheritance should be divided in a 3 : 4 ratios then the part of the sisters should be further divided into five. The above are only examples to provide guidance on calculating the legacy. Of course, a myriad of additional corrections may also play a role, all depending on the case and the agreement of the heirs.

The investigation of network dynamics is a major issue in systems and synthetic biology. One of the essential steps in a dynamics investigation is the parameter estimation in the model that expresses biological phenomena. Indeed, various techniques for parameter optimization have been devised and implemented in both free and commercial software. While the computational time for parameter estimation has been greatly reduced, due to improvements in calculation algorithms and the advent of high performance computers, the accuracy of parameter estimation has not been addressed. We propose a new approach for parameter optimization by using differential elimination, to estimate kinetic parameter values with a high degree of accuracy. First, we utilize differential elimination, which is an algebraic approach for rewriting a system of differential equations into another equivalent system, to derive the constraints between kinetic parameters from differential equations. Second, we estimate the kinetic parameters introducing these constraints into an objective function, in addition to the error function of the square difference between the measured and estimated data, in the standard parameter optimization method. To evaluate the ability of our method, we performed a simulation study by using the objective function with and without the newly developed constraints: the parameters in two models of linear and non-linear equations, under the assumption that only one molecule in each model can be measured, were estimated by using a genetic algorithm (GA) and particle swarm optimization (PSO). As a result, the introduction of new constraints was dramatically effective: the GA and PSO with new constraints could successfully estimate the kinetic parameters in the simulated models, with a high degree of accuracy, while the conventional GA and PSO methods without them frequently failed. The introduction of new constraints in an objective function by using differential elimination resulted in the drastic improvement of the estimation accuracy in parameter optimization methods. The performance of our approach was illustrated by simulations of the parameter optimization for two models of linear and non-linear equations, which included unmeasured molecules, by two types of optimization techniques. As a result, our method is a promising development in parameter optimization. The investigation of network dynamics is a major issue in systems and synthetic biology . In general, a network model for describing the kinetics of constituent molecules is first constructed with reference to the biological knowledge, and then the model is mathematically expressed by differential equations, based on the chemical reactions underlying the kinetics. Finally, the kinetic parameters in the model are estimated by various parameter optimization techniques , from the time-series data measured for the constituent molecules. While the computational time for parameter estimation has been greatly reduced, due to the improvement in calculation algorithms and the advent of high performance computers, the accurate numerical estimation of parameter values for a given model remains a limiting step. Indeed, the parameter values estimated by various optimization techniques are frequently quite variable, due to the conditions for parameter estimation, such as the initial values. In particular, we cannot always obtain the data measured for all of the constituent molecules, due to limitations of measurement techniques and ethical constraints. In this case, one of the issues we should resolve is that the parameters are estimated from the data for only some of the constituent molecules. Unfortunately, it is quite difficult to estimate the parameters in such a network model including unmeasured variables. Boulier and his colleagues developed differential elimination , derived from the Roselfeld-Gröbner base . Differential elimination rewrites a system of original differential equations into an equivalent system. The rewriting feature was applied to solve the parameter optimization issue, especially in network dynamics including unmeasured variables [3, 5], and in the applications, the equations rewritten by differential elimination were utilized to estimate the initial values for the parameter optimization, by Newton-type numerical optimization. Here, we propose a new method for optimizing the parameters, by using differential elimination . Our method partially utilizes a technique from a previous study , regarding the introduction of differential elimination into parameter optimization in a network including unmeasured variables. Instead of using differential elimination for estimating the initial values for the following parameter optimization, the equations derived by differential elimination are directly introduced as the constraints into the objective function for the parameter optimization. To validate the effectiveness of the constraint introduction, we performed simulations in two models of linear and nonlinear differential equations, where we assumed that the data for only one molecule among them were measured, by using two kinds of evolutionary optimization techniques. The accuracy of the parameter values estimated by the objective functions with and without the new constraints was compared. Finally, we discussed merits and pitfalls of our method in terms of its extension to more realistic and complex models. We first describe a perspective of our method, and then the two models are analyzed to illustrate its performance. The two models were chosen from representative kinetic models for biological phenomena at the molecular level: one model (Model 1) is composed of two variables, analogous to molecular binding and dissociation, such as affinity binding in an antibody cross-link, and the other model (Model 2) is composed of four variables, analogous to a molecular reaction cascade, such as phosphorylation in signal transduction. Notably, we assumed that only one variable is measured among the variables in the two models. Overview of present method The key point of this study is the introduction of new constraints obtained by differential elimination into the objective function, to improve the parameter accuracy. Following an explanation of differential elimination, the method of introducing the constraints is briefly described. Differential algebra aims at studying differential equations from a purely algebraic point of view [6, 7]. Differential elimination theory is a sub theory of differential algebra , based on Rosenfeld-Gröbner . The differential elimination rewrites the inputted system of differential equations to another equivalent system according to ranking (order of terms). Here, we provide an example of differential elimination, as shown below, according to Boulier [3, 5]. Assume a model of two variables, x1 and x2, in Fig. 1, which is described by the following system of parametric ordinary differential equations, where k12, k21, ke and Ve are some constants. Here, two molecules are assumed to bind according to Michaelis-Menten kinetics. The differential elimination then produces the following two equations equivalent to the above system. When we define the left sides of the above system as C1,t and C2,t, C2,t is composed of x1, its derivatives, and the parameters obtained by eliminating x2, and C1,t is composed of x1, its derivatives, the parameters and x2. Note that x2 in C1,t can be expressed by x1, its derivatives and the parameters in C2,t. Then, the values of C1,t and C2,t can be calculated, if we have time-series data of x1, and they would be zero, if all parameters were exactly estimated. Thus, C1,t and C2,t can be regarded as a kind of error function that expresses the difference between the measured and estimated data. In general, the typical objective function for evaluating the reproducibility of an experimentally measured time-series for a parameter set is the total relative error, E. The parameter set is then estimated when the total relative error falls below a given threshold. However, the immense searching space of parameter values frequently hinders correct parameter estimation. To overcome this problem, we introduce the constraint between the estimate obtained by differential elimination (DE constraints), C, into the objective function, i.e., where α is a weighting factor, which is approximately estimated by Pareto optimal solutions for E and C, and then is manually modified (see details in Methods). We analyzed a network model for the binding and dissociation of two molecules (Figure 2A). According to the kinetics of the model (see also Methods), the reference curve of one variable, xAB, was generated (Figure 2B), and two optimization techniques, genetic algorithm (GA) and particle swarm optimization (PSO), were applied to it to evaluate the effect of the introduction of differential elimination constraints (DE constraints) (see details in Additional File 1) into the objective function. Overall, the introduction of DE constraints into the objective function was highly effective for correctly estimating the parameter values in both GA and PSO (Figure 3). By using GA (Figure 3A), kp and km were correctly estimated with the introduction, while the estimation of kp failed without the introduction. Indeed, the most frequent values estimated with the introduction (right side of Figure 3A) were found in the bins corresponding to the range between 0.045 and 0.055 for kp and between 0.45 and 0.55 for km . In contrast, the most frequent values estimated without the introduction (left side of Figure 3A) were found in the range between 0.065 and 0.075 for kp , while those for km were correctly estimated. By using PSO (Figure 3B), km was correctly estimated with the introduction, but kp failed, while the estimations of both parameters failed without the introduction. Furthermore, another difference between the estimations with and without the introduction is the distribution form of the estimated values, although the numbers of trial successes in the optimization were different with and without the introduction (see details in Methods). As seen in Figure 3A and 3A, the values with the introduction were sharply distributed, while those without the introduction were widely distributed. The introduction of DE constraints contracted the parameter space to facilitate the estimation of the correct values. As a result, the parameter accuracy was improved by the new objective function with the introduction of DE constraints in Model 1. Figure 4 clarifies the contraction of parameter space with the introduction of DE constraints into the objective function. Indeed, the estimated values with the introduction by using GA and PSO were concentrated around the correct values (right side of Figure 4). In contrast, the estimated values without the introduction by using the two optimization techniques were broadly distributed (left side). Although the numbers of estimated parameter sets were different with and without the introduction (see details in Methods), the distributions by using the two techniques without the introduction show weak positive correlations. This indicates that the ratio of estimated parameter sets was approximately kept, but the correct estimations failed, without the introduction. We analyzed a network model for the molecular cascade reaction of four molecules (Figure 5A). According to the kinetics of the model (see also Methods), the reference curve of one variable, x1, was generated (Figure 5B), and GA and PSO were applied to it to evaluate the effect of the introduction of DE constraints (see details in Additional File 2) into the objective function. In Model 2, the introduction of DE constraints into the objective function was also highly effective for correctly estimating the parameter values in both GA and PSO (Figure 6). By using GA (Figure 6A), all three parameters were correctly estimated with the introduction (right side), while the estimations of k31 and k41 failed without the introduction (left side). By using PSO (Figure 6B), all three parameters were also correctly estimated with the introduction (right side), while the estimations of k41 failed without the introduction (left side). Furthermore, the features of the distribution forms of the estimated values were similar to those in Model 1 (Figure 3). As seen in Figure 6A and 6B, the distribution of the estimated values with the introduction was sharp (right side), while that without the introduction was wide (left side). As a result, the parameter accuracy was also improved by the new objective function to contract the parameter space with the introduction of DE constraints in Model 2. The contraction of parameter space with the introduction of DE constraints into the objective function is shown more clearly in Figure 7. The features of the parameter space in Figure 7 are similar to those in Figure 4. Indeed, the estimated values with the introduction by using GA and PSO were concentrated around the correct values (right side of Figure 7), while the estimated values without the introduction were broadly distributed (left side). In addition, the distributions by using the two techniques without the introduction also show weak positive correlations, similar to the case in Figure 4. Without the introduction, the ratio of estimated parameter sets was approximately maintained, but the correct estimations failed. The introduction of DE constraints into the objective function clearly improved the parameter accuracy. Indeed, the parameter value sets were correctly estimated by the introduction of DE constraint into the objective function, while they were falsely estimated without the introduction. Furthermore, the parameter sets with the introduction were sharply distributed near the correct values in all cases, in contrast to the wide distribution without the introduction. In general, the derivatives included the information on the curve form of the measured time-series data, such as slope, extremal point and inflection point. This indicates that the new objective function estimates the difference of not only the values but also the forms between the measured and estimated data, while the standard objective function estimates only the value difference. Note that the DE constraint is rationally reduced from the original system of differential equations for a given model in a mathematical sense. Thus, our approach is expected to be a general approach in parameter optimization for improving the parameter accuracy. To further test the performance of the present constraints in more realistic situations, we estimated the same parameters sets in Models 1 and 2 in the case of the simulated data with noise (see Methods). The reference curves for Models 1 and 2 were generated (Additional file 1), and the parameter sets were estimated by using GA with the same procedure as the case of the data without noise (Fig. 8). In both Models 1 and 2, the new constraints were also effective to improve the accuracy of parameter estimations. As the same as in Figures 4 and 7, the estimated values with the introduction were concentrated around the correct values (right side of Figure 8), while the estimated values without the introduction were broadly distributed (left side). However, the distribution ranges of parameter values in both models were widened in the data with noise, in comparison with those in the data without noise. Thus, our method may be more effective to the data curve obtained by some pre-processing methods than intact data, in the application of the present method to real data including noise. As expected, the new objective function requires more computational time, in comparison with an objective function with only a standard error function, due to the increase of the functions in DE constraints. Indeed, the computational time of our method was larger than that of the standard method in Models 1 and 2; the computational times for the standard method and our method were 0.4 and 2.3 hours in Model 1, and 0.03 and 0.22 hours in Model 2 (32 CPU’s of Intel(R) Xeon(R) X5550 2.67GHz). In addition to the computational time, a pitfall of our method is the equation size of DE constraints. In the equivalent systems, the number of terms frequently increases (see Additional file 3), and this may result in the difficulty of the application of our method to a complex or large model. Although we do not still reach a clear conclusion to overcome the difficulty, two ways can be considered. One way is an approximation method and the other is a mathematical manipulation method. As for the former method, in the DE constraints, the terms with a higher order of derivatives in the differential equations appeared frequently in the equivalent system (see Additional files 2 and 3). The magnitude of the estimated values of the higher order derivatives was relatively smaller than those of the lower order derivatives. If the estimation of terms with higher order derivatives can be neglected, then the computational time will be reduced. As for the latter method, we can use some equation-simplification methods by symbolic computation (personal communication from Drs. A. Sedoglavic, F. Lemaire and F. Boulier of Lille University). Indeed, the size of DE constraints for the negative feedback model with oscillation was reduced from 7.4MB obtained by the pure differential elimination in present procedure to 0.1MB after the equation simplification by symbolic computation (data not shown). Further studies will be needed to shorten the computational time by the combination of the approximation and the simplification of the DE constraints. Furthermore, more local minima in the objective function appeared by introducing the DE constraints, also due to the increase in the functions. Indeed, the number of successful estimations by GA in our method was less than that of the standard method in Model 1. To further survey the effects of the landscape of DE constraints on the parameter estimation, we performed parameter optimization by using a gradient method, the modified Powell method [8, 9]. While the evolutionary optimization techniques, such as GA and PSO, equip the algorithm to jump from the trap of local minima, the gradient method generally stop to estimate the parameter values in the valley of the local minima. The parameter values for the two models obtained by using the objective functions with and without the DE constraints are shown in Fig. 9. In Model 2, the situation was similar to the case where the evolutionary techniques were adopted in Figs. 6 and 7. Indeed, the parameter space was clearly contracted under the influence of the introduction of DE constraints. In contrast, in Model 1, our method failed to estimate the parameter values, due to the lack of an error function below a given threshold, while the standard method succeeded with the broad parameter space. This indicates a pitfall, in that the risk of being trapped by local minima increases in the objective function with DE constraints, in comparison with the risk in the objective function without DE constraints. Thus, the introduction of DE constraints into the objective function is more suitable for the evolutionary optimization techniques than the gradient based techniques. One possible use of our method is its application to network inference without known structure. Since the present method is designed with the assumption of a known network structure, the application range of our method to network inference is naturally restricted. However, our method can select the most possible network structure among the networks with similar structures. Indeed, we designed a similar procedure for evaluating the network structures with measured data . In our previous approach, we adopted the transformation of a system of differential equations into the equivalent system of algebraic equations by Laplace transformation. In this case, the system must be linear, due to the Laplace transformation. Furthermore, the numeric optimization in the previous approach frequently faces difficulties, due to the existence of the pole in the Laplace domain. In contrast, these pitfalls are overcome in the present method, by introducing the constraints by differential elimination. This supports the application of the present method to the model selection issue. Various models for describing biological phenomena are available . In particular, several feedback models are important for describing the biological phenomena [12, 13]. Although the performance of our approach for the two representative models in biological phenomena was tested in this study, further tests for the performance of the DE constraint introduction remain for the models that are important in systems and synthetic biology. In the near future, we will report the evaluation of our approach in the cases of various models, in addition to the reduction of computational time and the trials of model selection. The introduction of the constraints by using differential elimination was effectively improved the parameter accuracy in two models of linear and nonlinear equations, especially when we assumed that unmeasured variables were included, by two optimization techniques. This clearly indicates that the ability of our method for estimating the parameter values was far superior to that of various methods with the standard error function. Although the present study focused on two simple models, our method is a feasible approach for parameter estimation in network dynamics. The system of differential equations in Model 1 is expressed as follows: We assume that the model expresses the binding and dissociation between two molecules, and that only one complex, xAB , can be measured. The system of differential equations in Model 2 is expressed as follows: We assume that the molecules, x2, x3, and x4, activate x1 with linear relationships, and that only one molecule, x1, can be measured. The data with noise were generated by Box-Muller method . Each of data, Xe(t), is expressed as follows: where X(t) is a value at time t in original curve of Figures 2 and 5, Rn is random variable according to the standard normal distribution, and c was set to 0.666. Two well-known parameter optimization techniques, the genetic algorithm (GA) [15–19] and the particle swarm optimization (PSO) [20, 21], were used. In the parameter optimization, two thresholds were set to stop the optimization: the average value of the error function over time points, E/T, and the number of generations per optimization. In this study, we performed the optimization 200 times in both techniques, and the thresholds of E/T were set to 0.01 for Model 1 and 0.001 for Model 2, and the threshold for generation number was set to 2000. As a result, the numbers of successes by 200 trials were 200 without DE constraints and 51 by GA and 11 by PSO with DE constraints, for Model 1, and 200 for all cases for Model 2. Introduction of the new constraints into the objective function The objective function in this study is composed of two terms: one is the standard error function between the estimated and measured data, and the other is the constraints obtained by differential elimination. The error function is defined as follows: Suppose that xci,t is the time-course data at time t of xi calculated by using the estimated parameter values, and xmi,t represents the measured data at time t. The sum of the absolute value of the relative error between xci,t and xmi,t gives the total relative error, E, as a standard error function, i.e., where N and T are the number of variables and the time points, respectively: N was 2 for Model 1 and 4 for Model 2, and T was 100. Next we define the constraints obtained by differential elimination. In general, differential elimination rewrites the original system of differential equations into an equivalent system, which means that the number of equations is equal in both systems. Thus, we can express the constraint by differential elimination, CDE, as the linear combination of the equations in the equivalent system, as follows: where L and T are the numbers of equivalent equations and time points, respectively: L was 2 for Model 1 and 5 for Model 2. Finally, we introduce CDE into the objective function, OF, in combination with E, as: where α the a weight of two functions, which is approximately estimated by a Pareto optimal solutions for E and C and then is manually modified. In the present study, α was set to 0.1 in Model 1 and 0.9999999 in Model 2. As a result, our computational task is to determine a set of parameter values that minimize to OF. Implementation of differential elimination All of the symbolic computations for the differential elimination were performed using the diffalg package of MAPLE 10. In the performance of differential elimination, the ranking of variables was: xA≻xB≻xAB in Model 1 and P(Pool) ≻x4≻x3≻x2≻x1 in Model 2. Subsequently, we converted the form of the polynomial equations derived by differential elimination to the Java code by using the CodeGeneration feature in Maple 10. Kitano H: System Biology: A Brief Overview. Science. 2002, 1662-1664. Yung-Keun Kwon, Kwang-Hyun Cho: Quantitative analysis of robustness and fragility in biological networks based on feedback dynamics. Bioinformatics. 2008, 24 (7): 987-994. 10.1093/bioinformatics/btn060 Jonikow CZ, Michalewicz Z: An Experimental Comparison of Binary and Floating Point Representations in Genetic Algorithms. Proceedings of the Fourth International Conference on Genetic Algorithms. 1991, 31-36. Eberhart R, Kennedy J: A New Optimizer Using Particle Swarm Theory. Proc. of Sixth International Symposium on Micro Machine and Human Science (Nagoya Japan), IEEE Service Center, Piscataway, NJ. 1995, 39-43. This work was partly supported by a project grant, ‘Development of Analysis Technology for Gene Functions with Cell Arrays’, from The New Energy and Industrial Technology Development Organization (NEDO). KH was partly supported by a Grant-in-Aid for Scientific Research on Priority Areas "Systems Genomics" (grant 20016028) and for Scientific Research (A) (grant 19201039) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. In particular, the authors would like to express their gratitude to Drs. Alexander Sedoglavic, Francois Lemaire, and Francois Boulier of Lille University, for valuable discussions during the course of this work. This article has been published as part of BMC Systems Biology Volume 4 Supplement 2, 2010: Selected articles from the Third International Symposium on Optimization and Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/1752-0509/4?issue=S2 Authors and Affiliations Computational Biology Research Center, National Institute of Advanced Industrial Science and Technology, 2-4-7 Aomi, Koto-ku, Tokyo, 135-0064, Japan Masahiko Nakatsui & Katsuhisa Horimoto Institute of Systems Biology, Shanghai University, Shangda Road 99, Shanghai, 200444, China Laboratory for Bioinformatics, Graduate School of Systems Life Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka, 812-8581, Japan Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, Tokyo, 113-8656, Japan Yasuhito Tokumoto & Jun Miyake Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka, 560-8531, Japan The authors declare that they have no competing interests. MN performed the implementation and the calculations, and participated in the design of the study. KH conceived of the study, participated in its design and coordination, and drafted the manuscript. MO, YT, and JM participated in the design of the study, and helped to draft the manuscript. All authors read and approved of the final manuscript. Additional file 1: According to the kinetics of the models for Models 1 and 2, the reference data of one variable, xAB (A), and that of one variable, x1 (B), were generated under the same conditions as those in Figures 2 and 5. (PDF 119 KB) Additional file 3: The equivalent equations for Model 2 were derived from the system of differential equations by differential elimination. (PDF 40 KB) Rights and permissions This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nakatsui, M., Horimoto, K., Okamoto, M. et al. Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions. BMC Syst Biol4 (Suppl 2), S9 (2010). https://doi.org/10.1186/1752-0509-4-S2-S9

Access the full text. Sign up today, get DeepDyve free for 14 days. This article addresses the problem of air traffic service (ATS) pricing over a domestic air transportation system with either private or public ATS providers. In both cases, to take into account feedback effects on the air transportation market, it is considered that the adopted pricing approaches can be formulated through optimization problems where an imbedded optimization problem is concerned with the supply of air transportation (offered seat capacity and tariffs for each connection). Under mild assumptions in both situations the whole problem can be reformulated as a mathematical program with linear objective function and quadratic constraints. A numerical application is performed to compare both pricing schemes when different levels of taxes are applied to air carriers and passengers. Keywords Quadratic optimization, Flows optimization in networks, Pricing, Air traffic services Paper type Original Article 1. Introduction During the last decades, many studies in the field of Operations Research have been dedicated to the air transport sector by considering problems of planning, operation and pricing. In general, these studies consider the immediate effects of the decisions without taking into account indirect effects such as feedbacks which can be set up between the various actors of air transport. This has led to strategies which on the long term revealed to be largely suboptimal due to the unexpected reaction of other involved economic agents [1,2]. Thus, in this study a global approach including air traffic control, airports, airlines and passengers is developed for the pricing of the air navigation services as well as airport services considered as a whole as Air Traffic Services (ATS). The main ATS charges (en-route, approach and aerodrome charges) are collected for both air navigation and airport services. These different charges may also include provisions to reduce nuisances in the vicinity of airports as well as © Rabah Guettaf and Felix Mora-Camino. Published in Applied Computing and Informatics. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode Publishers note: The publisher wishes to inform readers that the article “Pricing schemes for air traffic services through multi-level approaches” was originally published by the previous publisher of Applied Computing and Informatics and the pagination of this article has been subsequently changed. There has Applied Computing and been no change to the content of the article. This change was necessary for the journal to transition from Informatics the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To pp. 90-107 access and cite this article, please use Guettaf, R., Mora-Camino, F. (2020), “Pricing schemes for air traffic Emerald Publishing Limited e-ISSN: 2210-8327 services through multi-level approaches”, New England Journal of Entrepreneurship. Vol. 17 No. 1, p-ISSN: 2634-1964 DOI 10.1016/j.aci.2018.07.001 pp. 90-107. The original publication date for this paper was 19/07/2018. other environmental impacts. These charges have been established in general on an empirical Pricing basis to cover broadly ATS costs without taking into consideration the reaction of the schemes for air different actors of air transportation (Figure 1). traffic services Here, two new pricing mechanisms are developed according to some fundamental assumptions: – the main objective of airlines when defining their air transportation supply is to maximize their profit over an annual exercise; – the main objective of a public ATS provider is to promote air transportation measured in flows of transported passengers while covering their operations costs; – the main objective of a private ATS provider is to maximize his own profit while taking into account the profitability of the airline sector. The revenue of the airlines depends directly of the price of their air tickets rates which should cover their operating costs which includes beyond fuel, crew and maintenance costs, their ATS charges. These ATS charges represent today between 10% and 20% of the operational costs of airlines. Thus, the pricing of ATS services has a global influence on ticket prices and affects the levels of demand for air transport (transport of passengers and freight). In this study is considered the frequent case of a domestic air traffic area, presenting some international connections, being under the control of a single ATC provider. In general, air cargo and business aviation represent small shares of air transportation with activity levels rather inelastic with respect to the variation of ATC charges, so the focus is here on commercial flights operated by local airlines to transport passengers through the domestic network of air routes linking the different airports. The cases where ATS operators are private or public are analyzed and particularized in this study. In both cases it seems essential to take into account the profit maximization behavior of the airlines when dimensioning their air transportation service levels (flights with associated frequencies and capacities) and setting their pricing policy. In order to limit the complexity of this study, the airlines sectors are considered as a whole without taking into account competition among them. This leads to the formulation of two-levels nonlinear optimization problems [3–5] which can be treated using already well established bi-level programming techniques [6–8]. For many high traffic airspaces, ATS pricing has been considered a potential tool to cope with saturation . Recently, research studies considering the modulation of air navigation charges to cope with capacity and peak-loading pricing through a bi-level approach have been published, [10–12], where short term feedback effects between the ATS pricing and the airlines demand for ATS services are analyzed. Using a general bi-level pricing scheme , other studies have introduced the network dimension of ATS pricing, [14,15]. Optimization of ATS providers’ fees πu, fu, φu vu Optimization air Figure 1. Airline Transportation Offer Two-level Decision- Making Process. (Follower) In the present study, the adopted long term analysis allows to introduce a third decision ACI level which takes into account the final users (i.e. the passengers) reaction to ATS charges 17,1 through their demand function. 2. Current pricing practice for ATC/ATM Air navigation services (ATC/ATS service providers) finance in general their activities by charging airlines using their airspace. OACI publishes periodically updated general guidelines for pricing, . The air navigation charges represent a significant portion of the cost of a flight for an airline , which has to increase ticket prices to cover them. These charges often represent between 10% and 20% of the cost of a flight, . The Chicago Convention of 1947 which founded ICAO, has given the basis of the current charging systems for air navigation services. A detailed formula for the calculation of air traffic charges was not proposed at that time, but it was recommended to the states to establish a method to calculate the amount of charges to cover the costs of using specialized manpower and equipment (computers, radars and communication systems) to ensure the safety of air traffic. Different charges are collected today for air navigation (route, approach and aerodrome charges) and other airport services. In the case of Europe, in the context of the Single European Sky operation , the central office for en-route charges (CRCO) of Eurocontrol is in charge of computing and collecting the charges paid by airspace users and of reassigning them to member countries traffic services. The following empirical formula has been used to compute the charges R received by each state from a given flight: rffiffiffiffiffi D M R ¼ T 3 (1) i i 100 50 where n is the number of considered states, T is the unit rate adopted by state i, D is i i the distance flown in kilometers by that flight in the airspace of the state i, and M is the maximum take-off weight in tons of the aircraft used in that flight. This unit rate varies in European countries from 22 (Ireland) to 90 (Belgium) Euros. In United States, the Federal budget covers all operations and investment costs related with ATS/ATM since there are today no effective fees or charges for the users of the US airspace. The airspace and its resources are free for any plane of any size that conforms to the Federal Administration rules . However, the air tickets comprise a set of taxes related with this use and which are collected by airlines. The exception is with flights that transit the US controlled airspace without departing or landing in United States. In that case the overflight charges consider different rates for the en-route and oceanic components of a flight. Different rates expressed per 100 nautical miles measured along the great circle distance between the entry and the exit points in the US-controlled airspace are applied. The charges are calculated with the formula which does not consider the mass or size of the aircraft: R ¼ r * DE =100 þ r * OD =100 (2) ij E ij O ij where Rij is the total fee charged to aircraft flying between entry point i and exit point j, DE is the total distance flown through each segment of en-route airspace between entry ij point i and exit point j, DO is the total distance flown through each segment of oceanic ii airspace between entry point i and exit point j, r and r are the en-route and oceanic E O rates, respectively around 60 and 25 US$. The FAA review these rates at least once every two years and adjust them to reflect the current cost and volume of the services provided. With the perspective of privatization of air navigation services in United States, a system of charges should be implemented for all users of the US airspace. Given the monopolistic nature of air navigation services, its charges should be regulated in order to avoid unfair pricing and to allow aviation users to pay the cost of the services to the air navigation service provider. From the above it appears that the current practice to establish charges for air navigation Pricing services does not take into account important factors related with the offer and the demand schemes for air for air transportation: traffic services – the structure of the operated air traffic network as a whole or with respect to each airline, – the influence of air navigation charges on ticket fares and the level and structure of demand. To take into account these two factors, in this study it is considered that the pricing approach adopted by public ATS providers can be formulated as an optimization problem through a bi- level optimization structure. This approach allows the consideration of the interactions between the different economic agents involved in air transportation. Then, reactive levels for supply by the airline companies and for demand by users as a result of the variation of ATS charges and air tickets prices can be taken into account when defining the air navigation charges. 3. Definitions and assumptions Let U be the set of local connections and let E be the set of international connections. Let π be the mean price for a seat on local connection u∈ U and let π be the mean price of a seat on an international connectione∈ E. Then the potential demands for local connections u∈ U are supposed to be given by d ¼ D ðπ; f Þ where f is the annual flow of aircraft on local u u u 2jUj ∞ þ connection u. Here D is a demand function which is supposed to be of class C from ðR Þ to R where marginal variations of frequency and prices parameters are such that: for u∈ U : vD =vπ ≤ 0; for v∈ U; v≠ u : vD =vπ ≥ 0 and jvD =vπ j >> vD =vπ (1-a) u u u v u u u v for u∈ U : vD =vf ≥ 0; for v∈ U; v≠ u : vD =vf ≤ 0 and vD =vf >> jvD =vf jÞ (1-b) u u u v u u u v The potential demands on international connections e∈ E are supposed to be given by ∞ þ þ d ¼ D ðπ ; f Þ where D is supposed to be a C function from ðR Þ to R where: e e e e e vD =vπ ≤ 0 and vD =vf ≥ 0 (2-a) e e e e It is also supposed that the fields ðD ; u∈ UÞ and ðD ; e∈ EÞ are invertible with respect to u e ðπ ; u∈ UÞ and ðπ ; e∈ EÞ. Let f be the satisfied demand for local connection u∈ U and let u e f be the satisfied demand on international connection e∈ E, they should meet capacity and potential demand constraints given in equation (3): f f ≤ minfK $f ; D ðπ; f Þg and ≤ minfK $f ; D ðπ ; f Þ g (3) u u u e e e e u e e where K is the mean seat capacity of flights on local connection u, f is the annual flow of u e aircraft and K is the mean seat capacity of flights on international connection e. Let us define here different parameters to allow the quantification of the annual revenue of the ATS provider and the airline sector: – Let v be the ATS fee applied on a flight operating the local connection u∈ U and v be u e the ATS fee applied on flights operating the international connectionse∈ E. – Let C be the fixed cost associated with ATS in the considered area, σ be the variable ATS average cost associated with ATS for a local connection u∈ U and σ be the variable average cost associated with ATS for an international connectione∈ E. – Let λ be the tax rate applied to the users of air transport on local connection u∈ U and let α be the part of this tax transferred to the ATS providers. int – Let λ be the tax rate applied to the users of international flights along e∈ E and α be E ext ACI the part of this tax transferred to the ATS providers. 17,1 The annual economic return for the ATS operators, R , is then given by: ATS X X R ¼ ððv σ Þf þ α λ f π Þþ ððv σ Þf þ α λ f π Þ C (4) ATS u u u int U u e e e ext E e u e ATS u∈U e∈E 94 F Let C be the fixed cost of the airlines sector operating U and a part of E, let c be the ALS average operating cost for a flight along connection u∈ U and c be the average operating cost for a flight along connectione∈ E. Then, the annual economic return for the airlines sector, R , is given by: ALS X X R ¼ ðπ ð1 λ Þf ðc þ v Þf Þþ μ$ ðπ ð1 λ Þf ðc þ v Þf Þ C (5) ALS u U u u u e E e e e u e ALS u∈U e∈E where μ∈ ½0; 1 is the proportion of international traffic operated by local airlines. Given the total fleet of aircraft, the adopted theoretical fleet capacity for network U, F , is such as for any frequency distribution ff ; u∈ Ug: L f ≤ F (6) u u U u∈U where L is the block time associated to connection u∈ U. 4. ATS pricing through multi-level approach In this study it is considered that the definition of ATS charges must take into account the reaction of the airlines sector since these charges constitute a noticeable part of their operational costs. In the case of a public ATS provider, the final objective when fixing ATS charges is supposed to be the maximization of the total volume of passenger flows while considering the maximizing profit behavior of the airline sector and insuring budget equilibrium for the ATS provider. The main objective of a private ATS provider is in general to maximize its profit while providing acceptable conditions to the airlines sector to continue or develop its air transport activity. Here it is supposed that international flights f ∈ E are fixed by international agreements while π ∈ E are fixed by the international market. Once the ATS fees ðν ; u∈ U; ν ; e∈ EÞ have been fixed, the airlines sector is supposed u e to fix his air transport supply to solve the following domestic problem with respect to fðf ; π ; f Þ; u∈ Ug: u u max ðπ f ððc þ ν Þf Þ (7) u u u u f ; π ; f u u u u∈U with 0≤ f ≤ K $f f ≤ D ðπ; f Þu∈ U (8) u u u u u L f ≤ F (9) u u U u∈U π ≥ 0; f ≥ 0 u∈ U (10) u u and * * Let f ðνÞ; π ðνÞ; f ðνÞ be the solution of the above problem, then this solution will be profitable for the airline sector if: X X * * F Pricing π ð1 λ Þ f ðc þ v Þf Þþ μ$ ðπ ð1 λ Þf ðc þ v Þf Þ C ≥ 0 (11) U u u e E e e e u u u e ALS schemes for air u∈U e∈E traffic services In the case of a public ATS provider, the problem of optimization of the pricing of ATS can be formulated in the following way: X X max f ðνÞþ f (12) u e v ;u∈U u∈U e∈E with: * * min R f ðνÞ; f ðνÞ; v ≥ R (13) ATS ATS * * * min R π ðνÞ; f ðνÞ; f ðνÞ; ν ≥ R (14) ALS ALS ν ≥ 0 u∈ U (15) and min min where R is the minimum acceptable economic return for the ATS operator and R is the ATS ALS minimum acceptable economic result for the airlines sector. In this study, constraints on the economic result of the international airline companies are not considered. It is also supposed that ATS/ATM costs remain lower than a certain percentage of the revenue on an international connection: v f ≤ η f π with 0 < η < 1 e∈ E (16) e e e e e e In the case of a private ATS provider, the problem of optimization of the pricing of ATS can be formulated in the following way: * * max R f ðνÞ; f ðνÞ; v (17) ATS v ;u∈U with: * * * min R π ðνÞ; f ðνÞ; ðνÞ; ν ≥ R (18) ALS ALS and ν ≥ 0 u∈ U (19) min where R is the minimum acceptable economic result for the airlines sector. In this case, ALS v ; e∈ E are such as: v f ¼ η f π with 0 < η < 1 e∈ E (20) e e e e e e In both cases, the ATS pricing problem configure a bi-level optimization problem, where the leader is the ATS supplier and the follower is the airlines sector. This leads to the two-level scheme: 5. Air transport supply optimization by the airlines sector If it is supposed that on each link supply is not chosen superior to potential demand, conditions (8) can be rewritten as: ¼ K $f and K $f ≤ D ðπ; f Þ u∈ U (21) u u u u u since any overcapacity over a link will be an additional cost for the airlines and then the ACI effective transported flow along link u will be equal to the offered capacity on that link. Then 17,1 the airlines sector problem (7)–(10) can be rewritten as: max ðπ K ðc þ ν ÞÞf (22) u u u u u π ;f u u u∈U under constraints (9), (10) and (21). It appears that for any given feasible frequency distribution, maximizing the profit of the airlines sector will lead to increase π on each link. According to properties (1a) and (1b)of the demand functions, this will lead to a diminution of demand which will end when: K $f ¼ D ðπ; f Þ u∈ U (23) u u u Then, here it is considered that an efficient behavior for airlines will be to provide a supply no greater than the expected demand while all expected profitable demand should be satisfied. This leads to the equilibrium conditions: f ¼ K $f ¼ D ðπ; f Þ∀u∈ U (24) u u u Considering the invertibility property of ðD ; u∈ UÞ with respect to ðπ ; u∈ UÞ, from (24),a u u jUj jUj bijective mapping F from π ∈ R to f ∈ R can be defined, so that: f ¼ FðπÞ and f ¼ K$FðπÞ (25) where K ¼ diagfK ; ; K g. 1 jUj Then problem (7)–(10) is replaced by: max ðπ $K ðc þ ν Þ$ÞF ðπÞ (26) u u u u u π ;u∈U u∈U under L $F ðπÞ≤ F (27) u u U u∈U and π ≥ 0 u∈ U (28) In this study we consider particularly the case in which F is an affine function with respect to π such as: FðπÞ¼ f Φ$π (29) where Φ is a square matrix of dimensions jUj. Then we get for the airlines sector the following linear quadratic optimization problem: t t jUj max π Q π þ P π þ R with S π ≥ Tandπ ∈ ðRþÞ (30) π ;u∈U where Q ¼ KΦ; P ¼ − Φ ðc þ νÞþ Kf Þ; (31) t t R ¼ðc þ νÞ f ; S ¼ ΦL and T ¼ L Φπ 0 Problem (30) is a linear quadratic problem which can be numerically solved easily by using Pricing algorithms such as [22,23]. However, adopting some rather credible assumptions, the schemes for air solution of this problem can be turned analytic. For instance, taking into account relations traffic services (1a): Φ > 0 ∀u∈ U; Φ ≤0and jΦ j << Φ ∀u; v∈ U; u≠ v (32) uu uv uv uu and considering that K is a diagonal matrix, it is expected that matrix Q given by s t Q ¼ðQ þ Q Þ 2 will be a definite positive symmetric matrix. Here it is also useful to assume that: Φ L > Φ L ∀u∈ U (33) uu u uv v v≠u so that S is a positive vector and that: F ≥ L f (34) U 0 Then S is a positive vector and T is a positive scalar see (27), (29), so that the capacity constraint vanishes and the solution of problem (30) is given by: −1 π¼ ðQ Þ P (35) or −1 s t π¼ ðQ Þ ðΦ ðc þ vÞþ Kf Þ (36) 6. Optimal mean ticket rates by the airlines sector Now assuming that the whole airlines sector adopts as reference price a mean ticket rate per flown hour p, so ticket rates are defined such as: π ¼ L pu∈ U (37) u u problem (26)–(28) can be rewritten as a scalar optimization problem: max gðν; pÞ under hðpÞ≤ F (38) p∈R where: t t gðν; pÞ¼ p$L K$FðL$pÞ and hðpÞ¼ L $FðL$pÞ (39) In the case in which F is an affine function, see relation (29), and that Φ is such as (32) then the optimization criteria of problem (38) becomes: max αp þ βp þ γ; p≥ δ (40) p∈R with t t t α ¼L KΦL; β ¼ðL Kf þðc þ νÞ ΦLÞ (41) and t t t γ ¼ðc þ νÞ :f ; δ ¼ðL f F Þ L ΦLÞ (42) ACI 0 0 U 17,1 Since α is expected to have a negative value, the general solution of this problem is given by: n o p ¼ max δ; (43) 2α Or t t L f F L Kf þðc þ νÞ ΦLÞ 0 u p ¼ max ; (44) t t L ΦL 2L KΦL Then, the ATS fees will have an influence on the transportation fares chosen by the airlines sector if the maximum of (44) is given by the second term, this can be written: ν ΦL≥ ε (45) with t t L KΦL L KΦL t t ε ¼ L 2 K f 2 F c ΦL (46) 0 U t t L ΦL L ΦL The optimal ticket rate p* is given in that case by: * t p ¼ p þ ρ $ν (47) with t t L ðΦLÞ p ¼ Kf þ Φ c and ρ ¼ (48) 0 0 t t 2L KΦL 2L KΦL and the optimal frequencies and expected demand are given by: * * * f ðνÞ¼ g G $ν and f ðνÞ¼ Kf ðνÞ (49) with * t g ¼ f p ΦL and G ¼ ΦLρ (50) 0 0 ν The optimal frequency and expected demand are such as: ΦL L ΦL * * f ¼ 1 f þ F and f ¼ Kf (51) 0 U t t L ΦL L ΦL 7. Pricing of ATS with a public supplier Here it is supposed that the market conditions for international connections are already established so that their economic return for the airlines sector and the ATS provider are already known. According to relations (12)–(15), when considering that condition (45) holds, the problem of optimization solved by the public ATS provider to choose a level for the ATS fees such as demand is promoted can be rewritten under the form: t Pricing min K G ν (52) schemes for air traffic services with the constraints: v N v þ M v þ z ≥ 0 (53) T T v N v þ M v þ z ≥ 0 (54) A A A ν ≥ 0 ∀u∈ U (55) where t t N ¼G þ α λ G K Lρ ; T ν int U (56) t * t M ¼ð1 α λ ρL KÞg þ G ðσ þ α λ K Lp Þ T min U int U 0 F t * t * min z ¼ ððv σ Þf þ α λ f π Þ C α λ L Kg σ g R (57) T e e e ext E e e int U ATS ATS 0 0 e∈E t t N ¼ð1 ð1 λ Þρ L KÞG ; A U ν (58) t * t t * M ¼ð1 λ ÞL Kðg ρ p KG Þþ c G g A U 0 ν ν 0 0 t t * z ¼ðð1 λ Þp L K c Þ$g þ μ$ ðπ ð1 λ Þf A U 0 e E e e∈E (59) F min ðc þ v Þf Þ C R e e e ALS ALS In general, N is definite negative, whereas N is definite negative. Solution methods can be T A found in [21,22]. Figure 2 illustrates the two-dimensional case (two air links operated in a single sector of air traffic control) (Figure 3): The feasible region is represented by the area which is the intersection of the profitability areas of the airlines sector (ALS) and of the ATS (ATC). Here the demand level lines are straight lines parallel to Δ, whereas the optimal solution is at point A. In the case in which the solution of (52)–(55) does not satisfy condition (45), the volume of demand is fixed and given by (51). Then the optimum problem reduces to finding a feasible solution to the linear set of constraints with respect to ν given by: X X X *t * t * F min f $ν þ α λ f π σ $f þ ððv σ Þf þ α λ f π Þ C R ≥ 0 int U e e e ext E e u u e ATS ATS u∈U e∈E e∈E (60) X X *t * * * t f :v þ p L ð1 λ Þ$f c $f þ μ$ ðπ ð1 λ Þf u U e E u e u∈U e∈E ! (61) F min ðc þ v Þf Þ C R ≥ 0 e e e ALS ALS ν ΦL≤ ε (62) * * with (55) where p is given by (44), f and f are given by (51). ACI 17,1 Figure 2. Solution for public ATS supplier (bi- dimensional case). Figure 3. Solution for private ATS supplier (bi- dimensional case). Now considering that ATS fees are established on a flown time basis, an ATS rate per flown hour v can be introduced such as: ν ¼ L vu∈ U (63) u u which is solution when (45) is satisfied of the scalar optimization problem: min K G Lv (64) v∈R under the constraints: t 2 ðL N LÞv þðM LÞv þ z ≥ 0 (65) T T T t 2 ðL N LÞv þðM LÞv þ z ≥ 0 (66) A A ðL ΦLÞv≥ ε (67) If the feasible set associated to constraints (65)–(67) is empty, the ATS supplier can adopt the solution of the scalar optimization problem: *t max f $ν (68) v∈R under the constraint t Pricing v ΦL≤ ε (69) schemes for air * * with (44) where p is given by (51), f and are given by (51). traffic services 8. Pricing of ATS with a private supplier According to relations (17)–(19), considering that condition (35) holds, the problem of optimization solved by the public ATS provider can be rewritten under the form: max v N v þ M v þ z (70) T T jUj v∈ðR Þ under the constraints: v N v þ M v þ z ≥ 0 (71) A A ðL ΦLÞv≥ ε (72) Here also, solution methods can be found in [21,22]. The feasible region is represented by the area which is the intersection of the profitability areas of the airlines sector (ALS) and of the ATS (ATC). Here the demand level lines are not represented, the profit level lines are parameterized by the profit level, whereas the optimal solution is at point B. If the feasible set associated to constraints (57) and (58) is empty, the ATS supplier can adopt the solution of the scalar optimization problem (68), (69) with (55) where p is given by (44) while f and f are given by (51). Here also, if we are interested in the ATS rate per flown hour v, the problem of optimization solved by the public ATS provider can be rewritten under the scalar form: t 2 max ðL N LÞv þðM LÞv þ z (73) T T T jUj v∈ðR Þ under the constraints: t 2 ðL N LÞv þðM LÞv þ z ≥ 0 (74) A A A ðL ΦLÞv≥ ε (75) 9. General solution algorithm The problem considered in Section 5 with a linear criterion and quadratic constraints can be considered to be a special case of a non convex linear program with LMI constraints such as: t 1 2 min c zunderM ðzÞ≥ 0 and; M ðzÞ≥ 0 (76) where c∈ R is given and where: j j j M ðzÞ¼ M þ z M j ¼ 1; 2 (77) 0 i i¼1 M j; i ¼ 0to m; j ¼ 1; 2 are symmetric matrices. i Observe also that problem (70)–(72) can be rewritten in this formalism by replacing (70) by: ACI t t 17,1 max w with v N v þ M v þ z w≥ 0 (78) T T where w is the level of the objective function. General non convex problem (76) and (77) can be solved through an ellipsoid algorithm which has been developed in the field of LMI’s. At start it is supposed that an ellipsoid E in R contains the feasible set and hence the optimal solution. A cutting plane crossing the center z ð0Þ of this ellipsoid is chosen so that the optimal solution lies in one of the half spaces of R given by: m t fz∈ R : νð0Þ ðz z ð0ÞÞ≤ 0g (79) where νð0Þ is a non zero vector of R . Then an ellipsoid E with minimum generalized volume and containing the half ellipsoid given by: E ∩ fz∈ R : νð0Þ ðz z ð0ÞÞg (80) is constructed. The size of this ellipsoid is smaller than the one of the previous ellipsoid and contains the solution. This process can be repeated until a required accuracy is achieved. Given an ellipsoid E given by: m −1 fz∈ R : ðz z ðkÞÞ A ðz z ðkÞÞ≤ 1g (81) c c where A is a symmetric definite positive matrix, the minimum volume ellipsoid E k kþ1 containing the half ellipsoid: E ∩ fz∈ R : νðkÞ ðz z ðkÞÞ≤ 0g (82) is given by m −1 fz∈ R : ðz z ðk þ 1ÞÞ A ðz z ðk þ 1ÞÞ≤ 1g (83) k c where: z ðk þ 1Þ¼ z ðkÞ A w (84) c k k m þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w ¼ v ν A v (85) k k k k and m 2 A ¼ A A w w A (86) kþ1 k k k m 1 m þ 1 Then considering at step k a point ν in R , two cases can be considered: 1 2 – either M ðy Þ≥ 0 and M ðy Þ≤ 0, in that case one takes ν ¼ c and the half space: k k m t fz∈ R : ν ðz y Þ > 0g (87) t t can be deleted since there c z ¼ c y and points z cannot be solution of the optimization problem. n 1 2 – or M ðy Þ < 0or M ðy Þ > 0, there exists a non zero vector u of R such that according Pricing k k to the case: schemes for air m m X X traffic services t 1 1 t 2 u ðM þ z M Þ < 0or u ðM þ z M Þ > 0 (88) 0 i i 0 i i¼1 i¼1 then choosing: ν ¼ −u M u; i ¼ 1to m (89) ki i m t we have for every z∈ R such that ν ðz− y Þ≥ 0: t 1 t 1 t t 2 t 2 t u M ðzÞu ¼ u M ðy Þu ν ðz y Þ < 0or u M ðzÞu ¼ u M ðy Þu ν ðz y Þ > 0 k k k k k (90) The feasible set will be in the half space: m t fz∈ R : ν ðz y ð0ÞÞ < 0g (91) and ν allows to define the cutting plane at point ν . Then the whole process is repeated until k k the size of the ellipsoid becomes sufficiently small to insure accuracy of the solution. It can be shown that convergence is exponential. 10. Numerical application To illustrate the proposed approach, we consider the case of the air traffic area represented in Figure 4. Table 1 provide the adopted values for the main parameters of the considered air traffic network: Figure 4. The considered air traffic network. Link u 123 45 6 7 Table 1. L (hours) 2 1 1 2 1 1 2 u Adopted values for air K (seats) 100 100 100 100 100 100 200 u links delays, capacity c 8000 4000 4000 8000 4000 4000 16000 and costs. u (Euros) Here demand is supposed to depend only on the average price of tickets. Relation (92) ACI display the average daily demand for each link. 17,1 2 3 2 32 3 100 0:010 0:000 0:000 0:000 0:000 0:000 0:0000 π 6 7 6 76 7 80 0:000 0:010 0:000 0:000 0:000 0:000 0:000 π 6 7 6 76 7 6 7 6 76 7 60 0:000 0:000 0:013 0:000 0:000 0:000 0:000 π 6 7 6 76 7 6 7 6 76 7 Dð;Þ¼ 100 0:000 0:000 0:000 0:012 0:000 0:000 0:000 π (92) 6 7 6 76 7 104 6 7 6 76 7 40 0:000 0:000 0:000 0:000 0:0012 0:000 0:000 π 6 7 6 76 7 4 5 4 54 5 70 0:000 0:000 0:000 0:000 0:000 0:010 0:000 π 120 0:000 0:000 0:000 0:000 0:000 0:000 0:007 π F min F min C ¼ 85000; R ¼ 130000; C ¼ 1020000; R ¼ 1550000 ATS ATS ALS ALS Minimum returns for the ATS and the ALS have been taken equal to 1550 000 Euros and 130 000 Euros respectively while fixed costs for ATS and ALS have been taken equal to 85 000 Euros and 1 020 000 Euros respectively. Tables 2 and 3 displays the obtained results for different values of σ and λ given in %, unit for the ATS service rate v is Euros per flight hour, units for mean tickets prices π are F F Euros. Decreases of fuel costs and other expenses which are included in C and C have ALS ATS min min been considered allowing to introduce higher levels for R and R . With the following ALS ATS F min F min values, C ¼ 82000Euros, R ¼ 132000Euros, C ¼ 1000000Euros, R ¼ 1560000 ATS ATS ALS ALS Euros, the resulting pricings are given in Table 4 and 5. 11. Discussion of the results In all the considered numerical cases, the adopted solution algorithm (Section 9) produced the optimal solution in a reduced number of iterations. This has allowed to consider large range of variations for the average ATS costs (σ) and for airlines tax rate (λ) while sensitivity analysis with respect to other relevant parameters could be performed. A global view of prices at the network level is obtained which is of interest for both the ATS and the airlines sector. In the considered demand structure (relation 92) no competition has been introduced between destinations but this situation could have been tackled easily by the proposed approach. In σ λ νπ π π π π π π 0 1 2 3 4 5 6 7 20 0.10 108.06 216.12 108.06 108.06 216.12 108.06 108.06 216.12 0.15 105.14 210.28 105.14 105.14 210.28 105.14 105.14 210.28 0.20 102.22 204.44 102.22 102.22 204.44 102.22 102.22 204.44 0.25 99.3 198.6 99.3 99.3 198.6 99.3 99.3 198.6 30 0.10 118 236 118 118 236 118 118 236 0.15 115 230 115 115 230 115 115 230 0.20 112 224 112 112 224 112 112 224 0.25 109 218 109 109 218 109 109 218 40 0.10 128 256 128 128 256 128 128 256 0.15 125 250 125 125 250 125 125 250 0.20 122 244 122 122 244 122 122 244 0.25 119 238 119 119 238 119 119 238 50 0.10 138 276 138 138 276 138 138 276 0.15 135 270 135 135 270 135 135 270 Table 2. 0.20 132 264 132 132 264 132 132 264 Pricing results for the public ATS case. 0.25 129 258 129 129 258 129 129 258 Pricing σ λ νπ π π π π π π 0 1 2 3 4 5 6 7 schemes for air 20 0.10 136.5 273 136.5 136.5 273 136.5 136.5 273 traffic services 0.15 142 284 142 142 284 142 142 284 0.20 149 298 149 149 298 149 149 298 0.25 155 310 155 155 310 155 155 310 30 0.10 136.5 273 136.5 136.5 273 136.5 136.5 273 0.15 142 284 142 142 284 142 142 284 0.20 149 298 149 149 298 149 149 298 105 0.25 155 310 155 155 310 155 155 310 40 0.10 136.5 273 136.5 136.5 273 136.5 136.5 273 0.15 142 284 142 142 284 142 142 284 0.20 149 298 149 149 298 149 149 298 0.25 155 310 155 155 310 155 155 310 Table 3. 50 0.10 136.5 273 136.5 136.5 273 136.5 136.5 273 Pricing results for the 0.15 142 284 142 142 284 142 142 284 0.20 149 298 149 149 298 149 149 298 private ATS case in 0.25 155 310 155 155 310 155 155 310 Europe. σ λ νπ π π π π π π 0 1 2 3 4 5 6 7 20 0.10 107 214 107 107 214 107 107 214 0.15 104 208 104 104 208 104 104 208 0.20 102 204 102 102 204 102 102 204 0.25 99 198 99 99 198 99 99 198 30 0.10 117 234 117 117 234 117 117 234 0.15 114 228 114 114 228 114 114 228 0.20 112 224 112 112 224 112 112 224 0.25 109 218 109 109 218 109 109 218 40 0.10 127 254 127 127 254 127 127 254 0.15 124 248 124 124 248 124 124 248 0.20 122 244 122 122 244 122 122 244 0.25 119 238 119 119 238 119 119 238 50 0.10 137 274 137 137 274 137 137 274 Table 4. 0.15 134 268 134 134 268 134 134 268 0.20 132 264 132 132 264 132 132 264 Pricing results for the 0.25 129 258 129 129 258 129 129 258 public ATS case. σ λ νπ π π π π π π 0 1 2 3 4 5 6 7 20 0.10 121 242 121 121 242 121 121 242 0.15 127 254 127 127 254 127 127 254 0.20 134 268 134 134 268 134 134 268 0.25 140 280 140 140 280 140 140 280 30 0.10 121 242 121 121 242 121 121 242 0.15 127 254 127 127 254 127 127 254 0.20 134 268 134 134 268 134 134 268 0.25 140 280 140 140 280 140 140 280 40 0.10 121 242 121 121 242 121 121 242 0.15 127 254 127 127 254 127 127 254 0.20 134 268 134 134 268 134 134 268 0.25 140 280 140 140 280 140 140 280 Table 5. 50 0.10 121 242 121 121 242 121 121 242 Pricing results for the 0.15 127 254 127 127 254 127 127 254 0.20 134 268 134 134 268 134 134 268 private ATS case in 0.25 140 280 140 140 280 140 140 280 Europe. the considered numerical case developed in this section, public ATS provides in general ACI better results for travellers (lower ticket fares) than private ATS, airlines results remain 17,1 stable (lower fares compensated by higher demand) while ATS results are lower in the public case. According to tax levels, fares can be modified (þ or ) up to 15% and demand can be modified (þ or ) up to 10%. However, adopting different values for the parameters as well as considering different network structures, could lead to different conclusions. 12. Conclusion In this article we addressed the complex problem of ATS pricing at network level by integrating within a new multilevel framework the behavior of the different involved economic agents. Then, it has been possible to take into account the reactivity of supply by the airline companies and of demand by users as a result of the variation of ATS charges and air tickets prices. The proposed framework allows in particular to tackle the issue of having either a public or a private ATS provider by introducing differentiated objectives depending of the nature of the ATS provider and leading to different optimization problems. This has resulted in the formulation of two different multilevel programming problems with a common lower level problem associated with the profit maximization behavior of the airline sector. This lower problem has been tackled on a multidimensional basis. It has been shown that when mean spatial rates are considered, it is possible under mild assumptions to solve analytically this problem. The higher problems associated to the behavior of a private or a public ATS provider, result in quadratic constrained optimization problems which can be easily solved numerically using a specialized version of the ellipsoid algorithm. The proposed approach allows for different sets of cost and demand parameters, the extensive comparison of the optimal solutions in terms of expected aircraft/passengers flows for the whole network and in terms of economic returns for the ATS provider and the airline sector. The complexity of the considered issue has been tackled by designing a multilevel solution approach which produces, through the successive resolution of reduced numerical problems, a sound basis for decision by public authorities to pursue efficiency and fairness at network level for ATS pricing. References M.W. Tretheway, Cost Allocation Principles for ATC, Conference on Air Traffic Economics, Belgrade (2009). United States Government Accountability Office, Characteristics and performances of selected international air navigation service providers and lessons learned from their commercialisation, United States Government Accountability Office, Report to Congressional requesters, July 1-34 (2005). A. A. Oumarou and F. Mora-Camino, Pricing in air transportation systems: a multilevel approach, XIV Congreso Panamericano de Ingenieria de Transito y Transporte, September 20-23, Las Palmas de Gran Canaria, Spain, 2006. R. Guettaf, M. Larbani, F. Mora-Camino, Pricing of ATC/ATM Services with a Private Provider, VIII SITRAER, Sao Paulo, 2009. R. Guettaf, C. Mancel, M. Larbani, F. Mora-Camino, Pricing of ATC/ATM services through bilevel programming approaches, J. Braz. Air Trans. Res. Soc. (2010). J.F. Bard, Practical Bi-level Optimization: Algorithms and Applications. Kluwer Book Series: Non Convex Optimization and its Applications 30 (1998). S. Dempe, Foundations of Bi-Level Programming, Kluwer Academie Publishers, Dordrecht, 2002. Pricing G. Savard, J. Gauvin, The steepest descent direction for the nonlinear bilevel programming schemes for air problem, Oper. Res. Lett. 15 (1994) 265–272. traffic services Marianne Raffarin, Congestion in European airspace a pricing solution?, J. Trans. Econ. Policy 38 (1) (2004) 109–125. A. Ranieri, L. Castelli, Pricing schemes based on air navigation service charges to reduce en-route ATFM delays, Third International Conference on Research in Air Transportation, 2008, Fairfax, VA. T. Bolic, L. Castelli, D. Rigonat, Peak-load pricing for the European air traffic management system using modulation of en-route charges, Eur. J. Trans. Infrastruct. Res. 17 (1) (2017) 136–152. R. Jovanovic, V. Tosic, M. Cangalovic, M. Stanojevic, Anticipatory modulation of air navigation charges to balance the use of airspace network capacities, in: Transportation Research Part A, Policy and Practice, Elsevier, 2014, pp. 84–99. M. Labbe, A. Violin, Bilevel programming and price setting problem, in: Annals OR, Springer, 2016, pp. 141–169. A. Violin, M. Labbe, L. Castelli, En route charges for ANSP revenue Maximization, 4th International Conference on Research in Air Transport, 2010. Budapest. L. Castelli, M. Labbe, A. Violin, A Network Pricing Formulation for the Revenue Maximization of European Air Navigation Service Providers, ORP3 Meeting, Cadiz, Sept. 13-17 (2011). ICAO, Manual on Air Navigation Services Economics, Doc 9161, 2013 Edition. P. Holder, Airline Operation Costs, prepared for: Managing Aircraft Maintenance Costs Conference, Brussels 22 (2003). ICAO, Tariffs for Airports and Air Navigation Services, 2010 Edition. Eurocontrol, Guidelines for the Implementation of Single European Sky Legislation, the Military Released Issue, ed. 1.0, 14/07/2009. United States Government Accountability Office, Assigning Air Traffic Control Costs to Users Elements of FAA’s (2010). R.M. Freund, Solution Methods for Quadratic Optimization, Massachusetts Institute of Technology, 2004. E.M. Gertz, S.J. Wright, Object-oriented software for quadratic programming, ACM Trans. Math. Software 29 (1) (2003) 58–81 2003. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, 1994. B. Dane, Improved, ellipsoid algorithm for LMI feasibility problems, Int. J. Control, Automation Syst. 7 (6) (2009) 1015–1019. Corresponding author Felix Mora-Camino can be contacted at: email@example.com For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: firstname.lastname@example.org Applied Computing and Informatics – Emerald Publishing Published: Jan 4, 2021 Keywords: Quadratic optimization; Flows optimization in networks; Pricing; Air traffic services Access the full text. Sign up today, get DeepDyve free for 14 days.

