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http://erkdemon.blogspot.com/2009/12/snowflake-fractal.html

math

If you want something that looks more like a snowflake than the previous hexagonal carpet, you could always use the "Koch Snowflake" fractal, which is gotten by repeatedly adding triangles to the sides of other triangles. But every single general text on fractals seems to include the Koch. I mean, don't get me wrong, it's a fairly pleasant shape, but after the nth "fractals" text slavishly copying out exactly the same fractal set-pieces, you start to think ... guys, could we have a little bit of variation pleeeeaaase? So here's a different snowflake. This one's built from hexagons. Each hexagonal corner forms a nucleation site that attracts a cluster of three smaller hexagons, and their free corners in turn attract clusters of three smaller ... you get the idea. The sample image has been drawn with about six thousand hexagons. The resulting "snowflake" outline is really very similar to the Koch, but the internal structure's a bit more spicy. A suitable design for Christmas cards for mathematicians, perhaps.

s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125654.80/warc/CC-MAIN-20170423031205-00384-ip-10-145-167-34.ec2.internal.warc.gz

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https://www.eeeguide.com/line-to-line-fault/

math

Line to Line Fault (LL): Figure 11.7 shows a Line to Line Fault at F in a power system on phases b and c through a fault impedance Zf. The phases can always be relabelled, such that the fault is on phases b and c. The currents and voltages at the fault can be expressed as The symmetrical components of the fault currents are from which we get The symmetrical components of voltages at F under fault are Writing the first two equations, we have From which we get Substituting Ib from Eq. (11.15) in Eq. (11.14), we get Equations (11.11) and (11.16) suggest parallel connection of positive and negative sequence networks through a series impedance Zf as shown in Figs. 11.8a and b. Since Ia0 = 0 as per Eq. (11.12), the zero sequence network is unconnected. In terms of the Thvenin equivalents, we get from Fig. 11.8b From Eq. (11.15), we get Knowing Ia1,we can calculate Va1 and Va2 from which voltages at the fault can be found. If the fault occurs from loaded conditions, the positive sequence network can be modified on the lines.

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https://www.ellibs.com/book/9780470055458/regression-analysis-by-example

math

Regression analysis is a conceptually simple method for investigating relationships among variables. Carrying out a successful application of regression analysis, however, requires a balance of theoretical results, empirical rules, and subjective judgement. Regression Analysis by Example, Fourth Edition has been expanded and thoroughly updated to reflect recent advances in the field. The emphasis continues to be on exploratory data analysis rather than statistical theory. The book offers in-depth treatment of regression diagnostics, transformation, multicollinearity, logistic regression, and robust regression. This new edition features the following enhancements: Regression Analysis by Example, Fourth Edition is suitable for anyone with an understanding of elementary statistics. Methods of regression analysis are clearly demonstrated, and examples containing the types of irregularities commonly encountered in the real world are provided. Each example isolates one or two techniques and features detailed discussions of the techniques themselves, the required assumptions, and the evaluated success of each technique. The methods described throughout the book can be carried out with most of the currently available statistical software packages, such as the software package R. An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

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https://www.math.ias.edu/seminars/abstract?event=119315

math

|Homological Mirror Symmetry Seminar| |Topic:||Riemann-Hilbert correspondence revisited| |Affiliation:||Kansas State University| |Date:||Wednesday, November 2| |Time/Room:||10:45am - 12:00pm/West Building Lecture Hall| Conventional Riemann-Hilbert correspondence relates the category of holonomic $D$-modules (de Rham side) with the category of constructible sheaves (Betti side). I am going to reconsider this relationship from the point of view of deformation quantization (on the de Rham side) and Fukaya categories (on the Betti side). Besides of useful re-interpretations of some classical results (e.g. Deligne-Malgrange Riemann-Hilbert correspondence for irregular connections on curves), this point of view allows us to conjecture some new results, e.g.the Riemann-Hilbert correspondence for difference equations (e.g. quantum spectral curves). Contents of the talk is a part of a bigger project called "Holomorphic Floer theory", joint with Maxim Kontsevich.

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http://www.economist.com/comment/1521780

math

Data Isorare Sep 25th 2013 22:21 GMT Been looking for years but still no good answer to data is/are. Your post raises another question. I thought periods and commas belong inside a closing quotation mark rather than outside. "statistics." "statistics". Can you check this out? Data Isorare in reply to Data Isorare Sep 25th 2013 23:52 GMT Never mind, methinks I understand it now "English English". vs. "American English."