If you need to convert a unit between fractional, milesimal inch and millimeter go to the page: Calculate linear dimensions in millimeter, fractional and milesimal inch and solve your problem, later come back here to learn how to do it. This Interactive conversor of millimeter to inch Interativo de millimeter para polegada (and vice-versa) was created to facilitate your comprehension, to allow you to evaluate your results and antecipate them. It's utilization is very simple, roll down the page until the conversor, drag and drop the red line in the horizontally, by aligning the values on the scales (millimeter 'mm' and inch 'in'). Just below them, the values appear in millimeters, fractional inch (fractioned) and corresponding milesimal inch and the formula of calculation. After interacting, rol down the page to understand the mechanics of the calculation. Before going on on this page, I recommend you to interact with the Fraction topic on page: Using the fractional inch understanding and measuring , later come back here, it's worthing! Inch ( polegada / polegadas in portuguese). Your abreviation is in , sometimes we see " (double quotes) or pol . It is a linear measure unit of English System, Imperial System, among others. Was based on the dimensions of the human body and the fractional number system (fractionated). Some people maintain that its controversial origin is associated with the extent of the human thumb, another argues that its origin comes from the Latin word uncia that means "one twelfth" so, inch is an uncia , or 1/12, the foot. Later, beyond the fractional system, it became to be divided by the decimal system, this way we have two division systems of inches that we have to know about: the fraction (eg.: 1/128") and milesimal (eg.: 0.025in). The milesimal system is also known as milesimal of inch (mil) the notation of the previous example is 25mil (twenty-five thousandths of an inch). Nowadays, and by definition, the yard was standardized as 0.9144m. A yard is aproximatelly to 3 foot -table 1, a inch, that is 1/12 of a foot, and it's aproximatelly to 2.54cm or 25.4mm (unit used in the metrology, mechanical engeneering and in the metal mechanic ofices). 0.9144/(3*12) Thus, 1in = 25.4 mm. So we have 25.4 mm/in (25.4 millimeters by "for each" inch)-Figure 1 (note that in Interactive conversor (simulador) there are two parallel horizontal lines of the same length. They are equivalent to 1.000in and 25.4mm - 1 ") (bounded by 0 and 1). Transformation of inch to millimeter? Nothing more easy! The fractional inch is the one represented as a fraction, for example: 1/2" (half inch); 1/4" (a quarter of inch); 7/16"(seven sixteenths inch); 3.5/128" (three inches five hundred twenty eight cents). To convert to milesimal inch form simply divide the fraction and add the integer part. See: Just multiply the measure in milesimal inch by 25.4mm/in. See: Just transform the fractional inch in milesimal inch and multiplicate the meausre in milesimal inch by 25.4mm/in. See this example: 1.3/8" (or 1.3/8in or also a three-eighths inch). We will divide three by eight, to transform the fraction in mils of inch, will add the integer part and multiply by 25.4 mm/in. other example: we will transform 5.1/8in (five and one-eighth inch) in millimeter: we will divide one by eight (1/8 = 0.125), and add five and multiplicate by 25.4mm/in-figure 2. It's a little bit more difficult, but it's also easy, just divide the measure in millimeter by 25.4 mm/in. See this example: 70 mm. We will divide 70mm by 25.4mm/in obtainning 2.756in (or two thousand seven hundred fifty-six mils) already is the milesimal inch. To convert to fractional inch, subtract the integer part and keep it: 2.756in - 2in=0.756in. Multiply this value by 128/128 (that is the lower fraction of inch used in the mechanical engineering, besides being equivalent to one -that os the neutral element of multiplication) and obtain as result 96.77/128. Round the numerator 97/128in and add the whole number that was reserved: 2.97/128in. With no intention to make anymore decorate, the fact that the numerator is an odd number indicates that it is necessary to simplify the fraction. If the numerator is even simplify the fraction - Figure 3. See! There is nothing more than this. practise a little bit with the Interactive Conversor and check the results at: Program to calculate linear dimensions in millimeter, fractional inch and milesimal Any problems? take a look in the page: Use of measures in fractional inch - understanding and applying . Meter , from greek metron = measure. Your abreviation is m (minuscule). That is a unit of linear measure of International System. Was based on the size of the earth and the decimal number system. His thousandth part is millimeter abreviation mm (minuscules), widely used in mechanical engineering. The utilization is very simple, drag and drop the vertical line in the horizontal, aligning the values in the scales (millimeter 'mm' and inch 'in'). Just below them, t will appear the corresponding values in millimeters, fractional inch and milesimal inch and the formula to the calculation. This Interactive Conversor of millimeter to inch (and vice-versa) was designed to facilitate comprehension, allow you to evaluate your results and anticipate. Use of meausres in fractional inch - understanding and applying Use of paquimeter in fractional inch - measuring and interpreting Table of conversion between fractional inch, milesimal inch, foot and millimeter Calculate linear dimensions in millimeter, fractional inch and milesimal inch Conversion program between linear units: transformation between Millimeter, fractional Inch and Milesimal

• Largest democracy and 6th largest country in the world. • India is a 34,000 years old country • India is the world's largest, oldest, continuous • India is the third largest economy in the world with an ever growing GDP of 9.2%. • Exports software to 90 countries etc etc. • Sanskrit is the mother of all the European languages. • World's first university was in Takshashila in 700 BC. • Ayurveda is the earliest school of medicine. • Sushruta is the father of surgery. • Aryabhatta invented the Number System. • 42% doctors, 34% software engineers and 24% Scientists (In Europe and USA) - We are not Indian (Except during Diwali and religious festivals) ‘India is, the cradle of the human race, the birthplace of human speech, the mother of history, the grandmother of legend, and the great grand mother of tradition. Our most valuable and most instructive materials in the history of man are treasured up in India only.’ ‘If there is one place on the face of earth where all the dreams of living men have found a home from the very earliest days when man began the dream of existence, it is India.’ • Health care Budget just 0.9% of GDP • Average amount a person spends on health care $82 • 45% of Children under five are malnourished • Yet it has become a Hub for Medical Tourism • An angioplasty that takes $50,000 in US costs just $11,000 in India. • 35.6 % reproductive age women have low BMI • 30 % women (25-49) gave birth before 18 years • 58 % women are married before legal age of 18 • 56 % married women have anaemia • 69.5 per cent of the children (6-59 mnth) have anaemia. • Just over 50 % mothers receive 3 or more ANC visits • Only 44 % children are fully immunized • 25% MPs have criminal records. • 99% MPs were elected despite getting less than • We have one of the lowest voter turn out at 58%. • There are 173 registered political parties in India. • 40 richest Indians worth $170 Billion • 26% population still lives below official poverty line. • Orissa - Poverty rate is 43%, Bihar - Poverty rate is 41% (Below Malawi and Ghana : one of the poorest countries in the world) • Haryana - Poverty rate is 5.7 % Punjab - Poverty rate is 2.4 % (Better than Costa rica, one of the most developed countries of South • 100 millions Indians are unemployed. • 50 million child labour. • Our manufacturing sector 1/5th Contributes only 17% to GDP. • 60% of country is still employed in Agriculture Contribute only 22% to GDP. • 33% of all illiterates of world live in India. In china this figure is just 11%. • There are 200 millions children in India 50 millions do not go to School. • 80,000 Schools are without Blackboard. • 1,44,000 Schools have just one teacher. • Literacy rate is 67% which was just 14% in 1947 • 100 million people live in slums. • 30 millions house are required for lower and middle • In Human Development Report of 177 nations, India is at 127th (China is at 85th and Sri Lanka at 93rd Law & order • We have 13 Judges per million of population. • There are 3 Crore (30 million) pending cases in our 25.5 millions in District & Sub-district court 3.36 lakhs in High Court 39,000 in Supreme Court. • We have one of the lowest conviction rate just 42.4% Japan & Russia have 99% conviction rate. • CO2 emission per ton per capita : India - 0.9, US - 19.5, China - 2.7 and Russia - 10.7 • Yet India generated 29,129 million litres per day of sewage out of which only 6190 MLD could be • Over 50% of Urban India has no access to sanitation. • Over 6 million people are getting connected every • Total usage 100 million, more than entire population largest cellular market behind China, US, Japan & • At independence, there were 1.1. Million telephones. Now this number has reached 218 millions. • 33% of electricity is lost or stolen during transmission and distribution. This theft rate is among highest in the world. • The total power production is 12% short of demand. • Only 44% rural household has access to power. • Rural household did not have power for 13-17 hours • India system - corrupt system. • Cost of corruption for India? Rs 2.5 lakh crores for 2009! • Bribes for basic health services Rs. 8824 crores (India Corruption study 2008). • 3 crores households had to pay bribes during 2008. • Agriculture, Health, Education, Law and Order • Beurocratic system !! Why so ? • High population growth rate • Agrarian form of economy • Primitive agricultural practices • Unemployment and underemployment • Caste based politics • Urban rural divide • Social iniquity and discrimination etc etc. How to combat corruption ? • Acts and Laws Right to Information Act 2005 - Use it • Agencies involved in combating corruption Government of India Agencies, Voluntary bodies • Organized Crime, Corruption and Indian Knowledge and Action Corruption and Citizens Pressure on social and political bodies Still we have time… • To enjoy IPL matches • On Miss World and Universe shows • What’s going on in Abhi-Ash personal life • Launch of new Mercedes Benz • Green card dream and Flying abroad for job Is desk ka kuch nahi ho sakta …. • How many times we have cared for our own country and our own problems ? • No nation is born great - we have to make it great ‘So long as the millions live in hunger and ignorance, I hold every man a traitor who, having been educated at their expense, pays not the least heed to them.’ Any thought !!

SEMITIP, VERSION 4, DOCUMENTATION A program for computing the electric potential and tunnel current due to a probe tip in proximity to a semiconductor, with circular symmetry. Prolate spheroidal coordinates are used in the vacuum, and a carefully chosen updating scheme is used to ensure stability of the iterative solution. Includes capability for a user specified distribution of surface states. Version 4.0 - written by R. M. Feenstra, Carnegie Mellon University, Version 4.1 - posted Dec 13, 2010 Version 4.2 - posted Jan 13, 2011 All routines are written in standard FORTRAN. A complete description of the background theory of this program is contained in Refs. 1-4. Some examples of running the program are provided in the SEMITIP V4 Introduction. Also, a user should carefully study the documentation for VERSION 2, and VERSION 3 of the program. The major change in VERSION 4 compared to VERSION 3 is that computations of the tunnel current are now possible. Additionally, VERSION 4 incorporates several changes to the input and output from the program, to permit upwards compatibility to VERSION 5. Specific changes in VERSION 4 compared to VERSION 3 include: The number of entries in the charge density table is now user specified, line 40 of FORT.9 (in prior versions this value was set in the program to 50000). In the output file FORT.10, the fourth entry is now the contact potential, rather than the opening angle of the tip shank. Fort.14 and FORT.15 have been added, containing values for the tunneling current and conductance, respectively. The conductance is computed according to the value of the modulation voltage on line 19 of FORT.9, being the difference between the currents at sqrt(2)xVmod and -sqrt(2)Vmod (the modulation voltage is thus interpreted as the rms value for a lock-in amplifier). Multiple input sample bias voltages can now be entered; the number of biases in listed on line 41 of FORT.9, and the full list of biases is on line 42. (The bias voltages can extended on to additional lines if needed. Also, the values can be separated by either commas or blank spaces.) Computations of current are for three valence bands (light hole, heavy hole, and split-off band) and a single conduction band. All of the relevant effective masses are entered in FORT.9, on lines 12-15, along with the spin-orbit splitting on line 16. Lines 22-30 of FORT.9 contains information on possible surface states, whose charge densities can be included in the computation. Either a uniform or Gaussian-distributed bands of surface states are permitted as standard input (or, with program modifications, any user specified distribution of states is allowed). Two separate distributions of surface states are permitted as input example 3, in which Gaussian bands are used to describe intrinsic states and a uniform band is used for extrinsic states). Given the individual charge-neutrality levels of the distributions, the program searches for the overall charge-neutrality level of the combined distribution. Each specified The distribution is assumed to be uniform if the specified parameter for the Gaussian FWHM is zero. For a specified Gaussian distribution, two Gaussian bands are assumed, with the center of each band separated from the charge-neutrality level by plus or minus the centroid energy. The input densities (lines 23 and 27) now refer to the integrated densities over the band. Line 22 is a indicator of whether the temperature dependence of the surface state occupation should be included (this dependence is not included in VERSION 3 or prior versions). Including this term necessitates additional computations for determining the table of surface charge densities, prior to each voltage point computed. In most cases including this temperature dependence is not necessary, since a very similar effect can be achieved by some small shift of the parameters used to describe the surface states themselves (See example 1 for further discussion). Lines 31 of FORT.9 contains a value for the electron affinity of the semiconductor (energy difference between vacuum level and conduction band minimum). This value is needed, along with the contact potential and bias voltage, in order to construct the vacuum barrier and compute the wavefunction decay through the barrier. Lines 32 of FORT.9 contain a value for the Fermi-energy of the tip, relative to the bottom of the metallic band of the tip (typically 8 eV). The theory assume that this value is relatively large, in which case there is not dependence on this value other than as a multiplier in the pre-factor for the tunneling current. Lines 41-44 of FORT.9 contains values for parameters involved in the computation of the tunnel current. Line 41 is the number of parallel wavevector values to sum over (same number of values are used for in each of kx and ky), and line 42 is the number energies in the energy integral. Line 43 refers to the step size in the integration of Schrödinger's equation performed to obtain the wavefunctions. Line 44 is the depth into the semiconductor that this integration is performed over; e.g. a value of 0.75 means that the integration extends over three-quarters of the grid points inside the semiconductor. The grid points become further and further apart as the depth into the semiconductor increases, whereas for the integration of Schrödinger's equation it is necessary to maintain some fixed (suitably small) spacing. Hence, limiting the integration such that it stops after, say, 0.5 or 0.75 of the grid points (i.e. at which point the potential is negligible) provides some saving in the computation time required. Lines 48-50 of FORT.9 contain information on the V-shaped s(V) ramp used in the spectroscopic measurement. If no such ramp was used, then these parameters can all just be set to zero. A PARAMETER statement at the beginning for the main program defines the array dimensions. (This type of statement might be compiler specific, and if necessary it can easily be changed to explicit numerical array dimensions and assignment statements for the variables NRDIM, NSDIM, etc. as in VERSION 3 or earlier). Changes to VERSION 4.1 compared to VERSION 4.0 are: In VERSION 4.0, usage was made of Kane's two-band model for obtaining the decay length of the wavefunction at energies throughout the forbidden gap region (E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957)). This model describes the continuous evolution in a state from electron-character (near the conduction band edge) to hole-character (near the valence-band edge). The light-hole band is assumed to be connected in this sense to the lowest lying conduction band, whereas the heavy-hole and split-off valence bands are connected to higher lying conduction bands. See Y.-C. Chang, Phys. Rev. B 25, 605 (1982) for a description of the relevant complex band structures. Thus, in VERSION 4.0, values for the energy gaps between the heavy-hole band and its connected conduction band and betwen the split-off band and its connected conduction band were input to the program, and these values were employed in the computation of the tunnel current. The results of the computations are generally very insensitive to these two parameters, except when observable current occurs for energies deep within the band gap. That situation might occur, e.g., when probing the current very near the onset of a band edge in the presence of large band bending, in which case the two-band solution for the decay length can produce an observably different current. The two-band solution for the decay length can also play a role when inversion occurs. But, additional considerations are needed to properly model that situation; when the two-band model is used and the maximal z parameter value in line 44 of FORT.9 is too large, then the inversion current will erroneously appear in both the computations of the VB current and the CB current. Since it might be difficult for a non-expert user to correctly set the value of the maximal z parameter, in VERSION 4.1 we have eliminated usage of the two-band model. This was accomplished by commenting out lines 54-55 of the main program and lines 548, 556, 656, 664, 768, 783, 877, and 892 in intcurr.f. Inversion currents can still be computed, but they will occur only in the localized states of the relevant band, as is appropriate. The commented-out lines can be added back in if one wants to evaluate the influence of the two-band effects (but again, care must be taken in setting the value of the maximal z parameter for inversion situations for inversion situations). In that case, one would also add two lines following line 16 of FORT.9 giving two additional band gaps between the heavy-hole band and its connected conduction band and betwen the split-off band and its connected conduction band. For GaAs, the values are: 4.55 heavy hole energy gap (eV) 4.71 split-off energy gap (eV) Output files FORT.91 - FORT.94 have been added, provided separately the currents and conductances from the valence band and conduction band. Changes to VERSION 4.2 compared to VERSION 4.1 are: A new grid for the z coordinates in the semiconductor has been implemented. These values, which we denote by s, are now given by: This spacing is analogous to that used for the r coordinates (as described in Ref. 2), with the first s value spaced from the surface by only one-half Δs. This smaller spacing for the first point provides an improved description of the potential near the surface, where it is varying rapidly. Regarding the value of the potential at the final value of s, it is not zero, but rather it is given by the solution to Poisson's equation employing a boundary condition that, in effect, corresponds to the Von Neumann condition (zero slope of the potential). This same condition is used at the largest r value as well. 1. R. M. Feenstra, Electrostatic Potential for a Hyperbolic Probe Tip near a Semiconductor, published in J. Vac. Sci. Technol. B 21, 2080 (2003). For preprint, see 2. R. M. Feenstra, S. Gaan, G. Meyer, and K. H. Rieder, Low-temperature tunneling spectroscopy of Ge(111)c(2x8) surfaces , Phys. Rev. B 71, 125316 (2005). For preprint, see 3. R. M. Feenstra, Y. Dong, M. P. Semtsiv, and W. T. Masselink, Influence of Tip-induced Band Bending on Tunneling Spectra of Semiconductor Surfaces, Nanotechnology 18, 044015 (2007). For preprint, see 4. Y. Dong, R. M. Feenstra, M. P. Semtsiv and W. T. Masselink, Band Offsets of InGaP/GaAs Heterojunctions by Scanning Tunneling Spectroscopy, J. Appl. Phys. 103, 073704 (2008). For preprint, see

Squaring the circle is the age old problem of constructing a square with the same area(or perimeter) as the circle. Greek mathematicians attempted to solve this ancient riddle using a ruler and compass only. Due to the transcendental nature of π we can only approximate this geometry. Earth, Moon, the Great Pyramid, and Stonehenge all encode this great philosophical quandary. Stonehenge encodes the squared circle through its bluestones, named so because when it rains they turn blue. If we draw a square around this circle of stones it will have the same perimeter as the sarsen ring of stones on the very outside. The ancients preserved esoteric gnosis in their sacred buildings as a way to ensure that the information would never be lost or occulted by the greed of mortal men. “ A tradition which has been credited by many learned men over the centuries is that the Ancients encoded their knowledge of the world in the dimensions of their sacred monuments.” – John Michell The Great Pyramid’s height is in relationship to its base sides as a circle’s radius is to its circumference, and thus it ‘squares the circle’. Put another way, the perimeter of the base equals the circumference of a circle whose radius is equal to the height of the pyramid. This is only achieved due to the slope angle being 51 degrees and 51 minutes. (or 51.84 degrees since there are 60 arc minutes in 1 degree) I noticed that in the six days it supposedly took to create heaven and earth, there are exactly 518,400 seconds, which resonates with the decad (the decimal system), and the slope angle. And just recently my friend Dayne Herndon pointed out the fact that 5184 also resonates with the canonical value for the Precession of the Equinoxes of 25,920 years. As 25,920 x 2 = 51,840 What else could the Great Pyramid possibly encode? How about the Prime Meridian as suggested by Carl Munck? This is a future post but for now check out The Code. How about Pi and Tau? “Squaring the circle” is the alchemical process of transferring an airy concept from the mental plane to the physical dimension so that objective conception and birth become a demonstrative reality. – Dr. John Munford The Earth-Moon relationship and the Great Pyramid of Giza both encode the secret to the mystery. Both of these correlations are well over 99.9% accurate. This is what Leonardo Da Vinci’s “Vitruvian Man” is all about. Based on measurements of Leonardo da Vinci’s Vitruvian Man we can see that the sizes of the square and circle aren’t quite the correct size to actually square the area of the circle. (in red) But we can also square the circle with equal perimeters (in blue). Vitruvius’ square (in yellow) is almost exactly halfway between ‘squaring the circle’ with equal area, and ‘squaring the circle’ with equal perimeters. It’s slightly closer to the latter. The square that Da Vinci used is a consolidation of two distinct solutions to ‘squaring the circle’, or obtaining the unobtainable. Did Leonardo encode the solution of this ancient philosophical mystery by suggesting that man is the mean between two transcendental impossibilities? “The workings of the human body are an analogy of the workings of our universe” – Leonardo da Vinci Squaring the Circle Methods Square with area of 4 has width of 2 Circle with area of 4 has width of 2.256758.. 9/8 = 1.125 2.256 / 2 = 1.128379 1.125 / 1.128379 = .9970 A 9:8 ratio squares the area of the circle to 99.70% accuracy In music theory, a 9 to 8 ratio is the whole step, otherwise known as the whole tone, or major second. Square with perimeter of 4 has width of 1 Circle with circumference of 4 has width of 1.2732396.. Interestingly, our Moon takes 27.32166 days to make 1 revolution about Earth. (A 99.998% accurate correlation according to NASA) 14/11 = 1.27272727… 1.2732396 / 1 = 1.2732396 1.272727 / 1.2732396 = .9996 A 14:11 ratio squares the circle with equal perimeters to 99.96% accuracy Earth and Moon solve the first riddle to a very high degree of accuracy. Depending on which measures you use, equatorial, polar and mean diameters, it is at least 99.9% accurate. Square the Circle/ Earth-Moon Correlations: (NASA 2014 measurements in miles) Earth-Moon radii / Earth’s radius = Square the Circle with equal perimeters 5043.175 / 3963.17 = 1.27251039 1.27251039 / 1.2732396 = 99.943% accuracy 5028.63 / 3949.93 = 1.27309 1.27309 / 1.2732396 is 99.988% accuracy 5038.7 / 3958.75 = 1.27280 As you can see you don’t have to cherry pick measurements to achieve a very high degree of accuracy. Earth is a living and breathing organism. Its measurements change over time so this could never be 100% accurate. “The circle is a symbol of spirit, of heaven, of the unmanifest, the immeasurable and the infinite, while the square is the symbol of the material, the Earth, the measurable and the finite,” …the symbolic essence of the problem was the reconciliation of seemingly opposing principles, and the resolution of dualities – “a sacred, cosmological act.” -Daniel Pinchbeck The Earth and Moon are the perfect size to solve the riddle. This only works because the Moon is huge. It’s larger than any other of the solar system in proportion to its planet. Some say we have a double planet system. “The squaring of the circle is a stage on the way to the unconscious, a point of transition leading to a goal lying as yet unformulated beyond it. It is one of those paths to the centre.” – Carl G. Jung Bert Janssen found the squaring of the circle encoded into certain crop circles. His website is definitely worth checking out. The alchemical marriage between Earth and Moon reveal profound geometrical symbols we use to understand our reality. These relationships are a hint into the mystery of our existence. Perhaps we are more than mere coincidence and product of time and chance. The Divine Universal Architect may have hid these clues in plain sight for all to see and cherish, however it is only the few who possess such esoteric knowledge that have the eyes to see these Secrets in Plain Sight. An exercise in contemplative geometry from Robert Lawlor’s Sacred Geometry, 1982. “There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as ‘Squaring of the Circle’. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it. Nevertheless the Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible world. When a near equality is drawn between the circle and the square, the infinite is able to express its dimensions or qualities through the finite” (p74). The derivation begins with an initial circle (within the square) of radius unity. Along its horizontal diameter are drawn two tangent circles, each with radius one half. Observe that the total circumference of the smaller circles equals the circumference of the initial circle, but the total area of the smaller circles is one half that of the initial circle: “One has become Two” (p73), an image of the primary duality, of yin-yang. Next are drawn two arcs from the ends of the initial circle’s vertical diameter with radius tangent to the far sides of the smaller circles. This radius is φ, the golden ratio, dividing the vertical radius of the initial circle into the golden section of lengths 1/φ and 1/φ2. The two arcs meet to create a vesica that encloses the primary duality — the mouth of Ra, the Word, the vibrating string. Around the initial circle is drawn a tangent square, with side 2, perimeter 8; and, finally, a large circle is drawn with diameter equal to the width of the vesica, 2√φ, giving a circumference of 2π√φ = 7.993, approximately equal to 8. The circle is squared. “From the domed Pantheon of ancient Rome, if not before, architects have fashioned sacred dwellings after conceptions of the universe, utilizing circle and square geometries to depict spirit and matter united. Circular domes evoke the spherical cosmos and the descent of heavenly spirit to the material plane. Squares and cubes delineate the spatial directions of our physical world and portray the lifting up of material perfection to the divine. Constructing these basic figures is elementary. The circle results when a cord is made to revolve around a post. The right angle of a square appears in a 3:4:5 triangle, easily made from a string of twelve equally spaced knots.1 But “squaring the circle”—drawing circles and squares of equal areas or perimeters by means of a compass or rule—has eluded geometers from early times.2 The problem cannot be solved with absolute precision, for circles are measured by the incommensurable value pi (S = 3.1415927…), which cannot be accurately expressed in finite whole numbers by which we measure squares.3 At the symbolic level, however, the quest to obtain circles and squares of equal measure is equivalent to seeking the union of transcendent and finite qualities, or the marriage of heaven and earth. Various pursuits draw from the properties of music, geometry and even astronomical measures and distances.”

We made a bourgeois dirt cake by frosting it with chocolate frosting, covering it in cake crumbs from the discarded cake pieces! Cut out the center and slice off a tiny bit of the upper left corner. Then, connect the pound cake to the cut corner. This is the easiest number. We made a 6 stage alarm fire truck, so we covered it in red frosting, made a ladder out of pretzels, oreos for wheels, and a licorice hose! Just kidding, THIS is the easiest one! Cut into three slices, cut the middle slice in half diagonally. Discard the right slice. Flip the left slice up horizontally and connect the other two slices beneath it. To decorate the ocean, we added in swedish fish and Teddy grahams with lifesaver gummy inner tubes. To decorate the beach, we cut fruit roll ups into squares and used them as sunbathing towels for the Teddy grahams. Cut out the centers of the circle cakes, and then slice two edges. Connect the cakes. Life Stage Numerology We made an 8 race track, so we covered the cake in black frosting, piped some green grass, and used Good N Plenty candy to make the lines in the road. This one is the 6 cake flipped upside down. Because we at So Yummy acknowledge that Pluto is a planet ohanameansfamily , we decided to make a galaxy cake out of the number nine to honor all nine planets. We started with a base of black frosting, then swirled in blue and purple frosting and speckled it with white food coloring. Add some candy planets and star sprinkles for an out of this world cake! Slice the pound cake in half. Cut the center out of the circle cake and then slice it in half. Separate the circle halves and fill in the gaps with the pound cake halves. We decided to make a pineapple cake by piping yellow frosting through a small star tip all over the cake. Then just pipe on a smiley face and some leaves to finish it up! Come here often? Subscribe to our newsletter. Yummy Recipes. By: So Yummy. Click here to load earlier stories. So Yummy. Lisa Marie Basile. - What Your Karmic Debt Number Means. - match making horoscope free; - Happy Birthday - Numbers 6:24-26; - Birthday Number 6 in Numerology - Your Strengths & Weaknesses. - The Numerology meaning of the 6 birthday. Gift Guides. Deirdre Durkan. Karen Belz. Read Just One More. This yields. Therefore, the expression above is not only an approximation, but also an upper bound of p n. The inequality. Solving for n gives. Now, ln 2 is approximately Therefore, 23 people suffice. Mathis cited above. This derivation only shows that at most 23 people are needed to ensure a birthday match with even chance; it leaves open the possibility that n is 22 or less could also work. In other words, n d is the minimal integer n such that. The classical birthday problem thus corresponds to determining n The first 99 values of n d are given here:. A number of bounds and formulas for n d have been published. In general, it follows from these bounds that n d always equals either. The formula. Conversely, if n p ; d denotes the number of random integers drawn from [1, d ] to obtain a probability p that at least two numbers are the same, then. This is exploited by birthday attacks on cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for all practical purposes, inevitable. The theory behind the birthday problem was used by Zoe Schnabel under the name of capture-recapture statistics to estimate the size of fish population in lakes. The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an arbitrary number of types. Shared birthdays between two men or two women do not count. The probability of no shared birthdays here is. A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? The answer is 20—if there is a prize for first match, the best position in line is 20th. In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability q n that someone in a room of n other people has the same birthday as a particular person for example, you is given by. Another generalization is to ask for the probability of finding at least one pair in a group of n people with birthdays within k calendar days of each other, if there are d equally likely birthdays. Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a week of each other. The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals . In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday. If we consider the probability function Pr[ n people have at least one shared birthday], this average is determining the mean of the distribution, as opposed to the customary formulation, which asks for the median. How Common is Your Birthday? This Visualization Might Surprise You The problem is relevant to several hashing algorithms analyzed by Donald Knuth in his book The Art of Computer Programming. An analysis using indicator random variables can provide a simpler but approximate analysis of this problem. An informal demonstration of the problem can be made from the list of Prime Ministers of Australia , of which there have been 29 as of [update] , in which Paul Keating , the 24th prime minister, and Edmund Barton , the first prime minister, share the same birthday, 18 January. An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair. Voracek, Tran and Formann showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given. The reverse problem is to find, for a fixed probability p , the greatest n for which the probability p n is smaller than the given p , or the smallest n for which the probability p n is greater than the given p. Some values falling outside the bounds have been colored to show that the approximation is not always exact. How Common is Your Birthday? This Visualization Might Surprise You | The Daily Viz A related problem is the partition problem , a variant of the knapsack problem from operations research. Some weights are put on a balance scale ; each weight is an integer number of grams randomly chosen between one gram and one million grams one tonne. The question is whether one can usually that is, with probability close to 1 transfer the weights between the left and right arms to balance the scale. In case the sum of all the weights is an odd number of grams, a discrepancy of one gram is allowed. If there are only two or three weights, the answer is very clearly no; although there are some combinations which work, the majority of randomly selected combinations of three weights do not. If there are very many weights, the answer is clearly yes. - President Nelson’s 95th Birthday Celebration. - Some Birthdays Are More Common Than Others? - Guess a Birthday? The question is, how many are just sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as it is to be impossible? Often, people's intuition is that the answer is above Most people's intuition is that it is in the thousands or tens of thousands, while others feel it should at least be in the hundreds. The correct answer is The reason is that the correct comparison is to the number of partitions of the weights into left and right. Arthur C. Clarke 's novel A Fall of Moondust , published in , contains a section where the main characters, trapped underground for an indefinite amount of time, are celebrating a birthday and find themselves discussing the validity of the birthday problem. As stated by a physicist passenger: "If you have a group of more than twenty-four people, the odds are better than even that two of them have the same birthday. The reasoning is based on important tools that all students of mathematics should have ready access to. The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer. What calculators do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories. From Wikipedia, the free encyclopedia. For yearly variation in mortality rates, see birthday effect. For the mathematical brain teaser that was asked in the Math Olympiad, see Cheryl's Birthday. Main article: Birthday attack. In particular, many children are born in the summer, especially the months of August and September for the northern hemisphere , and in the U. In Sweden 9. See also: Murphy, Ron. Retrieved International Journal of Epidemiology.

If a whole thing is divided into equal parts then each part is said to be a fraction. In other words, a fraction is a part of the whole. A fraction is formed up of two parts - numerator and denominator. Expressed as - numerator/denominator. The numerator tells us how many of the total parts are taken and the denominator tells us the total number of equal parts. For example, in ¼ the top number ‘1’ is said to be the numerator and ‘4’ is said to be the denominator. Here we can say that 1 is taken from 4 equal whole parts. Fractions are very important to understand as it is used in our daily life. Depending on the numerator and denominator, fractions are divided into different types. These types are given as below: Here let us understand the concept of Like fractions and Unlike fractions in detail. What is like a fraction? In two or more fractions or a group of fractions when the denominator is exactly the same then they are said to be like fractions. Or you can say that fractions have the same number at the bottom. Like fraction example - 2/4, 6/4, 8/4, 10/4. Here we can see that the denominator of all the fractions is 4, so these fractions are called as like fractions. Fractions like 2/8, 25/20, 9/12, 8/32 are also like fractions though they have different denominators because when they are written in the simplest form such as ¼, 5/4, ¾, ¼, we get the same denominator here 4. Fractions like 4/2, 4/6, 4/10, 4 /13 are not like fractions. Here the numerators are the same but the denominators are different. Natural numbers like 2, 5, 6, 8 are called fractions because they have the same denominator 1. They are written as 2/1, 5/1, 6/1, 8/1. Like fraction example Mathematical operations like addition and subtraction can be carried out easily with Like fractions. As the denominator is the same we have to just add or subtract the numerator accordingly. When the denominators of two or more fractions are different then they are said to be unlike fractions. We can define it as fractions with different denominators are called,unlike fractions. Or you can say that fractions that have different numbers in the denominator are called unlike fractions. Unlike fraction example, ⅜, 1/13, 5/16, are called unlike fractions. Unlike fraction example Mathematical operations like addition and subtraction are not as easy as like fractions. Because the denominator of unlike fractions are different. To carry out addition and subtraction with unlike fractions first we have to convert unlike fractions into like fractions. There are two methods to convert unlike fractions to like fractions they are: Cross multiplication method. Cross multiplication method To perform addition or subtraction of two unlike fractions, first, we have to simplify the fraction into simplest form and make the denominators as coprime or relatively prime, then we have to follow the steps explained below in cross multiplication method. Step 1: Multiply the numerator of the first fraction by the denominator of the second fraction. Step 2: Multiply the numerator of the second fraction by the denominator of the first fraction. Step 3: Multiply the denominators of both fractions and take it as a common denominator for the results of step 1 and step 2. Step 4: After simplification, we will get the fractions with the same denominators and now we can carry out the given operation. Example: Add 2/5 and 4/3 2/5 + 4/3 Applying cross multiplication method, we get; = [(2 x 3) + (4 x 5)]/5 x 3 = (6 + 20)/15 To perform addition or subtraction of two unlike fractions, first, check if denominators of the fractions are not coprime (there is a common divisor other than 1), then we have to apply this method. We have to follow these steps to use the LCM method. Step 1: Find the least common multiple of the denominators of the given fractions. Step 2: Using the least common multiple, make all the fractions like fractions. Step 3: Now the denominator of all the fractions will be the same. So we can carry out the necessary operations. Example: Add 5/8 and 9/12 5/10 + 9/12 Now take the LCM of 10 and 12, we get; LCM (10, 12) = 2 x 2 x 5 x 3 = 60 Now multiply the given fractions to get the denominators equal to 60, such that; =[(5 x 6)/(10 x 6)] + [(9 x 5) + (12 x 5)] =(30/60) + (45/60) Add 2/21 and ⅓. 2/21 + ⅓ Now take the L.C.M of 21 and 3, we get L.C.M(21, 3) = 3 x 7 = 21 Now multiply the numbers to get the denominators equal to 21, such that, =[(2 x 1)/(21 x 1) + [(1 x 7)/(3 x 7)] =(2/21) + (7/21) Subtract 2/7 from 5/7. 5/7 - 2/7 Now the denominators of these numbers are already equal. So we can subtract 2/7 from 5/7 very easily. = 5/7 - 2/7 i] 30/15 and 31/15 ii] 6/11 and 7/33 i] 10/8 from 21/8 ii] ½ from 5/8 How to convert unlike fractions to like fractions? To carry out the mathematical operations like addition and subtraction we need to convert the unlike fractions to like fractions. Let us convert 1/2, 3/4, 5/9, and 7/12 into like fractions. Steps for conversion: Find the LCM of the denominators 2, 4, 9,12 we get 36. Calculate the fractions to make the denominators the same. 1/2 = (1×18)/(2 x 18) = 18/36 3/4 = (3 x 9)/(4 x 9) = 27/36 5/9 = (5 x 4)/(9 x 4) = 20/36 7/12 = (7 x 3)/(12 x 3) = 21/36 ½, 3/4, 5/9, 7/12 which are unlike fractions can be represented as 18/36, 27/36, 20/36, and 21/36 which are like fractions. How to find LCM of two numbers? LCM stands for the least common multiple. The least common multiple is the smallest number that is the common multiple of all the given numbers. To find the LCM of given numbers first we have to write all the factors of the given numbers. Now multiply a each factor the maximum number of times it occurs in either number. You will get the LCM of the numbers. example: Let us find the LCM of 30 and 50 First, calculate the prime factors 30 = 2 x 3 x 5 50 = 2 x 5 x 5 Now, LCM = 2 x 5 x 3 x 5

RELATED APPLICATION DATA This application claims priority from U.S. Provisional Patent Application Ser. No. 60/178,482, titled “NETWORK SEARCHING WITH SCREEN SNAPSHOT SEARCH RESULTS,” filed Jan. 27, 2000, U.S. Provisional Patent Application Ser. No. 60/199,781, titled “NETWORK SEARCHING WITH SCREEN SNAPSHOT SEARCH RESULTS AND RICH CONTENT FILES WITH SELF-CONTAINED VIEWER,” filed Apr. 26, 2000, and U.S. Provisional Patent Application Ser. No. 60/230,043, titled “IMAGE COMPRESSION TECHNIQUE FOR USE WITH ANIMATED IMAGE FILES,” filed Sept. 1, 2000. - FIELD OF THE INVENTION A portion of the disclosure of this patent document contains material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. - BACKGROUND OF THE INVENTION This invention pertains to compression techniques on a computer, and more particularly to a lossless compression technique usable with digital color images, including animated images. File compression technology has recently experienced a resurgence. Originally, file compression was necessary because disk space was limited. To maximize the number of files that could be stored on a disk, it was occasionally necessary to compress files. More recently, hard disk space has become very cheap, and users have been able to store massive quantities of data. The need for compression to save disk space has diminished. At the same time that disk space has become cheap, however, another bottleneck has arisen: throughput. Although people enjoy the freedom the Internet gives them, in terms of research and file transfer, most people use limited throughput connections to the Internet. For example, at 56 Kbps, to transfer a 1 MB file takes approximately 2 minutes and 26 seconds. A single image file, storing a 1024×768 image in true color, taking up 2.25 MB of space, requires 5 and a half minutes to download. Multiply that time by several files, and the transfer times become a serious problem. One technique used to reduce the size of the file is to limit the number of colors used in the image. There are two reasons why including a large number of colors in an image is impractical or unnecessary. First, the computer hardware on which the image is displayed (i.e., the monitor and video card) might be limited in the number of colors that can be displayed at one time. Second, the human eye is limited in the number of colors it can distinguish when looking at an image. To address these concerns, a typical image uses a color palette, which includes either a subset of the colors in the image or approximations of the colors in the image. The number of entries in the color palette determines the number of different colors that occur in the image. In the preferred embodiment of the invention, the color palette of the image includes 256 colors, but a person skilled in the art will recognize that this number can vary. The Median Cut or a similar algorithm can be used to select the colors stored in the color palette. The specifics of how the colors are selected for the color palette is not relevant to the invention and will not be discussed here. Using a color palette begins the process of compressing the image. For example, if the image is stored using 24-bit color, it takes three bytes to store the color for each pixel. If only 256 colors are used and stored in the color palette, the color for each pixel can be represented using only one byte: the index into the color palette. This reduces the size of the image file by two thirds. Further compression is also possible. For example, instead of using one byte to identify the index into the color palette for a pixel, a Huffman coding can be applied to the indices into the color palette for the pixels. In a Huffman coding, the frequencies for each symbol (in this case, the different colors in the color palette) in the message (in this case, the image) are calculated. The entire image is scanned, and the number of times each color is counted is scanned. The frequency for each color can then be determined by dividing the number of occurrences of each color by the total number of pixels in the image. Once the frequencies of each symbol in the message are known, a Huffman tree can be constructed. FIG. 10 shows the construction of the Huffman tree. In FIG. 10, there are four symbols, “A,” “B,” “C,” and “D,” with the respective frequencies of 0.10, 0.20, 0.30, and 0.40. The frequencies start out as leaves 1005, 1010, 1015, and 1020 in a to-be-constructed tree. The two smallest frequencies are assigned a common parent node in the tree, and the parent node is assigned a frequency equal to the sum of its children. In FIG. 10, the two smallest frequencies are 0.10 and 0.20, which combine to a parent node frequency of 0.30. The process then repeats, using the parent node's frequency in place of its two children, until only a single (root) node remains. Once the Huffman tree is constructed, the two children of each parent node are assigned a “0” or a “1”, depending on whether they are a “left” or a “right” branch from the parent node. (A person skilled in the art will recognize that the determination of which branch is “left” and which is “right” is arbitrary.) The Huffman coding for each symbol is the sequence of branches from the root node of the Huffman tree to the leaf for that symbol. For example, the Huffman coding for symbol “D” is “1”, whereas the Huffman coding for symbol “B” is “001.” The advantage of Huffman coding is that symbols that occur frequently are assigned shorter codes than symbols that occur infrequently. As can be seen from the example of FIG. 10, the symbol “D” occurs 40% of the time in the message, whereas symbol “B” occurs only 20% of the time. Because there are more occurrences of the symbol “D,” a shorter code for the symbol “D” as compared with symbol “B” will result in a shorter message. There are two problems with using a Huffman coding as described above. First, the image must be scanned twice: once to determine the Huffman codes, and once to compress the image. Huffman coding cannot be determined while scanning the image. Second, because the coding is necessary to determine the appropriate color for each pixel, the coding must be stored in the compressed image file. Other techniques exist to compress images: for example JPEG (Joint Photographic Expert Group) and MPEG (Motion Picture Expert Group) compression. These techniques allow for fast compression and decompression. But JPEG and MPEG compression techniques are lossy: that is, they achieve high fast compression rates by losing information. Typically, the loss is imperceptible: for example, with still images compressed using JPEG compression, the lost information is typically below the level of perception of the human eye. But often, the user cannot afford to lose information from the image that needs to be compressed. For such images, JPEG and MPEG compression is useless. - SUMMARY OF THE INVENTION BRIEF DESCRIPTION OF THE DRAWINGS Accordingly, a need remains for a way to compress digital images that addresses these and other problems associated with the prior art. FIG. 1A shows a computer system designed to compress an image file according to an embodiment of the invention. FIG. 1B shows a computer system designed to decompress an image file according to an embodiment of the invention. FIG. 1C shows two computers as shown in FIGS. 1A and 1B connected via a network for transferring a compressed image file. FIG. 2 shows a close-up of an image file being compressed by the computer system of FIG. 1A. FIG. 3 shows five probability models used to compress the image of FIG. 2 on the computer system of FIG. 1A. FIG. 4A shows the image file of FIG. 2 being divided into boxes for compression on the computer system of FIG. 1A. FIG. 4B shows an image file divided into two different tessellations on the computer system of FIG. 1A for compression. FIGS. 5A and 5B show an animated image file being analyzed to determine a distance frame to compress the animated image file on the computer system of FIG. 1A. FIGS. 6A and 6B show the procedure used to compress the image file of FIG. 2 on the computer system of FIG. 1A. FIG. 7 shows the procedure used to divide the image of FIG. 2 into boxes for compression on the computer system of FIG. 1A. FIG. 8 shows a structure for an image file compressed according to the preferred embodiment of the invention. FIGS. 9A and 9B show the procedure used to decompress the image file of FIG. 2 on the computer system of FIG. 1A. FIG. 10 shows a Huffman coding tree according to the prior art. - DETAILED DESCRIPTION Appendix A shows an implementation of the compression algorithm described herein, implemented in the C programming language and including comments. FIG. 1A shows a computer system 105 in accordance with the invention. Computer system 105 includes a computer 110, a monitor 115, a keyboard 120, and a mouse 125. Computer 110 includes hardware components, such as a processor 105, a memory 130, and a branch prediction apparatus (not shown). Computer system 105 may also include other equipment not shown in FIG. 1A, for example, other input/output equipment or a printer. Computer system 105 stores image 130. Typically, image 130 is loaded into the memory of computer system 105. A person skilled in the art will recognize that image 130 can be accessed in various ways: for example, over a network connection or from a scanner (not shown). Image 130 includes color palette 135, which specifies the colors used to display image 130. Software 140 is installed in computer system 105. Software 140 includes probability set 145, update module 150, compressor 155, distance frame generator 160, block locator 165, and size estimator 170. Probability set 145 includes at least one model, and can possibly include multiple models, which can be used to predict the likelihood of the next pixel having a color that matches its left or upper neighbors. Update module 150 is responsible for updating probability set 145 based on the actual colors of the pixel and its left and upper neighbors. Update module 150 can also select a different model from probability set 145, if needed. (Probability set 145 and update module 150 will be discussed further with reference to FIG. 3, below.) Compressor 155 is responsible for encoding/decoding the color of the current pixel. Distance frame generator 160 is responsible for generating distance frames between two frames of an animated image file. (Distance frame generator 160 is not used when an image file is static.) Block locator 165 is responsible for locating blocks within image 130 that are different from the background color of the image. Size estimator 170 is responsible for estimating the size of compressed image file 175 using different blocks located by block locator 165. Compressor 155 uses the size estimations calculated by size estimator 170 to select the tessellation of the image (or the image frame) that will produce the smallest file size. Compressor 155 then uses compresses each block in the tessellation with the smallest file size for the image (or image frame). This compression uses the probability values in probability set 145. In FIG. 1B, computer system 105 includes decompressor 180. Decompressor 180 is responsible for decompressing compressed image file 175. Since the contents of compressed image file 175 contain all the data of the image, decompressor 180 can completely reconstruct original image file 130 from compressed image file 175. FIG. 1C shows two computers connected via a network for transferring a compressed image file. In FIG 1C, computer system 105 stores a compressed image file. Other computer systems, such as computer systems 185A, 185B, and 185C, are like computer system 105 and can make requests for the image file from computer system 105. The compressed image file is then transferred over network 190 to the requesting computer, which can then decompress compressed image file 175 to reproduce original image file 130. A person skilled in the art will recognize that, although a network is shown in FIG. 1C, there are other ways of transferring the compressed image file from computer system 105 to computer systems 185A, 185B, and 185C. For example, the compressed image file can be placed on a computer-readable medium, such as a floppy disk or compact disc (CD), and physically transferred to the destination computer system. In addition, there are many different types of networks over which the compressed image file can be transferred: for example, local area networks (LANs), wide-area networks (WANs), a global internetwork, wireless networks, and so on. FIG. 2 shows a close-up of image file 130 being compressed by the computer system of FIG. 1A. In FIG. 2, image 130 includes a rectangular array of pixels. Close-up section 205 shows some of the pixels in image 130 more closely. For example, pixel 210 has left neighbor 215 and upper neighbor 220. Left neighbor 215 and upper neighbor 220 have the same color (represented by the cross-hatch pattern), and pixel 210 has the same color. In contrast, pixel 225 has the same color as its upper neighbor 230, but a different color than its left neighbor 210. Pixel 235 has a color different than both its left neighbor 225 and its upper neighbor 240. A person skilled in the art will recognize other combinations of colors for a given pixel and its respective left and upper neighbors. FIG. 3 shows why it matters what the colors of the current pixel and its left and upper neighbors are. FIG. 3 shows a probability set which includes five probability models used to compress the image of FIG. 2 on the computer system of FIG. 1A. In FIG. 3, there are five probability models 305, 310, 315, 320, and 325, but any number of models can be used. Because the details of each are similar, only the details of probability model 305 are shown in detail. Each of the probabilities 305-1, 305-2, 305-3, 305-4, and 305-5 reflects a combination of the colors of the current pixel and its left and upper neighbors; the probabilities are determined by the ratio of the individual probability relative to the sum of all probabilities with the same color combinations for the left and upper neighbors. Thus, probabilities 305-1 and 305-2 are determined relative to all pixels whose left and upper neighbors have the same color, and probabilities 305-3, 305-4, and 305-5 are determined relative to all pixels whose left and upper neighbors have different colors. Probability 305-1 reflects the probability that the current pixel has the same color as both its left and upper neighbors (10/11). Probability 305-2 reflects the probability that the current pixel has a different color than the left and upper neighbors, which have the same color (1/11). Probability 305-3 reflects the probability that the current pixel has the same color as its left neighbor but a different color from its upper neighbor (10/21). Probability 305-4 reflects the probability that the current pixel has the same color as its upper neighbor but a different color from its left neighbor (10/21). Probability 305-5 reflects the probability that the current pixel has a different color than either its left or upper neighbors, which have different colors (1/21). The values shown for probabilities 305-1, 305-2, 305-3, 305-4, and 305-5 are the initial probabilities for each color combination, and are updated as the image is compressed. For example, probabilities 305-1 and 305-2 define the probabilities that the current pixel has the same color as its left and upper neighbors given that the left and upper neighbors have the same color. For example, given that the left and upper neighbors have the same color, it is initially assumed that the current pixel is ten times as likely as not to have the same color as its left and upper neighbors. As the model is updated, this probability changes. It is important to distinguish between the terms “probability set,” “probability model,” and “probability.” “Probability set” refers to the set of all probability models used in the compression. There can be one or more probability models in each probability set; typically, there will be only one probability set used to compress a single image file. “Probability model” refers to the set of probabilities (which can also be called probability values) used in a single probability model. Typically, each probability model will include probabilities for similar conditions. “Probability” refers to the individual probability of a given condition happening in a single probability model. So, in FIG. 3, probability set 145 includes five probability models 305, 310, 315, 320, and 325, and each probability model (such as probability model 305) includes five probabilities 305-1, 305-2, 305-3, 305-4, and 305-5, shown as initial relative values 10, 1, 10, 10, and 1, respectively. The different probability models allow for additional history to be used in predicting the current pixel's color. In general, the behavior of the next pixel will be similar to the behavior of other pixels that have similar color matches between the current pixel and its left and upper neighbors. Thus, if the current pixel has the same color as its upper neighbor but a different color from its left neighbor, it is expected that the next pixel will be colored relatively similarly to other pixels whose previous pixel has the same color as its upper neighbor but a different color from its left neighbor. Just as the colors of the left and upper neighbors select which probability to use within a probability model, the colors of the left and upper neighbors can be used to select the next probability set to use. For example, given that the left and upper neighbors of the current pixel are the same color and the current pixel has the same color, probability set 305 can be used in determining the probability of the next pixel's color. Or, given that the left and upper neighbors of the current pixel have different colors and the current pixel has the same color as the upper neighbor, probability set 320 can be used in determining the probability of the next pixel's color. By changing probability models, each probability model tends to become focused on one probability value, which improves compression. In the preferred embodiment, one model is used after one of the five probabilities occurs: that is, probability model 305 is used after the current pixel has the same color as both its left and upper neighbors, probability model 310 is used after the left and upper neighbors have the same color, but the current pixel has a different color, probability model 315 is used after the left and upper neighbors have different colors, and the current pixel has the same color as its left neighbor, probability model 320 is used after the left and upper neighbors have different colors, and the current pixel has the same color as its upper neighbor, and probability model 325 is used after the left and upper neighbors have different colors, and the current pixel has a different color than either its left or upper neighbor. Update module 150 from FIG. 1 updates the individual probabilities of the probability models in the probability set. For example, assume that probability model 305 is currently being used, and the current pixel has the same color as its left neighbor, but not its upper neighbor. In this case, update module 150 updates probability 305-3. Update module 150 can also select the probability model to use with the next pixel: in this case, probability model 315. FIG. 4A shows the image file of FIG. 2 being divided into boxes for compression on the computer system of FIG. 1A. In the example shown in FIG. 4A, image 130 includes two curved shapes and a rectangle. First, image 130 is scanned to determine a background color. This is done by analyzing the border of the image. If there is a color that predominates the border of the image, this color is selected as the background color. Then, to divide image 130 into boxes, the image is scanned to determine vertical stripes in which pixels appear that differ from the background color. Two such stripes are present in FIG. 4A: stripe 405 and stripe 410. Each stripe is then divided into boxes, such that each horizontal row of pixels in the box includes at least one pixel with a color different from the background color. This forms boxes 415 and 420. Boxes 415 and 420 can then be compressed individually, with the remaining pixels in image 130 colored with the background color. A person skilled in the art will recognize that, although the stripes were first located vertically and the boxes then formed horizontally, this is not the only way the boxes can be formed. For example, horizontal stripes can be formed first, and then boxes by analyzing the horizontal stripes. In addition, although there are advantages to rectangular image shapes, the boxes do not have to be rectangular in shape. Indeed, any shape can be used to define a box. The advantage of the rectangle is its simplistic definition: only a starting point and dimensions are required to define a rectangular box. In the preferred embodiment, boxes are located by analyzing the image for stripes both horizontally and vertically. The size of the compressed file is estimated using both horizontal and vertical striping, and the striping that produces the smaller file size is selected. This comparison is discussed further below with reference to FIG. 4B. FIG. 4B shows an image divided into two different tessellations on the computer system of FIG. 1A for compression. In FIG. 4B, image 425 includes one L-shaped object 430. (Object 430 can be thought of as an approximation of the elliptical objects in box 415 of FIG. 4A.) Although object 430 could be enclosed by box 432, there are other ways to divide image 425. Instead, image 425 can be tessellated into rectangular boxes, and each box compressed separately. Two different tessellations are shown in images 435 and 440, respectively showing the boxes found when the first stripes are located vertically and horizontally. (The shading in images 435 and 440 are only used to show the different tessellations, and are not meant to represent different colors.) Depending on which tessellation from in images 435 and 440 are selected, either boxes 445-1 and 445-2, or boxes 450-1 and 450-2, can be compressed according to the preferred embodiment of the invention. The tessellations shown in images 435 and 440 may result in a smaller file size for the compressed image file. For example, if object 430 includes only one color (ignore for the moment dot 455), compressing box 432 would include two colors, and compression according to the preferred embodiment of the invention would be necessary. On the other hand, boxes 445-1 and 445-2, or boxes 450-1 and 450-2, would each include only one color, and thus can be specified with only a location, size, and color. Although when dot 455 is ignored the two tessellations would result in compressed files of identical size, including dot 455 can make a difference. The box that includes dot 455 is compressed using the preferred embodiment of the invention, rather than just storing a location, size, and color. When dot 455 is included, image 435 has an advantage, since box 445-2, which includes dot 455, is smaller than box 450-2 in image 440, and hence box 445-2 would compress to a smaller file size. FIGS. 5A and 5B show an animated image file being analyzed to determine a distance frame to compress an animated image file on the computer system of FIG. 1A. In FIG. 5A, image frames 130-1 and 130-2 are consecutive frames from the animated image file. Although each frame can be analyzed as described above with reference to FIG. 4A and compressed, this approach might not result in the best possible compression. For example, consider the change between image frames 130-1 and 130-2. Careful inspection will reveal that the only change between image frames 130-1 and 130-2 is that rectangle 505 has moved downward slightly, as indicated by arrow 507. By “subtracting” the image frame 130-1 from image frame 130-2, a distance frame can be computed. This difference is shown in FIG. 5B as distance frame 130-3. Box 510 has changed from the color of rectangle 505 to the background color of the animated image, and box 515 has changed from the background color to the color of rectangle 515. It should be apparent that compressing boxes 510 and 515 of distance frame 130-3 would require less space than compressing image frame 130-2. When it becomes necessary to display image frame 130-2, distance frame 130-3 can be decoded, and only the pixels that have changed from image frame 130-1 need to be redisplayed. Where animated images are used, the preferred embodiment analyzes each frame four different ways. First, as discussed above, the size of each compressed frame is estimated using horizontal and vertical striping. Then the distance frame between the current frame and the previous frame of the animated image is calculated, and size estimations are calculated using horizontal and vertical striping on the distance frame. FIGS. 6A and 6B show the procedure used to compress the image file of FIG. 2 on the computer system of FIG. 1A. At step 605, the color of the current pixel is compared with the colors of the current pixel's left and upper neighbors. Depending on the colors of the current pixel and its left and upper neighbors, the appropriate probability value (and, if necessary, the color of the current pixel) is encoded at the applicable one of steps 610, 615, 620, 625, or 630 (FIG. 6B). At step 635 (FIG. 6A) the model is updated. If multiple models are being used, at step 640 a new model is selected. If there are pixels remaining to be compressed, the procedure returns to step 605 for another pixel. Otherwise, at step 645, the indices of the color palette entries are compressed according to the model(s). The encoding performed at the applicable one of steps 610, 615, 620, 625, or 630 is performed using a range encoder. The range encoder is a variation of an arithmetic coder: the difference between an arithmetic coder and a range coder are primarily technical in nature and the distinction need not be explained here. Arithmetic coding operates by assigning each possible symbol a range of probability values between 0% and 100%. Initially, the arithmetic coding covers the entire range from 0% to 100%. Then, as an individual symbol is encountered, the range is narrowed to include only the probabilities covered by that individual symbol. The process is then repeated, applied to the sub-range established by the previous symbol. An example can help to clarify how the compression works, using a range coder with probability already known: i.e., a static model. Consider the message “DCDACBDCDB.” The letter “A” occurs once, for a probability of 10%. Similarly, the letters “B,” “C,” and “D” have probabilities 20%, 30%, and 40%, respectively. The arithmetic coder can assign to the letter “A” all values between 0.0 (0%) and 0.1 (10%). Similarly, the letter “B” can be assigned all values between 0.1 and 0.3, the letter “C” values between 0.3 and 0.6, and the letter “D” values between 0.6 and 1.0. (The border between ranges for adjacent symbols can be dealt with a technical fix: for example, each range is defined to be exclusive of its upper limit, and so the value 0.1 is assigned only to the letter “B.” A corollary of this definition is that the value 1.0 is excluded. A person skilled in the art will also recognize other ways this problem can be addressed.) When the range encoder encounters the first letter (“D”), it narrows the range of acceptable encodings from 0.0 through 1.0 to 0.6 through 1.0 (since the letter “D” is assigned the range 0.6 to 1.0). Since the second letter is “C,” the range is further narrowed to 0.72 through 0.84. This range is calculated by multiplying the range for the new symbol (“C”) by the size of the range calculated so far (0.4, which gives the relative range of 0.12 through 0.24), and adding the resulting values to the low end of the previous range (resulting in 0.72 through 0.84). When the next symbol (“D”) is encountered, the range is narrowed to 0.792 through 0.84. Table 1 shows the range of acceptable encodings after each symbol in the message is encountered. | ||TABLE 1 | | || | | || | | ||Message ||Range | | || | | ||D ||0.6 though 1.0 | | ||DC ||0.72 though 0.84 | | ||DCD ||0.792 though 0.84 | | ||DCDA ||0.792 though 0.7968 | | ||DCDAC ||0.79344 though 0.79488 | | ||DCDACB ||0.793584 though 0.793872 | | ||DCDACBD ||0.7937568 though 0.793872 | | ||DCDACBDC ||0.79379136 though 0.79382592 | | ||DCDACBDCD ||0.793812096 though 0.79382592 | | ||DCDACBDCDB ||0.7938134784 though 0.7938162432 | | || | After the entire message is processed, the final range produced is 0.7938134784 though 0.7938162432. By using the single number 0.7938134784, the entire message is represented. Decoding is accomplished by reversing the process. Again, the probabilities of the symbols in the message are known in advance. Then, the number is examined. Since it falls between 0.6 and 1.0, the first character of the message is “D.” The low value for the range of the determined character (0.6) is then subtracted from the encoded value, resulting in the value 0.1938134784. This value is then divided by the size of the range for the determined character (0.4), which produces 0.484533696. Since the new value falls between 0.3 and 0.6, the next character of the message is “C.” The low value for the range for the character “C” (0.3) can be subtracted, resulting in 0.184533696, and this can be divided by the size of the range for the character “C” (0.3), which produces 0.61511232. This process can be repeated, until all the characters of the message have been identified. Table 2 shows the complete message as it is decoded. |TABLE 2 | |Encoded Value ||Message | |0.7938134784 ||D | |0.484533696 ||DC | |0.61511232 ||DCD | |0.0377808 ||DCDA | |0.377808 ||DCDAC | |0.25936 ||DCDACB | |0.7968 ||DCDACBD | |0.492 ||DCDACBDC | |0.64 ||DCDACBDCD | |0.1 ||DCDACBDCDB | The reader may wonder how this compresses the message, since 10 characters were needed to represent the encoding. This is explained by the fact that there are very few characters in the message. A longer message would show that the encoded message is shorter than the original message. The above example shows how a message can be encoded using static probability values. As pointed out, it was assumed that the probabilities of the individual symbols in the message were known in advance. When dynamic probability values are used in the present invention enabling single pass compression, this assumption can be discarded. Some initial probability values are assigned to each symbol. One possibility is that each symbol is equally likely. In the preferred embodiment of the invention, certain symbols (symbols that match either their left or upper neighbors) are considered more likely than others. Then, as symbols are encountered, the probability values for the symbols are dynamically updated to reflect the changing probabilities. For example, consider again the message “DCDACBDCDB” above. Initially, each of the four symbols “A,” “B,” “C,” and “D” can be assigned the same probability of 25%. This can be accomplished in many ways: for example, it can be preliminarily assumed that each symbol was encountered once for purposes of defining the initial probabilities. Then, as the symbols are encountered, the counts for the symbols can be updated. So, after the first symbol (“D”) is encountered, its probability value can be updated to 40% (2 out of 5), with each of the other symbols reduced to 20% (1 out of 5). After the next symbol (“C”) is encountered, the probability values for symbols “A” and “B” reduces to 16.67% (1 out of 6 for each), and the probability values for symbols “C” and “D” are changed to 33.33% (2 out of 6 for each). And so on. Using dynamic probability values improves the speed of the encoding technique. To compute the static probability values requires either selecting random probability values that may have no bearing on the actual message, or scanning the message to determine the probability values of the individual symbols in the message. Selecting random probability values can result in poor encoding; scanning the message requires performing two passes over the message (one to determine the probability values, and one to encode the message). With dynamic probability values, both disadvantages are avoided. The probability values become accurate for the message, but only one pass is needed over the message (both to encode the message and to update the dynamic probability values). FIG. 7 shows the procedure used to divide the image of FIG. 2 into boxes for compression on the computer system of FIG. 1A. At step 705, a background color is determined for the image. This is usually done by analyzing the colors of the pixels on the border of the image. At step 710 the image is divided into blocks. As described above with reference to FIG. 4A, in the preferred embodiment rectangular blocks are used. But any tessellation (division of the plane) can be used, provided that no pixel is included in more than one block, and that each pixel with a color different from the background color is included in a block. Steps 715 and 720 are applicable only for animated images that include multiple frames. Step 715 computes the distance frame between two frames in the animated image, and step 720 computes tessellations of the distance frame. In both steps 710 and 720, multiple tessellations can be considered to find a tessellation that results in the smallest compressed image size. Once multiple tessellations have been analyzed, at step 725 the tessellation that results in the smallest compressed file size is selected, and then each block in the tessellation is compressed. Note that only the blocks need to be compressed: the remaining pixels include only the background color, which can be very easily identified. FIG. 8 shows a structure for compressed image file 175 according to the preferred embodiment of the invention. In FIG. 8, the structure used for storing and for transmitting compressed image file 175 begins with preliminary information 805. Preliminary information 805 is sent once per image, and specifies the size of the image (typically in two pixel dimensions: for example, 640×480), the color palette (typically as a table or a list of indices to color values), and the background color of the image (typically as an index into the color palette). Next comes block information 810. Block information 810 specifies the location, size, and contents of a particular block in the compressed image file. The contents of the block are the compressed indices into the color palette for each pixel in the block. (In the preferred embodiment, pixels in each block are examined row by row, from the top of the block to the bottom, and within each row from left to right. However, a person skilled in the art will recognize that pixels can be scanned in other orders as well.) For example, referring back to FIG. 4A, block information 810 can specify the location, size, and contents of box 415. Block information 810 can be repeated if there are multiple blocks in the image. If the image is an animated image, then frame information 815 can be provided. Frame information 815 specifies the type of frame (for example, the frame can be completely redrawn without reference to the prior frame, or the frame can be a distance frame, as described above) and the information about the frame. The information about the frame can include one or more blocks, as described above. Preliminary information need not be resent until the next image. However, a person skilled in the art will recognize that some preliminary information (for example, a new color palette) can be sent, if desired. Note that in nowhere in FIG. 8 is the structure of compressed image file 175 described as including the model(s) used in compression. The reason the models are not included in compressed image file 175 is that the models do not need to be stored. Instead, the models can be reconstructed as compressed image file 175 is read. For example, if when the image is compressed a particular pixel has the same color as its left and upper neighbors, the index into the color palette stored in the compressed image file will reflect this. Thus, the models can be reconstructed when compressed image file 175 is read, and do not need to be stored in compressed image file 175. This is an advantage over Huffman codes, which must be stored with the image file. FIGS. 9A and 9B show the procedure used to decompress compressed image file 175 of FIG. 2 on the computer system of FIG. 1A. At step 905 (FIG. 9A), the probability value for the current pixel is decoded. At step 910, the probability value is analyzed to determine whether the color of the current pixel is supposed to be the same as either the left or upper neighbor of the current pixel. If the current pixel is the same color as either the left or upper neighbor of the current pixel, then at step 915 the color of the current pixel is copied from the left or upper neighbor, as appropriate, of the current pixel. Otherwise, at step 920, the color of the current pixel is decoded from the compressed image file. At step 925 (FIG. 9B), the probability model is updated, and if necessary, at step 930 a new probability model is selected. Then, if more pixels remain to be decoded, the process returns to step 905. Otherwise, at step 935, the decompressed image is displayed. Regarding step 930, as discussed above with reference to FIG. 3, the use of different probability models helps to improve the compression by focusing each probability model on a different probability value. For example, consider probability model 305, and assume that the current pixel and its left and upper neighbors have the same color. Because the current pixel had the same color as its left and upper neighbors, the compression technique expects that the next pixel will have the same color as its left and upper neighbors. If this expectation is satisfied, probability value 305-1 will be further increased, as opposed to any of probability values 305-2, 305-3, 305-4, and 305-5. In range coding, the bigger the available range, the fewer the number of bits necessary to compress the message. Thus, by focusing each probability model on a different combination of the colors of the current pixel and its left and upper neighbors, the probability models are able to compress the image using fewer bits, resulting in better compression than would otherwise occur. The decompressor is able to decompress compressed image file 175 without reading the probability models from compressed image file 175 because the decompressor is able to recreate the probability models on the fly. Like the compressor, the decompressor starts with initial probability models, the same as shown in FIG. 3. As the decompressor reads the compressed information from compressed image file 175, it can update the probability models in the same way as the compressor did when the compressor compressed image file 130. In this way, the compressor/decompressor resembles Lempel-Ziv compression. In the Lempel-Ziv compression, the compressor and decompressor build dictionaries as the file is read for compression/decompression. Although the instant invention does not use a dictionary, the probability models can be built by both the compressor and decompressor as they read the image file and the compressed file, respectively. A person skilled in the art will recognize that the method and apparatus for compression described herein provides for lossless compression. That is, the image file is compressed with no loss of information. When compressed image file 175 is decompressed, the resulting file contains the same information as was in image file 130, before the compression occurred. Appendix A shows an implementation of the compression algorithm described herein. In Appendix A, the comments describe a preferred embodiment of the algorithm implemented in the source code. The comments in Appendix A can be thought of as pseudocode. Although exemplary, a person skilled in the art will recognize that other implementations and variations on the implementation shown are possible. Having illustrated and described the principles of our invention in an embodiment thereof, it should be readily apparent to those skilled in the art that the invention can be modified in arrangement and detail without departing from such principles. We claim all modifications coming within the spirit and scope of the accompanying claims.

- When would you use a histogram? - What are the advantages of histogram? - What is the difference between Pareto chart and histogram? - How do you determine your class size? - What is a histogram chart? - What type of data is a histogram? - How do you know what Class A histogram is? - How do you interpret a histogram? - Where are histograms used in real life? - Is a histogram qualitative or quantitative? - What is class width on a histogram? - How a histogram should look? - What is histogram and its types? - What is the difference between a histogram and a bar chart? - What type of data is best displayed in a histogram? - How do you analyze data from a histogram? - What are the 8 possible shapes of a distribution? - How many types of histograms are there? - Is a bar graph qualitative or quantitative? - Is a histogram discrete or continuous? When would you use a histogram? When to Use a Histogram Use a histogram when: The data are numerical. You want to see the shape of the data’s distribution, especially when determining whether the output of a process is distributed approximately normally.. What are the advantages of histogram? The main advantages of a histogram are its simplicity and versatility. It can be used in many different situations to offer an insightful look at frequency distribution. For example, it can be used in sales and marketing to develop the most effective pricing plans and marketing campaigns. What is the difference between Pareto chart and histogram? A histogram is a bar graph that illustrates the frequency of an event occurring using the height of the bar as an indicator. A Pareto chart is a special type of histogram that represents the Pareto philosophy (the 80/20 rule) through displaying the events by order of impact. How do you determine your class size? In inclusive form, class limits are obtained by subtracting 0.5 from lower limitand adding 0.5 to the upper limit. Thus, class limits of 10 – 20 class interval in the inclusive form are 9.5 – 20.5. Class size: Difference between the true upper limit and true lower limit of a class interval is called the class size. What is a histogram chart? A histogram is a chart that shows frequencies for. intervals of values of a metric variable. Such intervals as known as “bins” and they all have the same widths. The example above uses $25 as its bin width. What type of data is a histogram? The histogram is a popular graphing tool. It is used to summarize discrete or continuous data that are measured on an interval scale. It is often used to illustrate the major features of the distribution of the data in a convenient form. How do you know what Class A histogram is? We begin this process by finding the range of our data. In other words, we subtract the lowest data value from the highest data value. When the data set is relatively small, we divide the range by five. The quotient is the width of the classes for our histogram. How do you interpret a histogram? Left-Skewed: A left-skewed histogram has a peak to the right of center, more gradually tapering to the left side. It is unimodal, with the mode closer to the right and greater than either mean or median. The mean is closer to the left and is lesser than either median or mode. Where are histograms used in real life? The primary use of a Histogram Chart is to display the distribution (or “shape”) of the values in a data series. For example, we might know that normal human oral body temperature is approx 98.6 degrees Fahrenheit. Is a histogram qualitative or quantitative? The main visual difference between a bar graph (qualitative data) and a histogram (quantitative data) is that there should be no horizontal spacing between numerical values along the horizontal axis. In other words, rectangles touch each other in a histogram. What is class width on a histogram? The “class width” is the distance between the lower limits of consecutive classes. The range is the difference between the maximum and minimum data entries. … Find the class limits: You can use the minimum data entry as the lower limit of the first class. To get the lower limit of the next class, add the class width. How a histogram should look? In an ideal world, the graph should just touch the left and right edges of the histogram, and not spill up the sides. The graph should also have a nice arch in the center. This is how an ideal histogram might look, evenly distributed, edge to edge, not up the sides. This is a histogram for a dark subject. What is histogram and its types? A histogram is used to summarize discrete or continuous data. In other words, it provides a visual interpretation. This requires focusing on the main points, factsof numerical data by showing the number of data points that fall within a specified range of values (called “bins”). It is similar to a vertical bar graph. What is the difference between a histogram and a bar chart? Histograms are used to show distributions of variables while bar charts are used to compare variables. Histograms plot quantitative data with ranges of the data grouped into bins or intervals while bar charts plot categorical data. What type of data is best displayed in a histogram? It is similar to a Bar Chart, but a histogram groups numbers into ranges . The height of each bar shows how many fall into each range….Histograms are a great way to show results of continuous data, such as:weight.height.how much time.etc. How do you analyze data from a histogram? Make a histogram using Excel’s Analysis ToolPakOn the Data tab, in the Analysis group, click the Data Analysis button.In the Data Analysis dialog, select Histogram and click OK.In the Histogram dialog window, do the following: … And now, click OK, and review the output table and histogram graph: What are the 8 possible shapes of a distribution? Classifying distributions as being symmetric, left skewed, right skewed, uniform or bimodal. How many types of histograms are there? Bimodal: A bimodal shape, shown below, has two peaks. This shape may show that the data has come from two different systems. If this shape occurs, the two sources should be separated and analyzed separately. Skewed right: Some histograms will show a skewed distribution to the right, as shown below. Is a bar graph qualitative or quantitative? Pie charts and bar graphs are used for qualitative data. Histograms (similar to bar graphs) are used for quantitative data. Line graphs are used for quantitative data. Scatter graphs are used for quantitative data. Is a histogram discrete or continuous? Histograms are a special form of bar chart where the data represent continuous rather than discrete categories. This means that in a histogram there are no gaps between the columns representing the different categories.