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http://rafimoor.com/english/GRE1.htm

math

The Principles of General Relativity Once when Einstein was preparing for a review of his (not yet called special) theory of relativity, he thought about the fact that a man falling from the roof of a building doesn't feel his own weight. This thought which he later described as "The happiest thought of my life", was the seed from which the theory of General Relativity grew. The idea of general relativity is not very hard to understand. The mathematics of it is quite complicated and involves curved space geometry that is not easy to comprehend. Einstein had struggled with the mathematics of his theory for several years before he got to the correct version of his famous field equation. Though looks quite simple, this equation actually includes 10 different differential equations, and cannot be used in practice as it is. Einstein did not expect exact solutions for his equation to come soon. Surprisingly the first solution for the equation was found by Karl Schwarzschild a few months after Einstein published his final version of the General theory of Relativity in 1915. This solution describes the gravity field around a massive static spherical body. No other solutions were found until the sixties when new mathematical tools where developed and computers became available. If a space-ship is in a free fall, everything in it seems weightless. A man inside a closed spaceship would not be able to tell whether his spaceship is freely falling or cruising at a constant speed in the interstellar space where no significant gravity exists. Any mechanical experiment he might do would give the same result in both cases. Also, the man in the closed spaceship would not be able to tell whether his spaceship is parking on the surface of a planet or accelerating at a constant acceleration in the interstellar space. Einstein suggested that this is not just a similarity in the behavior but actually the same physical states. In other words, a freely falling frame in a gravity field is equivalent to an inertial frame with the absence of gravity. Also, a static frame in a gravity field is equivalent to an accelerating frame with the absence of gravity. But wait! What about light? If this equivalence is true than in a gravity field, light pulse move in a straight line relative to a freely falling frame but relative to a static frame it is bent downwards (Fig. 1). Is light also affected by gravity? General Relativity predicts that it does, and this prediction has been confirmed by observations of star light and by experiments. Einstein said that gravity can be looked at as curvature in space-time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and not a force, but the question what gravity really is, is a philosophic question, not a physical one and we will not get into it here.) How can we explain the behavior of objects in a gravity field by curvature of space-time? To understand this let us look at a curved system that is known to all of us - the surface of Earth. First we have to find an equivalent to a strait line on our curved surface. We can define such line as the shortest path between two points, or alternatively, a line that as we go along it we don't turn relative to the curved surface (we only turn with the curvature of the surface). Such line is called a geodesic. On a spherical surface it will be a great circle, that is, a circle that its center is at the center of the sphere. Longitudes, for instance, are geodesics. Now, imagine two ships starting at two points on the equator and moving at the same velocity strait to the north. At first they sail parallel to each other. But after some time they start getting closer and closer in the east-west direction. If time-space is curved in such way that two geodesics in time direction get closer and closer in space, we will see two bodies that start at rest relative to each other (moving only in time) beginning to move towards each other just as if some force attracts them. What if we want to keep a constant distance between two bodies in a gravity field? Let's look again on the model of the Earth surface. The distance between two latitudes is constant. But latitudes, with the exception of the equator, are not geodesics. Going along a latitude means continuously turning relative to Earth's surface. If it is hard for you to see it, think about a latitude very close to one of the poles; say a circle with a radius of 10 meters around it. Obviously, if one wants to follow this circle he must continuously turn. So, if two sailing ships need to keep a constant distance between them, at least one of them must turn away from the other all the time. Similarly, in a gravity field a constant force is needed to keep one body at a constant distance from the other. We can now change a little the first law of Newton and instead of saying that without any force acting on them bodies are moving at a constant speed along straight lines, we say that without any force acting on them bodies are moving along geodesics in space-time. Where no gravity is present, time-space is flat (in this context flat means not curved), and these geodesics are straight lines in time-space or a constant speed along straight lines in space. With the presence of massive bodies, time-space is curved and geodesics can be a constant acceleration along straight lines in space (free-fall), an orbit around the center of mass of a heavy body, or any other known behavior of bodies in a gravity field. Gravity field is not a force field but curvature of time-space caused by the presence of mass and energy. Let's look at the following scenario: A very large spaceship is accelerating in the interstellar space in a direction we call upwards. On the floor of the spaceship there is a light source. While the light travels from the floor to the ceiling, the spaceship gains some velocity. When the light gets to the ceiling, the spaceship is moving away from the source of the light as it was when the light was transmitted. Because of the Doppler effect, an observer at the ceiling will see the light frequency shifted to red (lower frequency) compared to an observer near the light source. Similarly, an observer on the floor will see light coming from a source near the ceiling shifted to blue (higher frequency). The equivalence principle tells us that the same must be true for a stationary spaceship on the surface of a planet. So, a man on the top of a tower will see light coming from the ground red-shifted and a man on the ground will see light coming from the top of the tower blue-shifted. This phenomenon is called gravitational redshift. Now suppose there are two identical laser sources one on the ground and the other on the top of a high tower. An observer on the top of the tower has an instrument that counts the cycles of light waves. He activates the instrument for exactly one second, and compares the number of cycles of light from the source near him to the number of cycles of the light