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Eberhard Frederich Ferdinand Hopf Bloomington, Indiana, USA BiographyEberhard Hopf, an Austrian mathematician who made significant contributions in topology and ergodic theory, was born in Salzburg. Most of his scientific formation, however, was in Germany, where he received a Ph.D. in Mathematics in 1926 and, in 1929, his Habilitation in Mathematical Astronomy from the University of Berlin. In 1930 Hopf received a fellowship from the Rockefeller Foundation to study classical mechanics with Birkhoff at Harvard in the United States. He arrived Cambridge, Massachusetts in October of 1930 but his official affiliation was not the Harvard Mathematics Department but, instead, the Harvard College Observatory. While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory. In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory. Another important contribution from this period was the Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener from the Massachusetts Institute of Technology. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. Other work which he undertook during this period was on stellar atmospheres and on elliptic partial differential equations. On 14 December 1931, with the help of Norbert Wiener, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology accepting the position of Assistant Professor. Initially he had a three years contract but this was subsequently extended to four years (1931 to 1936). While at MIT, Hopf did much of his work on ergodic theory which he published in papers such as Complete Transitivity and the Ergodic Principle (1932), Proof of Gibbs Hypothesis on Statistical Equilibrium (1932) and On Causality, Statistics and Probability (1934). In this 1934 paper Hopf discussed the method of arbitrary functions as a foundation for probability and many related concepts. Using these concepts Hopf was able to give a unified presentation of many results in ergodic theory that he and others had found since 1931. He also published a book Mathematical problems of radiative equilibrium in 1934 which was reprinted in 1964. In addition of being an outstanding mathematician, Hopf had the ability to illuminate the most complex subjects for his colleagues and even for non specialists. Because of this talent many discoveries and demonstrations of other mathematicians became easier to understand when described by Hopf. In 1936, at the end of the MIT contract, Hopf received an offer of full professorship in the University of Leipzig. As a result of this Hopf, with his wife Ilse, returned to Germany which, by this time, was already being ruled by the Nazi party. In Leipzig Hopf undertook research on quantic mechanics (1937), Geodesics on manifolds of negative curvature (1939), Statistik der geod (1939) and on the influence of curvature of a closed Riemannian manifold on its topology (1941). One important event from this period was the publication of the book Ergodentheorie Ⓣ (1937), most of which was written when Hopf was still at the Massachusetts Institute of Technology. In that book containing only 81 pages, Hopf made a precise and elegant summary of ergodic theory. In 1940 Hopf was on the list of the invited lecturers to the International Congress of Mathematicians to be held in Cambridge, Massachusetts. Because of the start of World War II, however, this Congress was cancelled. In 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, Hopf was appointed to a professorship at the University of Munich. He held this post until 1947 by which time he had returned to the United States, where he presented the definitive solution of Hurewicz's problem. On 22 February 1949 Hopf became a US citizen. He joined Indiana University as a Professor in 1949, a position he held until he retired in 1972. In 1962 he was made Research Professor of Mathematics, staying in that position until his death. An important publication from this period was An inequality for positive linear integral operators (1963) which appeared in the Journal for Mathematics and Mechanics. This paper is concerned with some extensions of Jentzsch's theorem on the existence of a positive eigenfunction for a positive integral operator. In 1971 Hopf was the American Mathematical Society Gibbs Lecturer. Coming out of this lecture was a paper Ergodic theory and the geodesic flow on surfaces of constant negative curvature which he published in the Bulletin of the American Mathematical Society. Hopf wrote in the introduction to that paper:- Famous investigations on the theory of surfaces of constant negative curvature have been carried out around the turn of the century by F Klein and H Poincaré in connection with complex function theory. The theory of the geodesics in the large on such surfaces was developed later in the famous memoirs by P Koebe. This theory is purely topological. The measure-theoretical point of view became dominant in the later thirties after the advent of ergodic theory, and the papers of G A Hedlund and E Hopf on the ergodic character of the geodesic flow came into being. The present paper is an elaboration of the author's Gibbs lecture of this year and at the same time of the author's paper of 1939 on the subject, at least of its part concerning constant negative curvature.Hopf was never forgiven by many people for his moving to Germany in 1936, where the Nazi party was already in power. As a result most of his work to ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the dropping of Hopf's name from the discrete version of the so called Wiener-Hopf equations, which are currently referred to as "Wiener filter". In Icha summarises Hopf's mathematical achievements:- His interests and principal achievements were in the fields of partial and ordinary differential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis. Hopf's work is also of the greatest importance to the hydrodynamics, theory of turbulence and radiative transfer theory. - P M Anselone, In honor of Professor Eberhard Hopf on the occasion of his seventieth birthday, Applicable Anal. 3 (1973), 1-5. - M Denker, Eberhard Hopf: 04-17-1902 to 07-24-1983, Jahresber. Deutsch. Math.-Verein. 92 (2) (1990), 47-57. - M Frank, Eberhard Hopf : The ergodic theorist who went back to Germany in 1936, American Mathematical Society Meeting (special session on history of mathematics), Washington, DC, (January, 2000). - A Icha, Andrzej, Eberhard Hopf (1902-1983), Nieuw Arch. Wisk. (4) 12 (1-2) (1994), 67-84. - In memoriam Eberhard Hopf: 1902-1983, Indiana Univ. Math. J. 32 (6) (1983), i-ii. Additional Resources (show) Written by J J O'Connor and E F Robertson based on a biography submitted by Osvaldo de Oliveira Duarte which in turn made substantial use of . Last Update March 2001 Last Update March 2001

The Hamermesh study: what percentage of pitches are biased? The Hammermesh study on racial bias among umpires (which I posted about here) concluded that “a given called pitch is approximately 0.34 percentage points more likely to be called a strike if the pitcher and umpire match race/ethnicity.” But I don’t think they actually say what percentage of calls are biased. I’m tried to figure this out for myself, without using regression. It depends on the assumptions you make; under the assumptions I’ll show you in a bit, I get 0.13%. I’m going to start with a simple example. Suppose you have only white umpires and black umpires, and only white pitchers. The white umpires call 31% strikes, but the black umpires only call only 30% strikes. We don’t know yet whether this is bias or just random fluctuation. But we can say that for whatever reason, the pitchers are advantaged by 1 percentage point with a white umpire, relative to a black umpire. But suppose it were bias. What percentage of (called) pitches would be affected? Your first inclination might be to say that 1% of pitches are affected when there’s a white umpire, but none are affected when there’s a black umpire. But, wait. We don’t know what direction the bias goes. It’s possible that all the bias is from the black umpire. He should be calling 31% strikes, but he’s only calling 30%. In that case, 1% of pitches are affected when there’s a black umpire, but 0% when there’s a white umpire. There’s still another case. Perhaps the pitcher should actually 30.5% strikes, and both umpires are biased by 0.5%. In that case, 0.5% of pitches are affected when there’s a black umpire, and 0.5% are affected when there’s a white umpire. Indeed, there is an infinity of possibilities. Maybe the black umpire is biased 0.3% and the white 0.7%. Maybe the black umpire is biased 2%, and the white umpire is biased negative 1%. And so on. So we can’t say what percentage of pitches are biased for which umpire. We can’t even say what percentage of total pitches are biased. In the last case I suggested, where the black umpire was biased 2% and the white umpire was biased 1% in the other direction, there would be 1.5% of total pitches biased (assuming the white and black umpires called equal numbers of pitches). But in all the other cases I used as examples, it would be only 0.5%. So it all depends on your assumptions of where the bias is. OK, now let’s move to a real-life example, one where the umpires don’t call equal numbers of pitches. In fact, I’ll take the “white pitcher” column of the paper’s Table 3. Here it is: White umpires: 32.06% strikes. Hspnc umpires: 31.91% strikes. Black umpires: 31.93% strikes. Now, again, let’s suppose all the differences between umpires are bias. That means we want equal numbers for all umpires. Since the overall strike rate for white pitchers was 32.05% (from Table 2), we might choose that. So we *really* want the table to look like: White umpires: 32.05% strikes. Hspnc umpires: 32.05% strikes. Black umpires: 32.05% strikes. How do we make that happen? Well, we can subtract called strikes from the first case, and add them to the second and third cases: White umpires: subtract 0.01 from the original 32.06 Hspnc umpires: add 0.14 to the original 31.91 Black umpires: add 0.12 to the original 31.93. This will make all the percentages equal to 32.05%. Since we assumed the differences represented bias, that means that 0.01% of white umpires’ calls were biased; 0.14% of hispanic umpires’ calls were biased; 0.12% of black umpires’ calls were biased. Those numbers depended on our choice of 32.05% as the baseline. Another choice we can make is to use 32.06% as the baseline instead, which assumes that both hispanic and black umpires should call the same 32.06% strikes as the white umpires. In that case, you need to add 37 strikes to the hispanic umpires, and 61 strikes to the black umpires. That means 0.00% of white umpire calls are biased 0.15% of hspnc umpire calls are biased 0.13% of black umpire calls are biased Which do you like? Either: there’s no real reason to choose one over the other, and no way to prove which is more accurate. We could go extreme the other way, by assuming that the hispanic umpires are correct, and the “real” percentage is 31.91%. In that case, 0.15% of white umpire calls are biased 0.00% of hspnc umpire calls are biased 0.02% of black umpire calls are biased Do you like that better? It’s arbitrary still. Arbitrary or not, my feeling is that we should find a pattern where all three groups of umpires seem about equally biased. That way, we don’t have to single out one group. Maybe we can choose 31.99 as our estimate of the “true” value. That would mean: 0.07 of white umpire calls are biased 0.08 of hspnc umpire calls are biased 0.06 of black umpire calls are biased. This is my favorite, because it spreads the blame around; in the absence of evidence that one group is “guiltier” than another, this seems like the ethical default assumption. Now, back to the original question: what percentage of *all pitches* are biased? From this last distribution of bias, it looks like about 0.07% of pitches are biased (regardless of who the umpire is). That’s one out of every 1,400 pitches. I’ll quickly do the Hispanic and Black umpires too. For Hispanic, it looks like 31.15% might be a pretty good stab at the “real” strike percentage, the one that makes all the umpires look equally biased. That means that for Hispanic pitchers, 0.32% of white umpire calls are biased 0.35% of hspnc umpire calls are biased 0.28% of black umpire calls are biased. Since white umpires are an overwhelming majority, the overall average of these numbers is probably 0.32%. For black pitchers, let’s use 30.69% as the base: 0.08% of white umpire calls are biased 0.08% of hspnc umpire calls are biased 0.09% of black umpire calls are biased That’s about 8% overall for the black pitchers. Summarizing the three groups of pitchers: White pitchers: 0.07% of calls are biased Hspnc pitchers: 0.32% of calls are biased Black pitchers: 0.08% of calls are biased Hispanic pitchers are 25% of the total, and black pitchers about 5%. If we weight the three groups accordingly, we get: Overall: 0.13% of all umpire calls are biased. That’s about 1 pitch in 750. Again, this is subject to assumptions, not all of which might be true: -- we assume that all umpires have an equal propensity to be biased for a particular race of pitcher. -- we assume that 100% of the discrepancies actually seen represent bias. This is almost certainly not true, because there is inherent random variation in what kinds of pitches umpires will see. -- more importantly, we assume that bias exists. I’m still not convinced that the amounts of variation seen in the study are statistically significant. But it appears that if you do accept the above assumptions, it follows that at most 1 pitch in 700 will be biased. Unless I’ve screwed up the logic somewhere.

1 Perfect Competition Chapter CHAPTER OUTLINE 1. Explain a perfectly competitive firm s profit-maximizing choices and derive its supply curve. A. Perfect Competition B. Other Market Types C. Price Taker D. Revenue Concepts E. Profit-Maximizing Output F. Marginal Analysis and the Supply Decision G. Exit and Temporary Shutdown Decisions H. The Firm s Short-Run Supply Curve 2. Explain how output, price, and profit are determined in the short run. A. Market Supply in the Short Run B. Short-Run Equilibrium in Good Times C. Short-Run Equilibrium in Bad Times 3. Explain how output, price, and profit are determined in the long run and explain why perfect competition is efficient. A. Entry and Exit 1. The Effects of Entry 2. The Effects of Exit B. Change in Demand C. Technological Change D. Is Perfect Competition Efficient? E. Is Perfect Competition Fair? CHAPTER ROADMAP What s New in this Edition? The material dealing with external economies and diseconomies of scale has been eliminated. In its place is a dis 2 324 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE cussion of the efficiency of perfect competition and the fairness of perfect competition. Where We Are In this chapter, we examine the profit maximizing decisions made by a perfectly competitive firm in the short run and the long run. To do so, we use the groundwork on firms costs laid in the previous chapter. Where We ve Been In Chapter 13 we use the foundation built in Chapter 12, which studied firms production and costs. The cost material covered in Chapter 12 remains important throughout not only Chapter 13 but also Chapters 14, 15, and 16. Where We re Going After this chapter, we continue studying firms behavior by looking at the demand and marginal revenue curves for monopolies, oligopolies and monopolistically competitive firms. By combining the cost, demand, and revenue curves, we will see operating decisions faced by these firms and we can compare them with those made by perfectly competitive firms. IN THE CLASSROOM Class Time Needed This chapter is very important. Perfect competition is the standard against which other industries are compared, so do not rush through this material. You should plan on spending at least two and a half class sessions and possibly even three. An estimate of the time per checkpoint is: 13.1 A Firm s Profit Maximizing Choices 60 to 80 minutes 13.2 Output, Price, and Profit in the Short Run 30 to 50 minutes 13.3 Output, Price, and Profit in the Long Run 30 to 40 minutes 3 Chapter 13. Perfect Competition 325 CHAPTER LECTURE 13.1 A Firm s Profit-Maximizing Choices Perfect competition exists when Many firms sell identical products to many buyers There are no restrictions on entry into the industry Established firms have no advantage over existing ones Sellers and buyers are well informed about prices Other market types are: Monopoly, a market for a good or service that has no close substitutes and in which there is one supplier that is protected from competition by a barrier preventing the entry of new firms. Monopolistic competition, a market in which a large number of firms compete by making similar but slightly different products. Oligopoly, a market in which a small number of firms compete. Have the students consider the markets for goods for which they are familiar to see if any meet the strict criteria for perfect competition. The markets that come closest are agricultural markets, though others such as lawn service, laundromats, fishing, plumbing, and so on, come close. Students sometimes worry that these markets are not exact examples of perfect competition. Reassure them that the model of perfect competition gives us a great deal of understanding into the workings of extremely competitive real world markets and the real world firms in the markets. A firm s objective is to maximize economic profit, which is the difference between total revenue (the price of the firm s output multiplied by the quantity sold) and its total cost of production. Part of the total cost is the normal profit. Price Taker Perfectly competitive firms are price takers, a firm that cannot influence the market price and so it sets its own price equal to the market price. Revenue Concepts Because the firm is a price taker, its marginal revenue which is the change in total revenue that results in a one unit increase in the quantity sold is equal to the market price and remains constant as output sold increases. The firm s demand is perfectly elastic and the firm s demand curve is a horizontal line at the market price. Profit-Maximizing Output The firm produces the quantity of output for which the difference between total revenue and total cost is at its maximum because this difference is its economic profit. Marginal analysis can be used to determine the profit maximizing quantity. The firm compares the marginal revenue (which remains constant with output) to the marginal cost (which changes with output) of producing different levels of output. 4 326 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE When MR > MC, then the extra revenue from selling one more unit exceeds the extra cost of producing one more unit, so the firm increases its output to increase its profit. When MR < MC, then the extra cost of producing one more unit exceeds the extra revenue from selling one more unit, so the firm decreases its output to increase its profits When MR = MC, then the extra cost of producing one more unit equals the extra revenue from selling one more unit, so the firm s profit is maximized at this level of output. In the figure, the firm maximizes its profit by producing q. The Firm s Short-Run Supply Curve The firm will temporarily shut down in the short run when price falls below the price that just allows it to cover its total variable cost. The minimum AVC is the lowest price at which the firm will operate because if it operated with a lower price, the firm s loss would be greater than if it shut down. The loss when the firm shuts down is equal to its fixed cost. As long as the firm remains open, it produces where MR = MC. So the firm s supply curve is its MC curve above the minimum AVC. At prices below the minimum AVC, the firm shuts down and supplies zero. Monday is typically the slowest day in the restaurant industry. So why do restaurants stay open on Monday? The answer is that even if a restaurant incurs an economic loss on Monday, it still might increase its total profit by remaining open. The point is that as long as the restaurant can cover all its variable costs the cost of the food, the cost of the servers, and so on it likely will be able to pay some of its fixed costs using the revenue left over after paying its variable costs. As long as the restaurant can pay some of its fixed costs on Monday, its total profit by staying open exceeds what its total profit would be if it closed. So losing money on Monday might be good business! 5 Chapter 13. Perfect Competition Output, Price, and Profit in the Short Run Market Supply in the Short Run The market supply curve in the short run shows the quantity supplied by the industry at each possible price when the number of firms is fixed. The quantity supplied in the industry at any price is the summation of all quantities supplied by each firm at that price. Short-Run Equilibrium in Good Times There are three possible profit outcomes an economic profit, zero economic profit, and an economic loss. If the price exceeds the ATC, the firm earns an economic profit. The figure illustrates a perfectly competitive firm that is earning an economic profit. The firm produces 4 units, has a price of $3 per unit, and earns an economic profit equal to the area of the darkened rectangle. Short-Run Equilibrium in Bad Times If the price is less than the ATC, the firm incurs an economic loss. The figure illustrates a perfectly competitive firm that is suffering an economic loss.. The firm produces 3 units, has a price of $3 per unit, and incurs an economic loss is equal to the area of the darkened rectangle. 6 328 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE 13.3 Output, Price, and Profit in the Long Run If the price equals the ATC, the firm earns zero economic profit. In this case, the firm earns a normal profit. Entry and Exit Changes in market demand influence the output and the entry or exit decisions made by firms. An increase in market demand shifts the demand curve rightward and raises the market price. Each firm in the industry responds by increasing its quantity supplied. The higher price now exceeds each firm s minimum ATC and the firms in the industry earn an economic profit. The economic profit motivates firms to enter the industry, thereby increasing the market supply. The market supply curve shifts rightward and the market price falls. Eventually the price falls to equal the minimum ATC for each firm in the industry. At this price, firms in the industry no longer earn an economic profit. The effects of a decrease in market demand are the opposite of those outlined above: The price falls, firms incur an economic loss, some firms exit thereby decreasing the supply and so the price rises until the surviving firms earn a normal profit. When demand for a good increases so that the existing firms in an industry earn an economic profit, the economic profit indicates that consumers are willing to pay a higher price for the good than they were willing to pay before the demand increased. The economic profit for the firms is a signal from the consumers to the owners of firms in other industries that society now values the availability of the good more highly than the availability of goods from those other industries. These self interested firm owners choose to enter the industry in order to earn an economic profit. Their self interested decisions promote the social interest by using more resources to produce those goods that are more highly valued by society. The dynamic behavior of a perfectly competitive market characterizes the invisible hand coined by Adam Smith. Change in Technology New, cost saving technologies typically require new plant and equipment. So it takes time for new technology to spread throughout an industry. Firms that adopt the new technology lower their costs and their supply curves shift rightward. The price of the good falls, so that firms using the old technology incur economic losses. Old technology firms either adopt the new technology or else exit the industry. In the long run, all the firms use the new technology and earn zero economic profit. Is Perfect Competition Efficient? Resource allocation in a market is efficient when society values no other use of the resources more highly. Resource use is efficient when production is such that the marginal benefit of the good equals the marginal cost of the good. 7 Chapter 13. Perfect Competition 329 A firm s supply curve for a good is its marginal cost curve and so the market supply curve for a good is the society s marginal cost curve. The demand curve is the marginal benefit curve. In a competitive equilibrium, the quantity demanded equals the quantity supplied. The demand curve is the same as the marginal benefit curve and the supply curve is the same as the marginal cost curve, so at the competitive equilibrium, the marginal benefit equals the marginal cost. Resource use is efficient. Because resources are used efficiently, at the competitive equilibrium there is no other allocation of resources that will generate greater net benefits to society. The figure shows this outcome, where resource use is efficient at the equilibrium quantity of 3,000 units. Is Perfect Competition Fair? Perfect competition allows anyone to enter the market and the process of competition brings the maximum benefits for consumers. So fairness of opportunity and fairness as equality of outcomes are achieved in perfect competition in the long run. 8 330 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE Lecture Launchers 1. Launch your lecture by drawing a spectrum of market types noting the four market structures to be studied in this and the next chapters. Let your students know that you will be comparing how a firm in each of these market structures chooses its equilibrium price and equilibrium quantity. Putting this diagram on the board provides a good foundation for the following chapters. 2. For Chapters 13, 14, 15, and 16, the students, with the instructor s guidance, can create a chart similar to the one below. They would, for each market structure, suggest real world examples of industries that fit the market structure, tell how prices are set, what problems and what benefits the consumer and producer face in each market structure, and what role the government might play in each market structure. You also can add other topics is there an economic profit in the long run?; can the firm price discriminate?; and so forth. The information could put by the instructor on the course web site, or assembled for a handout to be given at the end of presentation of Chapter 16. Perfect competition Monopoly Monopolistic competition Oligopoly Examples Setting price Consumer advantages Consumer drawbacks Producer advantages Producer drawbacks Role of government 3. Once you discuss the characteristics that define perfect competition (many firms selling an identical product to many buyers, no restrictions on entry, established firms have no cost advantage over new firms, and sellers and buyers are well informed about prices) it is natural to give examples of perfectly competitive markets. The examples that always spring to mind are ag 9 Chapter 13. Perfect Competition 331 ricultural in nature. Often students, particularly those in urban areas, wonder why they will spend so much of their time studying agriculture. You need to combat the natural view that the model of perfect competition applies only to farms. There are two, complementary paths you can take: First, tell your students that although agriculture certainly meets all the requirements of perfect competition, a lot of other industries come close. If you have a mall near by, you can assign your students to walk through the mall and take note of the different types of businesses and list those that they think are closest to perfect competition. Businesses such as shoe stores, jewelry, toy stores, book stores, hair salons, and so forth are all commonly found in malls and are all relatively close to being in perfectly competitive markets. For instance, you can point out to the students that one jewelry store s products aren t identical to those of any other jewelry store, but they are very close substitutes. So, although the jewelry market does not exactly meet the definition of a perfectly competitive market, nonetheless it is likely close enough so that if we want to understand the forces that affect firms within this industry, perfect competition is a reasonable starting point. Second, you can use a physical analogy. Ask your students how many of them have taken physics and encountered the assumption of a perfect vacuum. A perfect vacuum cannot exist and our world is not close to being a perfect vacuum. Yet physicists often use the model of a perfect vacuum to understand our physical world. For example, to predict how long it will take a 50 pound steel ball to hit the ground if it is dropped from the top of the Empire State Building, you will be very close to the actual time if you assume a perfect vacuum and use the formula that applies in that case. Friction from the atmosphere is obviously not zero, but assuming it to be zero is not very misleading. In contrast, if you want to predict how long it will take a feather to make the same trip, you need a fancier model! Economists use the model of perfect competition in a similar way to understand our economic world. Emphasize to students that although no real world industry meets the full definition of perfect competition, the behavior of firms in many real world industries and the resulting dynamics of their market prices and quantities can be predicted to a high degree of accuracy by using the model of perfect competition. 4. Every term you probably have students who ask, Do firms really choose the output that maximizes profit? To answer this question, perhaps before it is asked, it is useful to explain to your students that many big firms routinely make tables using spreadsheets of total revenue, total cost, and economic profit. But most firms, and certainly most small firms, don t make such calculations. Nonetheless, they do make their decisions at the margin. They can figure out how much it will cost to hire one more worker and how much output that worker will produce. So they can figure out their mar 10 332 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE ginal cost wage rate divided by marginal product. They can compare that number with the price. They are choosing at the margin as our model of perfect competition assumes. Land Mines 1. Show what is meant by the term price taker by drawing the market supply and market demand curves and the resulting equilibrium price on the left side of the board and then draw the firm s demand and marginal revenue curves on a separate graph on the right side of the board. Draw a dotted line across from the market graph to the firm graph. Really emphasize the fact that the market demand differs from the firm s demand because the firm is such a small part of the market. Students consistently confuse the difference between the market demand and the firm s demand, so the more time you spend clearly explaining this distinction, the better. 2. You always will have students asking why the firm bothers to produce the precise unit of output for which MR = MC. Indeed, it is simply amazing how many students worry about this one particular unit of output! Try the following: Draw the conventional upward sloping MC curve and horizontal MR curve. Make sure to draw these so that the firm will produce a good deal of output. Then, starting at 0, move a bit to the right along the horizontal axis and stop at a point. Tell the students that this point measures 1 unit of output and ask them if this unit should be produced. The answer ought to be yes, because you have arranged matters so MR > MC. Pick some numbers say, MR = $10 and MC = $1. Ask your students what the profit is for this unit and what the firm s total profit is if it produces only 1 unit. The answers are $9 and $9. Below the x axis, label two rows, one called profit on the unit and the other total profit. Put $9 and $9 in each space under your 1 unit of output. Then move your finger a bit more along the horizontal axis until you come to where you will define the second unit of output. Ask your students if this second unit should be produced. Again, the answer ought to be yes, because you have arranged matters so MR > MC. Pick another number for MC, say $2. Ask your students what the profit is on this unit and what the firm s total profit is if it produces 2 units. The answers are $8 and $17. Stress that the total profit is what interests the firm and the total profit equals the sum of the profit from the first unit plus the profit from the second unit. Pick a couple of more units and use numbers until you fell it is safe to generalize that the firm produces a unit of output as long as MR > MC. Then, slide your finger to the right, stopping at closer and closer intervals, asking the class if that particular unit should be produced. Always stress that the firm s total profit continues to increase, albeit more and more slowly. As you get closer to the magical MR equal to MC point, make your stopping intervals even closer. Finally, when you reach 11 Chapter 13. Perfect Competition 333 MR = MC, tell the students that although this specific unit yields no profit, to have stopped anywhere before it means that the firm would have lost some profit. So, only by producing where MR = MC will the firm obtain the maximum total profit. 3. Students need repeated reminders that to determine whether a firm can increase profit by changing output, price and marginal cost are the only things to consider. Questions that throw average total cost into the mix often cause confusion. 4. Students are often skeptical that a zero economic profit is an acceptable outcome for an entrepreneur. The key is to reinforce the meaning of normal profit. A rational decision is one that is based on a weighing of the full opportunity cost of each alternative against its full benefits. Opportunity cost includes the benefits from forgone opportunities as well as explicit costs. One of these forgone opportunities for the entrepreneur is pursuing his or her next best activity. The value of this forgone opportunity is normal profit. So, when a firm earns zero economic profit, the entrepreneur earns normal profit and enjoys the same benefits as those available in the next best activity. There is no incentive to change to the next best activity. 5. Explaining whether a firm exits, temporarily shuts down, or produces even though it has an economic loss is difficult because the last two topics are tough for the students to understand. Exit is the easiest for them to understand because they have seen firms fail throughout their life. But, temporary shutdown is harder to explain. You can help them with the intuition by pointing out that the rationale for temporary shutdown isn t confined to perfect competition and that they can see the phenomenon right around the corner. Many restaurants close on Sunday evening and Monday. Many hairdressers close on Sunday and Monday. Why? Your students will easily figure out that total revenue is less than total variable cost and equivalently that price is less than average variable cost. The mechanics of the shutdown analysis will be a lot easier to explain once the students have thought about these real situations with which they are familiar. Students can have a hard time understanding why operating at an economic loss can be the best action. I use a story to help them see this point, Wally s Wiener World hot dog cart. Wally has four costs: his variable costs for his hot dogs, buns, and mustard and his fixed cost for the interest he pays for the loan he used to buy his cart. (If you choose, you can make up numbers for each of these costs.) When price is greater than average variable cost, P>AVC, Wally can pay for his hot dogs, buns, and mustard, and part of the interest cost, his fixed cost. I show that because he can pay part of his fixed cost, he should stay open. But if P < AVC, Wally can t even pay for all the dogs, buns, and mustard, much less pay for the interest on his loan. In this circumstance, Wally is better off by shutting down. 12 334 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE ANSWERS TO CHECKPOINT EXERCISES CHECKPOINT 13.1 A Firm s Profit-Maximizing Choices 1a. Paula s total revenue equals the price multiplied by the quantity produced, which is 800 boxes $15 a box = $12,000. 1b. The market is perfectly competitive, so Paula s marginal revenue equals the price. So, Paula s marginal revenue is $15 a box. 1c. Paula is not maximizing profit because marginal cost ($18) is greater than marginal revenue ($15). She should decrease her production until marginal cost falls to $15 because then her marginal cost equals her marginal revenue and she would maximize her profit. 1d. If the price is $12 a box, Paula s marginal revenue is $12 a box. Paula is maximizing profit by selling 500 boxes a week because marginal cost equals the marginal revenue. Paula is incurring an economic loss if she has any fixed costs because in that case price is less than average total cost. 1e. One point on Paula s supply curve is a price of $12 and quantity of 500 boxes a week. The combination of a price of $15 and quantity of 800 boxes a week is not on Paula s supply curve because at that combination she is not maximizing her profit. So this combination is only temporary because Paula will change her production in order to maximize her profit. CHECKPOINT 13.2 Output, Price, and Profit in the Short Run 1a. If the market price is $20 a tattoo, the marginal revenue is $20 a tattoo. The marginal cost of the fifth tattoo is the total cost of 5 tattoos minus the total cost of 4 tattoos, which is $110 $90 = $20. So, Tom sells 5 tattoos an hour because the marginal cost of the fifth tattoo, $20, equals the marginal revenue. 1b. The average total cost of a tattoo equals the total cost divided by the quantity of tattoos, or $110 5 tattoos = $22 a tattoo. The price of a tattoo is $20 a tattoo, so Tom makes an economic profit of $20 minus $22, which is $2 a tattoo. Tom sells 5 tattoos an hour, so his total economic profit equals $2 5 tattoos, which is $10. Tom incurs an economic loss of $10 an hour. 2a. If the market price falls to $15 a tattoo, the marginal revenue is $15 a tattoo. If Tom remains open, the marginal cost of the fourth tattoo is the total cost of 4 tattoos minus the total cost of 3 tattoos, which is $90 $75 = $15. So, Tom sells 4 tattoos an hour because the marginal cost of the fourth tattoo, $15, equals the marginal revenue. Alternatively, Tom might shut down and produce 0 because the price, $15, is the same as the minimum average variable cost. 2b. If Tom remains open, the average total cost of a tattoo equals the total cost divided by the quantity of tattoos, which is $90 4 tattoos = $22.50 a tattoo. 13 Chapter 13. Perfect Competition 335 The price of a tattoo is $15, so Tom makes an economic profit of $15.00 minus $22.50, which is $7.50 a tattoo. Tom sells 4 tattoos an hour, so his total economic profit equals $ tattoos, which is $30. Tom incurs an economic loss of $30 an hour. Alternatively, if Tom shuts down, then Tom s economic loss equals his fixed cost, $30. Tom s loss is the same regardless of whether he remains open or shuts down. 3a. The price at which Tom shuts down equals minimum average variable cost. Average variable cost equals total variable cost divided by the quantity produced. And, total variable cost equals total cost minus fixed cost. Tom s total fixed cost is $30, his total cost when zero tattoos are produced. When Tom produces 4 tattoos, his total variable cost is $90 $30, which is $60 and so his average variable cost is $60 4 = $15. This is the minimum average variable cost. As a result, Tom shuts down at any price less than $15 a tattoo. 3b. When Tom shuts down, his loss equals his total fixed cost, which is $30. CHECKPOINT 13.3 Output, Price, and Profit in the Long Run 1a. If the market price is $20 a tattoo, marginal revenue equals $20 a tattoo. The marginal cost of the fifth tattoo is the total cost of 5 tattoos minus the total cost of 4 tattoos, which is $110 $90 = $20. So, Tom sells 5 tattoos an hour because the marginal cost of the fifth tattoo, $20, equals the marginal revenue. The average total cost of a tattoo equals the total cost divided by the quantity of tattoos, which is $110 5 tattoos = $22 a tattoo. The price of a tattoo is $20 a tattoo, so Tom makes an economic profit of $20 minus $22, which is $2 a tattoo. Tom sells 5 tattoos an hour, so his total economic profit equals $2 5 tattoos, which is $10. Tom incurs an economic loss of $10. 1b. Because firms are incurring an economic loss, tattoo firms exit the industry. 1c. In the long run, the price is equal to minimum average total cost, which is $22 a tattoo. 1d. In the long run, Tom sells five tattoos an hour because at this quantity price equals minimum average total cost. 1e. In the long run, Tom makes zero economic profit. 14 336 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE ANSWERS TO CHAPTER CHECKPOINT EXERCISES 1a. Wheat is sold in a perfectly competitive market. 1b. Jeans are sold in a monopolistically competitive market. Each firm produces a similar but slightly different type of jean. 1c. The film market is an oligopoly. There are four major firms: Kodak, Fuji, Agfa, and Konica. 1d. The toothpaste market is monopolistically competitive with a large number of similar but not identical brands. 1e. The taxi rides are provided by a monopoly. 2. The firm cannot choose its price because it produces only a small portion of the entire market output and its good has perfect substitutes. If the firm increases its production, it has no impact on the market price. If the firm raises the price of its good above the market price, no one buys from it, switching instead to cheaper, perfect substitutes. And, if the firm lowers its price below the market price, the firm does not maximize its profit because it does not pick up any sales beyond what it could have gained even if it did not lower its price. The firm s horizontal demand curve shows it can sell as much output as it wants at the market price. 3a. Lin s Fortunate Cookies operates in a perfectly competitive market because Lin can sell all the fortune cookies he produces at a price of $50 a batch. 3b. Lin s marginal revenue equals the price, $50 a batch. 3c. Lin s marginal revenue equals price because Lin is a price taker. 4a. Lin s cost curves are in Figure Lin produces the quantity at which marginal revenue equals marginal cost. The marginal revenue is $50.00 a batch of cookies. So Lin produces 6.0 batches a day. Lin s economic profit is zero. Firms do not enter or exit the industry. 4b. Lin produces the quantity at which marginal revenue equals marginal cost. The marginal revenue is $35.20 a batch of cookies. So Lin produces 5.0 batches a day. Lin s average total cost is $52, so Lin incurs an economic loss of $16.80 per batch of cookies, which means she incurs a total economic loss of approximately $ Firms exit the industry. 4c. Lin produces the quantity at which marginal revenue equals marginal cost. The marginal revenue is $83 a batch of cook 15 Chapter 13. Perfect Competition 337 ies. So Lin produces 7.5 batches a day. Lin makes an economic profit of $225. Firms enter the industry. 5. At a price of $35.20 a batch of cookies, Lin produces about 5.0 batches; at a price of $40 a batch of cookies, Lin produces 5.5 batches; at a price of $57 a batch of cookies, Lin produces 6.5 batches; and, at a price of $83 a batch, Lin produces 7.5 batches. At any price less than $35.20 a batch, Lin produces no cookies because her minimum average variable cost is $ Figure 13.2 shows Lin s supply curve. 6. Lin s supply curve is the part of the marginal cost curve that is above minimum average variable cost. At prices below minimum average variable cost, Lin produces no cookies. For prices less than minimum average variable cost, $35.00 a batch, Lin incurs a smaller economic loss by shutting down and producing nothing than by remaining open. So, at these low prices, the firm s supply curve runs along the vertical axis. 7a. The short run equilibrium price is $57 a batch of cookies. 7b. The short run equilibrium quantity is 6,500 batches of cookies. 7c. The economic profit per batch of cookies equals the price of a batch of cookies minus the average total cost of a batch of cookies. The price is $57 a batch of cookies. Each firm produces 6.5 batches of cookies, so the average total cost is $50.50 a batch. Economic profit per batch of cookies equals $57.00 per batch of cookies minus $50.50 a batch of cookies, which is $6.50 a batch of cookies. Each firm makes an economic profit of $6.50 a batch of cookies. Each firm produces 6.5 batches of cookies, so the total economic profit is $6.50 a batch of cookies 6.5 batches of cookies, which equals $ d. Existing firms are making an economic profit, so firms enter the industry. 7e. In the long run equilibrium, the price equals minimum average total cost. So the price equals $50 a batch of cookies. 16 338 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE 7f. At a price of $50 a batch of cookies, the quantity demanded is 7,000 batches. At this price, each firm produces 6.0 batches of cookies, so the total number of firms is 1, Joe s Diner doesn t shut down because the price of a meal is greater than average variable cost. By staying open, even though Joe does not have many customers, he can pay all of the variable costs of remaining open and some of his fixed cost. 9a. Other firms entered the post it note market because 3 M was earning an economic profit. 9b. As years go by, fewer firms enter the market because the economic profit disappears. 9c. If the demand for post it notes decreases, the price falls and some firms incur an economic loss. At some point, these firms leave the market. 10a. Figure 13.3 shows the market for wooden tennis rackets. The initial demand curve is D0 and the supply curve is S. The initial equilibrium price is $80 a racket and the initial equilibrium quantity is 600 rackets an hour. Figure 13.4 shows an individual producer of rackets. The firm initially produces 5 rackets an hour and has a normal profit because the price, $80 a racket, equals average total cost. 10b. The demand for wooden tennis rackets decreases and the demand curve shifts leftward to D1. In the short run, illustrated in Figure 13.3, the equilibrium price of a tennis racket falls to $40 a racket and the equilibrium quan 17 Chapter 13. Perfect Competition 339 tity decreases to 400 rackets an hour. The marginal revenue of the individual producer falls from $80 a racket, MR0, to $40 per racket, MR1. This firm responds by decreasing the quantity of rackets it produces to 3 1/2 rackets an hour. 10c. In the short run, illustrated in Figure 13.4, firms incur an economic loss because price is less than average total cost. As time passes, firms exit the market and the supply decreases. In the case of wooden tennis rackets, the demand decreased sufficiently so that eventually all the firms exited the wooden tennis racket market because today no tennis rackets are made of wood. 11a. Figure 13.5 shows the market for personal computers. The demand curve is D and the initial supply curve is S0. The initial equilibrium price is $2,000 a computer and the initial equilibrium quantity is 400,000 computers per day. Figure 13.6 shows an individual producer of computers. The initial marginal cost curve is MC0, the initial average total cost curve is ATC0, and the initial marginal revenue curve is labeled MR0. The firm initially produces 300 computers a day and has a normal profit because the price, $2,000 a computer, equals average total cost. 11b. The fall in costs shifts the firm s average total cost and marginal cost curve downward, as shown in Figure 13.6 by the shift from ATC0 to ATC1 and from MC0 to MC1. At the initial price of $2,000 a computer, the firms earn an economic profit. So, new firms enter the market and the market supply of computers increases. In the long run, the supply increases and the sup 18 340 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE ply curve shifts rightward from S0 to S1 in Figure This increase in supply lowers the equilibrium price of a computer, to $1,000 a computer, and increases the equilibrium quantity, to 600,000 computer a day. The fall in the price shifts each firm s marginal revenue curve downward, as illustrated in Figure When the marginal revenue curve falls to MR1, the firm produces 500 computers a day. The firm earns only a normal profit, so there is no longer an incentive for new firms to enter the market. 12a. A typical figure showing the cost and revenue curves is Figure Before the fad began, the scooter firms are in long run equilibrium, earning a normal profit. Figure 13.7 shows a typical firm, producing 40 scooters a day and selling them for $60 each. The firm earns zero economic profit because the price, $60 a scooter, equals the average total cost of producing a scooter. 12b. When the scooter fashion is two years old, the scooter firms are earning an economic profit. A few new firms have entered the market and other firms might be considering entry, but these firms are considering entry only if the existing firms are earning an economic profit. Figure 13.8 illustrates a typical scooter firm that is earning an economic profit. The firm is producing 50 scooters a day and selling them for a price of $80. The firm is earning an economic profit because the price, $80 a scoter, exceeds the average total cost. 19 12c. After the scooter fashion has faded, the price returns to its long run equilibrium. Presuming that the costs of the surviving firms have not changed, the price returns to its initial level. The scooter firms earn only a normal profit. Figure 13.9 illustrates this outcome. In the figure, the firm is producing 40 scooters a day and selling them for $60 each. The firm is earning a normal profit. The price, $60 a scooter, equals the average total cost of producing a scooter. Chapter 13. Perfect Competition 341 20 342 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE Critical Thinking 13a. Airport security providers earned an economic profit in The demand for their services increased substantially, so the price rose. The price exceeded the firms average total cost and so they earned an economic profit. 13b. In the future, the price of airport security services will fall. New security firms will enter the market because there currently is an economic profit available. As the new firms enter, the supply increases which leads to a fall in the price. 14a. The maple syrup market is an example of perfect competition because there are many firms selling identical maple syrup to many buyers, there are no restrictions on entry into or exit from the market, established producers have no advantage over new producers, and sellers and buyers are well informed about prices. 14b. Figure shows the market for maple syrup. The initial demand curve in 1980 is D0 and the initial supply curve is S0. The equilibrium price is $8 a half gallon can and the equilibrium quantity is 4 million gallons of maple syrup a year. Figure shows the situation for an individual producer. The firm s marginal revenue is MR and the firm produces 4,000 gallons of maple syrup a year. 14c. Technological change lower the firm s cost curves so that the firm earns an economic profit. The economic profit leads to entry by new firms so that the supply of maple syrup increases. In Figure 13.10, the supply curve shifts rightward from S0 to S1. Independently of the increase in supply, the 21 Chapter 13. Perfect Competition 343 demand also increased over the years so that in Figure 13.10, the demand curve shifts rightward from D0 to D1. (Demand as well as supply must increase because the quantity increases but the price does not change. If only the supply increased, the price would fall.) The equilibrium price in 2003 is $8 a half gallon can and the equilibrium quantity is 8 million gallons a year. The technological change has not lowered the cost curves in the long run because the maple syrup industry must be characterized by external diseconomies so that as new firms enter, the costs of the firms increase and the cost curves shift (back) upward. As a result, the marginal cost curve and average cost curve in 2003 are the same so that the firms are earning a normal profit and entry no longer occurs. Firms that did not adopt the new technology exited the market. 14d. The invention of a great tasting artificial syrup decreases the demand for maple syrup. The demand curve shifts leftward, as illustrated in Figure by the shift from D0 to D1. The equilibrium price falls to $4 a half gallon and the equilibrium quantity decreases to 6 million gallons a year. Figure shows the effect on an individual producer. The firm s marginal cost curve shifts downward from MR0 to MR1. This firm decreases its production from 4,000 gallons a year to 2,000 gallons a year. The firm incurs an economic loss because the price is less than average total cost. In the long run, the economic loss leads to some firms exiting the market. As firms exit, the supply of maple syrup decreases and the supply curve shifts leftward. The equilibrium price of maple syrup rises. Eventually enough 22 344 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE firms exit so that the price rises to equal the firms average total cost. At this point, the firms earn a normal profit and exit ceases. 15. The markets your student choose likely are different than the markets discussed here. But the effects sketched here should be similar in your students answers. An example of a market that will expand is the market for DVDs. As more people own DVD players, the demand for DVDs increases. The firms producing DVDs earn an economic profit in the short run. But as time passes, more firms enter this market. The price of a DVD falls and in the long run, the firms producing DVDs earn a normal profit. An example of a market that will contract is the market for live attendance at sporting events because the quality of watching the events at home increases. The demand for attending a live sporting event decreases, especially second tier live sporting events. The price of a ticket to live sporting events falls and the firms producing live sporting events incur economic losses. As time passes, some of the firms exit the market and, in the long run, the remaining firms earn a normal profit. 23 Chapter 13. Perfect Competition 345 Web Exercises 16a. The wheat market is likely a perfectly competitive market because: many farms sell an identical product to many buyers, there are no restrictions on entry into (or exit from) the market, established farms have no advantage over new farms, and sellers and buyers are well informed about prices. 16b. The price of wheat shows an upward trend through 1996, and a downward trend through the remainder of the decade. 16c. The quantity fluctuated around a falling trend through 1994, then it increased through 1997, after which it decreased. 16d. The main influences on demand have been rising incomes and rising population. 16e. The main influences on supply have been advances in seed and fertilizer technology that have increased yields and the supply. The weather also has played a role and it has brought fluctuations in output. 16f. Before 1996, if anything, there might have been some entry because of rising prices. But, output was falling during these years, and so it is difficult to determine if there was any entry. Regardless, if there was entry or exit it was probably small. After 1997, with sharply falling prices and falling production, most likely exit has occurred. 24 346 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE ADDITIONAL EXERCISES FOR ASSIGNMENT Questions CHECKPOINT 13.1 A Firm s Profit-Maximizing Choices 1. Roy is a potato farmer, and the world potato market is perfectly competitive. The market price is $25 a bag. Roy sells 40 bags a week, and his marginal cost is $20 a bag. 1a. Calculate Royʹs total revenue. 1b. Calculate Royʹs marginal revenue. 1c. Is Roy maximizing profit? Explain your answer. 1d. The price falls to $18 a bag, and Roy cuts his output to 25 bags a week. His average variable cost and marginal cost fall to $18 a bag. Is Roy maximizing profit? Is he making an economic profit or incurring an economic loss? 1e. What is one point on Roy s supply curve? CHECKPOINT 13.2 Output, Price, and Profit in the Short Run 2. Lisa s Lawn Company is a lawn mowing business in a perfectly competitive market for Quantity (lawns per hour) lawn mowing services. The table sets out Lisa s costs. If the market price is $30 a lawn: 2a. How many lawns an hour does Lisaʹs Lawn Company mow? 2b. What is Lisaʹs profit in the short run? 3. In Exercise 2, if the market price falls to $20 a lawn: 3a. How many lawns an hour does Lisaʹs Lawn Company mow? 3b. What is Lisaʹs profit in the short run? 4. In Exercise 2: 4a. At what market price will Lisa shut down? 4b. When Lisa shuts down, what will be her economic loss? CHECKPOINT 13.3 Output, Price, and Profit in the Long Run 5. Lisaʹs Lawn Company is a lawn mowing business in a perfectly competitive market for Quantity (lawns per hour) lawn mowing services. The table sets out Lisaʹs costs. 5a. If the market price is $30 a lawn, what is Lisa s economic profit? 5b. If the market price is $30 a lawn, do new firms enter or do existing firms exit the industry in the long run? Total cost (dollars per lawn) Total cost (dollars per lawn) 25 Chapter 13. Perfect Competition 347 5c. What is the price of the lawn service in the long run? 5d. What is Lisa s economic profit in the long run? 6. A catfish farmer is operating in a perfectly competitive market. The market price of catfish is $5 a pound and each farmer produces 1,000 pounds a week. The average total cost is $7 a pound. 6a. What is a catfish farmer s profit in the short run? 6b. What happens to the total number of farms in the long run? 6c. If technology reduces the cost of catfish farming while the demand for catfish increases, what happens to price in the long run? Answers CHECKPOINT 13.1 A Firm s Profit-Maximizing Choices 1a. Roy s total revenue equals the quantity multiplied by the price, which is 40 $25 = $1,000. 1b. Roy is a perfect competitor, so his marginal revenue equals his price. Roy s marginal revenue is $25 a bag. 1c. Roy is not maximizing profit because marginal cost, $20, is less than marginal revenue, $25. He should sell more bags until marginal cost rises to $25 because then his marginal cost equals his marginal revenue. 1d. If the price is $18, Roy s marginal revenue is $18. Roy is maximizing profit by selling 25 bags because the marginal cost of the 25th bag is $18. Roy is incurring an economic loss because price is less than average total cost. 1e. One point on Roy s supply curve is a price of $18 and a quantity of 25 bags. CHECKPOINT 13.2 Output, Price, and Profit in the Short Run 2a. If the market price is $30 a lawn, the marginal revenue is $30 a lawn. Lisa mows 5 lawns because the marginal cost of mowing the fifth lawn is $30. 2b. Average total cost equals total cost divided by output, so Lisa s average total cost is $130 5, which is $26 a lawn. The economic profit per lawn is the price minus the average total cost, which is $30 a lawn $26 a lawn = $4 a lawn. So Lisa s total profit is $4 a lawn 5 lawns, which is $20. 3a. If the market price is $20 a lawn, marginal revenue is $20 a lawn. Lisa mows 3 lawns because the marginal cost of mowing the third lawn is $20. 3b. Average total cost equals total cost divided by output, so Lisa s average total cost is $75 3, which is $25 a lawn. The economic profit per lawn is the price minus the average total cost, which is $20 a lawn $25 a lawn = $5 a lawn. Lisa incurs an economic loss of $5 a lawn. So her total economic loss is $5 loss per lawn 3 lawns, which is $15. 4a. Lisa s shutdown point is at $10 a lawn and 1 lawn mowed. Mowing 1 lawn has a marginal cost of $10, so when the price is $10, mowing 1 lawn sets marginal cost equal to marginal revenue. Lisa s total variable cost at this 26 348 Part 5. PRICES, PROFITS, AND INDUSTRY PERFORMANCE level of output is $10 because her total fixed cost is $30. (Her total fixed cost is her total cost when she mows zero lawns.) When Lisa mows 1 lawn, her average variable cost is $10 1, which is $10. Lisa s minimum average variable cost is $10, so when the price is $10, Lisa is at her shutdown point. 4b. When Lisa shuts down, her loss equals her total fixed cost, which is $30. CHECKPOINT 13.3 Output, Price, and Profit in the Long Run 5a. If the market price is $30, marginal revenue equals $30. Lisa mows 5 lawns because that is the quantity for which marginal cost equals marginal revenue. The average total cost of cutting a lawn when 5 are cut is $26 so Lisa earns an economic profit of $4 a lawn. She cuts 5 lawns so her total economic profit is $4 5 = $20. 5b. Because firms are earning an economic profit, new lawn mowing firms enter the market. 5c. In the long run, the price is $25, which is minimum of average total cost. 5d. In the long run, Lisa s economic profit is zero. 6a. Because the price is less than average total cost ($5 versus $7), each farmer incurs an economic loss of $2 a pound and a total economic loss of $2,000. 6b. Because some firms are incurring an economic loss, they exit the market. Supply decreases and the market price rises. In the long run, the number of farms decreases. 6c. New technology shifts the cost curves downward. This effect decreases the price in the long run because in the long run price equals the minimum average total cost. 27 Chapter 13. Perfect Competition 349 USING EYE ON THE U.S. ECONOMY Entry in Personal Computers, Exit in Farm Machines The story provides good examples of economic profit motivating firms to enter a market (personal computers) and economic loss motivating a firm to exit a market (farm machines). For both of the stories, you can draw short run and longrun scenarios. Use the PC market to discuss the transition from one firm (IBM) earning large economic profit in the short run to many firms eking out normal profits in the long run. You can note that the short run and IBM s economic profit lasted several years until the new firms could produce a reliable clone. And, with the entry of Dell, etc., eventually IBM essentially exited by selling its name to a Chinese firm with whom IBM has a co venture. The International Harvester portion of the article provides a good example of a firm exiting a market because of economic loss. As the structure of the agriculture industry changed (fewer family farms and more efficient, conglomerate farms), less equipment was needed. As a result, the demand for and the price of farm equipment decreased. The fall in price means firms incurred economic losses. While more firms than International Harvester were suffering losses, they chose not to exit. Navistar s decision allowed itself, and the surviving firms, the opportunity to earn a normal profit. USING EYE ON THE GLOBAL ECONOMY The North American Market in Maple Syrup The article highlights the perfectly competitive wholesale maple syrup market. You can use this story to show short run and long run profit scenarios graphically. Show how technology led to some firms exiting the market while other, larger firms decided to adopt the technology. Also show how the demand curve shifted rightward as demand increased. Perfect Competition Chapter 10 CHAPTER IN PERSPECTIVE In Chapter 10 we study perfect competition, the market that arises when the demand for a product is large relative to the output of a single producer. Chapter 6 Competitive Markets After reading Chapter 6, COMPETITIVE MARKETS, you should be able to: List and explain the characteristics of Perfect Competition and Monopolistic Competition Explain why a Chapter 11 PERFECT COMPETITION Competition Topic: Perfect Competition 1) Perfect competition is an industry with A) a few firms producing identical goods B) a few firms producing goods that differ somewhat Chapter 6 The Two Extremes: Perfect Competition and Pure Monopoly Learning Objectives List the four characteristics of a perfectly competitive market. Describe how a perfect competitor makes the decision Practice Questions Week 8 Day 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The characteristics of a market that influence the behavior of market participants Chapter 7 Monopoly, Oligopoly and Strategy After reading Chapter 7, MONOPOLY, OLIGOPOLY AND STRATEGY, you should be able to: Define the characteristics of Monopoly and Oligopoly, and explain why the are Perfectly Competitive Markets It is essentially a market in which there is enough competition that it doesn t make sense to identify your rivals. There are so many competitors that you cannot single out ANSWERS TO END-OF-CHAPTER QUESTIONS 23-1 Briefly indicate the basic characteristics of pure competition, pure monopoly, monopolistic competition, and oligopoly. Under which of these market classifications MBA 640 Survey of Microeconomics Fall 2006, Quiz 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A monopoly is best defined as a firm that 3 Demand, Supply, and Market Equilibrium The price of vanilla is bouncing. A kilogram (2.2 pounds) of vanilla beans sold for $50 in 2000, but by 2003 the price had risen to $500 per kilogram. The price Page 1 1. Supply and demand are the most important concepts in economics. 2. Markets and Competition a. Market is a group of buyers and sellers of a particular good or service. P. 66. b. These individuals Practice Multiple Choice Questions Answers are bolded. Explanations to come soon!! For more, please visit: http://courses.missouristate.edu/reedolsen/courses/eco165/qeq.htm Market Equilibrium and Applications CHAPTER 8 PROFIT MAXIMIZATION AND COMPETITIVE SUPPLY TEACHING NOTES This chapter begins by explaining what we mean by a competitive market and why it makes sense to assume that firms try to maximize profit. CHAPTER 11 PRICE AND OUTPUT IN MONOPOLY, MONOPOLISTIC COMPETITION, AND PERFECT COMPETITION Chapter in a Nutshell Now that we understand the characteristics of different market structures, we ask the question CHAPTER 9: PURE COMPETITION Introduction In Chapters 9-11, we reach the heart of microeconomics, the concepts which comprise more than a quarter of the AP microeconomics exam. With a fuller understanding 1 How to Study for Chapter 18 Pure Monopoly Chapter 18 considers the opposite of perfect competition --- pure monopoly. 1. Begin by looking over the Objectives listed below. This will tell you the main Chap 13 Monopolistic Competition and Oligopoly These questions may include topics that were not covered in class and may not be on the exam. MULTIPLE CHOICE. Choose the one alternative that best completes Managerial Economics & Business Strategy Chapter 8 Managing in Competitive, Monopolistic, and Monopolistically Competitive Markets I. Perfect Competition Overview Characteristics and profit outlook. Effect Chapter 8. Competitive Firms and Markets We have learned the production function and cost function, the question now is: how much to produce such that firm can maximize his profit? To solve this question, Principles of Microeconomics, Quiz #5 Fall 2007 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question on the accompanying scantron. 1) Perfect competition ECON 103, 2008-2 ANSWERS TO HOME WORK ASSIGNMENTS Due the Week of June 23 Chapter 8 WRITE Use the demand schedule that follows to calculate total revenue and marginal revenue at each quantity. Plot Chapter 7: Market Structures Section 1 Key Terms perfect competition: a market structure in which a large number of firms all produce the same product and no single seller controls supply or prices commodity: Chapter 14 Monopoly 14.1 Monopoly and How It Arises 1) A major characteristic of monopoly is A) a single seller of a product. B) multiple sellers of a product. C) two sellers of a product. D) a few sellers Principles of Microeconomics Fall 2007, Quiz #6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question on the accompanying scantron. 1) A monopoly is Name: Class: Date: ID: A Economics Chapter 7 Review Matching a. perfect competition e. imperfect competition b. efficiency f. price and output c. start-up costs g. technological barrier d. commodity h. In this chapter, look for the answers to these questions: How is similar to perfect? How is it similar to monopoly? How do ally competitive firms choose price and? Do they earn economic profit? In what chapter 7 >> Making Decisions Section : Making How Much Decisions: The Role of Marginal Analysis As the story of the two wars at the beginning of this chapter demonstrated, there are two types of decisions: Chapter 12 PERFECT COMPETITION Answers to the Review Quizzes Page 275 1. Why is a firm in perfect competition a price taker? One firm s output is a perfect substitute for another firm s output and each Chapter 12 Production and Cost 12.1 Economic Cost and Profit 1) The primary goal of a business firm is to A) promote fairness. B) make a quality product. C) promote workforce job satisfaction. D) maximize Econ 202 Exam 3 Practice Problems Principles of Microeconomics Dr. Phillip Miller Multiple Choice Identify the choice that best completes the statement or answers the question. Chapter 13 Production and Chapter 8 Production Technology and Costs 8.1 Economic Costs and Economic Profit 1) Accountants include costs as part of a firm's costs, while economists include costs. A) explicit; no explicit B) implicit; Demand, Supply and Elasticity CHAPTER 2 OUTLINE 2.1 Demand and Supply Definitions, Determinants and Disturbances 2.2 The Market Mechanism 2.3 Changes in Market Equilibrium 2.4 Elasticities of Supply and Pre-Test Chapter 21 ed17 Multiple Choice Questions 1. Which of the following is not a basic characteristic of pure competition? A. considerable nonprice competition B. no barriers to the entry or exodus Price Theory Lecture 6: Market tructure Perfect Competition I. Concepts of Competition Whether a firm can be regarded as competitive depends on several factors, the most important of which are: The number N. Gregory Mankiw Principles of Economics Chapter 15. MONOPOLY Solutions to Problems and Applications 1. The following table shows revenue, costs, and profits, where quantities are in thousands, and total 1 How to Study for Chapter 14 Costs of Production (This Chapter will take two class periods to complete.) Chapter 14 introduces the main principles concerning costs of production. It is perhaps the most chapter 9 The industry supply curve shows the relationship between the price of a good and the total output of the industry as a whole. Perfect Competition and the >> Supply Curve Section 3: The Industry Microeconomics Topic 7: Contrast market outcomes under monopoly and competition. Reference: N. Gregory Mankiw s rinciples of Microeconomics, 2 nd edition, Chapter 14 (p. 291-314) and Chapter 15 (p. 315-347). Perfectly Competitive Markets A firm s decision about how much to produce or what price to charge depends on how competitive the market structure is. If the Cincinnati Bengals raise their ticket prices Lab 12: Perfectly Competitive Market 1. Perfectly competitive market 1) three conditions that make a market perfectly competitive: a. many buyers and sellers, all of whom are small relative to market b. Monopoly 18 The public, policy-makers, and economists are concerned with the power that monopoly industries have. In this chapter I discuss how monopolies behave and the case against monopolies. The case Chapter 9: Perfect Competition Perfect Competition Law of One Price Short-Run Equilibrium Long-Run Equilibrium Maximize Profit Market Equilibrium Constant- Cost Industry Increasing- Cost Industry Decreasing- Final Exam Microeconomics Fall 2009 Key On your Scantron card, place: 1) your name, 2) the time and day your class meets, 3) the number of your test (it is found written in ink--the upper right-hand corner Name: Date: 1. Most electric, gas, and water companies are examples of: A) unregulated monopolies. B) natural monopolies. C) restricted-input monopolies. D) sunk-cost monopolies. Use the following to answer Final Exam Economics 101 Fall 2003 Wallace Final Exam (Version 1) Answers 1. The marginal revenue product equals A) total revenue divided by total product (output). B) marginal revenue divided by marginal Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The four-firm concentration ratio equals the percentage of the value of accounted for by the four Masaryk University - Brno Department of Economics Faculty of Economics and Administration BPE_MIC1 Microeconomics 1 Fall Semester 2011 Final Exam - 05.12.2011, 9:00-10:30 a.m. Test A Guidelines and Rules: Understanding Economics 2nd edition by Mark Lovewell and Khoa Nguyen Chapter 5 Perfect Competition Chapter Objectives! In this chapter you will: " Consider the four market structures, and the main differences The Cost of Production 1. Opportunity Costs 2. Economic Costs versus Accounting Costs 3. All Sorts of Different Kinds of Costs 4. Cost in the Short Run 5. Cost in the Long Run 6. Cost Minimization 7. The Equilibrium of a firm under perfect competition in the short-run. A firm is under equilibrium at that point where it maximizes its profits. Profit depends upon two factors Revenue Structure Cost Structure WSG8 7/7/03 4:34 PM Page 113 8 Market Structure: Perfect Competition and Monopoly OVERVIEW One of the most important decisions made by a manager is how to price the firm s product. If the firm is a profit (Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion Principles of Economics: Micro: Exam #2: Chapters 1-10 Page 1 of 9 print name on the line above as your signature INSTRUCTIONS: 1. This Exam #2 must be completed within the allocated time (i.e., between EXAM TWO REVIEW: A. Explicit Cost vs. Implicit Cost and Accounting Costs vs. Economic Costs: Economic Cost: the monetary value of all inputs used in a particular activity or enterprise over a given period. 1. The supply of gasoline changes, causing the price of gasoline to change. The resulting movement from one point to another along the demand curve for gasoline is called A. a change in demand. B. a change Microeconomics Topic 3: Understand how various factors shift supply or demand and understand the consequences for equilibrium price and quantity. Reference: Gregory Mankiw s rinciples of Microeconomics, Experiment 8: Entry and Equilibrium Dynamics Everyone is a demander of a meal. There are approximately equal numbers of values at 24, 18, 12 and 8. These will change, due to a random development, after Econ 101: Principles of Microeconomics Chapter 14 - Monopoly Fall 2010 Herriges (ISU) Ch. 14 Monopoly Fall 2010 1 / 35 Outline 1 Monopolies What Monopolies Do 2 Profit Maximization for the Monopolist 3 CHAPTER 10 MARKET POWER: MONOPOLY AND MONOPSONY EXERCISES 3. A monopolist firm faces a demand with constant elasticity of -.0. It has a constant marginal cost of $0 per unit and sets a price to maximize I. Learning Objectives In this chapter students will learn: A. The significance of resource pricing. B. How the marginal revenue productivity of a resource relates to a firm s demand for that resource. Chapter 6 - Markets in Action - Sample Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The short-run impact of the San Francisco earthquake Chapter 7 AGGREGATE SUPPLY AND AGGREGATE DEMAND* Key Concepts Aggregate Supply The aggregate production function shows that the quantity of real GDP (Y ) supplied depends on the quantity of labor (L ), CHAPTER 12 MARKETS WITH MARKET POWER Microeconomics in Context (Goodwin, et al.), 2 nd Edition Chapter Summary Now that you understand the model of a perfectly competitive market, this chapter complicates RELEVANT TO ACCA QUALIFICATION PAPER F1 / FOUNDATIONS IN ACCOUNTANCY PAPER FAB Introduction to microeconomics The new Paper F1/FAB, Accountant in Business carried over many subjects from its Paper F1 predecessor, a CHAPTER 6: ELASTICITY, CONSUMER SURPLUS, AND PRODUCER SURPLUS Introduction Consumer responses to changes in prices, incomes, and prices of related products can be explained by the concept of elasticity. In this chapter, look for the answers to these questions: Why do monopolies arise? Why is MR < P for a monopolist? How do monopolies choose their P and Q? How do monopolies affect society s well-being? Chapter 13 MONOPOLISTIC COMPETITION AND OLIGOPOLY Key Concepts Monopolistic Competition The market structure of most industries lies between the extremes of perfect competition and monopoly. Monopolistic CHAPTER 1 The Central Idea CHAPTER OVERVIEW Economic interactions involve scarcity and choice. Time and income are limited, and people choose among alternatives every day. In this chapter, we study the Chapter 04 Firm Production, Cost, and Revenue Multiple Choice Questions 1. A key assumption about the way firms behave is that they a. Minimize costs B. Maximize profit c. Maximize market share d. Maximize LECTURE NOTES ON MACROECONOMIC PRINCIPLES Peter Ireland Department of Economics Boston College email@example.com http://www2.bc.edu/peter-ireland/ec132.html Copyright (c) 2013 by Peter Ireland. Redistribution A2 Micro Business Economics Diagrams Advice on drawing diagrams in the exam The right size for a diagram is ½ of a side of A4 don t make them too small if needed, move onto a new side of paper rather than COST THEORY Cost theory is related to production theory, they are often used together. However, the question is how much to produce, as opposed to which inputs to use. That is, assume that we use production CHAPTER 18 MARKETS WITH MARKET POWER Principles of Economics in Context (Goodwin et al.) Chapter Summary Now that you understand the model of a perfectly competitive market, this chapter complicates the CHAPTER 13 MARKETS FOR LABOR Microeconomics in Context (Goodwin, et al.), 2 nd Edition Chapter Summary This chapter deals with supply and demand for labor. You will learn about why the supply curve for This file includes the answers to the problems at the end of Chapters 1, 2, 3, and 5 and 6. Chapter One 1. The economic surplus from washing your dirty car is the benefit you receive from doing so ($6) N. Gregory Mankiw Principles of Economics Chapter 14. FIRMS IN COMPETITIVE MARKETS Solutions to Problems and Applications 1. A competitive market is one in which: (1) there are many buyers and many sellers THIRD EDITION ECONOMICS and MICROECONOMICS Paul Krugman Robin Wells Chapter 19 Factor Markets and Distribution of Income WHAT YOU WILL LEARN IN THIS CHAPTER How factors of production resources like land, CHAPTER 6 MARKET STRUCTURE CHAPTER SUMMARY This chapter presents an economic analysis of market structure. It starts with perfect competition as a benchmark. Potential barriers to entry, that might limit Demand and Supply Chapter CHAPTER CHECKLIST Demand and supply determine the quantities and prices of goods and services. Distinguish between quantity demanded and demand, and explain what determines demand. Name Eco200: Practice Test 2 Covering Chapters 10 through 15 1. Four roommates are planning to spend the weekend in their dorm room watching old movies, and they are debating how many to watch. Here is

Recall what Keynes wrote (p. 207): "(2) There is the possibility, for the reasons discussed above, that, after the rate of interest has fallen to a certain level, liquidity-preference may become virtually absolute in the sense that almost everyone prefers cash to holding a debt which yields so low a rate of interest. In this event the monetary authority would have lost effective control over the rate of interest. But whilst this limiting case might become practically important in future, I know of no example of it hitherto. Indeed, owing to the unwillingness of most monetary authorities to deal boldly in debts of long term, there has not been much opportunity for a test. Moreover, if such a situation were to arise, it would mean that the public authority itself could borrow through the banking system on an unlimited scale at a nominal rate of interest." So Scott is right - this really does seem like a problem for this particular passage, at least. One interesting thing to note is that the last sentence Keynes has here seems right whether the first part of the paragraph holds true or not (in other words - the case for fiscal policy still stands just fine - a point that Krugman makes, and argues is independent of all those "Keynesian things"). The question, of course, is why is he wrong? I don't feel like I have guidance on that from Scott. Why would investors prefer a negative yield to a zero yield assuming both assets have negligible risk? This isn't a Keynesian question that deserves an answer - it's a basic rationality question that deserves an anwer. My first thought roamed into international economics (which makes me nervous beause I don't feel like I have a good intuition for it). If foreigners are buying a lot of Treasuries and there is some exchange rate risk involved, the yield on cash may not be zero. But that doesn't seem to work because the exchange rate risk is going to impact demand for Treasuries (which are denominated in dollars) as well. Perhaps transaction costs associated with working with bonds is lower for foreigners than cash? I don't know. The only other thing I could think of is that building a cash reserve must be limited somehow, so while a zero yield would be preferable its not a viable option. Is this true? That would seem to explain it. If the reason why interest rates can transcend the lower bound of the liquidity trap is that the supply of cash is constrained, you would want policy that creates more cash: like public works that create jobs or more bond purchases. If you'd like people to get jobs before banks get cash to sit on, you might prefer the former. If you'd like banks to get cash to sit on before the jobs start flowing, you might prefer the latter. I also wanted to point out that Barkley Rosser is talking about liquidity traps this morning, and he is particularly taking economists to task for allegely rejeting the idea of negative prices. Again, like the Sumner post, the discussion is interesting but it doesn't get at the real question that's bugging me: fine, I accept prices can be negative (although can't that just be thought of as a flip-flop of the role of buyer and seller? No matter). The real confusing question for me is how this price could be negative when the price of an otherwise equal product is zero? Barkley's example (brides who alternatively trade for dowries or bride prices) isn't relevant here because the brides aren't the same quality product. The whole problem with a liquidity trap is that cash and government bonds presumably are. Barkley's first commenter shares my concern for the real question here: "So what's the mechanism by which negative interest rate bonds come about?". He comes to the same conclusion about the limited supply of cash, and he is more amenable to the foreign flight to quality issue (I'm still not so sure... flying to quality is sensible, but why fly to American bonds rather than American cash??). I feel too dumb to answer this, but I feel smart enough to confidently say that Scott and Barkley and no one else I've seen has answered it either. There is another disconcerting implication of all of this. On page 202 of the General Theory, Keynes points out that "...if the general view as to what is a safe level of r is unchanged, every fall in r reduces the market rate relatively to the "safe" rate and therefore inreases the risk of illiquidity; and, in the second place, every fall in r reduces the current earnings from illiquidity, which are available as a sort of insurance premium to offset the risk of loss on capital account, by an amount equal to the difference between the squares of the old rate of interest and the new. For example, if the rate of interest on a long-term debt is 4 per cent., it is preferable to sacrifice liquidity unless on a balance of probabilities it is feared that the long-term rate of interest may rise faster than by 4 per cent. of itself per annum, i.e. by an amount greater than 0.16 per cent. per annum. If, however, the rate of interest is already as low as 2 per cent., the running yield will only offset a rise in it of as little as 0.04 per cent. per annum. This, indeed, is perhaps the chief obstacle to a fall in the rate of interest to a very low level. Unless reasons are believed to exist why future experience will be very different from past experience, a long-term rate of interest of (say) 2 per cent. leaves more to fear than to hope, and offers, at the same time, a running yield which is only sufficient to offset a very small measure of fear." We usually hear the "more to fear than to hope" logic as we approach the zero lower bound, right? But these calculations squaring the interest rate are symmetric. What does it say about expectations for the future path of interest rates if interest rates are currently negative? This sort of thinking explodes quickly. Part of me wants to paraphrase Herbert Stein and say that path is not sustainable, so it won't be sustained. There must be something driving this other than expectations of future interest rate paths, because that answer is the explosive answer. Another part of me thinks that even if the explosive solution doesn't come about, the fact that investors aren't afraid of a negative yield says something real about their expectations of future yields. I minor pet peeve on the workings of the internet - no need to comment on this in the comment section - why in the world are there more Austrian links in the reference list of the Wikipedia entry on liquidity traps than Keynesian links? It's really a shame that people who try to learn about economics from the internet get a such a distorted sense of the field, simply because old Mises Institute blog posts are so accessible relative to the actual economics literature.

Productivity really is demand constrained To find out if productivity is really demand constrained, let’s look to see what happens when productivity is up against the effective demand limit. We will find that productivity stops and sits for a number of years. First we get the data and build the model. The data will come from this graph at FRED. The graph shows real GDP (Y), effective demand (E) and total labor hours (L) all indexed to 2005=100. The light orange line is labor hours (nonfarm business sector), which are still at the same level of 15 years ago. Yet, real GDP (blue line) has risen over that time. So the basic story is that we have been more productive with the same labor hours. Here is a graph to show that real GDP used to rise with increased labor hours. However, since the 1990’s, real GDP has increased even though labor hours have not (total plot 1967 to 2013). OK… then how can productivity be demand constrained when real GDP keeps rising in spite of the fact that labor is not increasing their hours to earn more income? In other words, wouldn’t a demand constraint be dependent upon labor hours? Well, no… Productivity is normally determined by dividing real GDP output (Y) by total labor hours (L). Productivity = Y / L Now to show when productivity is constrained by effective demand, we divide effective demand (E) by total labor hours (L) to get the effective demand limit per labor hour. The reason is that what is produced in an hour cannot surpass the hourly effective demand limit. We will check the data to see if this reasoning holds up. Real hourly effective demand = E / L (Note: Effective demand has the equation, E = Y * e/T …………. e = effective labor share, T = TFUR, total factor utilization rate (employment rate * capital utilization) Real hourly effective demand, E / L = Y / L * e/T Real hourly effective demand, E / L = productivity * e/T Real hourly effective demand allows us to compare effective demand with hourly measurements, like productivity, real hourly compensation, capital used per hour, etc. Now, what happens if we graph productivity (Y/L) against real hourly effective demand (E/L)? (Data in graph is given by quarters from 1967 to 2013.) This graph is a scatter plot using the data from graph #1. Let’s first look at the red line which represents the effective demand limit. A basic principle of effective demand is that real GDP is constrained below effective demand. Thus the red line sets the theoretical effective-demand limit for productivity. The blue line is how productivity has moved with real hourly effective demand. Productivity is how much production is produced (in real 2005 dollar terms) per hour. Real hourly effective demand is the potential demand per hour for hourly production. In theory, productivity per hour should be limited by the hourly effective demand limit; Production would not go over the demand constraint. Thus, the plot in graph #3 should stay below the red line. In other words, productivity should stay below the effective demand constraint. And what do we see in graph #3? The plot of productivity does in fact stay below the effective demand limit (red line). Productivity will bounce along the effective demand limit. Using effective demand gives a wonderful way to view the behavior of productivity. Normally real GDP is plotted against effective demand. However, productivity moves differently than real GDP because of the variability in labor hours. Yet, the most interesting part of this graph is how productivity behaves when it is close to the effective demand limit. Productivity stalls for a number of years… 3 to 4 years. We find that during those 3 to 4 years, productivity does not increase much at all. Effective demand will not move much either. It is impossible to see in the graph, but there are numerous dots all bunched together in the spots where productivity stalls at the effective demand limit. I only count the dots up against the red line. You cannot see them all in the graph. For example, between 1994 and 1997, you see what looks like two dots peaking over the red line. There are actually 16 dots in that little space between those two dots; that is 16 quarters… 4 years of productivity being completely stationary and demand constrained. If you look at the spot of 1977 to 1979, you will see a line that heads straight toward the effective demand limit and comes straight back. There are in fact 20 dots in that little line that sticks out. That is 20 quarters or 5 years worth of data. In other words, productivity and hourly effective demand moved in a perfectly straight little line to the effective demand limit and back over 5 years. This relationship between productivity and effective demand has never been seen before. It certainly is interesting. When productivity increases up and toward the left in the graph (meaning hourly effective demand is declining while productivity increases), productivity eventually hits the effective demand limit and stops. Productivity sits against the effective demand limit for a number of years until the tide turns and effective demand starts to increase. Then productivity can start increasing. In effect, effective demand has to start increasing first in order for productivity to start increasing. Then, hourly effective demand and productivity will increase together moving in the direction of the effective demand limit (red line). Lots of words to describe a simple process. Let me boil it down. Productivity is often constrained by effective demand. Let’s look at current data at the end of the plot. We see that productivity has come close to the effective demand limit again and has stalled out in the same spot for 2 years. Productivity itself has stalled for over 3 years. I will state three conclusions… - Productivity is being compressed by the effective demand limit again. - Productivity is not going to be increasing soon unless effective demand reverses its decline. - For those, like Ray Dalio, who say that productivity will increase as the economy recovers, they will be disappointed. The implication is that the economy instead will have to grow upon an increase in credit-fueled consumption. Other relate posts at Effective Demand blog Demand determined output: Getting the definition correct Mostly I agree; however given the changes seen on Wall Street with investments, does it seem reasonable that the investments can be profitable without Labor? Or productivity gains impacting GDP wihout Labor input at the same time Labor hits the wall? We might call such gambling on Wall Street. The investments can still be profitable, but overall economic growth is dependent upon more labor hours because of the demand constraint. The explanation is that productivity has been constant for 4 years. Productivity = real GDP / total labor hours In order for productivity to stay constant, real GDP and labor hours have to rise at the same rate. Thus real GDP growth is dependent on an equal growth in labor hours. This is how the demand constraint can be stable as real GDP rises. More labor hours will allow an equal increase in production, but productivity per hour is ultimately constrained by effective demand per hour.

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . . How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . . If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . . Show that among the interior angles of a convex polygon there cannot be more than three acute angles. Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in A huge wheel is rolling past your window. What do you see? These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts. The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why! A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . . This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not. . . . The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . . Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200. Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat. . . . Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation? ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm. Replace each letter with a digit to make this addition correct. Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs. Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . . Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits. Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . . This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . . Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation. Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . . A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself. There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children? Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . . Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle. The sums of the squares of three related numbers is also a perfect square - can you explain why? Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . . Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total. Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions? Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle. Which hexagons tessellate? A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter. This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . . Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches. Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance? A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree. . . . Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

W3150 / W4150 - Fall 2002 The Cell as a Machine: Cell Biophysics and Biosystems Engineering Dr. Michael Sheetz TA: Volodymyr Nikolenko Cells are complex micron-sized machines that may best be understood through reverse systems engineering, i.e. through an understanding of the details of cellular functions and how they were optimized. To understand the functions, we will review basic solution physical chemistry, mechanics, and diffusion theory. The reference books are "Molecular Biology of the Cell" by Alberts et al., (3rd Edition) and "Biomechanics" by Y.C. Fung (1981) The assignments for undergrads will primarily involve the solution of practical cellular problems using equations given in class. Graduate students will read and critique original papers in addition to solving the problems. Prerequisites: calculus and physics Co-requisites: cell biology or biochemistry (Of course, these can also be taken in a prior semester) Outline of Course This is a general survey of a variety of cell functions, which emphasizes a problem-based approach to understanding the functions. Because cell functions are based upon a wide range of physical principles, the physical principles will only be introduced and references are provided in case you wish to delve further in any area. Many of the principles are discussed in Molecular Biology of the Cell (Alberts et al.) and in other Biophysics (e.g. Random Walks in Biology, Berg) and Physical Chemistry (Atkins) texts. Problems are meant to facilitate a quantitative understanding of cellular functions and lower the energy barrier to future back-of-the-envelope calculations that can determine if something is physically possible or worthless tripe. Cells are nanomachines that have an intermediate number of components and typically work in a Low Reynold’s number size regime. Thus, most cell functions are based upon stochastic processes that vary from cell to cell and over time. It is perhaps surprising that cells function properly under a wide variety of conditions and exhibit a “robustness” of function that belies highly developed feedback control systems. In other words, how can a machine that shows stochastic variability be programmed to reliably execute complex inter-related functions under widely varied environmental conditions? For a variety of cell functions, we will explore by quantitative analyses the magnitude of the tasks that trillions upon trillions of cells routinely perform. For example, human cells can faithfully replicate two meters of linear DNA in an hour within the confines of a five micrometer nucleus and then sort and delivery equal copies of DNA to two daughter cells. Grading will be on a curve with a median grade of a B. Half of points for the grade will come from the problems that are at the end of each lecture. Answers are to be submitted electronically within one week of the lecture, i.e. e-mail time stamped before the time of the lecture one week later. Collaboration on problems is allowed but one should understand how to solve the problems because the in-class tests will be based upon the take-home problems. The two in-class tests (mid-term and final) will be weighted differently (20% and 30% of the grade respectively) and notes from the lectures will be allowed. Thursday afternoons beginning at 3 PM, I will meet with people to discuss any questions that were submitted by e-mail (first priority) and raised at the time (second priority). E-mail questions to the TA can be submitted at any time and will be answered within the week. There is no one textbook for the course but Molecular Biology of the Cell (Alberts et al. 4th edition) is recommended as a resource for the Cell Biology and Physical Chemistry (Peter Atkins) is recommended for the Physical Chemistry. Outline of Lectures (tentative) overview and outline: Lecture 1 2. How Nano-BioMachines Work (diffusion and transport) (Lectures 2 & 3) overview and outline: Lecture 2 | Lecture 3 3. DNA Packaging and Replication (Lectures 4, 5 and 6) overview and outline: Lecture 4 | Lecture 5 | Lecture 6 4. RNA Transcription and Processing (Lectures 7 and 8) overview and outline: Lecture 7 | Lecture 8 5. Lipid Bilayer and Plasma membrane (hydrophobic effect, Guoy-Chapman potential, mechanics, and diffusion in 2-D) (Lectures 9, 10, and 11) overview and outline: Lecture 9 | Lecture10 | Lecture 11 6. Midterm (note that the course organization is different from last year) 7. Protein synthesis and processing (polymerases, processing (hybridization), ribosomes, and translocation) (Lectures 12, and 13) overview and outline: Lecture 12 | Lecture 13 8. Cytoskeleton, which defines a non-spherical shape and mechanical properties (polymer assembly, mechanics) (Lectures 14, 15, 16 and 17) overview and outline: Lecture 14 | Lecture 15 | Lecture 16 & 17 9. Glycoprotein and Secreted Protein Processing (Membrane transport, flow, and stabilization) (Lectures 18, and 19) overview and outline: Lecture 18 | Lecture 19 10. Endocytosis and Protein Degradation (Lecture 20) overview and outline: Lecture 20 11. Ion balance and volume regulation (membrane potential, osmotic pressure, and resealing) (Lecture 21) overview and outline: 12. Signal Integration (Biosystems Engineering) (Lecture 22) overview and outline: 13. Migration, Force generation and Chemotaxis (Lectures 23, 24 and 25) overview and outline: Lecture 23 | PowerPoint files :: Neural Crest | Lecture 25 (Force) 14. Final Exam (Will contain problems from whole course with emphasis on last half) Lectures 1-7 | Lectures 8-14 | Lectures 15-23 | Midterm

Finally, an analysis is made of the mechanics of the rigid body. Member abc is 6 meters long with point b being in the middle. We usually only deal with motion of objects in two dimensions, in which case the conditions for static equilibrium are f x 0, f y 0 6 2 0 63. Refer to support reactions section and refresh your memory. Any more supports than this will either not allow the system to be in. Member abc is 6 meters long with point b being in the middle determine all forces acting on member abc 450 450 irs 300 cas z q. They will come from the other objects with which the body is in contact supports, walls. So far, we have only considered translational motion. Note that the forces need to go in their correct positions i. Since the rigid body is in equilibrium, the sum of the moments of f 1, f 2, and f 3 about any axis. Equilibrium of rigid bodies a rigid body is said to be in equilibrium if. Statics is typically the first engineering mechanics course taught in universitylevel engineering programs. The concept of equilibrium is used for determining unknown forces and moments of forces that act on or within a rigid body or system of rigid bodies. Equilibrium of a rigid body torques and rotational. Determine the tension in the cable and the reactions at a and b if 45. Then we place three ball bearings on the disk, and then a second steel disk on top of the ball bearings. Write down the equilibrium equations based on your fbds. Chapter 5 equilibrium of a rigid body alvick lau 53. For a rigid body to be in equilibrium, the net force as well as the net moment about any arbitrary point o must be equal to zero. Equilibrium of a rigid body introduction a rigid body is in equilibrium when it is not undergoing a change in rotational or translational motion. F 0 and m o 0 forces on a rigid body forces on a particle. Member abc is 6 meters long with point b being in the. Draw the free body diagram of the dumpster d of the truck, which has a weight of 5000 lb and a center of gravity at g. Equilibrium of a rigid body torques and rotational equilibrium. Equilibrium of a rigid body torques and rotational equilibrium overview when a system of forces, which are not concurrent, acts on a rigid object, these forces will tend to move the object from one position to another translation and may also produce a turning effect of the object around a given axis rotation. Count the number of equations and the number of unknowns. F 0 no translation and m o 0 no rotation forces on a rigid body we need to draw a free body diagram fbd showing all the external active and reactive forces. The first step in the analysis of the equilibrium of rigid bodies must be to draw the free body diagram of the body in question. It has four attached hooks and a protractor attached to the bar at each end so that the angle can be read. Conditions of equilibrium or motion are not affected by transmitting. There are two typical kinds of supports for structures. To evaluate the unknown reactions holding a rigid body in equilibrium by solving the equations of static equilibrium. In order for a rigid body to stay in equilibrium, the net force and the net torque on the object must both be zero. This assumption means that the object does not change shape when forces act on the object. For a twodimensional problem, a rigid body will have one pin joint and one roller support. The joist is a 3 force body acted upon by the rope, its weight, and the reaction at a. For an rigid body in static equilibrium, that is a nondeformable body where forces are not concurrent, the sum of both the forces and the moments acting on the body must be equal to zero. Static equilibrium of rigid bodies university of hawaii system. Draw the free body diagram show all active and reactive forces 3. Equilibrium of a threeforce body consider a rigid body subjected to forces acting at only 3 points. In contrast to the forces on a particle, the forces on a rigidbody are not usually concurrent and may cause rotation of the body due to the moments created by the forces. A problem that is statically indeterminate has more unknowns than equations of equilibrium. When performing static equilibrium calculations for objects, we always start by assuming the objects are rigid bodies. Select the extent of the free body and detach it from the ground and all other bodies. Forces on a particle for a rigid body to be in equilibrium, the net force as well as the net moment. If rotation is prevented, then the support exerts a couple moment on the body. A new aspect of mechanics to be considered here is that a rigid body under the action of. Equilibrium of rigid bodies request pdf researchgate. In order to create a rigid body in equilibrium, we begin with a force table, which is a horizontal disk supported by a tripod base. Therefore, the first step in the analysis of the equilibrium of rigid bodies must be to draw the free body diagram of the body. In each case the forces and torques exerted on the rigid body are determined from data collected. A correct freebody diagram and an appropriate application of the equilibrium conditions are the key to the solution of a coplanar or a spatial problem. This video screencast was created by dr terry brown with doceri on an ipad. In this tutorial, we will cover the previous exam questions related to equilibrium of rigid bodies 3d problems chapter 5. Introduce center of mass and the conditions for rotational equilibrium 3. Again, how can we make use of an idealized model and a free body diagram to answer this question. In rigid body dynamics we have two types of motion. Newtons first law given no net force, a body at rest will remain at restand a body moving at a constant velocity will continue to do so along a straight path. Indicate point of application, magnitude, and direction of external forces, including the. Solving rigid body equilibrium problems using the ramp system shown we may go through the following steps 1. Free body diagram first step in the static equilibrium analysis of a rigid body is identification of all forces acting on the body with a free body diagram. A pin joint prevents motion horizontally and vertically. General systems of forces, equilibrium of a rigid body. Expand the number of support conditions used in equilibrium problems. Determine the horizontal and vertical components of reaction at. When a rigid body with a fixed pivot point o, is acted upon by a force, there may be a rotational change in velocity of the rigid body. Imagine the pipe to be separated or detached from the system. To study the use of a balanced meter stick, the concept of torque and the conditions that must be met for a body to be in rotational equilibrium. Then we need to draw a free body diagram fbd showing all the external active and finally, we need to apply the equations of equilibrium to solve for any unknowns. It teaches you to think about how forces and bodies act and react to one another. Identify the force and moment reaction components on a 3d rigid free body, including but not limited to. When a rigid body with a fixed pivot point o, is acted upon by a force, there may be a. Equilibrium is the state when all the external forces acting on a rigid body form a system of forces equivalent to zero. Dan gable lesson 21 3d rigid body equilibrium monday, february 26, 2018 lesson objectives 1. In performing equilibrium analysis, only external forces that external and internal forces can act on a rigid body. The addition of moments as opposed to particles where we only looked at the forces adds another set of possible. Scalar equations of equilibrium six scalar equations are required to express the conditions for the equilibrium of a rigid body in the general threedimensional case. Many manmade structures are designed to achieve and sustain a state of equilibrium, and this, in turn, sets require. Mechanics map equilibrium analysis for a rigid body. To evaluate the unknown reactionsholding a rigid body in equilibrium by solving the equations of static equilibrium. The two conditions of equilibrium of a rigid body are as follows. We place a steel disk on the force table, thus forming a smooth flat surface. Assuming that their lines of action intersect, the moment of f 1 and f 2 about the point of intersection represented by d is zero. Then we need to draw a free body diagram showing all. Determine all the forces that are acting on the rigid body. If the bar is suspended from two spring scales, three forces are. Equipment and supplies meter stick standard masses with hook triplebeam balance fulcrum fulcrum holder discussion of equilibrium first condition of equilibrium. This also means that the object is considered unbreakable. Whenhisgrandmothertelephonedtoaskhowhewasanursesaid nochangeyet. Pdf chapter 5 equilibrium of a rigid body alvick lau. The process of solving rigid body equilibrium problems for analyzing an actual physical system, first we need to create an idealized model above right. To recognize situations of partial and improper constraint, as well as static indeterminacy, on the basis of the solvability of the equations of static equilibrium. This condition for equilibrium was extended to larger bodies under either of two possible conditions. Sketch a free body diagram showing all the forces acting on the meter stick. Lesson 4 rigid body statics when performing static equilibrium calculations for objects, we always start by assuming the objects are rigid bodies. Apply the equations of equilibrium to solve for the unknowns. A roller support prevents motion in one dimension only. Equilibrium of rigid bodies introduction most of the objects that one sees are in a state of equilibrium, that is, at rest or in a state of uniform motion. From newtons third law, support connection exerts equal and opposite reaction force on body. Top 15 items every engineering student should have. Es226 spring 2018 lesson 21 3d rigid body equilibrium i went undefeated for seven years, lost a match, and then i got good. Translational equilibrium rotational equilibrium lines of action two. The vector sum of the forces on the body must be zero. Static equilibrium of rigid bodies 2d learning objectives. Translational eguilibrium analyze translational equilibrium problems by identifying all forces, making a free body diagram, and applying the first condition of. How can you determine the force acting in each of the chains. A rigid body is in equilibrium when it is not undergoing a change in rotational or translational motion. Es226 spring 2018 lesson 21 3d rigid body equilibrium homework due wednesday. The addition of moments as opposed to particles where we only looked at the forces adds another set of possible equilibrium equations, allowing us to solve for more. In cases i and ii of the experiment a rigid body a meter stick is subjected to various combinations of forces in such a way that the body remains in equilibrium. Are statically indeterminate st ructures used in practice. The equations of equilibrium are the most useful equations in the area of statics, and they are also important in the dynamics and mechanics of materials. You may replace one or both of the force equations with moment equations. Rigidbody dynamics studies the movement of systems of interconnected bodies under the action of external forces. When a body has more supports than necessary to hold it in equilibrium, it becomes statically indeterminate. Expand the number of support conditions used in equilibrium. This equilibrium requires that two conditions must be met. F 0 no translation forces on a rigid body and m o 0 no rotation in contrast to the forces on a particle, the forces on a rigid body are not usually concurrent and may cause rotation of. To recognize situations of partialand improper constraint, as well as static indeterminacy, on the basis of the solvability of the equations of static equilibrium. Equilibrium of a rigid body balancing the forces question 3. When examining the equilibrium of rigid bodies, it is extremely important to consider all the forces acting on the body and keep out all the other forces that are not directly exerted on the body. Therefore, in order for the rigid body to be in equilibrium, both the resultant force and the resultant couple must be zero. Total virtual work done on the entire rigid body is zero since virtual work done on each particle of the body in equilibrium is zero. To evaluate the unknown reactions holding a rigid body in equilibrium by solving the. Elementary statics static equilibrium of a rigid body. Determine the horizontal and vertical components of reaction at the pin a. Since internal forces occur in pairs that are equal in magnitude opposite in direction, they are not considered in the equilibrium of rigid bodies. A net torque will cause a body, initially at rest, to undergo rotation. It is the study of objects that are either at rest, or moving with a constant velocity. The first conditionis related to the translational motion. Jul 07, 2017 no equilibrium problem should be solved without first drawing the free body diagram if a support prevents translation of a body in a particular direction, then the support exerts a force on the body in that direction. Rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces. Equations of equilibrium, free body diagram, reaction, static indeterminacy and partial constraints, two and three force systems. The car below has a mass of 1500 lbs with the center of mass 4 ft behind the front wheels of the caru what are the normal forces on the front and. This chapter shows us how to include rotation into the dynamics. Engineering statics online engineering courses online. Statics is important in the development of problem solving skills. Typical supports o supports are used to keep a body in static equilibrium, and to do so, they can apply forces andor couples to the body. If the forces acting on the body were concurrent, that is, if they.381 1171 351 1247 560 787 43 548 363 1122 391 809 1351 297 465 686 63 281 813 1136 65 422