from the ground. Since the light from the ground is red-shifted, he counts more cycles of the upper source then these of the ground source. But he knows that the two light sources are identical and thus, an observer on the ground would count the same number of cycles per second as he counts for his light source. He concludes that while one second passes for him, les than a second pass on the ground, or that time pass slower on the ground. This is called gravitational time dilation. Unlike time dilation due to relative velocity, here the two observers agree on whose clock runs slower. The rule says that clocks lower in the gravity field runs slower and not, as sometimes mistakenly said, that clocks run slower where gravity is stronger. It is true that in nature, lower in the gravity field means closer to the attracting body and thus stronger gravity, but gravitational time dilation is also true for hypothetical uniform gravity field. Gravitational time dilation may raise some questions. In Special Relativity we could put two observers in different reference frames at nearly the same point, and by that eliminate effects of the distance and the time difference between them. With gravitational time dilation there is always a distance between the observers and it always takes time for information to get from one point to the other. The carrier of the information (light for instance) is affected by gravity during this time. The question is then: Does the lower clock really run slower or dose it only look so to the distant observer? Once again, this is not a physical question. Physics deals with rules and equations that describe the behavior of nature. There is no doubt about the fact that the distant observer does see time dilation, and General Relativity can calculate and predict it. There is no way to compare the time of the two observer which is independent of the distance and the limit on the speed at which information can be transferred. The question whether there is some objective truth that is independent of our ability to measure it remains a philosophic one. The equivalence principle is absolutely true only in a uniform gravity field. But uniform gravity field does not exist in the real world. If a man in a freely falling spaceship releases two balls that are vertically distant from each other, the lower one is subject to a slightly stronger gravity then the upper and accelerates faster. So, as time progress the distance between the balls grows. On the other hand, if he releases two balls horizontally distant, there is a slight angle between the directions of their acceleration because they accelerate towards the center of mass of the planet. So, as time progress the distance between them gets smaller. If a cloud of small particles is freely falling towards a heavy body, it changes its shape with time. It stretches in the direction of the falling and shrinks in directions perpendicular to it. If a rigid body is freely falling it will be subject to forces that stretch it in the direction of the fall and press it in the directions perpendicular to it. These forces are called tidal forces. Of course, in a spaceship cruising in space without any gravity, there would be no tidal forces and a cloud of particles will retain its original shape. Thus, the equivalence principle in non uniform gravity field can be considered true only for a very small space in which the gravity field is practically uniform and tidal forces are negligible. It would be absolutely true only at a point. We say that the equivalent principle is only locally true. The presence of matter curves space-time in all directions. Space itself is also curved so that two strait (geodesic) lines in space that are parallel at some area get angled and closer or farther apart at a distance. But the effect of the curvature of space is usually minor and negligible. First because we usually talk about velocities that are much lower then light speed. For instance the speed of the Earth in its orbit around the Sun is about 1/10000 the speed of light. That is, while Earth moves one kilometer in say x direction it moves 10,000Km in the ct direction. Obviously the effect of the curvature on time (time dilation) is much more significant than this of the curvature in space directions. In small distances curvature is negligible and space can be considered flat. In addition when we talk about very long distances in space we may get far away from the body that cause the gravity field, and curvature in all directions of space-time gets very small. When we eliminate space curvature from the equation of General Relativity, it gets the form of Newton's law of universal gravitation. So Newton's low of gravitation is an approximation to General Relativity for low speeds and week gravity field, just as classic mechanics is an approximation to Special Relativity for low speeds. Space curvature gets significant when we deal with high speeds and very dense and massive bodies. Then, space curvature gets significant and classic gravitation equations do not predict the motion of bodies correctly. An example for such conditions is a small body in a close orbit around a neutron star. Its orbit is no longer elliptic and has a complex shape. There are some known experiments and observations that verify the theory of General Relativity: The bending of light by gravity has been shown by observations made during a total solar eclipse. At a total solar eclipse stars can be seen very close to the limb of the Sun. Measurements done on stars that appeared around the Sun during the eclipse showed that the distance between the stars looked greater then usual. The reason for this was that the light coming from the stars was bent by the gravity of the Sun. Gravitational redshift has been shown by examining the spectrum of light coming from heavy starts like white dwarfs and neutron starts. Also experiments done on Earth by sending laser beams up a high tower could show the slight expected redshift. Gravitational time dilation could be measured using precise clocks on planes and satellites. The Global Positioning System must take the gravitational time dilation into account to get accurate results. The elliptic orbit of the planet Mercury is turning slowly in the direction of the orbit. The amount of this change in the orbit could not be explained by Newtonian gravity but it matches the equations of GR. Since Mercury is the closest planet to the Sun it is in a relatively strong gravity field and it also moves faster the other planets in the solar system. The effect of space curvature on its orbit is noticeable. To make physical calculations in curved space we first need mathematical tools to describe a curved space. The field in mathematics that deals with it is called Differential Geometry. Once we have a mathematical description of the curved time-space we can calculate distances velocities and geodesics. The next step is to find out how the presence of matter and energy curve time-space around them. This is what Einstein's field equation does.