Following is a paper copyrighted by William Verhart of Madrid, Spain, received by AIP for review. We are publishing the piece here for the review of others. The sole copyright for the piece is with Mr. Verhart. The paper claims that the ancient Egyptians knew of the meter, a measure we thought was devised by Napoleon’s Enlightened savants based on their indirect calculation of the 1/10,000,000 part of a meridian through Paris. Apparently, as has happened so often, the deceit of modernity is trumped by the genius of the supposed ignorant races from the past. Mr. Verhart’s brilliant piece begins below. Almost all countries in Europe use the unit of length the meter. In the United States, Panama and the United Kingdom, the unit of length is the yard. We know the value of the yard in meters and vice versa. But is it possible to reflect these two units of length in a geometric figure or in a mathematical formula, so that people who do not know either the meter or the yard can know which units of length have been used? Well, it’s possible. We can construct a rectangular prism and through its measurements we can know which unit of length has been used and what value it has in meters. This prism corresponds to a specific statement that fixes its dimensions, both internal and external. It only works with units of length whose value in meters we know. So, in summary, we can say: Let’s build a rectangular prism whose measurements reveal the unit of length used and its value in meters without having to know the meter or the value of the unit of length used. The statement would be the following: 1. The outer length would be twice the outer width 2. The inner length corresponds to twice the unit of length used 3. The inner width is the inverse of the outer width 4. The outer height is also the inverse of the outer width 5. The inner volume is half the outer volume 6. The relationship between the outer height and the inner height is equal to the numerical value in meters of the outer length (Since the inside width is the inverse of the outside width, the numerical value of the outside width will be equal to or greater than 1 (one) but smaller or equal to the cube root of 2 (two), namely 1.25992104989 .., otherwise the value of the high interior would be greater than the outer height. This does not mean that the prism can be applied only to a small number of measures of length, since you only have to divide the measure used by a certain number to obtain a value that is located between 1 and 1.2599210 …) Each unit of length will thus have its own prism. You can build a rectangular prism with the first five points of the statement. This prism, by not including point 6, would not reflect the value in meters of the unit of length used. This prism would have this form: Table 1 — THE THEORETICAL PRISM |Measures||Value (of measure b)| |Outer length||2 ab| |Inner height||0,5 a2b| Where a is the numerical value of the chosen unit of length (if, for example, the outer width corresponded to 1.4 Cubits, then its numeric value would be 1.4) and b the unit of length chosen. If we include point 6, then the value in meters of the chosen unit of length would always be reflected in the relations of the prism. The formulas for this rectangular metric prism are obtained through the following equation: Outer length = outer height / inner height = = 2ab = (b/a) / 0.5 a2b therefore, 2a4b2 = 2b and therefore b = a4b2 from where it is deduced b = 1 / (root4)a We can obtain now the formulas that determine the six sides of the prism using any unit of length, whose value in meters we know: |Dimensions||Units of length b| |Outer length||2 / (root4)a| |Outer width||1 / (root4)a| |Inner height||1 / (2√ a)| where “a” is the value in meters of the unit of length chosen and “b” is the unit of length chosen. If we want, as an example, to combine the meter and the yard in this prism, the steps to follow would be the following: The English yard equals 0.9144 meters; therefore the numerical value of the outer width would be, in yards: 1 /(root4)a = 1 /(root4) 0,9144 = 1.022623916809237 The values of the prism would be, following the statement, then (in yards): Table 3 – Mathematical prism using the “yard” In this case, the outer capacity is equal to: 2.04524783361848 x 1.02262391680924 x 0.97787660112641 = 2.04524783361848 And the inner volume is equal to: 2 x 0.97787660112641 x 0.52287983761514 = 1.02262391680924 Exactly half of the outer volume. The product between the value of the outer width and the inner width is equal to 1.0000000000. In meters the values of the prism would be the following ones (we multiply the values of the previous table by the value in meters of the yard): But thanks to this mathematical form the relation between the outer height and the inner height is equal to the numerical value of the outer length in meters: 0.89417036406999 / 0.47812132351527 = 1.87017461906074 0.97787660112641 / 0.52287983761514 = 1.87017461906074 And the result of the division between the inner length with the outer length, to the fourth (exponent 4), gives as a result the value of the English yard. ————————— = 0.91440000000 meters (2 / 2.04524783361848)4 = 0.91440000000 meters With these characteristics, the unit of length used and its value in meters will always be present in this rectangular prism. Any unit of length can be recognized in a prism of these characteristics, as long as we know its value in meters. The statement is fulfilled only with the measures expressed in the unit of length chosen, since converted to the metric system, not all elements of the statement are met (3 and 4), as the value of the inner width is the inverse of the outer width. Such a prism could be very useful for future generations to know the units of measurement of our current civilization. And the ancient civilizations would leave a great legacy demonstrating to future civilizations their units of measurement in use. Did any civilization before ours know the mathematical properties of a prism like that? During the fourth dynasty of the Ancient Egyptian Empire, Hordjedef, son of Khufu and half-brother of Khafre,, had a sarcophagus built, with the shape of a rectangular prism, with the same characteristics as our rectangular prism before mentioned. The measures of the sarcophagus, which is currently in the Egyptian Museum in Cairo (catalogued with the number 54,938-6193), are the following (data of D. Manuel J. Delgado): Table 5 – Sarcophagus of Hordjedef (in meters) To know if this sarcophagus includes the statement shown, we will study these measures. The first observations are the following: - The outer length is almost twice the outer width (2.45/1.23 = 1.99186 …) – the difference of one centimetre is probably due to the possible wear of the stone after 4,500 years, to blows or a bad measurement - The outer volume is twice the inner volume: 2.45 x 1.23 / 2.09 x 0.72 = 2.0025. - The inner length corresponds to 4 Royal Cubits of 0.524 meters each - The relationship between the outer height and the inner height is equal to half the numerical value in meters of the outer length These 4 observations are almost conclusive to ensure that this sarcophagus is based on the same statement as the mathematical prism. To know the value in meters of the unit of length used (a), we will have to use the formula on page 4, taking into account that the inner length corresponds to 4 units of the Royal Cubit, instead of 2 Royal Cubits: 1 (root4)/ a = 1.23 / 2 a whose value, 2a, would have to be half the inner length 4a. The value of “a” would then be: 2a = 1.23 (root4) a = 1.045996… which would correspond almost exactly to two units of the unit of length used. The unit of length used is, as we have said, the Royal Cubit of 0.524 meters. Consequently, the builders of the sarcophagus supposedly used the unit of measurement “one Royal Cubit”, but they gave the definitive form of the sarcophagus multiplying all its measures by two, to have a prism in the form of a coffin where a Pharaoh could fit. Therefore, the measurements of the sarcophagus, for the Egyptians, would be the following: Table 6 – Mathematical prism using the unit “two Royal Cubits” |Dimensions||Unit “2 Royal Cubits”| These data confirm (despite the two decimals) that the prism meets the special characteristics for the statement to be fulfilled: 1) The outer length is twice the outer width and the outer volume is twice the inner volume. 2) The inner width is the inverse of the outer width (always in the original unit of measurement). 3) The inner length corresponds to 2 times the unit of measurement “Two Royal Cubits”. 4) The relationship between the outer height and the inner height is equal to the numerical value of the outer width in meters (being the unit of length “two” Royal Cubits, instead of “one” Royal Cubit, the ratio between the outer height and the inner height corresponds to the outer width in stead of the outer length). If the statement is fulfilled, then the builders of the Hordjedef sarcophagus knew the meter. Indeed, since the outer width gives the value in meters of the unit of measurement used, namely the Royal Cubit: (1 / 1.17) 4 = 0.524 meters And it is also verified when applying the relationship between the outer height and the inner height: 0.85 / 0.69 = 1.23 meters which gives the value in meters of the outer width. It is plausible that the Egyptians could have built a sarcophagus with the exposed statement, but it seems impossible that by “chance” they chose a value that would indicate exactly the value in meters of the Royal Cubit. The definition of the meter, introduced by the French in the 18th century, was a completely mathematical expression, and therefore, it is not surprising that the Egyptians themselves did the same in their time. Today we know that ancient civilizations had knowledge much higher than normally accepted. To reconstruct the sarcophagus of Hordjedef as accurate as possible, we are going to fix, for the moment, the value of the Royal Cubit in 0.524 meters, a value given by most scholars of the subject, including the Egyptologists. The values of the sarcophagus should then have been the following: Table 7 – Values of the sarcophagus of Hordjedef in meters |Outer length||2.4635 meter| |Inner length||2.0960 meter| |Outer width||1.2317 meter| |Inner width||0,8916 meter| |Outer height||0.8916 meter| |Inner height||0.7239 meter| But, the value in meters of the outer width of 1.2317 meters is very close to the value of 4 Egyptian or Geographic Feet of 0.3079 meters, each one, corresponding to the 1/100 part of the value of one second of arc in the latitude of the Great Pyramid (29.979 … degrees North). An Egyptian or Geographical Foot corresponds to 2/3 of a Geographical Cubit of 0.4618 meters. The exact value for a Geographical Foot at the latitude of the Great Pyramid is 0.3079235 .. meters. These data indicate that the Egyptians certainly knew the mathematical form of the Earth and its exact dimension (which would confirm their knowledge about the meter). The verification of the existence of an Egyptian Foot is very important. All monuments in Egypt, which are part of the Old Kingdom, with more than 4,200 years old, are not in their original state because of many factors. Many monuments have even disappeared over the course of time. This necessarily means that talking about measures from the old empire is a very sensitive issue. Even so, monuments, large and small, are preserved, which provide us with enough information to be able to ensure that the aforementioned measures of the Hordjedef sarcophagus are not invented but real measures. Different measurements of the sarcophagus of Hordjedef, are also found in the other three sarcophagi from the Giza complex (all from the Fourth Dynasty). We can observe the following coincidences (the value of the outer height of Menkaure is equal to the value of the outer height of Hordjedef, namely 0.891 meters, its value corresponds to two Small Cubits): Table 8 – Measures of the Sarcophagi of Khufu, Khafre and Menkaure* |Outer length||2.293 m||2.633 m||2.463 m| |Inner length||1.985 m||2.150 m||1.847 m| |Outer width||0.983 m||1.065 m||0.935 m| |Inner width||0.678 m||0.676 m||0.600 m| |Outer height||1.048 m||0.965 m||0.891 m| |Inner height||0.839 m||0.750 m||0.616 m| * Data obtained from the books of Flinders Petrie and André Pochan. 1) The outer length of Menkaure is equal to the outer length of Hordjedef (2.46 meters) 2) The outer height of Menkaure corresponds to the outer height of Hordjedef (0.89 meters = 2 Small Cubits) 3) The sum of the outer lengths of Khufu and Khafre is equal to the sum of the outer lengths of Hordjedef and Menkaure (2.293 + 2.633 = 2.463 + 2.463) 4) The inner height of Menkaure corresponds to two Geographical Feet (0.616 meters) 5) The value of the outer length of Khafre corresponds to 10 Geographical Feet minus 1 Small Cubit (3.079 – 0.4457 = 2.633 meters) 6) The outer width of Khafre corresponds to two Geographical Feet plus 1 Small Cubit (0.6158 + 0.4457 = 1.0615 meters) 7) The inner length of Menkaure corresponds to 6 Geographical Feet (1.847 meters) 8) The inner length of Khufu is one Geographical Foot inferior to its outer length (2.293 – 0.308 = 1.985) 9) The outer volume of Khafre is equal to the outer volume of Hordjedef (2.463 x 1.2316 x 0.8914 = 2.633 x 1.065 x 0.965 10. The measurement of the length of the Khufu sarcophagus (2.293 m.) is half the sum of the measurements of the Hordjedef sarcophagus for the calculation of its outer volume (outer length, outer width and outer height). With all these data, we can say that the four sarcophagi unequivocally enclose the Real Cubit and the Geographical Foot on their walls. Of the four sarcophagi mentioned, only in the coffin of Hordjedef we can recognize the unit the “meter”. The imaginary line parallel to the diagonal of the side face of the prism, from the corner of the projected inner bottom, intersects the upper edge of this side face at a point whose distance to the furthest corner is exactly one meter. (1 Royal Cubit = 0.5239 meter) The sarcophagus of Hordjedef, the Geographical Pie and the Royal Cubit constitute the proof that the Egyptians knew the meter. Did the Egyptians obtain this knowledge by their own means or did they have the help of beings from other worlds? Time will tell us. ================================= William Verhart, Madrid 04-01-2019 *Pochan A. – 1971 – Editions Robert Laffont S:A. – L’Énigme de la Grande Pyramide *Flinders Petrie W.M. – 1883/1990 – Histories & Mysteries of Man Ltd. – The Pyramids and Temples of Gizeh *Delgado M.J. – 1995 – “El Problema Matemático más Antiguo del Mundo”

Is the triangle law of vector addition?Asked by: Anabelle Simonis Score: 4.2/5 (37 votes) Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector. You can use this law in abuse as well as obtuse angles.View full answer Simply so, What are the laws of vector addition? Addition of vectors satisfies two important properties. 1. The Commutative law states that the order of addition doesn't matter, that is : A+B is equal to B+A. 2 The Associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first, that is: (A+B)+C=A+(B+C). In this manner, How do you prove the triangle law of vector addition?. Triangle Law of Vector Addition Derivation Consider two vectors →P and →Q that are represented in the order of magnitude and direction by the sides OA and AB, respectively of the triangle OAB. Let →R be the resultant of vectors →P and →Q . Above equation is the magnitude of the resultant vector. Secondly, What is the triangular law of vectors? A law which states that if a body is acted upon by two vectors represented by two sides of a triangle taken in order, the resultant vector is represented by the third side of the triangle. What is the triangle rule? The sides of a triangle rule asserts that the sum of the lengths of any two sides of a triangle has to be greater than the length of the third side. ... The sum of the lengths of the two shortest sides, 6 and 7, is 13. Is it possible to add two velocities using the triangle law? Yes. Your Velocities must be in vector form or each have a MAGNITUDE and DIRECTION. ... If the velocities did not have the same direction, the two velocities and the sum vector will form a triangle. The area A of a triangle is given by the formula A=12bh where b is the base and h is the height of the triangle. in triangle law of vector addition the third side of the triangle is the resultant but in parallelogram law of vector addition the diagonal is the resultant. According to the parallelogram law of vector addition if two vectors act along two adjacent sides of a parallelogram(having magnitude equal to the length of the sides) both pointing away from the common vertex, then the resultant is represented by the diagonal of the parallelogram passing through the same common vertex ... Vector algebra in geometric form Two vectors u and v are equal if they have the same magnitude (length) and direction. for all vectors v and u. for all vectors u, v and w and for all scalars s and t. We may then simply write u + v + w, without using brackets. A parallelogram is a four-sided, two-dimensional shape in which opposite sides are parallel and have equal length. ... A triangle is a two-dimensional shape with three sides and three angles. To find the area of a triangle, we take one half of its base multiplied by its height. Vectors having the same length as a particular vector but in the opposite direction are called negative vectors. ... For example, if a vector PQ points from left to right, then the vector QP will point from right to left. Since these directions are opposite, we say that PQ = –QP. Triangles can be used to make trusses. Trusses are used in many structures, such as roofs, bridges, and buildings. Trusses combine horizontal beams and diagonal beams to form triangles. Bridges that use trusses are called truss bridges. - The Pythagorean Theorem, a2+b2=c2, a 2 + b 2 = c 2 , can be used to find the length of any side of a right triangle. - The side opposite the right angle is called the hypotenuse (side c in the figure). The formula for the volume of a cylinder is V=Bh or V=πr2h . The radius of the cylinder is 8 cm and the height is 15 cm. ... Therefore, the volume of the cylinder is about 3016 cubic centimeters. As per the sin theta formula, sin of an angle θ, in a right-angled triangle is equal to the ratio of opposite side and hypotenuse. The sine function is one of the important trigonometric functions apart from cos and tan. Addition of two vectors can only be zero when there directions are opposite and their magnitude is additive inverse of each other. Adding velocities. Consider two objects. The first object moves with velocity v relative to the second object, while the second object moves with velocity u with respect to an observer. In Newtonian physics the observer would say that the velocity of the first object is the sum of the two velocities. Vector addition is the operation of adding two or more vectors together into a vector sum. The so-called parallelogram law gives the rule for vector addition of two or more vectors. Unlike the dot product, the vector product is a vector. The direction of the vector (A crossB) is defined by the so-called right-hand rule. ... The vector (A cross B) is then perpendicular to both A and B and points in the direction of one's thumb. A vector dotted into itself gives the square of the length of the vector. To add vectors, lay the first one on a set of axes with its tail at the origin. Place the next vector with its tail at the previous vector's head. When there are no more vectors, draw a straight line from the origin to the head of the last vector. This line is the sum of the vectors. Two vectors can be added together to determine the result (or resultant). This process of adding two or more vectors has already been discussed in an earlier unit. To add or subtract two vectors, add or subtract the corresponding components. Let →u=⟨u1,u2⟩ and →v=⟨v1,v2⟩ be two vectors. The sum of two or more vectors is called the resultant. Answer. Two vectors are equal if they have the same magnitude and the same direction. Just like scalars which can have positive or negative values, vectors can also be positive or negative.

2 edition of Quaternions and Projective Geometry found in the catalog. Quaternions and Projective Geometry Charles Jasper Joly by Published for the Royal Society of London by Dulau and Co. Written in English Quaternionic projective space of dimension n is usually denoted by and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line is homeomorphic to the 4-sphere. "Application of Quaternions to Projective Geometry" is an article from American Journal of Mathematics, Volume View more articles from American Journal of Mathematics. View this article on JSTOR. View this article's JSTOR metadata. This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of /5. of projective differential geometry, c-projective geometry, and almost quaternionic ge-ometry. Such geometries, which we call projective parabolic geometries, are abelian parabolic geometries whose flat model is an R-space G pin the infinitesimal isotropy representation Wof a larger self-dual symmetric R-space H q. We also give a classi-File Size: 1MB. Quaternions ∗ (Com S / Notes) ters 3–6 of the book by J. B. Kuipers, Sections 1 and 6 are partially based on the essay by S. Oldenburger who took the course, and Section 5 is based on . 1For the purpose of this course, you don’t really need to know what a . Octonions online links to other websites containing material about the octonions. Brougham bridge pictures of the bridge where Hamilton carved his definition of the quaternions. Integral octonions Integral octonions and their connections to geometry and physics. On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry my review of John Conway and Derek Smith's book. drink problem in Russia Borders for calligraphy New York law of accidents. Life and times of Sir Alexander Tilloch Galt. Trees on the farm Russian housing in the modern age Self-abandonment to divine providence Teachers manual for probability and statistics for business decisions. Quaternions and Projective Geometry Paperback – January 1, by Charles Jasper Joly (Author) See all 15 formats and editions Hide other formats and editions. Price New from Used from Hardcover "Please retry" $ $ Author: Charles Jasper Joly. This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. Excerpt: of xf + yf2, and as x and y vary, the locus of the transformed point is a Author: Charles Jasper Joly. Buy Quaternions and Projective Geometry on FREE SHIPPING on qualified orders Quaternions and Projective Geometry: Charles Jasper Joly: : Books Skip to main content. Quaternions and Projective Geometry Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb. Show more/5(2). Quaternions and Projective Geometry by Charles Jasper Joly. Publication date Publisher Published for the RoyalSociety of London byDulau and Co. Collection americana Digitizing sponsor Google Book from the collections of Harvard University Language English. Book digitized by Google from the library of Harvard University and uploaded to the. Additional Physical Format: Online version: Joly, Charles Jasper, Quaternions and projective geometry. London. Published for the Royal Society of London by Dulau and Co., Free 2-day shipping. Buy Quaternions and Projective Geometry at This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of by: Audio Books & Poetry Community Audio Computers, Technology and Science Music, Arts & Culture News & Public Affairs Non-English Audio Spirituality & Religion. Librivox Free Audiobook. Podcasts. Featured software All software latest This Just In Old School Emulation MS-DOS Games Historical Software Classic PC Games Software Library. The object of this paper is to include projective geometry within the scope of quaternions. The calculus, as established by Hamilton, was solely adapted to the treatment of metrical relations, but when we regard a quaternion as representing a weighted point, projective properties can be investigated with great facility. Tell readers what you thought by rating and reviewing this book. Rate it * You Rated it * 0. 1 Star Quaternions and Projective Geometry. by Charles Jasper Joly. Thanks for Sharing. You submitted the following rating and review. We'll publish them on our site once we've reviewed them. : Charles Jasper Joly. This work is as fluid, clear and sharp as can be on this general subject area -- rotations, quaternions and double groups -- and the related Clifford algebra, linear algebra, linear transformations, bilinear transformations, tensors, spinors, matrices, vectors and complex numbers -- and in relation to quantum physics and its by: The representation of physical motions by various types of quaternions posthumously-published book of Sir William Rowan Hamilton that he called Elements of Quaternions [ 3] that came almost a century after Euler inalthough the work was considering was actually projective geometry, not affine Euclidian geometry. ChaslesCited by: 3. Textbook for undergraduate course in geometry. Ask Question Asked 7 years, Michèle Audin wrote a very good book about affine, projective, curves and surfaces. It is aimed to future (French) high school teachers. SOC),and RP3 The Algebra M of Quaternions Quaternions and Rotations in SOC) Quaternions and Rotations in SOD) Quaternions and Projective Geometry. Joly, C Proceedings of the Royal Society of London (). – This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating 5/5(1). v modularforms: Algorithmsandarithmetic(DMS,July–August) andCAREER:Explicitmethodsinarithmeticgeometry(DMS,July– July ), and. Introducing The Quaternions The Complex Numbers I The complex numbers C form a plane. I Their operations are very related to two-dimensional geometry. I In particular, multiplication by a unit complex number: jzj2 = 1 which can all be written: z = ei gives a rotation: Rz(w) = zw by Size: KB. I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on projective geometry that starts out with a list of axioms, as in high school geometry book. I'm looking for something in more Needham-like style. Try the new Google Books. Check out the new look and enjoy easier access to your favorite features. Try it now. eBook - FREE. Quaternions and the onedimensional projective group. Projective Geometry, Volume 2. One of the most appealing aspects of this book is the author's way of introducing projective geometry to pique the reader's interest in modern algebra. Summing Up: Recommended. Lower-division undergraduates and above.” (J. A. Bakal, Choice, Vol. 56 (03), November, ).Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century and proof methods used by mathematicians from those.

A new walks-into-a-bar joke begins (thanks to reader Kip Hansen for the tip): A mathematician, a philosopher and a gambler walk into a bar. As the barman pulls each of them a beer, he decides to stir up a bit of trouble. He pulls a die from his pocket and rolls it ostentatiously on the bar counter: it comes up with a 1. The mathematician says: ‘The probability that 1 would come up is 1/6, and at the next throw it will be the same. If we roll the die infinitely many times, the relative frequency of the number 1 will converge to 1/6, that is, to one occurrence every six throws.’ The philosopher strokes her chin, and remarks: ‘Well, this doesn’t mean we won’t get the number at the next throw. Actually, it’s physically possible to have the same number on the next 1,000 throws, although that’s highly improbable.’ The gambler says: ‘I know you’re both right, but I wouldn’t bet on that number for the next throw.’ ‘Why not?’ asks the mathematician. ‘Because I trust mathematics, and so I expect that number to come up about once every six throws,’ the gambler answers. ‘Having the same number twice in a row is a rare event. Why would that happen right now?’ The joke, the article goes on to say, is on the gambler whose “argument is a mix of conceptual inadequacy, misinterpretation, irrelevant application of mathematics, and misleading use of language.” And in the spirit of gender parity, the gambler is a she. So is the authoress. The article continues with other sins of gamblers, such as the eponymous gambler’s fallacy, “where someone believes that a series of bad plays will be followed by a winning outcome, in order for the randomness to be ‘restored'”, plus there are cautions about serotonin and addiction. The authoress also wonders if exposing gamblers to naked mathematics will cure them of their bad habits and thinking. It hasn’t worked for the philosopher or mathematician, as we’ll see, so that’s unlikely. Now the authoress acknowledges “statistical models are grounded in probability theory, one of the fields in mathematics most open to philosophical debate”, which is true. But that’s because everybody hasn’t yet read—and assimilated—Uncertainty, where the true understanding of probability is given (stating it this way riles people). I don’t agree with anybody, really, but my sympathies are closest to the gambler’s. The mathematician and the philosopher have committed the Deadly Sin of Reification. The gambler alone sought to understand the cause of the roll, in a vague way, with his idea of a restorative force, a cause. The gambler was the only scientist among the three (where I use that word in its old-fashioned sense). The die had no probability whatsoever of coming up anything. The die was caused to come up 1. To say it has a probability is to reify a model of the die and say the model is reality itself. This is, as said, a deadly sin. Here is one possible model of a die, out of (as far as I know) an infinite number of them: “An object has six different sides, labeled 1-6, which when tossed has one side come up.” Given that model, the probability of a 1 is, as both the mathematician and the philosopher say, 1/6. Does that model apply, in real life, to real throws or real dice by bartenders on wine-soaked bars? Who knows? Nobody, that’s who. The only guide is to try it and see. The model has loose similarities to real dice, but real dice are rough and real; the model is infinitely smoother. Real dice are thrown on strange surfaces with varying amounts of force and spin. Real dice are never symmetric, except grossly. They wear through use. Throwing conditions are non-uniform. People know how to manipulate throws. On and on. Dice exist. Throws exist. The model does not. Are there other models that are better than our simple one, as the gambler thought? Why, yes. Yes, there are. The best model is the one which delineates all the causes of each particular throw, a model which gives “extreme probability”, i.e. 0 or 1, to each outcome. Since the causes depend on the milieu, which is ever-changing, this Reality model must change with every throw, too. It can be done. It’s just that real dice are sensitive to initial conditions, which makes measuring all the causes difficult. That’s why real dice are useful in gambling. Not knowing causes makes throws unpredictable to some degree. Casinos try to force both unpredictability of cause and symmetry of forces operating on dice in ways we all know. That enforcement brings the simple model above closer to Reality in some aspects, while never matching it. Experience with actual throws is what gives us a notion the simple model does an adequate job abstracting Reality in controlled conditions. The mathematician has the bartender throwing the die an infinite number of times, which is an impossibility. Not a light one, either since any finite number of throws is infinitely far from infinity. We should have been able to deduce from talking of infinite anything we’re dealing with a model and not Reality. No number of finite throws will match the model except by coincidence, and unless the real number of throws is divisible by 6, matching is impossible. The philosopher mixes up Reality of the “physically possible” with the simple model’s probabilities. Now you will hear some say “the dice have no memory” when discussing the so-called gambler’s fallacy. The gambler seems to think they do; or, if not the dice, than whatever causes are operating on the dice, material or spiritual, hence his idea of a restorative force. We can’t prove to him he’s wrong. Especially when the finite groupings of tosses he witnesses provide confirmatory evidence he’s right. These groupings will have distributions with wide departures from the model’s theoretical limit. The philosopher and mathematician also believe certain spiritual forces operate on the dice, which they call randomness. That force imbues the dice with a different kind of directing force, which ensures the relative frequency of actual tosses confirms to the model, which you recall they think is real. The randomness force is real to them, which is why they speak of tossing “fair” dice. What in the world could that be, except a die that matches the imagined simple model exactly, an impossibility in Reality. Yet they say that fairness is (or can be) a property of the dice, like its weight or ink spot color. Fairness is real but, strangely, cannot be measured. It’s in there somewhere, no one knows where. Or how. Or maybe it’s in the dice-throwing milieu somewhere. Again, no one knows where. Or how. If this doesn’t convince you everybody has a problem with reification, answer this question, “An unfair die is tossed. What is the probability it comes up 1?” I leave the answer to homework. Subscribe or donate to support this site and its wholly independent host using credit card or PayPal click here

Method of matched asymptotic expansions In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, each of which is valid (i.e. accurate) for part of the range of the independent variable, and then combining these different solutions together to give a single approximate solution that is valid for the whole range of values of the independent variable. In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains. In one of these, often the largest, the solution is accurately approximated by an asymptotic series found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero). The other subdomains consist of one or more small areas in which that approximation is inaccurate, generally because the perturbation terms in the problem are not negligible there. These areas are referred to as transition layers, and as boundary or interior layers depending on whether they occur at the domain boundary (as is the usual case in applications) or inside the domain. An approximation in the form of an asymptotic series is obtained in the transition layer(s) by treating that part of the domain as a separate perturbation problem. This approximation is called the "inner solution," and the other is the "outer solution," named for their relationship to the transition layer(s). The outer and inner solutions are then combined through a process called "matching" in such a way that an approximate solution for the whole domain is obtained. A simple exampleEdit Consider the boundary value problem where is a function of independent time variable , which ranges from 0 to 1, the boundary conditions are and , and is a small parameter, such that . Outer solution, valid for t = O(1)Edit Since is very small, our first approach is to treat the equation as a regular perturbation problem, i.e. make the approximation , and hence find the solution to the problem Alternatively, consider that when and are both of size O(1), the four terms on the left hand side of the original equation are respectively of sizes O( ), O(1), O( ) and O(1). The leading-order balance on this timescale, valid in the distinguished limit , is therefore given by the second and fourth terms, i.e. This has solution for some constant . Applying the boundary condition , we would have ; applying the boundary condition , we would have . It is therefore impossible to satisfy both boundary conditions, so is not a valid approximation to make across the whole of the domain (i.e. this is a singular perturbation problem). From this we infer that there must be a boundary layer at one of the endpoints of the domain where needs to be included. This region will be where is no longer negligible compared to the independent variable , i.e. and are of comparable size, i.e. the boundary layer is adjacent to . Therefore, the other boundary condition applies in this outer region, so , i.e. is an accurate approximate solution to the original boundary value problem in this outer region. It is the leading-order solution. Inner solution, valid for t = O(ε)Edit In the inner region, and are both tiny, but of comparable size, so define the new O(1) time variable . Rescale the original boundary value problem by replacing with , and the problem becomes which, after multiplying by and taking , is Alternatively, consider that when has reduced to size O( ), then is still of size O(1) (using the expression for ), and so the four terms on the left hand side of the original equation are respectively of sizes O( −1), O( −1), O(1) and O(1). The leading-order balance on this timescale, valid in the distinguished limit , is therefore given by the first and second terms, i.e. This has solution for some constants and . Since applies in this inner region, this gives , so an accurate approximate solution to the original boundary value problem in this inner region (it is the leading-order solution) is We use matching to find the value of the constant . The idea of matching is that the inner and outer solutions should agree for values of in an intermediate (or overlap) region, i.e. where . We need the outer limit of the inner solution to match the inner limit of the outer solution, i.e. which gives . To obtain our final, matched, composite solution, valid on the whole domain, one popular method is the uniform method. In this method, we add the inner and outer approximations and subtract their overlapping value, , which would otherwise be counted twice. The overlapping value is the outer limit of the inner boundary layer solution, and the inner limit of the outer solution; these limits were above found to equal . Therefore, the final approximate solution to this boundary value problem is, Note that this expression correctly reduces to the expressions for and when is O( ) and O(1), respectively. This final solution satisfies the problem's original differential equation (shown by substituting it and its derivatives into the original equation). Also, the boundary conditions produced by this final solution match the values given in the problem, up to a constant multiple. This implies, due to the uniqueness of the solution, that the matched asymptotic solution is identical to the exact solution up to a constant multiple. This is not necessarily always the case, any remaining terms should go to zero uniformly as . Not only does our solution successfully approximately solve the problem at hand, it closely approximates the problem's exact solution. It happens that this particular problem is easily found to have exact solution which has the same form as the approximate solution, by the multiplying constant. Note also that the approximate solution is the first term in a binomial expansion of the exact solution in powers of . Location of boundary layerEdit Conveniently, we can see that the boundary layer, where and are large, is near , as we supposed earlier. If we had supposed it to be at the other endpoint and proceeded by making the rescaling , we would have found it impossible to satisfy the resulting matching condition. For many problems, this kind of trial and error is the only way to determine the true location of the boundary layer. The problem above is a simple example because it is a single equation with only one dependent variable, and there is one boundary layer in the solution. Harder problems may contain several co-dependent variables in a system of several equations, and/or with several boundary and/or interior layers in the solution. It is often desirable to find more terms in the asymptotic expansions of both the outer and the inner solutions. The appropriate form of these expansions is not always clear: while a power-series expansion in may work, sometimes the appropriate form involves fractional powers of , functions such as , et cetera. As in the above example, we will obtain outer and inner expansions with some coefficients which must be determined by matching. Second-order differential equationsEdit A method of matched asymptotic expansions - with matching of solutions in the common domain of validity - has been developed and used extensively by Dingle and Müller-Kirsten for the derivation of asymptotic expansions of the solutions and characteristic numbers (band boundaries) of Schrödinger-like second-order differential equations with periodic potentials - in particular for the Mathieu equation (best example), Lamé and ellipsoidal wave equations, oblate and prolate spheroidal wave equations, and equations with anharmonic potentials. - R.B. Dingle (1973), Asymptotic Expansions: Their Derivation and Interpretation, Academic Press. - Verhulst, F. (2005). Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Springer. ISBN 0-387-22966-3. - Nayfeh, A. H. (2000). Perturbation Methods. Wiley Classics Library. Wiley-Interscience. ISBN 978-0-471-39917-9. - Kevorkian, J.; Cole, J. D. (1996). Multiple Scale and Singular Perturbation Methods. Springer. ISBN 0-387-94202-5. - Hinch, John (1991). Perturbation Methods. Cambridge University Press. - R.B. Dingle and H.J.W. Müller, J. reine angew. Math. 211 (1962) 11-32 and 216 (1964) 123-133; H.J.W. Müller, J. reine angew. Math. 211 (1962) 179-190. - H.J.W. Müller, Mathematische Nachrichten 31 (1966) 89-101, 32 (1966) 49-62, 32 (1966) 157-172. - H.J.W. Müller, J. reine angew. Math. 211 (1962) 33-47. - H.J.W. Müller, J. reine angew. Math. 212 (1963) 26-48. - H.J.W. Müller-Kirsten (2012), Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific, ISBN 978-9814397742. Chapter 18 on Anharmonic potentials.

Black hole entropy from the -invariant formulation of Type I isolated horizons A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly invariant formulation where the (effective) horizon degrees of freedom are described by an Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy. pacs:04.60.-m, 04.60.Pp, 04.20.Fy, 11.15.Yc Black holes are intriguing solutions of classical general relativity describing important aspects of the physics of gravitational collapse. Their existence in our nearby universe is by now supported by a great amount of observational evidence observ . When isolated, these systems are remarkably simple for late and distant observers: once the initial very dynamical phase of collapse is passed the system is expected to settle down to a stationary situation completely described (as implied by the famous results by Carter, Israel, and Hawking wald ) by the three extensive parameters (mass , angular momentum , electric charge ) of the Kerr-Newman family kerrnew . However, the great simplicity of the final stage of an isolated gravitational collapse for late and distant observers is in sharp contrast with the very dynamical nature of the physics seen by in-falling observers which depends on all the details of the collapsing matter. Moreover, this dynamics cannot be consistently described for late times (as measured by the infalling observers) using general relativity due to the unavoidable development, within the classical framework, of unphysical pathologies of the gravitational field. Concretely, the celebrated singularity theorems of Hawking and Penrose hawking imply the breakdown of predictability of general relativity in the black hole interior. Dimensional arguments imply that quantum effects cannot be neglected near the classical singularities. Understanding of physics in this extreme regime requires a quantum theory of gravity. Black holes (BH) provide, in this precise sense, the most tantalizing theoretical evidence for the need of a more fundamental (quantum) description of the gravitational field. Extra motivation for the quantum description of gravitational collapse comes from the physics of black holes available to observers outside the horizon. As for the interior physics, the main piece of evidence comes from the classical theory itself which implies an (at first only) apparent relationship between the properties of idealized black hole systems and those of thermodynamical systems. On the one hand, black hole horizons satisfy the very general Hawking area theorem (the so-called second law) stating that the black hole horizon area can only increase, namely On the other hand, the uniqueness of the Kerr-Newman family, as the final (stationary) stage of the gravitational collapse of an isolated gravitational system, can be used to prove the first and zeroth laws: under external perturbation the initially stationary state of a black hole can change but the final stationary state will be described by another Kerr-Newman solution whose parameters readjust according to the first law where is the surface gravity, is the electrostatic potential at the horizon, and the angular velocity of the horizon. There is also the zeroth law stating the uniformity of the surface gravity on the event horizon of stationary black holes, and finally the third law precluding the possibility of reaching an extremal black hole (for which ) by means of any physical process333The third law can only be motivated by a series of examples. Extra motivations comes from the validity of the cosmic censorship conjecture.. The validity of these classical laws motivated Bekenstein to put forward the idea that black holes may behave as thermodynamical systems with an entropy and a temperature where is a dimensionless constant and the dimensionality of the quantities involved require the introduction of leading in turn to the appearance of the Planck length , even though in his first paper beke Bekenstein states “that one should not regard as the temperature of the black hole; such identification can lead to all sorts of paradoxes, and is thus not useful”. The key point is that the need of required by the dimensional analysis involved in the argument called for the investigation of black hole systems from a quantum perspective. In fact, not long after, the semiclassical calculations of Hawking Hawking:1974sw —that studied particle creation in a quantum test field (representing quantum matter and quantum gravitational perturbations) on the space-time background of the gravitational collapse of an isolated system described for late times by a stationary black hole—showed that once black holes have settled to their stationary (classically) final states, they continue to radiate as perfect black bodies at temperature . Thus, on the one hand, this confirmed that black holes are indeed thermal objects that radiate at a the given temperature and whose entropy is given by , while, on the other hand, this raised a wide range of new questions whose proper answer requires a quantum treatment of the gravitational degrees of freedom. Among the simplest questions is the issue of the statistical origin of black hole entropy. In other words, what is the nature of the the large amount of micro-states responsible for black hole entropy. This simple question cannot be addressed using semiclassical arguments of the kind leading to Hawking radiation and requires a more fundamental description. In this way, the computation of black hole entropy from basic principles became an important test for any candidate quantum theory of gravity. In string theory it has been computed using dualities and no-normalization theorems valid for extremal black holes string . There are also calculations based on the effective description of near horizon quantum degrees of freedom in terms of effective -dimensional conformal theories carlip . In loop quantum gravity the first computations (valid for physical black holes) were based on general considerations and the fact that the area spectrum in the theory is discrete bhe0 . The calculation was later refined by quantizing a sector of the phase space of general relativity containing a horizon in ‘equilibrium’ with the external matter and gravitational degrees of freedom bhe1 . In all cases agreement with the Bekenstein-Hawking formula is obtained with logarithmic corrections in . In this work we concentrate and further develop the theory of isolated horizons in the context of loop quantum gravity. Recently, we have proposed a new computation of BH entropy in loop quantum gravity (LQG) that avoids the internal gauge-fixing used in prior works nous and makes the underlying structure more transparent. We show, in particular, that the degrees of freedom of Type I isolated horizons can be encoded (along the lines of the standard treatment) in an boundary connection. The results of this work clarify the relationship between the theory of isolated horizons and SU(2) Chern-Simons theory first explored in kiril-lee , and makes the relationship with the usual treatment of degrees of freedom in loop quantum gravity clear-cut. In the present work, we provide a full detailed derivation of the result of our recent work and discuss several important issues that were only briefly mentioned then. An important point should be emphasized concerning the logarithmic corrections mentioned above. The logarithmic corrections to the Bekenstein-Hawking area formula for black hole entropy in the loop quantum gravity literature were thought to be of the (universal) form amit . In majundar Kaul and Majumdar pointed out that, due to the necessary gauge symmetry of the isolated horizon system, the counting should be modified leading to corrections of the form . This suggestion is particularly interesting because it would eliminate the apparent tension with other approaches to entropy calculation. In particular their result is in complete agreement with the seemingly very general treatment (which includes the string theory calculations) proposed by Carlip carlip-log . Our analysis confirms Kaul and Majumdar’s proposal and eliminates in this way the apparent discrepancy between different approaches. The article is organized as follows. In the following section we review the formal definition of isolated horizons. In Section III we state the main equations implied by the isolated horizon boundary conditions for fields at a spherically symmetric isolated horizon. In Section IV we prove a series of propositions that imply the main classical part of our results: we derive the form of the conserved presymplectic structure of spherically symmetric isolated horizons, and we show that degrees of freedom at the horizon are described by an Chern-Simons presymplectic structure. In Section VI we briefly review the derivation of the zeroth and first law of isolated horizons. In Section V we study the gauge symmetries of the Type I isolated Horizon and explicitly compute the constraint algebra. In Section VII we review the quantization of the spherically symmetric isolated horizon phase space and present the basic formulas necessary for the counting of states that leads to the entropy. We close with a discussion of our results in Section VIII. The appendix contains an analysis of Type I isolated horizons from a concrete (and intuitive) perspective that makes use of the properties of stationary spherically symmetric black holes in general relativity. Ii Definition of isolated horizons The standard definition of a BH as a spacetime region from which no information can reach idealized observers at (future null) infinity is a global definition. This notion of BH requires a complete knowledge of a spacetime geometry and is therefore not suitable for describing local physics. The physically relevant definition used, for instance, when one claims there is a black hole in the center of the galaxy, must be local. One such local definition was introduced in ack ; better ; ih_prl with the name of isolated horizons (IH). Here we present this definition according to ih_prl ; afk ; abl2002 ; abl2001 . This discussion will also serve to fix our notation. In the definition of an isolated horizon below, we allow general matter, subject only to conditions that we explicitly state. Definition: The internal boundary of a history will be called an isolated horizon provided the following conditions hold: Manifold conditions: is topologically , foliated by a (preferred) family of 2-spheres and equipped with an equivalence class of transversal future pointing vector fields whose flow preserves the foliation, where is equivalent to if for some positive real number . Dynamical conditions: All field equations hold at . Matter conditions: On the stress-energy tensor of matter is such that is causal and future directed. Conditions on the metric determined by , and on its levi-Civita derivative operator : (iv.a) The expansion of within is zero. This, together with the energy condition (iii) and the Raychaudhuri equation at , ensures that is additionally shear-free. This in turn implies that the Levi-Civita derivative operator naturally determines a derivative operator intrinsic to via , tangent to . We then impose (iv.b) . Restriction to ‘good cuts.’ One can show furthermore that for some intrinsic to . A 2-sphere cross-section of is called a ‘good cut’ if the pull-back of to is divergence free with respect to the pull-back of to . As shown in abl2002 , every horizon satisfying (i)-(iv) above possesses at least one foliation into ‘good cuts’; this foliation is furthermore generically unique. We require that the fixed foliation coincide with a foliation into ‘good cuts.’ Let us discuss the physical meaning of these conditions. The first two conditions are rather weak. The third condition is satisfied by all matter fields normally used in general relativity. The fifth condition is a partial gauge fixing of diffeomorphisms in the ‘time’ direction. The main condition is therefore the fourth condition. (iv.a) requires that be expansion-free. This is equivalent to asking that the area 2-form of the 2-sphere cross-sections of be constant along generators . This combined with the matter condition (iii) and the Raychaudhuri equation implies that in fact the entire pull back of the metric to the horizon is Lie dragged by . Condition (iv.b) further stipulates that the derivative operator be Lie dragged by . This implies, among other things, an analogue of the zeroeth law of black hole mechanics: conditions (i) and (iii) imply that is geodesic — . The proportionality constant is called the surface gravity, and condition (iv.b) ensures that it is constant on the horizon for any given . Furthermore, if we had not fixed , but only required that an exist such that the isolated horizon boundary conditions hold, then condition (iv.b) would ensure that this is generically unique abl2002 . From the above discussion, one sees that the geometrical structures on that are time-independent are precisely the pull-back of the metric to , and the derivative operator . In fact, the main conditions (iv.a) and (iv.b) are equivalent to requiring and . For this reason it is natural to define as the horizon geometry. Let us summarize. Isolated horizons are null surfaces, foliated by a family of marginally trapped 2-spheres such that certain geometric structures intrinsic to are time independent. The presence of trapped surfaces motivates the term ‘horizon’ while the fact that they are marginally trapped — i.e., that the expansion of vanishes — accounts for the adjective ‘isolated’. The definition extracts from the definition of Killing horizon just that ‘minimum’ of conditions necessary for analogues of the laws of black hole mechanics to hold. Boundary conditions refer only to behavior of fields at and the general spirit is very similar to the way one formulates boundary conditions at null infinity. All the boundary conditions are satisfied by stationary black holes in the Einstein-Maxwell-dilaton theory possibly with cosmological constant. Note however that, in the non-stationary context, there still exist physically interesting black holes satisfying our conditions: one can solve for all our conditions and show that the resulting 4-metric need not be stationary on lew2000 . In the choice of boundary conditions, we have tried to strike the usual balance: On the one hand the conditions are strong enough to enable one to prove interesting results (e.g., a well-defined action principle, a Hamiltonian framework, and a realization of black hole mechanics) and, on the other hand, they are weak enough to allow a large class of examples. As we already remarked, the standard black holes in the Einstein-Maxwell-dilatonic systems satisfy these conditions. More importantly, starting with the standard stationary black holes, and using known existence theorems one can specify procedures to construct new solutions to field equations which admit isolated horizons as well as radiation at null infinity lew2000 . These examples, already show that, while the standard stationary solutions have only a finite parameter freedom, the space of solutions admitting isolated horizons is infinite dimensional. Thus, in the Hamiltonian picture, even the reduced phase-space is infinite dimensional; the conditions thus admit a very large class of examples. Nevertheless, space-times admitting isolated horizon are very special among generic members of the full phase space of general relativity. The reason is apparent in the context of the characteristic formulation of general relativity where initial data are given on a set (pairs) of null surfaces with non trivial domain of dependence. Let us take an isolated horizon as one of the surfaces together with a transversal null surface according to the diagram shown in Figure 1. Even when the data on the isolated horizon may be infinite dimensional (for Type II and II isolated horizons, see below), in all cases no transversing radiation data is allowed by the IH boundary condition. Roughly speaking the isolated horizon boundary condition reduces to one half the number of local degrees of freedom. Notice that the above definition is completely geometrical and does not make reference to the tetrad formulation. There is no reference to any internal gauge symmetry. In what follows we will deal with general relativity in the first order formulation which will introduce, by the choice of variables, an internal gauge group corresponding to local transformations (in the case of Ashtekar variables) or transformations (in the case of real Ashtekar-Barbero variables). It should be clear from the purely geometric nature of the above definition that the IH boundary condition cannot break by any means these internal symmetries. Isolated horizon classification according to their symmetry groups Next, let us examine symmetry groups of isolated horizons. A symmetry of is a diffeomorphism on which preserves the horizon geometry and at most rescales elements of by a positive constant. It is clear that diffeomorphisms generated by are symmetries. So, the symmetry group is at least 1-dimensional. In fact, there are only three possibilities for : Type I: the isolated horizon geometry is spherical; in this case, is four dimensional ( rotations plus rescaling-translations444In a coordinate system where the rescaling-translation corresponds to the affine map with constants. along ); Type II: the isolated horizon geometry is axi-symmetric; in this case, is two dimensional (rotations round symmetry axis plus rescaling-translations along ); Type III: the diffeomorphisms generated by are the only symmetries; is one dimensional. Note that these symmetries refer only to the horizon geometry. The full space-time metric need not admit any isometries even in a neighborhood of the horizon. In this paper, as in the classic works bhe1 ; ack , we restrict ourselves to the Type I case. Although a revision would be necessary in light of the results of our present work, the quantization and entropy calculation in the context of Type II and Type III isolated horizons has been considered in jon . Iii Some extra details for Type I isolated horizons In this section we first list the main equations satisfied by fields at an isolated horizon of Type I. The equations presented here can be directly derived from the IH boundary conditions implied by the definition of Type I isolated horizons given above. Most of the equations presented here can be found in ack . For completeness we prove these equations at the end of this section. As we shall see in Subsection III.2, some of the coefficients entering the form of these equations depend on the representative chosen among the equivalence class of null generators . Throughout this paper we shall fix an null generator by the requirement that the surface gravity matches the one corresponding to the stationary black hole with the same macroscopic parameters as the Type I isolated horizon under consideration. This choice makes the first law of IH take the form of the usual first law of stationary black holes (see Section VI). iii.1 The main equations When written in connection variables, the isolated horizon boundary condition implies the following relationship between the curvature of the Ashtekar connection at the horizon and the -form (in the time gauge) where is the area of the IH, the double arrows denote the pull-back to with a Cauchy surface with normal at , and null and normalized according to . Notice that the imaginary part of the previous equation implies that Another important equation is The previous equations follow from equations (3.12) and (B.7) of reference ack . Nevertheless, they also follow from the abstract definition given in the introduction. From the previous equations, only equation (5) is not explicitly proven from the definition of IH in the literature. Therefore, we give here an explicit prove at the end of this section. For concreteness, as we think it is helpful for some readers to have a concrete less abstract treatment, another derivation using directly the Schwarzschild geometry is given in Appendix A. The previous equations imply in turn that where is the Ashtekar-Barbero connection 555In our convention the isomorphism is defined by which implies that and . . iii.2 Proof of the main equations In this subsection we use the definition of isolated horizons provided in the previous section to prove some of the equations stated above. We will often work in a special gauge where the tetrad is such that is normal to and and are tangent to . This choice is only made for convenience, as the equations presented in the previous section are all gauge covariant, their validity in one frame implies their validity in all frames. Lemma 1: In the gauge where the tetrad is chosen so that (which can be completed to a null tetrad , and ), the shear-free and vanishing expansion (condition () in the definition of IH) imply Proof: The expansion and shear of the null congruence of generators of the horizon is given by where we have used the definition of the spin connection . Similarly we have As and form a non degenerate frame for , and from the definition of pull-back, the previous two equations imply the statement of our lemma. The previous lemma has an immediate consequence on the form of equation (4) for the component in the frame of the previous lemma. More precisely it says that . The good-cut condition () in the definition implies then that Another important consequence of the previous lemma is equation (3), also derived in ack . We give here for completeness and self consistency a sketch of its derivation. This equation follows from identity where is the Riemann tensor and , which can be derived using Cartan’s structure equations. A simple algebraic calculation using the null tetrad formalism (see for instance chandra page 43) with the null tetrad of Lemma 1, and the definitions and , where is the Ricci tensor and the Weyl tensor, yields where . An important point here is that the previous expression is valid for any two sphere embedded in spacetime in an adapted null tetrad where and are normal to . However, in the special case where (where with a Type I isolated horizon) it follows from spherical symmetry that with a constant on the horizon . Moreover, in the gauge defined in the statement of Lemma 1, the only non vanishing component of the previous equation is the component for which (using Lemma 1) we get with the area element of . Integrating the previous equation on one can completely determine the constant , namely from where equation (3) immediately follows. Lemma 2: For Type I isolated horizons for some constant . One can choose a representative from the equivalence class of null normals to the isolated horizon in order to fix by making use of the translation symmetry of IH along . By studying the stationary spherically symmetric back hole solutions one can show that this corresponds to the choice where the surface gravity of the IH matches the stationary surface gravity (see Appendix A). Proof: In order to simplify the notation all free indices associated to forms that appear in this proof are pulled back to (this allows us to drop the double arrows from equations). In the frame of lemma 1, where is normal to , the only non trivial component of the equation we want to prove is the component, namely: where and . Now, in that gauge, we have that for some matrix of coefficients . Notice that the left hand side of the previous equation equals . We first prove that is time independent, i.e. that . We need to use the isolated horizon boundary condition where is the derivative operator determined on the horizon by the Levi-Civita derivative operator . One important property of the commutator of two derivative operators is that it also satisfy the Leibnitz rule (it is itself a new derivative operator). Therefore, using the fact that the null vector is normalized so that we get where we have also used that . Evaluating the equation on the right hand side explicitly, and using the fact that 666 Here we used that which comes from the restriction to ‘good cuts’ in definition of Section II. More precisely, if one introduces a coordinate on such that and on some leaf of the foliation, then it follows—from the fact that is a symmetry of the horizon geometry , and the fact that the horizon geometry uniquely determines the foliation into ‘good cuts’—that will be constant on all the leaves of the foliation. As must be normal to the leaves one has , whence . we get where in the second line we have used the fact that plus the fact that as the Lie derivative for some (moreover, one can even fix if one wanted to by means of internal gauge transformations). Then it follows that where, in addition to previously used identities, we have made use of lemma 1, eq. (7). The previous equations imply that the left hand side of equation (19) is Lie dragged along the vector field , and since is also Lie dragged (in this gauge), all this implies that Now we must use the rest of the symmetry group of Type I isolated horizons. If we denote by () the three Killing vectors corresponding to the symmetry group of Type I isolated horizons. Spherical symmetry of the horizon geometry implies which,through similar manipulations as the one used above, lead to which completes the prove that is constant on . We can now introduce the dimensionless constant . Finally one can fix by choosing the appropriate null generator from the equivalence class . Iv The conserved presymplectic structure In this section we show in detail how the IH boundary condition implies the appearance of an Chern-Simons boundary term in the symplectic structure describing the dynamics of Type I isolated horizons. This result is key for the quantization of the system described in Section VII. iv.1 The action principle The conserved pre-symplectic structure in terms of Ashtekar variables can be easily obtained in the covariant phase space formalism. The action principle of general relativity in self dual variables containing an inner boundary satisfying the IH boundary condition (for asymptotically flat spacetimes) takes the form where and is the self-dual connection, and a boundary contribution at a suitable time cylinder at infinity is required for the differentiability of the action. No boundary term is necessary if one allows variations that fix an isolated horizon geometry up to diffeomorphisms and Lorentz transformations. This is a very general property and we shall prove it in the next section as we need a little bit of notation that is introduced there. First variation of the action yields from which the self dual version of Einstein’s equations follow as the boundary terms in the variation of the action cancel. iv.2 The classical results in a nutshell In the following subsections a series of technical results are explicitly proven. Here we give an account of these results. The reader who is not interested in the explicit proofs can jump directly to Section V after reading the present subsection. In this work we study general relativity on a spacetime manifold with an internal boundary satisfying the isolated boundary condition corresponding to Type I isolated horizons, and asymptotic flatness at infinity. The phase space of such system is denoted and is defined by an infinite dimensional manifold where points are given by solutions to Einstein’s equations satisfying the Type I IH boundary condition. Explicitly a point can be parametrized by a pair satisfying the field equations (28) and the requirements of Definition II. In particular fields at the boundary satisfy Einstein’s equations and the constraints given in Section III. Let denote the space of variations at (in symbols ). A very important point is that the IH boundary conditions severely restrict the form of field variations at the horizon. Thus we have that variations are such that for the pull back of fields on the horizon they correspond to linear combinations of internal gauge transformations and diffeomorphisms preserving the preferred foliation of . In equations, for and we have that where the arrows denote pull-back to , and the infinitesimal transformations are explicitly given by while the diffeomorphisms tangent to H take the following form where for any -form , and the first term in the expresion of the Lie derivative of can be dropped due to the Gauss constraint . So far we have defined the covariant phase space as an infinite dimensional manifold. For it to become a phase space it is necessary to provide it with a presymplectic structure. As the field equations, the presymplectic structure can be obtained from the first variation of the action (27). In particular a symplectic potential density for gravity can be directly read off from the total differential term in (27) cov . The symplectic potential density is therefore and the symplectic current takes the form Einstein’s equations imply . Therefore, applying Stokes theorem to the four dimensional (shaded) region in Fig. 1 bounded by in the past, in the future, a timelike cylinder at spacial infinity on the right, and the isolated horizon on the left we obtain Now it turns out that the horizon integral in this expression is a pure boundary contribution: the symplectic flux across the horizon can be expressed as a sum of two terms corresponding to the two-spheres and . Explicitly (see Proposition 1 proven below), the symplectic flux across the horizon factorizes into two contributions on given by Chern-Simons presymplectic terms according to which implies that the following presymplectic structure is conserved or in other words is independent of . The presence of the boundary term in the presymplectic structure might seem at first sight peculiar; however, we will prove in the following section that the previous symplectic structure can be written as where we are using the fact that, in the time gauge where is normal to the space slicing, when pulled back on . The previous equation is nothing else but the familiar presymplectic structure of general relativity in terms of the Palatini variables. In essence the boundary term arises when connection variables are used in the parametrization of the gravitational degrees of freedom. Finally, as shown in Section IV.4, the key result for the quantization of Type I IH phase space: the presymplectic structure in Ashtekar Barbero variables takes the form The above equation is the main result of the classical analysis of this paper. It shows that the conserved presymplectic structure of Type I isolated horizons aquires a boundary term given by an Chern-Simons presymplectic structure when the unconstrained phase space is parametrized in terms of Ashtekar-Barbero variables. In the following subsection we prove this equation. iv.2.1 On the absence of boundary term on the internal boundary Before getting involved with the construction of the conserved presymplectic structure let us come back to the issue of the differentiability of the action principle. In the isolated horizon literature it is argued that the IH boundary condition guaranties the differentiability of the action principle without the need of the addition of any boundary term (see afk ). As we show here, this property is satisfied by more general kind of boundary conditions. As mentioned above, the allowed variations are such that the IH geometry is fixed up to diffeomorphisms of and gauge transformations. This enough for the boundary term arising in the first variation of the action (26) to vanish. The boundary term arising on upon first variation of the action is First let us show that for as given in (30). We get where we integrated by parts in the first identity, the first term in the second identity vanishes due to Eisntein’s equations while the second term vanishes due to the fact that fields are held fixed at the initial and final surfaces and and so when evaluated at . Similarly we can prove that for as given in (31) with (this is the only difference) . We get where in the last line the first and second terms vanish due to Einstein’s equations, and the last term vanishes because variations are such that the vector field vanishes at . Notice that we have not made use of the IH boundary condition. iv.3 The presymplectic structure in self-dual variables In this section we prove a series of propositions implying that the presymplectic structure of Type I isolated horizons is given by equation (37). In addition, we will prove that the symplectic structure is real and takes the simple form (38) in terms of Palatini variables. Proposition 1: The symplectic flux across a Type I isolated horizon factorizes into boundary contributions at and according to for and . Lets start with transformations. Using (30) we get where in the first line we used the equations of motion and in the second line we used the IH boundary condition (3). We have therefore shown that Similarly, for diffeomorphisms we first notice that (31) implies that where and the explicit form of is defined as We have that where in the third line we used the vector constraint , while in last line we have used the equations of motion and equation (3). Notice that the calculation leading to equation (44) is also valid for a field dependent such as . This plus the linearity of the presymplectic structure lead to and concludes the proof of our proposition . The previous proposition implies that the presymplectic structure (37) is indeed conserved by evolution in . Now we are ready to state the next important proposition. Proposition 2: The presymplectic form given by is independent of and real. Moreover, the symplectic structure can be described entirely in terms of variables and taking the familliar form

« AnteriorContinuar » 10. If sixteen be added to a certain number, tha sum is three times sixteen; what is that number? Ans. 32. 11. Divide sixty so that the seventh of one part may equal the eighth of the remainder. Ans. 28 & 32. 12. A post was a certain distance in the water, twice as much was painted, and three times as much as this above was unpainted, the wbole was nine feet long; what was the length of each part separately ? Ans. 1 ft., 2 ft., 6 ft. 13. A father is four times as old as his son, foar years ago the father's age was six times that of his son; what are the respective ages of father and son ? Ans. 40 and 10. 14. Fifty added to five times a number is the same as fifty-six added to four times the number? What is the number? Ans. 6. 15. If I take away eleven from a certain number multiplied by sixteen, the remainder is the same as the sum of seventy and seven times the number; what is the number? Ans. 9. 16. Twenty-eight times the sum of a number and nine, is the same as twenty-seven times the difference of forty-six and the number; what is the number? Ans. 18. 17. A subscribes a certain sum to a charity, B subscribes half as much, and C one third; the total subscription paid by A, B, and C, is £11; what does each subscribe to the charity ? Ans. £6, £3, & £2. 18. One fifth of a post is in the water, one third is painted and outside, and there are seven feet above this unpainted; what is the length of the post? Ans. 15 ft. 19. If I take the half of a number after adding to it three; and the fourth of the number after adding to it five; and the third of the number after four has been added to it, I find the sum of these three quantities is 16; what is the number? Ans. 11. 20. If in a lottery of 1000 tickets, half the number of prizes, added to one quarter of the blanks, was 262, how many prizes were there in the lottery ? Ans. 48. 21. The ages of a man and his wife together amount to 75 years; and they are in the proportion of 7 to 8; what is each one's age? Ans. 40 and 35. 22. The ages of a man, woman, and child together amount to fifty-five years. The woman is four times as old as the boy, and the man six times as old; what is each one's age ? Ans. 5, 20, and 30. 23. The sum of two numbers is 27 and the difference 3; what are the numbers ? Ans. 12 & 15. 24. The difference of two numbers is six, and the quotient is lz; what are the two numbers ? Ans. 12 and 18. 25. The sum of two numbers is thirty-four, and eir product 280; what are the numbers ? 14 and 20. 26. A floor was relaid for £108, at the value of 5s. per square foot, the length of the room was 120 feet; what was the breadth ? Ans. 3.6 feet. 27. The walls of a room are papered at a cost of £720, being at the rate of ls. per square foot, the height of the room was 24 feet, and the breadth 64 feet; what was the length of the room ? Ans. 236 feet. 28. The content of a room was 1846 cubic feet, the length 16 feet, and the depth 13; what was the breadth of the room ? Ans. 8-9 29. A rectangular field measured 61874 square yards; its length was 6187 yards; what was its breadth ? Ans. 107145 T. 30. If the surface of a rectangular board be 9768 square inches, and the breadth be 84 inches ; what is the length ? Ans. 116. 31. Two men earn together £600; ono earns twice as much as the other; what is each man's earnings ? Ans. £200 and £400. 32. Two men put money into the savings' bank; one man puts in £150 more than the other, and their joint savings are £970; what does each invest? Ans. £410 and £560. 33. Three men together walk a distance of 600 miles; one walks twice and the other three times as far as the first; what is the distance travelled by each ? Ans. 100, 200, and 300 miles. 34. What is that number which, with 17 added to it, will make 55 ? Ans. 38. 35. What number is that whose half exceeds its third part by 17 ? Ans. 102. A. Determine a number which, when 365 is taken from it, shall leave 214. B. What is the number whose third part and fourth part added together make 91 ? 36. What number is that which is as much short of 100 as its double exceeds 100 ? Ans. 663. C. A farmer had three times as many cows as horses, and ten times as many sheep as cows. The whole number of his live stock (horses, cows, and sheep) was 306. How many horses had he ? 37. A boy being asked how many marbles he had, said he would auswer no such question, but that it was a matter of indifference to him whether he gained 15 more, or doubled what he had and then lost 10. Can you compute from that hint how many he actually had ? Ans. 25. D. Two tradesmen began business with equal sums of money. The first gained £142, and the second lost £150. The first was then twice as rich as the second. What sum had each at the commencement ? 38. Divide 1000 into two such parts, that 9 times the larger shall exceed 13 times the smaller by 970. Ans. 635 and 365. 39. The sum of two numbers is 154, and their difference 42. What re they? Ans. 98 and 56. E. Divide 100 into two such parts that 25 times the less may exceed 24 times the greater by 1. F. Divide 100 into two such parts, that half the one added to one-seventh of the other shall just make 20. 40. An election, at which there were only two candidates, was gained by Mr. Smith, who had a majority of 75 votes over Mr. Jones: the number of 'voters in the interest of Jones was just two-thirds of the number who supported Smith. What was the whole number of voters ? Ans. 375. G. A charitable gentleman distributed the contents of his purse, which contained 23 shillings, among a poor family, consisting of a man, his wife, and two children. He gave the man twice as much as his wife, and to the oldest child one-third of what he gave to his father. There was then a shilling over, which he gave to the youngest child: How much did each receive ? 41. In the House of Commons there are 654 members, Scotland sending half as many as Ireland and half a member more, and England contributing 29 short of 5 times the number of Irish members. What are the respective numbers ? Avs. 496, 105, and 53. H. Four persons engaged in a speculation requiring an outlay of £2400. Of that sum, A contributed twice as much as B, and £10 more; C contributed £20 less than A; and D only two-thirds of C. What sums did they separately contribute ? 42. Two boys, who made their living by selling nuts, commenced the week with the same sum; but, when they met on the Saturday night, the one found that, after paying for his maintenance, he had gained a half-crown; and the other that he had lost Is. 6d. The consequence was, that the former had now three times as much as the latter. What did they begin the week with ? Aus. 3s. 6d. each. I. Two fields were purchased at £43 and £28 per acre respectively. Their united area was 12 acres, and their united price £411. What were the sizes of the fields separately? 43. A gentleman, after travelling 12 hours without stopping, found that, if he had travelled 3 miles an hour faster, he would have accomplished the journey in two hours less of time. At what rate did he travel ? Ans. 15 miles hour. J. Two friends, living at Walton and Middleton, 24 miles apart, agreed to meet on an angling excursion between the two places. The one from Walton set out an hour after the other, but, having no encumbrance, got on at the rate of 4 miles an hour, while his friend, having undertaken to bring fishing-tackle and provisions for both, could only proceed at threefourths of that speed. At what point on the road did they meet ? K. A man departs on a journey, walking at the uniform rate of 3 miles an hour; and, two hours later, another sets out after him, riding, at 7 miles an hour. At what distance on the road will the latter overtake the former ? 44. If you divide a certain number by 9, and add together divisor, dividend, and quotient, their sum shall be 59. The number is required. Ans. 45. L. Divide the number 100 into four such parts that the first may be equal to half the second, but greater than the third by 6, and less than the fourth 45. Find three numbers, such that their sums, taken two by two, sball be 11, 12, and 13. Ans. 5, 6, and 7. M. A man and his wife were married at the respective ages of 40 and 20 years. How old will the man be when his wife's age becomes three-fourths of his own ?

The manuscript of the first edition was completed in 1944. In this revision, the first six chapters of the first edition have been reproduced, and the following chapters rewritten completely. The book is not the last word on the topics it deals with, but the newly written chapters include material which was not to be found except in scattered form, in the literature of the last fifteen years. - Foreword. This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals. The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics. The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points. This classic work (first published in 1947), in three volumes, provides a lucid and rigorous account of the foundations of modern algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties but geometrical meaning has been emphasized throughout. This first volume is divided into two parts. The first is devoted to pure algebra: the basic notions, the theory of matrices over a non-commutative ground field and a study of algebraic equations. The second part is in n dimensions. It concludes with a purely algebraic account of collineations and correlations. Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics and considered to be useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists. Proceedings of a Colloquium Held in Utrecht, August 1959 Author: Hans Freudenthal Algebraical and Topological Foundations of Geometry contains the proceedings of the Colloquium on Algebraic and Topological Foundations of Geometry, held in Utrecht, the Netherlands in August 1959. The papers review the algebraical and topological foundations of geometry and cover topics ranging from the geometric algebra of the Möbius plane to the theory of parallels with applications to closed geodesies. Groups of homeomorphisms and topological descriptive planes are also discussed. Comprised of 26 chapters, this book introduces the reader to the theory of parallels with applications to closed geodesies; groups of homeomorphisms; complemented modular lattices; and topological descriptive planes. Subsequent chapters focus on collineation groups; exceptional algebras and exceptional groups; the connection between algebra and constructions with ruler and compasses; and the use of differential geometry and analytic group theory methods in foundations of geometry. Von Staudt projectivities of Moufang planes are also considered, and an axiomatic treatment of polar geometry is presented. This monograph will be of interest to students of mathematics. An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles. This book is the first in a series of three volumes that comprehensively examine Mario Pieri’s life, mathematical work and influence. The book introduces readers to Pieri’s career and his studies in foundations, from both historical and modern viewpoints. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizations — one in arithmetic and one in geometry. The book combines an engaging exposition, little-known historical notes, exhaustive references and an excellent index. And yet the book requires no specialized experience in mathematical logic or the foundations of geometry. Mathematics by Bartel Eckmann L. Van der van der Waerden,Emil Artin,Emmy Noether Author: Bartel Eckmann L. Van der van der Waerden,Emil Artin,Emmy Noether Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben. This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books Mathematics by Bertrand Toen,Bertrand Toën,Gabriele Vezzosi This is the second part of a series of papers called ""HAG"", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, etale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category $C$, and prove that this notion satisfies the expected properties. Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring theory and algebraic geometry as well as varieties of arbitrary dimension and some elementary mathematics on curves. Upon finishing the text, students will have a foundation for advancing in several different directions, including toward a further study of complex algebraic or analytic varieties or to the scheme-theoretic treatments of algebraic geometry. 2015 edition.

Eszter Morvay 2017 Eszter joined CMT’s Summer Math Program for Young Scholars in the Summer of 2017. She applied to this program in order to improve her math skills, and learn new topics of math that aren’t covered in her school. Her favorite part of the program was the Number Theory class. She learned a lot of new theorems and ways of problem solving which helped grasp concepts much better. The classes and talks at this program gave her a better understanding of the kind of work that professional mathematicians do. She was surprised because most of the math fields they were in were very different from the kind of math that she was used to -- it was more “pure” math. She found this kind of math very interesting, and thinks she would go into a math field when she graduates from high school as well. Eszter is currently in the 10th grade at Hunter College High School. At her high school, she is participating in various extracurricular activities such as the math team, where she did various competitions each year, such as SBIMC, Purple Comet, and Math League. She also volunteered as a peer tutor in her school, tutoring students in grades below her in math. She also takes Saturday math classes at the program Math-M-Addicts. She recommends CMT’s Summer Math program for students looking to be introduced to college level math topics, learn interesting new types of math, and solve challenging math problems. Dora Woodruff 2017 Dora joined CMT’s Summer Math Program for Young Scholars in the Summer of 2017. Her desire to learn about advanced math topics was the main reason why she applied to the program. Before CMT, Dora had not had much exposure to advanced math topics such as number theory, group theory and graph theory. She assumed that people did math purely for fun at a high level, but she now understands that even subjects that at first seem very pure can end up having surprising uses in everyday life -- how math applies to the real world. Her most favorite moment during the program was when she was able to prove Euler’s theorem via group theoretic methods, namely Lagrange’s Theorem, in her number theory class. In her own words, “we found an equation for Euler’s phi function! It was such a beautiful result, and proving it using almost everything we had accumulated over the course of six classes was really exciting.” Dora is currently in the 10th grade at Horace Mann School. After she graduates from high school, she wants to continue studying math and ultimately become a research mathematician. In the past, she attended Summer Programs at Center for Talented Youth and next year she is planning to attend the Columbia Science Honors program. Moreover, she just started an internship at MoMATH (National Museum of Mathematics) this Fall 2017, which she plans to continue through the next three years. Dora has kept herself busy with extracurricular activities, including her school’s math team and a math magazine that she started called Prime. She definitely recommends CMT because not only did she get to improve her problem solving skills, but also was able to get exposure to higher level math topics, and got an idea of what math classes in college really look like. Saskia Van Horn 2016 Saskia participated in the CMT Summer Math Program for Young Scholars in the Summer of 2016. It was her curiosity in learning about other topics in mathematics outside of her classroom and desire to experience class in a university setting that led her to applying. Her favorite moment about the program was learning about graphing in the real world. In Saskia's own words, "I never knew a subway map was an actual graph. To me, not only was that mind-blowing but it was a lesson where I was exposed to seeing math used in a real world situation that people benefit from daily." Her math journey had just begun, and Saskia discovered that Math was not just a mere core subject where one just takes test, but rather something that has an effect on our whole world and lives. Her experience with this program "only further catapulted [her] desire and goal to attend college." She is in fact planning to attend NYU after graduating from high school. Saskia is currently a junior at the Energy Tech High School. She keeps herself very much active through various extracurricular activities. She is the founder and president of the college readiness program at her school, she is one of the school's ambassadors, she is in the Math Club and Women in STEM Club, and she is also a tutor to elementary school students . Her active leadership roles in these activities have earned her awards and have shaped the student morale of her school. Saskia highly recommends CMT Summer Math Program for Young Scholars to everyone "because students can learn a lot of math skills in which they can transfer into their high school work but also gain exposure of what it is like to be a university student and to sit in a classroom of one of the most respected universities in the country."

1.Refer to the above data. The marginal cost column reflects:a.the law of diminishing returnsb.the law of diminishing marginal utilityc.diseconomies of scaled.economies of scale2.A purely competitive firm?s short-run supply curve is:a.The upward slopping portion of its marginal cost curveb.The upward sloping portion of its average variable cost curve.c.Its marginal cost curve above average variable cost.d.Its average total cost curve.3.Long-run competitive equilibrium:a.Is realized only in constant- cost industries.b.Will never change once it is realizedc.Is not economically efficientd.Results in zero economic profits4.Which of the following statement is correct?a.Economic profits induce firms to enter an industry; losses encourage firm to leave.b.Economic profits induce firms to enter an industry; profits encourage firm to leave.c.Economic profits and losses have no significant impact on the growth or decline of an industry.d.Normal profits will cause an industry to expand.5.Suppose losses cause industry X to contract and, as a result, the prices of relevant inputs decline. Industry X is:a.A constant-cost industry.b.A decreasing-cost industry.c.An increasing-cost industry.d.Encountering X-inefficiency6.Resources are efficiently allocated when production occurs where:a.Marginal cost equals variable cost.b.Prices is equal to average revenuec.Prices is equal to marginal costd.Prices is equal to average variable cost7.For an imperfectly competitive firm:a.Total revenue is a straight, upsloping line because a firm?s sales are independent of product priceb.The marginal revenue curve lies above the demand curve because any reduction in price applies to all units sold.c.The marginal revenue curve lies below the demand curve because any reduction in price applies to all units soldd.The marginal revenue curve lies below the demand curve because any reduction in price applies only to the extra units sold.1.Refer to above diagram. The quantity difference between areas A and C for the indicated price reduction measures:a.Marginal costb.Marginal revenuec.Monopoly priced.A welfare or efficiency loss2.A nondiscrimination monopolist:a.Will never produce in the output range where marginal revenue is positive.b.Will never produce in the output range where demand is inelasticc.Will never produce in the output range where demand is elasticd.May produce where demand is either elastic or inelastic, depending on the level of production costs3.The vertical distance between the horizontal axis and any point on a nondiscriminating monopolist?s demand curve measures:a.The quantity demandedb.Product price and marginal revenuec.Total revenued.Product price and average revenue4.A pure monopolist:a.Will realize an economic profit if price exceeds ATC at the profit-maximizing/loss-minimizing level of outputb.Will realize an economic profit if ATC exceeds MR at the profit-maximizing/loss-minimizing level of outputc.Will realize an economic loss if MC intersects the downsloping portion of MRd.Always realizes an economic profit5.When a pure monopolist is producing its profit-maximizing output, price will:a.Be less than MRb.Equal neither MC nor MRc.Equal MPd.Equal MC6.The supply curve for a monopolist is:a.Perfectly elasticb.Upslopingc.That portion of the marginal cost curve lying above minimum average variable cost.d.Nonexistent7.Which of the following statement is correcta.The pure monopolist will maximize profit by producing at that point on the demand curve where elasticity is zerob.In seeking the profit-maximizing output the pure monopolist underallocates resources to its productionc.The pure monopolist maximizes profits by producing that output at which the differential between price and average cost is the greatestd.Purely monopolistic sellers earn only normal profits in the long run1.Refer to the above long-run cist diagram for a firm. If the firm produces output Q1 at an average total cost of ATC1, then firm is:a.Producing the profit-maximizing output, but is failing to minimize production costs.b.Incurring X-inefficiency, but is realizing all existing economies of scalec.Incurring X-inefficiency and is failing to realize all existing economies of scale.d.Producing that output with the most efficient combination of inputs and is realizing all economies of scale.2.The dilemma of regulation refers to the idea that:a.The regulated price which achieves allocative efficiency is also likely to result in persistent economic profits.b.The regulated price which results in a ?fair return? restricts output by more than would unregulated monopoly.c.Regulated pricing always conflicts with the ?due process? provision of the Constitution.d.The regulated price which achieves allocative efficiency is also likely to result in losses.3.The monopolistically competitive seller?s demand curve will become more elastic the:a.More significant the barriers to entering the industry.b.Greater the degree of product differentiation.c.Larger the number of competitors.d.Smaller the number of competitors.

Mathematical and physical papers Book file PDF easily for everyone and every device. You can download and read online Mathematical and physical papers file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with Mathematical and physical papers book. Happy reading Mathematical and physical papers Bookeveryone. Download file Free Book PDF Mathematical and physical papers at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF Mathematical and physical papers Pocket Guide. The papers are presented in chronological order across the volumes, enabling readers to understand their theoretical development and framing them in an accessible form for 'future historical interests'. Authorial notes and appendices are also included. This book will be of value to anyone with an interest in the word of Larmour, mathematics physics and the history of science. A Dynamical Theory of the Electric and Luminiferous. Note on the Complete Scheme of Electrodynamic Equations. The Methods of Mathematical Physics c e. On the Relations of Radiation to Temperature e e. Can Convection through the Aether be detected Electrically? Protection from Lightning and the Range of Protection afforded. Viscosity in Relation to the Earths Free Precession. Periodic Disturbance of Level arising from the Load of Neigh. On the Mathematical Expression of the Principle of Huygens. On the Ascertained Absence of Effects of Motion through. On the Constitution of Natural Radiation e e. Note on Pressure Displacement of Spectral Lines e. The Statistical and Thermodynamical Relations of Radiant. A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples. The first example is the oft-quoted one of planetary motion. Mathematical Modelling of Engineering Problems The laws of falling bodies became rather well established as a result of experiments carried out principally in Italy. These experiments could not be very accurate in the sense in which we understand accuracy today partly because of the effect of air resistance and partly because of the impossibility, at that time, to measure short time intervals. Nevertheless, it is not surprising that, as a result of their studies, the Italian natural scientists acquired a familiarity with the ways in which objects travel through the atmosphere. It was Newton who then brought the law of freely falling objects into relation with the motion of the moon, noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence. - Annales de l’Institut Henri Poincaré D; - DO VIKINGS WEAR GLASSES ?! - Mathematical Physical Papers - AbeBooks. - Donate to arXiv. Philosophically, the law of gravitation as formulated by Newton was repugnant to his time and to himself. Empirically, it was based on very scanty observations. The mathematical language in which it was formulated contained the concept of a second derivative and those of us who have tried to draw an osculating circle to a curve know that the second derivative is not a very immediate concept. Dicke, Am. Let us just recapitulate our thesis on this example: first, the law, particularly since a second derivative appears in it, is simple only to the mathematician, not to common sense or to non-mathematically-minded freshmen; second, it is a conditional law of very limited scope. It explains nothing about the earth which attracts Galileo's rocks, or about the circular form of the moon's orbit, or about the planets of the sun. The explanation of these initial conditions is left to the geologist and the astronomer, and they have a hard time with them. The second example is that of ordinary, elementary quantum mechanics. This originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say "if the mechanics as here proposed should already be correct in its essential traits. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless. Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we "got something out" of the equations that we did not put in. The same is true of the qualitative characteristics of the "complex spectra," that is, the spectra of heavier atoms. I wish to recall a conversation with Jordan, who told me, when the qualitative features of the spectra were derived, that a disagreement of the rules derived from quantum mechanical theory and the rules established by empirical research would have provided the last opportunity to make a change in the framework of matrix mechanics. In other words, Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas. The mathematical formalism was too dear and unchangeable so that, had the miracle of helium which was mentioned before not occurred, a true crisis would have arisen. Surely, physics would have overcome that crisis in one way or another. It is true, on the other hand, that physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom, which is perhaps the most striking miracle that has occurred in the course of the development of elementary quantum mechanics, but by far not the only one. In fact, the number of analogous miracles is limited, in our view, only by our willingness to go after more similar ones. Quantum mechanics had, nevertheless, many almost equally striking successes which gave us the firm conviction that it is, what we call, correct. The last example is that of quantum electrodynamics, or the theory of the Lamb shift. Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand. The preceding three examples, which could be multiplied almost indefinitely, should illustrate the appropriateness and accuracy of the mathematical formulation of the laws of nature in terms of concepts chosen for their manipulability, the "laws of nature" being of almost fantastic accuracy but of strictly limited scope. I propose to refer to the observation which these examples illustrate as the empirical law of epistemology. Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories. Without the laws of invariance the physical theories could have been given no foundation of fact; if the empirical law of epistemology were not correct, we would lack the encouragement and reassurance which are emotional necessities, without which the "laws of nature" could not have been successfully explored. Sachs, with whom I discussed the empirical law of epistemology, called it an article of faith of the theoretical physicist, and it is surely that. The empirical nature of the preceding observation seems to me to be self-evident. It surely is not a "necessity of thought" and it should not be necessary, in order to prove this, to point to the fact that it applies only to a very small part of our knowledge of the inanimate world. It is absurd to believe that the existence of mathematically simple expressions for the second derivative of the position is self-evident, when no similar expressions for the position itself or for the velocity exist. It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted. The ability of the human mind to form a string of conclusions and still remain "right," which was mentioned before, is a similar gift. Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions. The question which presents itself is whether the different regularities, that is, the various laws of nature which will be discovered, will fuse into a single consistent unit, or at least asymptotically approach such a fusion. International Journal of Mathematical Modelling & Computations Alternatively, it is possible that there always will be some laws of nature which have nothing in common with each other. At present, this is true, for instance, of the laws of heredity and of physics. It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the "ultimate truth," that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature. It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small.

how it works best if you load down the booklet where everything is described ... Do you want to discover the abc-code, then get a journal first and take a pendulum - a larger washer or a semi-precious stone donut will be enough to fasten a piece of twine - and ask for your personal sign for yes and no. For example, my clockwise circling is for yes, swinging left and right as shaking your head is the sign for no. Do not be afraid of the pendulum. It's nothing occult. It's also not the pendulum that gives the answer. As a human you are receptive to vibrations - like an instrument. The pendulum, however, makes the vibration visible, just as the pointer on a clock shows the time, or the screen on the computer what you are doing. Your inner interlocutor helps: You ask yourself, you yourself postulate answers and check them with the pendulum or with the inner ear for their correctness. I myself need the pendulum if the facts are complex and, accordingly, are far apart. But - let's start: If you now want to decode a number and interpret it, then you should boost your mathematical sense. With puzzle, tinkering, intuitive grasping you get to the goal. You remember - calculations were made at school. You had to calculate the result. The abc-code is the other way round. Here the number (you possibly dreamed) corresponds to the result. The task now is to reconstruct the corresponding calculation that led to this result. Of course, there are endless possibilities. So how do I find out which of these options is the right one now and today? - You ask until you have found the right term via the yes / no technic. This is actually very fast and is like riding a bicycle - at some point you can do it. But now to the order: Ask as follows: · Is the number the result of an calculation? If so, is it an addition or a subtraction? · A multiplication or a division? · A potency or a root? Now ask further, if the found parts of the calculation can be further dismantled, as far as the whole calculation stands in its individual limbs. So - write this down in your new book, which you got for your personal ABC messages. As an example (and each number is different!) let's take the number 5, an addition, here for example 2 + 3, this sum is further divided: 2 + 4 - 1. At most there is a third, fourth, fifth step , here: 2 + 2 x 2 - 1. When you are ready, you replace these single-digit numbers by the letters or their word meanings from the chapter "numbers 1 - 9". Are there still multi-digit numbers, ask if this means years, dates, ages or a number of parts of ... what do you hear? ... of something bigger. After this mathematical / logical part it now goes on to the decoding of name and word. Decode a word and interpret it? Maybe in the dream you had an encounter with a person named ... or you want to decode your own name or the name of someone you know. Maybe a word such as "teapot", "slotted ear", "nail file" or "what-the-hell" is stuck in your head and does not want to get out. Write down this word or name - preferably in lowercase or all in capital letters. Since each letter has four ways of being written, you proceed as follows and ask: · How many letters does the word have? · How many of these are capital letters? · Which? Put a dot under the letters. · How many are Greek? · Which? Place a dot over the letter. · Write the small letters blue, the big ones red, the Greek ones with dots on top. Back to the decoded number: Since you have replaced the individual numbers of the calculation with letters, you proceed as described above and ask in which of the four possible versions the letter is to be written. Now you ask if between the individual letters more letters - usually small and usually very few - are to be set. Write this down. Our example now has the form: 2 + 2 x 2 - 1 or replaced with letters from the first chapter: I + I x I - ONE (the "1" - wanted to be written in english) the check for capital letters resp. Greek gives the following: , I + I x, I - O'N'E Now let's write that down again: I + i x I - o'N'E Attention - many of the Greek capital letters are written the same. The point above is important. Now there are two more small letters added: I + ix I - o'N'E u, p Again, many lowercase Greek letters are spelled the same as in the German alphabet. The dot or apostrophe above is so important. Now the mathematical / logical phase has been played through and we come to the intuitive / interpretive phase: How do you get to a meaningful interpretation text? by writing the found term from the first phase vertically to each other so that there is - on the left side - room left for the basic meanings of the individual letters. Our example looks like this: mind I = give + u develop i = x mind I = send - 'p be o = love' N = inspiration 'E = Now you ask, to whom - mostly there are people, families, projects - refer to the basic meanings. At most you have to adjust the basic meaning of something. For example, the big T means BIRTH - but it can also be called start, new beginning, departure. The same applies to the other basic meanings. Make small marginal notes on who it is, e.g. the partner, a parent, the work colleague, the supervisor, a neighbor. Do not forget God, or if you don't like this term call him original creator of this cosmos. This usually turns on when you work with the abc-code. Give him a chance too. If you now intuitively try to formulate a text that contains all the basic meanings and additions in turn, you will receive a somewhat cryptic text that makes sense to you personally. Do not forget that you can not push yourself forward without asking yourself ... and that you yourself give the answers and explanations. The abc-code is like a vehicle whose operation I just have explained. But where you go ... ultimately, you decide yourself. My current example has meanwhile received the following meaning text: spirit I = the SPIRIT of the abc-code give + u give those, develop i = who develop themselves, spirit I = because their SPIRIT send – ‘p is sending for be o = being love ‘N = in LOVE inspiration ‘E = and INSPIRATION You want to receive a general message? So if you want to receive a - I call it ABC message - then grab your book or a piece of paper, colored pens and ask: How many digits does the message have? Mostly three, four or five, sometimes more. Then you ask each place, whether it contains a letter, a number, a color, a symbol or a component from another system, such as sequence or zodiac, for example. But most of the time it does not go that far. Ask each letter for capital or lowercase, and if they are in Greek. Recording numbers, you ask if there are results of calculations and derive them accordingly. Furthermore, intermediate links are still possible between the individual positions, usually small letters which serve as verbs for the connection of the individual nouns.

“Economics is the science which studies humanbehaviour as a relationship between given ends andscarce means which have alternative uses” (Robbins,1932)“Economics is a study of how people and society endup choosing with or without the use of money, toemploy scarce productive resources that could havealternate uses; it studies production of variouscommodities over time and their distribution forconsumption, now or in future, among various groupsin the society. It analyses costs and benefits ofimproving patterns of resource allocation.”(Samuelson, 1980) Health Economics• “The application of economic theory tophenomena and problems associatedwith health” (John Last) Why Health Economics?• Scarcity in Healthcare resources• Varied, ever increasing needs• Social, ethical and political aims• Allocating resource to fit the best need Scarcity in Healthcare Resources• Limited Budget• Human intensive industry• Expensive training• Technological progress• Distributional issues• Unexpected occurrence of diseases The Criterion for Economic EfficiencyEconomists definitions of efficiency encompasses 2 aspects:1. Technical Efficiency: the least costs method of achieving a given end2. Allocative Efficiency: Maximising the benefit obtained from available resources Areas of Health EconomicsExamples of Health Economics studies:• Cost of illness studies• Economic Evaluations• Health Impact Assessments• Health Technology Assessment• Health Financing• Equity, Priority Setting and Resource Allocations Cost of Illness/Disease Studies• Quantifies and Monetise the burden of a disease• Use to illustrate and present the burden of a disease in monetary term• Example: • Cost of UK Road Traffic Accidents • Cost of Smoking Cost of Illness Studies• Benefits: • Highlighting the magnitude of a problem • Compare diseases/problems using a common unit (monetary) • Provide information for economic evaluation studies• Issues: • Can be driven by: political, commercial, or other interest • Comparability between different studies • Do not offer explanation on how best to allocate resources Cost Analysis• Measure the range of costs within a particular aspect (or a disease)• Important aspect: • Cost perspective • Whose perspective is it? • NHS? • Employers? • Families? • Society? • Cost category: • Healthcare cost • Staff, Consumables, Overheads, Capital items, Other services • Transport cost • Out of pocket cost • Lost Production and earning • Measuring cost • Scope of costs: Width vs breadth • Costs vs Price/tariff • Time Horizon/Discounting • Currency for multi country setting• Analysing costs • Incomplete data • Outliers • Skewness (average or median cost?) Evidence Based Health Economics• So far, we have discussed Evidence in terms for effectiveness only. However, in reality, we are bound to many constraints – one of it is economic• And a range of competing alternatives• A specific clinically significant intervention does not necessarily means that we will automatically able to adopt it Economic Evaluations• Evaluating the costs and „benefit‟ of different programmes• The comparative analysis of alternative courses of action in terms of their costs and consequences.• A tool guide to resource allocation and priority setting in order to achieve efficiency Characteristics…• Economic evaluation has 2 characteristics 1. inputs and outputs (costs and consequences) 2. choice between at least 2 alternatives ConsequencesA Programme A CostsA Choice CostsB Comparator B ConsequencesB Cost in Economic Evaluation• The cost of a programme is defined as the opportunity cost• Benefit that is given up or forgone by making one choice over another Cost estimations• How Should Costs Be Estimated? Values for non-market items – volunteer time Capital outlays - Opportunity costs of funds tied up in the capital asset( discounting) Depreciation – best method – to annuitize the intitial capital outlay over the useful life of the asset – the equivalent annual cost. Types of economic evaluationType of Analysis Costs Consequences Result Identical in allCost Minimisation Money respects. Least cost alternative. Different magnitude of a Cost per unit of common measure eg.,Cost Effectiveness Money LY‟s gained, blood consequence eg. cost pressure reduction. per LY gained. Single or multiple effects Cost per unit of not necessarily common. Cost Utility Money Valued as “utility” eg. consequence eg. cost QALY per QALY. As for CUA but Net £ Cost Benefit Money valued in money. cost: benefit ratio. Cost Minimisation Analysis• Decision rule: • Find the programme with the least cost• Involve only cost analyses of programmes• Issue: Comparability between programmes: • Perspective • Cost widths and breadths • Time horizon • Discounting Cost Effectiveness Analysis • Find the programme that cost the least for the same level of output • When output are measured in a „natural scale‟ (e.g. death prevented, case prevented, etc) and are comparable across programmes • Example: • Cost effectiveness analysis of cervical cancer screening • Output measure: cost per potentially curable cancer detected • Issues? • Measuring cost: • Approaches: • top down • bottom up costing • Cost analyses • The cost of providing the alternative intervention • The cost of providing the usual care incremental cost (the difference of the cost required to implement the intervention to the usual care) • Measuring benefits • In „natural scale‟ • Death prevented • Subjects immunised • Etc Cost Effectiveness Analysis• Decision will be based on Incremental Cost Effectiveness Ratio Cost ICER Effectiven ess ICER Plane Incremental cost Intervention is more effective Intervention is less effective but more costly and more costly (COST EFFECTIVE?) (EXCLUDED) Incremental effect Intervention is less effective Intervention is more effective but less costly and less costly (QUESTIONABLE) (DOMINANT) Cost Utility Analysis• Find the best programme which cost the least for the same level of “health utility” unit• Measure output in „utility‟ terms• Measure the “quality of life” as well as life years gained• Note: Some would still call this type of studies as Cost Effectiveness Analyses Cost Utility Analysis• Methods of measuring Quality of life: • Survey: e.g. EQ-5D, SF36, SF • Visual analogue scale • Time trade off • Standard gamble EuroQoL EQ5D• Measure 5 dimensions • Mobility • Self Care • Usual Activities • Pain/Discomfort • Anxiety/Depression• The combination of answers are validated with the „population norms‟ to produce their Quality of Life Score Visual analogue scale Best imaginable health state Your Own Health State Today Worst imaginable health state Time Trade Off• A choice between two health states • A particular health state for a given number of years Or • Full health for a shorter period• When you are indifferent within the two choices, then you can calculate the Quality of life of a certain health state Standard Gamble• A choice of two health states with probabilities • A health state with certainty or • Perfect health with the probability of X (or death otherwise) QALY• Quality Adjusted Life Years QALYHealth Quality Index 1 10 x 1 = 10 QALYs 10 Years of Life QALYHealth Quality Index 1 0.5 10 x 0.5 = 5 QALYs 10 Years of Life Intervention that increase QALYs Health Quality Index 1 0.75 12 x 0.75 = 9 QALYs 0.5 10 12 Years of Life Intervention that improves quality of life and increasing life expectancy QALYHealth Quality Index 10.750.50.4 15 x 0.4 = 6 QALYs 10 12 15 Years of Life Intervention that reduces quality of life but increasing life expectancy Cost Benefit Analysis• Find the programme that has the highest rate of financial return• Measuring all output in monetary values• Natural units output needs to be converted to monetary values• Final analysis: find the highest Benefit Cost ratio Issue: How to monetise output• Example: • How much does saving one live worth? • Or How much does one live cost?• Approaches: • Human Capital Approach • Revealed Preference • Willingness to Pay • Willingness to Accept Valuing Human Life• How much does a human life cost?• How much does your right index finger cost?• How much does a painful back conditions „cost‟ you? Human Capital Approach• The amount of money that would be gained by a person if s/he do not have the health situation• Normally measured in salary received / loss of employment• Issue: • Productivity Gained • Gender • Efficiency CASP Economic Evaluation:A. Is the economic evaluationvalid? CASP Economic Evaluation:1. Was a well-defined question posed?Is it clear what the authors are trying to achieve?• What is the perspective of the analysis? Societal (gold standard), NHS and Social Care, households, financial organisations• How many options are compared? Against doing nothing? Usual care?• Are both costs and consequences considered?• What is the time horizon? Comparability, discounting 2. Are at least 2 alternatives compared? 1. Are both costs (inputs) and consequences (outputs) examined? NO YES Examines only Examines only consequences costs NO 2 PARTIAL EVALUATION 1A PARTIAL EVALUATION 1B • Outcome • Cost description. • Cost-outcome description. description. 3A PARTIAL EVALUATION 3B 4 FULL ECONOMIC EVALUATION • Efficacy or • Cost analysis. YES effectiveness • Cost-minimisation analysis. evaluation. • Cost-effectiveness analysis. • Cost-utility analysis. • Cost-benefit analysis. CASP Economic Evaluation:2. Was a comprehensive description ofthe competing alternatives given?Is there a clear decision tree (or similar informationgiven)Can you tell: who did what, to whom, where andhow often? CASP Economic Evaluation:3. Does the paper provide evidencethat the programme would be effective(i.e. would the programme do moregood than harm)?• Consider if an RCT or systematic review was used; if not, consider how strong the evidence was.• Economic evaluations frequently have to integrate different types of knowledge stemming from different study designs CASP Economic Evaluation:4. Were the effects of the interventionidentified, measured and valuedappropriately?• Effects can be measured in natural units (e.g. years of life) or more complex units (e.g. years adjusted for quality of life such as QALYs) or monetary CASP Economic Evaluation:B. How were consequences andcosts assessed and compared? CASP Economic Evaluation:5. Were all important and relevantresources required and healthoutcome costs for each alternativeidentified, measured in appropriateunits and valued credibly?• Identification of relevant costs and other relevant resource use (bear in mind the perspective being taken)• Measured accurately in appropriate units prior to evaluation? Appropriate units may be hours of nursing time, number of physician visits, years-of-life gained etc.• Credible valuation of the resource use: • Are the values realistic? • How have they been derived? • Have opportunity costs been considered? CASP Economic Evaluation:6. Were costs and consequencesadjusted for different times at whichthey occurred (discounting)?• Time value of money• Time horizon• Discounting both the costs and consequences CASP Economic Evaluation:7. What were the results of theevaluation?What• What is the bottom line?• What units were used (e.g. cost/life year gained, cost/QALY, Net benefit) CASP Economic Evaluation:8 Was an incremental analysis of theconsequences and costs ofalternatives performed?• Was an incremental analysis of the consequences and costs of alternatives performed? CASP Economic Evaluation:9. Was an adequate sensitivityanalysis performed?• Consider if all the main areas of uncertainty were considered by changing the estimate of the variable and looking at how this would change the result of the economic evaluation? CASP Economic Evaluation: C. Will the results help in purchasing for local people? CASP Economic Evaluation:10. Is the programme likely to beequally effective in your context orsetting?• Consider whether: • a) the patients covered by the review could be sufficiently different to your population to cause concern • b) your local setting is likely to differ much from that of the review. CASP Economic Evaluation:11. Are the costs translatable to yoursetting? CASP Economic Evaluation:12. Is it worth doing in your setting? Policy Context of Economic Evaluation• NICE – threshold value of a cost per QALY – £20,000- £30,000• Quality Agenda DH ( 1998) A First Class Service: Quality in the New NHS.• DH(2010) The NHS Outcomes Framework References• Donaldson, C., Mugford M. and Vale, L.(2002) Evidence– based health economics. From effectiveness to efficiency on systematic review. London: BMJ Books• Drummond, M. et.al. (2005) Methods for the economic evaluation of health care programmes - 3rd edition. Oxford: Oxford University Press• Wonderling, D., Gruen, R. and Black, N. (2005) Introduction to Health Economics. Maidenhead: Open University Press

- How do I start electrical engineering? - Do you have to be smart to be a electrician? - Do engineers actually use math? - Does electrical engineering require math? - How hard is the math in electrical engineering? - How is algebra used in electrical engineering? - Can I do engineering if I’m bad at math? - Are electrical engineers happy? - What is the hardest year of engineering? - Is electrical engineering hard? - Which engineering has highest salary? - Can I be an electrician if I’m bad at math? - Is being an electrician hard on your body? - Do electrical engineers use calculus? - What kind of math do electricians use? - What is the hardest subject in electrical engineering? - Is it worth becoming an electrical engineer? - Which engineering is the easiest? How do I start electrical engineering? Steps to Becoming an Electrical EngineerEarn a bachelor’s degree or higher. Take the Fundamentals of Engineering exam. Land an entry-level job and gain experience. Take Professional Engineer exam.. Do you have to be smart to be a electrician? Electrical installations require a measure of intelligence to do them correctly. When you get into designing installations, something Master Electricians do, then above average intelligence is a must because of circuit designs and knowledge of the Electrical Code. It’s a different kind of “smart”. Do engineers actually use math? All engineers take a few semesters of calculus, plus they often take other higher math classes like Ordinary Differential Equations or Linear Algebra. However, once they get into their career, there are many engineers that NEVER USE HIGHER MATH. … Personally, I use basic math and a ton of algebra. Does electrical engineering require math? Electrical engineering requires an enormous amount of applied math. Possibly more than the average engineering program. An undergraduate program generally requires three semesters of calculus, differential equations, linear algebra, and probability/statistics. Many students take a real analysis course. How hard is the math in electrical engineering? The mathematics part is not that hard. It just uses concepts of calculus,vector, linear algebra, differential equations etc. I can assure you that math is a major component of electrical engineering. But it’s not that tough. How is algebra used in electrical engineering? Linear algebra is useful for almost all kinds of engineering including EE. It’s useful for circuit analysis, linear systems analysis (in the controls and dynamical systems vein) and linear optimization. Probability theory is useful in things like information theory if you study communication systems. Can I do engineering if I’m bad at math? You don’t have to be good at math to be an engineer. However, you have to be able to pass math classes in engineering school to be an engineer. Once you get into the field of engineering, it is very possible to never touch calculus, differential equations, linear algebra etc. Are electrical engineers happy? Electrical engineers are below average when it comes to happiness. … As it turns out, electrical engineers rate their career happiness 3.1 out of 5 stars which puts them in the bottom 40% of careers. What is the hardest year of engineering? Originally Answered: What is the hardest year of engineering? Sophomore year may be considered the most difficult at your school because that is likely the year you begin taking “real engineering” classes and not just math, science, and other general requirements. Is electrical engineering hard? The electrical engineering major is considered one of the most difficult majors in the field, and these are the common reasons students list to explain why it is hard: There is a lot of abstract thinking involved. Which engineering has highest salary? In terms of median pay and growth potential, these are the 10 highest paying engineering jobs to consider.Computer Hardware Engineer. … Aerospace Engineer. … Nuclear Engineer. … Systems Engineer. … Chemical Engineer. … Electrical Engineer. … Biomedical Engineer. … Environmental Engineer.More items… Can I be an electrician if I’m bad at math? You can have a successful career as an electrician knowing nothing more than basic arithmetic and, maybe, very elementary algebra. Don’t be intimidated by the algebra, it’s not as difficult as it looks. The higher you go the more math you will need. … requires nothing more than basic arithmetic but lots of it. Is being an electrician hard on your body? The good news is that being an electrician isn’t too hard on your body – there are numerous jobs, including some trade jobs, that take a far harsher physical toll on those who do them. … Do electrical engineers use calculus? Calculus. As we move beyond resistor circuits and start to include capacitors and inductors, we need calculus to understand how they work. Think of calculus as a corequisite in parallel with electrical engineering. You don’t need to have a complete calculus background to get started, but it is helpful before too long. What kind of math do electricians use? Electricians use simple math, like addition, subtraction, multiplication, and division, to perform routine measurements and calculations at work. You’d be working with fractions, percentages, and decimals to figure out things like room dimensions, wiring lengths, convert watts to kilowatts, and calculate loads. What is the hardest subject in electrical engineering? Hardest Engineering MajorsElectrical Engineers are primarily focused on the physics and mathematics of electricity, electronics, and electromagnetism. … Students consider electrical engineering to be the toughest major mostly because of the abstract thinking involved.More items…• Is it worth becoming an electrical engineer? Yes, it will take a couple more years, but at the end you should have more employment opportunities and hopefully a much higher income. If you enjoy math and physics, it may be a better career for you. Electrical engineering is not for everyone, so it is up to you to make the call. … If so, it may be a good fit. Which engineering is the easiest? Civil/environmental engineering, industrial engineering, software engineering are all easier. Note that there is usually a distinction between Computer Engineering and Software Engineering. The former is more hardware focused (more akin to EE) and is more difficult.

Best answer: 3: s p d there is one s subshell that can contain 2e-s there are 3 d subshells for 6e-s there are 5 f subshells so 3 types and 9 total. 3 sub shells rector 1 decade ago 0 thumbs up 1 thumbs down how many subshells are there in the n = 3 principal shell more questions how many subshells are there in the n = 3 principal shell quantum numbers -- electron shells and subshells answer questions chemistry. In chemistry and atomic physics, an electron shell, or a principal energy level, shell has two subshells, called 2s and 2p the third shell has 3s, 3p, however, there are a number of exceptions to the rule for example palladium (atomic number 46). N=4 and higher: s, p, d and f the sublevels have various numbers of orbitals, which are regions of probability of finding an electron, and each orbital can have a maximum of two electrons. If they give shell# then you always have n^2 orbitals since it's 4th shell we have total of (4)^2 = 16 orbitals now, each orbital has 2 electrons so 162 = 32 electrons. Study 45 test 8-11 flashcards from charles b on studyblue how many subshells are there in the n = 2 principal shell 2 the subshell that has five orbitals and can hold up to ten electrons is the: the d subshell how many subshells are there in the n = 4 principal shell 4. Number of shubshell is equal to principle quantum numbern=1 -- 1 subshell n= 2--- 2 subshell n= p ---- p subhshell similarly n=4 --- 4 subshells ie 4s 4p 4d 4f. Subshells are the names for the different orientation for the shells an atom has in order to hold its electrons as you might have learned there are 4 quantum numbers, n, l, ml, and ms. Start studying chem exam 3 learn vocabulary, terms, and more with flashcards, games, and other study tools how many subshells are there in n=4 principal shell 4 (n) and specifies the principal shell of the orbital ground state lowest energy state excited state. In my textbook, it says that the maximum number of electrons that can fit in any given shell is given by 2n² this would mean 2 electrons could fit in the first shell, 8 could fit in the second shell, 18 in the third shell, and 32 in the fourth shell. Solution (a) there are four subshells in the fourth shell, corresponding to the four possible values of l (0, 1, 2, and 3) (b) these subshells are labeled 4 s, 4 p, 4 d, and 4 f the number given in the designation of a subshell is the principal quantum number, n the following letter designates the value of the azimuthal quantum number, l. For the principal quantum number n = 4,we have the 4th shell of whichthe orbital quantum numbers are l = 0, 1, 2, shell,orbitals,are,equals,how,many,in,the,how many orbitals are in the n equals 4 shell related are there different types of shadow people. 155 shells, subshells, and orbitals - bohr's model predicted that energy levels (called shells) were enough to describe completely how electrons were arranged around an atom but there's more to it shell: equivalent to bohr's energy levels electrons in the same shell are all the same distance from the nucleus they all have similar (but not. In chemistry and atomic physics, an electron shell, or a principal energy level, shell has two subshells, called 2s and 2p the third shell has 3s, 3p, however, there are a number of exceptions to the rule for example palladium. The k shell can hold up to 2 electrons, the l shell can hold up to 8 electrons, the m shell can occupy up to 18 electrons subshells each shell is composed of one or more subshells. The first shell has just the one orbital, the 1s the second shell has two sub-shells, the 2s and the 2p there are three p orbitals in the 2p sub-shell each orbital can hold two electrons, so there are eight electrons maximum in the second shell. The n=___ principal shell is the lowest that may contain a d-subshell 23)the n=____ principal shell is the lowest that may contain a d-subshell more questions. There are three subshells in principal quantum level \(n=3\) since \(l=3\) refers to the f subshell, the type of orbital represented is 4f (combination of the principal quantum number n and the name of the subshell. Inside every shell there is one or more subshells (s, p, d, f, etc) inside every subshell there is one or more orbitals the best way to explain this is using the quantum numbers: where, n: principal quantum number l: angular momentum quantum number the s subshell contains 1 orbital. What is the change in temperature of a 250 l system when its volume is reduced to 175 l if the initial temperature was 298 k a) 209. Principal quantum number (n): n = 1, 2, 3,, ∞ specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot. The n = 3 shell, for example, contains three subshells: the 3s, 3p, and 3d orbitals possible combinations of quantum numbers there is only one orbital in the n = 1 shell because there is only one way in which a sphere can be oriented in space. Question: how many subshells are in the n=4 shell how many orbitals are in the n=3 shell show transcribed image text how many subshells are in the n=4 shell how many orbitals are in the n=3 shell expert answer 100 % (12 ratings) get this answer with chegg study view this answer or. Atomic orbitals starting with the energy level/shell closest to the nucleus electrons fill the different shells in order of increasing energy. How many subshells exist on energy level shell n plus equals 4 for the case of n=4 the available orbitals include 1s 3p and 5d, a total of 9 electron orbitals which can occupy 18 electrons.

One erg converted into joule equals = 0.00000010 J 1 erg = 0.00000010 J The user must fill one of the two fields and the conversion will become automatically. 1 Joule is equal to 107 erg. E) 10 7 ergsc 2 joule to erg = 20000000 erg Median response time is 34 minutes and may be longer for new subjects. The answer is: 1 J equals 10,000,000.00 dyn cm. In 1922, chemist William Draper Harkins proposed the name micri-erg as a convenient unit to measure the surface energy of molecules in surface chemistry. How to convert energy of 1 kilojoules (kJ) to joules (J). ›› Quick conversion chart of joule to erg. 1 joule to erg = 10000000 erg. To distinguish derived units, they recommended using the prefix "C.G.S. More information from the unit converter. Physics. 1 mJ is equal to 0.001 joule. One erg is equal to the work done by a force of one dyne, when its point of application moves 1cm in the direction of action . One watt converted into erg per second equals = 10,000,000.00 erg/sec 1 W = 10,000,000.00 erg/sec Its name is derived from ergon (ἔργον), a Greek word meaning 'work' or 'task'. The answer is: 1 erg/s equals 0.00000010 J/s. Ask your question. *Response times vary by subject and question complexity. An erg (short for ergon, a Greek word meaning "work/task") is a unit of energy and mechanical work equal to 10?7 joules From kingofmates' reference Aug 27 2014, 8:59 AM It is defined as the amount of work done by a force of one dyne exerted for a distance of one centimeter. is that joule is in the international system of units, the derived unit of energy, work and heat; the work required to exert a force of one newton for a distance of one metre also equal to the energy of one watt of power for a duration of one second symbol: j while erg is the unit of work or energy, being the amount of work done by a dyne working through a distance of one centimeter equal to 10 −7 joules or erg can be … How many mJ in 1 joule? C) 10 2 ergs. In 1873, a committee of the British Association for the Advancement of Science, including British physicists James Clerk Maxwell and William Thomson recommended the general adoption of the centimetre, the gramme, and the second as fundamental units (C.G.S. One Joule is 1 Newton Metre, ie the work done or energy transfered to an object when a one Newton force acts on it over one metre. A) 10 –1 ergs. U is for Energy. The joules per second unit number 0.00000010 J/s converts to 1 erg/s, one erg per second. Q: How many Foot-Pound Force in 10000 Joules? So 1 kilojoule (kJ) is equal to 1000 joules (J): 1 kJ = 1000 J . m). unit of ..." and requested that the word erg or ergon be strictly limited to refer to the C.G.S. erg joule . The erg is not an SI unit. Exchange reading in ergs unit erg into joules unit J as in an equivalent measurement result (two different units but the same identical physical total value, which is also equal to their proportional parts when divided or multiplied). It is equal to the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of the force's motion through a distance of one metre (1 newton metre or N⋅m). Conversion of 1 joule into erg by a very simple method. Log in. For the micro-electroretinogram, see, British Association for the Advancement of Science, General Conference of Weights and Measures, "S is for Entropy. Filed under: Main menu • energy menu • Ergs conversion, * Whole number, decimal or fraction ie: 6, 5.33, 17 3/8 * Precision is how many digits after decimal point 1 - 9, Convert erg (erg) versus joules (J) in swapped opposite direction from joules to ergs, Or use utilized converter page with the energy multi-units converter. Conversion of 1 joule into erg by a very simple method. On the surface of the Earth, it takes about 0.98 ergs to lift a 1 milligram object by 1 centimeter. Pagkakaiba ng pagsulat ng ulat at sulating pananaliksik? The erg has not been a valid unit since 1 January 1978 when the EEC[clarification needed] ratified a directive of 1971 which implemented the International System (SI) as agreed by the General Conference of Weights and Measures. 0.00000010 J/s is converted to 1 of what? One erg is equal to the work done by a force of one dyne, when its point of application moves 1cm in the direction of action ... 10000000. Exchange reading in watts unit W into ergs per second unit erg/sec as in an equivalent measurement result (two different units but the same identical physical total value, which is also equal to their proportional parts when divided or multiplied). erg = J _____ 0.00000010000. A joule is a unit of energy in the International System of Units and named after the British physicist James Prescott Joule. Additionally, what do you mean by 1 Joule? The unit name “joule” is in honor of the English physicist James Prescott Joule. ", "Journal of the American Chemical Society - Issues for 1898-1901 include Review of American chemical research, v. 4-7; 1879-1937, the society's Proceedings", "Are ergs commonly used in astrophysics? Do a quick conversion: 1 ergs = 1.0E-7 joules using the online calculator for metric conversions. Why don't libraries smell like bookstores? The SI derived unit for energy is the joule. The SI derived unit for energy is the joule. Ergs to Joules (erg to J) conversion calculator of Energy measurement, 1 erg = 1.0E-7 joules. [verification needed], An erg is the amount of work done by a force of one dyne exerted for a distance of one centimetre. If so, is there a specific reason for it? TNT equivalent is a convention for expressing energy, typically used to describe the energy released in an explosion. This tool converts erg to joules (erg to j) and vice versa. B) 10 ergs. The joule is the S.I. Log in. Furthermore, what is the CGS unit of 1 Joule equal to? ", https://en.wikipedia.org/w/index.php?title=Erg&oldid=996460345, Short description is different from Wikidata, Articles containing Ancient Greek (to 1453)-language text, Wikipedia articles needing factual verification from July 2020, Wikipedia articles needing clarification from September 2019, Creative Commons Attribution-ShareAlike License, This page was last edited on 26 December 2020, at 19:09. It has the symbol erg. In the CGS base units, it is equal to one gram centimetre-squared per second-squared (g⋅cm2/s2). It is thus equal to 10−7 joules or 100 nanojoules (nJ) in SI units. 10,000,000.00 dyn cm is converted to 1 of what? Q: How many Joules in 1 Foot-Pound Force? The erg is not an SI unit. The symbol for joule is J. The erg is a unit of energy equal to 10−7 joules (100 nJ). System of Units). It is the EQUAL energy value of 1 joule … Joules to Ergs table. Technical units conversion tool for energy measures.

This episode explains the difference between independent and dependent variables in psychology experiments written by chris mayhorn produced by bypass publi. Just as including multiple dependent variables in the same experiment allows one to answer more research questions, so too does including multiple independent variables in the same experiment. Difference between independent and dependent variables are examples of variables in research for easier understanding of what is discussed in the paper. The two main variables in an experiment are the independent and dependent variable an independent variable is the variable that is changed or controlled in a scientific experiment to test the effects on the dependent variable a dependent variable is the variable being tested and measured in a . The dependent variable is what is affected by the independent variable-- your effects or outcomes for example, if you are studying the effects of a new educational program on student achievement, the program is the independent variable and your measures of achievement are the dependent ones. I need help in identifying and defining my variable from a research question and defining my variable from a research and define my dependent/independent . Dependent and independent variables it is important for scientists to control variables when conducting research this online lab will give you the opportunity to view how the european corn borer affects the yield of corn plants. Writing an apa paper 1 running head: writing an apa report 1989) methods in behavioral research identify the independent and dependent variables and the . This video covers identifying independent and dependent variables it includes four examples. Dependent variables in entrepreneurship research as an independent research domain, sections of the paper — of phenomena and related dependent variables . When writing a research paper, the research question or hypothesis is the spine relationship between the independent and dependent variables for example . Independent and dependent variables (independent and dependent variables are related to one another the dependent variables in the analysis of the data encompass ‘ten specific self-reported offenses and an offense index’ (field & mears, 2002, p22). Learn about the different types of variables in research the different types of variables you may come what dependent and independent variables . The independent variable, also known as the manipulated variable, is the factor manipulated by the researcher, and it produces one or more results, known as dependent variables this article is a part of the guide:. What is the relationship between qualitative research and variables be based on identify and exam the impact of independent variables on the dependent variable chosen analysis in a paper . The second variable changes in response to the purposeful change this is the dependent variable or responding variable for example, if students change the wing shape of a paper airplane and measure the resulting time that the plane stays in flight, the independent variable would be the wing shape and the dependent variable would be the . Research project paper and feasibility paper: week 2 as preparation for the final research paper, formulate a theory about the correlation between measurable independent variables (causes) and one measurable dependent variable (the effect). Research variables: dependent, independent, control, extraneous & moderator test your ability to understand how different variables in research can impact a study's outcome quiz questions . For any research article, be able to correctly identify the independent and dependent variables, and for each independent variable, correctly determine the number of groups for that variable, whether each independent variable was an active or attribute variable. Dependent and independent variables acting as a buffer against unknown research variables, might involve some children eating a food type some research papers . Sample paper for learning research proposal research summary to the next as you write your introduction include your independent and dependent variables in . Sampling & variables of sampling techniques commonly used in research projects and will discuss dependent and independent variables of a research paper. Research question for illegal immigration i need to do a research paper about illegal immigration i need a dependent variable and a couple independent variables. Sociologists classify social phenomena being studied as independent and dependent variables understanding social research requires knowledge of the difference . In this section, we will focus on how to identify and distinguish independent from dependent variables, and the roles these variables play in a research study independent variables in experimental research, an investigator manipulates one variable and measures the effect of that manipulation on another variable. Research papers will mention a variety of different variables, and, at first, these technical terms might seem difficult and confusing but with a little practice, identifying these variables becomes second nature because they are sometimes not explicitly labeled in the research writeup, it is . Organizing your social sciences research paper: independent and dependent variables the purpose of this guide is to provide advice on how to develop and organize a research paper in the social sciences.

We found this to be accurate. Once the appropriate equation has been selected, the physics problem becomes transformed into an algebra problem. Velocity is a vector and therefore has both a direction and a magnitude. Three common kinematic equations that will be used for both type of problems include the following: Advanced Senior Physics Edited by N. A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. Pool A pool ball leaves a 0. We calculated the value of gravity to be Solving Projectile Problems To illustrate the usefulness of the above equations in making predictions about the motion of a projectile, consider the solution to the following problem. An organized listing of known quantities as in the table above provides Projectile motion essay for the selection of the strategy. Because horizontal and vertical information is used separately, it is a wise idea to organized the given information in two columns — one column for Projectile motion essay information and one column for vertical information. The following procedure summarizes the above problem-solving approach. In this part of Lesson 2, we will focus on the first type of problem — sometimes referred to as horizontally launched projectile problems. Thus, it would be reasonable that a vertical equation is used with the vertical values to determine time and then the horizontal equations be used to determine the horizontal displacement x. This value agrees with the value of given in our PHYS lab workbook of 9. We calculated the value of the acceleration due to gravity to be The two sets of three equations above are the kinematic equations that will be used to solve projectile motion problems. Usually, if a horizontal equation is used to solve for time, then a vertical equation can be used to solve for the final unknown quantity. The true value of this acceleration is 9. After we had manipulated the data to calculate the vertical velocity during each time period we produced a relatively linear graph see Figure 2. While problems can often be simplified by the use of short procedures as the one above, not all problems can be solved with the above procedure. The solution of this problem begins by equating the known or given values with the symbols of the kinematic equations — x, y, vix, viy, ax, ay, and t. Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. Predictable unknowns include the time of flight, the horizontal range, and the height of the projectile when it is at its peak. This was as expected as we needed a linear graph to calculate the value of gravity. Identify the unknown quantity that the problem requests you to solve for. Equations for the Horizontal Motion of a Projectile The above equations work well for motion in one-dimension, but a projectile is usually moving in two dimensions — both horizontally and vertically. This is due to the direction of the ball. References All results were taken from my PHYS Lab Book 2Calculation of vertical velocity uncertainties I was unsure as to how to draw the error bars on the vertical velocity as the uncertainty in time was stated to be negligible. Examples of this type of problem are a. With the time determined, use one of the other equations to solve for the unknown. The value we calculated was found to be A soccer ball is kicked horizontally off a Determine the time of flight, the horizontal distance, and the peak height of the football. The sole reliance upon 4- and 5-step procedures to solve physics problems is always a dangerous approach. This graph is only showing the vertical velocity so half of the projectile motion movement will be negative due to the ball falling. For convenience sake, make a table with horizontal information on one side and vertical information on the other side. This was an optimal result and complied with what we expected. Thus, the three equations above are transformed into two sets of three equations. Select either a horizontal or vertical equation to solve for the time of flight of the projectile. Determine the initial horizontal velocity of the soccer ball.Projectile motion is a form of motion where gravitational acceleration influences the object thrown to travel in a curved path or trajectory. The projectile consists of two motions, which are the horizontal direction that does not change (velocity remain constant) and the vertical direction that is 3/5(1). Lab projectile motion Essay Sample. 1. Research to find equations that would help you find g using a pendulum. Design an experiment. Projectile Motion Purpose: Apply the concepts of two-dimensional kinematics (projectile motion) to predict the impact point of an object as its velocity increases. Introduction: The most common example of an object that is moving in two dimensions is a projectile. Theory: Projectile motion according to Dr. James S. Walker is defined as, “the motion of objects that are initially launched –or “projected”- and that then continue moving under the influence of gravity alone” (82). Free Essay: Freefall and Projectile Motion Introduction and Objectives This lab experiment was done to determine the characteristics of free fall and. Projectile Motion Experiment # 4 Introduction: Projectile Motion exists commonly in our everyday lives and is particularly evident in the motion or flight of objects which are projected from a particular height. The key to working with projectile motion is recognizing that when an object with mass is flying through the air, its motion is a combination of .Download

- Stability spectrum model theory, a branch of mathematical logic, a complete first-order theory "T" is called stable in λ (an infinite cardinal number), if the Stone space of every model of "T" of size ≤ λ has itself size ≤ λ. "T" is called a stable theory if there is no upper bound for the cardinals κ such that "T" is stable in κ. The stability spectrum of "T" is the class of all cardinals κ such that "T" is stable in κ. For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability. The stability spectrum theorem for countable theories Theorem.Every countable complete first-order theory "T" falls into one of the following classes: * "T" is stable in λ for all infinite cardinals λ. – "T" is totally transcendental. * "T" is stable in λ exactly for all cardinals λ with λ ≥ 2ω. – "T" is superstable but not totally transcendental. * "T" is stable in λ exactly for all cardinals λ that satisfy λ = λω. – "T" is stable but not superstable. * "T" is not stable in any infinite cardinal λ. – "T" is unstable. The condition on λ in the third case holds for cardinals of the form λ = κω, but not for cardinals λ of cofinality ω (because λ < λcof λ). Totally transcendental theories A complete first-order theory "T" is called totally transcendental if every formula has bounded Morley rank, i.e. if RM(φ) < ∞ for every formula φ("x") with parameters in a model of "T", where "x" may be a tuple of variables. It is sufficient to check that RM("x"="x") < ∞, where "x" is a single variable. For countable theories total transcendence is equivalent to stability in ω, and therefore countable totally transcendental theories are often called ω-stable for brevity. A totally transcendental theory is stable in every λ ≥ |"T"|, hence a countable ω-stable theory is stable in all infinite cardinals. Every uncountably categorical countable theory is totally transcendental. This includes complete theories of vector spaces or algebraically closed fields. The theories of groups of finite Morley rank are another important example of totally transcendental theories. A complete first-order theory "T" is superstable if there is a rank function on complete types that has essentially the same properties as Morley rank in a totally transcendental theory. Every totally transcendental theory is superstable. A theory "T" is superstable if and only if it is stable in all cardinals λ ≥ 2|"T"|. A theory that is stable in one cardinal λ ≥ |"T"| is stable in all cardinals λ that satisfy λ = λ|"T"|. Therefore a theory is stable if and only if it is stable in some cardinal λ ≥ |"T"|. Most mathematically interesting theories fall into this category, including complicated theories such as any complete extension of ZF set theory, and relatively tame theories such as the theory of real closed fields. This shows that the stability spectrum is a relatively blunt tool. To get somewhat finer results one can look at the exact cardinalities of the Stone spaces over models of size ≤ λ, rather than just asking whether they are at most λ. The uncountable case For a general stable theory "T" in a possibly uncountable language, the stability spectrum is determined by two cardinals κ and λ0, such that "T" is stable in λ exactly when λ ≥ λ0 and λμ = λ for all μ<κ. So λ0 is the smallest infinite cardinal for which "T" is stable. These invariants satisfy the inequalities *κ ≤ |"T"|+ *κ ≤ λ0 *λ0 ≤ 2|"T"| *If λ0 > |"T"|, then λ0 ≥ 2ω When |"T"| is countable the 4 possibilities for its stability spectrum correspond to the following values of these cardinals: *κ and λ0 are not defined: "T" is unstable. *λ0 is 2ω, κ is ω1: "T" is stable but not superstable *λ0 is 2ω, κ is ω: "T" is superstable but not ω-stable. * λ0 is ω, κ is ω: "T" is totally transcendental (or ω-stable) Spectrum of a theory title=A course in model theory. An introduction to contemporary mathematical logic|series= Universitext|publisher=Springer|place= New York|year= 2000|pages=xxxii+443 |isbn= 0-387-98655-3 Translated from the French *Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Classification theory and the number of nonisomorphic models | origyear=1978 | publisher=Elsevier | edition=2nd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-70260-9 | year=1990 Wikimedia Foundation. 2010. Look at other dictionaries: Stability radius — The stability radius of a continuous function f (in a functional space F ) with respect to an open stability domain D is the distance between f and the set of unstable functions (with respect to D ). We say that a function is stable with respect… … Wikipedia Spectrum Saloon Car (SSC) — The Spectrum Saloon Car (SSC), or Spectrum Patrol Car (SPC), is a fictional vehicle from Gerry Anderson s Science fiction television series Captain Scarlet and the Mysterons (1967). Appearance in Captain Scarlet and the Mysterons Accessible only… … Wikipedia Full Spectrum Warrior — North American cover Developer(s) Pandemic Studios Publisher(s) THQ … Wikipedia Political spectrum — A political spectrum (plural spectra) is a way of modeling different political positions by placing them upon one or more geometric axes symbolizing independent political dimensions.Most long standing spectra include a right wing and left wing,… … Wikipedia Emotional spectrum — The emotional spectrum is a fictional concept in the DC Comics shared universe, primarily within the Green Lantern mythos. Within the comics, it is the power source for the power rings of the Green Lantern Corps. The spectrum is divided into the… … Wikipedia USAF Stability and Control DATCOM — The United States Air Force Stability and Control DATCOM is a collection, correlation, codification, and recording of best knowledge, opinion, and judgment in the area of aerodynamic stability and control prediction methods. It presents… … Wikipedia List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia Stable theory — For differential equations see Stability theory. In model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be… … Wikipedia List of mathematical logic topics — Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic … Wikipedia Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity. Computer scientist Manindra Agrawal of the… … Universalium

Some of the equipment’s like tower or reactors are very critical equipment in refineries due to very high temperature and pressure. The temperature is also very high and coke tower works in the creep range for some duration of one full cycle. In such cases, creep along with fatigue plays a very important role in the failure of coke drums. We will consider the example of coke tower having design temperature as 471 Deg. C and working temperature of 149 Deg. C at the top portion and 441 Deg. C at the bottom portion. Below is the histogram for the thermal loading for the top and the bottom portion of the code tower – Thermal Histogram – The skirt to shell junction i.e. Y forging is a critical portion of coke tower because it is highly susceptible to creep and fatigue simultaneously. API 579-1/ASME FFS-1 2007 is used for the creep-fatigue interaction in the present case. A non-linear transient thermal analysis is coupled with the elastic-plastic structural analysis for calculation of stresses and strains. These stresses are further used in deducing a creep damage factor and strains are used for deducing permissible cycles for fatigue as per the approach based on API 579-1. An MPC Omega method code is developed in the Finite Element Analysis Software ANSYS 17.2 for calculation of the creep damage factor and permissible cycles for fatigue. MPC Omega method code validated through manual calculation. Now we will see analysis steps to calculate the creep-fatigue life of the equipment for thermal & pressure variation – Transient Thermal Loading Conditions: Loading for Thermal Loading – Figure 1 – Thermal loading as per temperature histogram Table 1 – Convection calculations for shell Table 2 – Convection calculations for the skirt Transient Structural Analysis – Loading & Boundary conditions for Transient Structural Analysis: Figure 2- Imported Body Temperature Figure 3- Coupled transient thermal + transient structural analysis layout Figure 4 – Self-weight & Internal Pressure Figure 5 – Cyclic pressure for top/outlet section of tower Figure 6 – Cyclic pressure for bottom/inlet section of tower Figure 7 – Nozzle Thrusts Figure 8 – Nozzle Process Loads Figure 9 – Moment loads on Nozzles Figure 10 – Nozzle Force & Moment Figure 11 – Nozzle Force for Nozzle N4 – Detailed view of nozzle load loading Figure 12 – Moment loads on Nozzles – Detailed view of moment loading Figure 13 –Boundary Condition-Fixed Support Transient Thermal Results • Temperature Graph at all time step: – • Temperature @ 3600 sec • Temperature @ 7200 sec • Temperature @ 30600 sec • Temperature @ 37800 sec • Temperature @ 50400 sec • Temperature @ 54900 sec • Temperature @ 72000 sec Coupled Transient Structural Results • Transient Equivalent von mises Stress plot for cycle • Maximum Equivalent Von Mises Stress @ 3600 • Minimum Equivalent von Mises Stress @ 30600 Fatigue Life Calculation for Data Cases according to ASME Section VIII, Division 2, Part 5. According to design data, design number of cycles are 400/year for cyclic temperature & pressure cycle, considering 20 years of design life, the total number of design cycles will be 8000. From FEA maximum and minimum stress for the cycle are 305.42 MPa & 67.579 MPa respectively occurred at the skirt to dished end junction as shown in above figure, So the component stress range, ΔSP, k is 237.84 MPa. According to point number 22.214.171.124 (ASME Section VIII, Division 2, Part 5, Point 126.96.36.199) the effective alternating stress amplitude for the kth cycle. ΔSP, k =237.84 N/mm2 ————(The component stress range between two-time point) Kf = Fatigue Strength reduction factor = 1.2 (ASME Section VIII, Div. 2, Table 5.11) Ke, k = Fatigue Penalty Factor = 1.0 (since ΔSn,k < SPS ) After solving this we get, Salt, k = 142.70 N/mm2 To calculate the design no. of cycles following formula is used Where ET = Young’s modulus for material. The coefficients C1, C2… are calculated from table 3.F.1 – Coefficient of fatigue curves. After solving the above equations, we get, *Et is considered at average cycle temperature. The fatigue damage factor(Df,k) is 0.13339 which is much less than unity. As fatigue damage factor is much less than unity, the design is safe. Creep-Fatigue damage factor calculation as per API 579 Cl 10.5.2.4 (c) For Skirt to dish end junction The cycle starts with 177 °C & after reaching 441°C in one hour, the cycle will hold at 441°C for next 8000 hours (assuming it as a maximum time of creep range in entire 8000 cycles). Additional 15 min is included at starting the cycle to stabilize temperature from ambient to 177°C. Pressure (0.054917 MPa) is considered throughout the cycle Loading & Boundary Condition – • Transient Thermal load – Temperature • Transient Thermal load – Convection • Transient Structural Load – Internal Pressure • Transient Structural boundary condition Note: In displacement boundary condition, X-directional displacement is constrained and Y-directional displacement is Free. • Transient Thermal plot for the cycle • Temperature counter plot at the end of the cycle • Transient Equivalent Von Mises Stress plot for cycle • Maximum Equivalent Von Mises Stress plot at the end of the cycle • Accumulated inelastic (creep) strain plot at the end of the cycle Creep damage factor calculation as per API 579 Cl 10.5.2.4 (C) Solver Output Result (MPC Omega Method Output from Finite Element Analysis Software): Creep damage factor(Dc) is 3.058E-05 at the end of the cycle and total accumulated inelastic (creep) strain is 7.88E-06. So, as per API 579 CL 10.5.2.4 (g) for total creep damage(3.058E-05) & total accumulated inelastic strain(7.88E-06) are satisfied the limit DcAllow=1 & the equivalent total accumulated inelastic strain (0.005) respectively. As the design is acceptable for this loading conditions. As per 10.5.3 Creep-Fatigue Interaction, Creep damage factor (3.058E-05) and fatigue damage factor (0.13339) are lying down in the chart of figure 10.29 (creep-fatigue damage criteria) for carbon steel. Thus, the design is acceptable for this loading conditions.