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https://www.jiskha.com/display.cgi?id=1328572354

math

posted by Patience . A researcher predicts that watching a film on institutionalization will change students’ attitudes about chronically mentally ill patients. The researcher randomly selects a class of 36 students, shows them the film, and gives them a questionnaire about their attitudes. The mean score on the questionnaire for these 36 students is 70. The score for a similar class of students who did not see the film is 75. The standard deviation is 12. Using the five steps of hypothesis testing and the 5% significance level (alpha), does showing the film change students’ attitudes towards the chronically mentally ill? What is your null hypothesis? Alternate hypothesis? Is this a one-tailed or two-tailed hypothesis? What is your obtained z? What is the critical value for z? Do you reject or fail to reject the null hypothesis? State in words what you have found. Ho: Mean1 = mean2 Ha: Mean1 ≠ mean2 Z = (mean1 - mean2)/standard error (SE) of difference between means SEdiff = √(SEmean1^2 + SEmean2^2) SEm = SD/√n If only one SD is provided, you can use just that to determine SEdiff. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. This should start you out. waht is the null and alternate hypothesis of a researcher prdicts that watching a film institutionalization will change students attitudes about chronically mentally ill patients

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https://www.wyzant.com/Tutors/TX/Austin/8423864/subjectdetail?sid=174

math

As an SAT math tutor, I specialize in helping students review content and identify strategies and shortcuts that will allow them to solve problems methodically. I have worked with several students on comprehensive SAT math. I am especially well-versed in teaching the geometry and probability problems. In my view, the most important strategy to use on the SAT math sections is to maximize the number of questions answered correctly. To that end, I think the best test strategy is to not rush but to identify the problems that can be solved most easily. After the "easy" problems are done, I encourage my students to tackle more time-consuming problems, with the caveat that an educated or "estimated" guess is better than no answer at all.

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https://www.jsjkx.com/CN/Y2011/V38/I11/264

math

计算机科学 ›› 2011, Vol. 38 ›› Issue (11): 264-266. • 图形图像 • 摘要： 基于形状轮廓上的采样点到形状质心的距离，提出了一种距离比上下文形状描述符，用于形状识别和检索。该描述符计算简单，能有效区分不同形状，本质上具有平移、缩放不变性，且在一定程度上能杭部分遮挡和形变。用动态规划算法度量形状比上下文之间的距离，解决了对起始轮廓点的选择问题。在kimia' s-99形状图像数据库中的实验结果表明，该方法在单目标封闭轮廓的形状图像检索中取得了良好的效果。 Abstract: We suggested a shape descriptor named Distance Ratio Context(DRC) which is based on the distance betwecn sampled points and the centroid of the object. hhis descriptor has the properties of invariant to scaling and translation essentially and can be calculated easily. It's also invariant to deformation and distortion in some ways and can discriminate different shapes effectively. The dynamic programming algorithms were used to measure the distance between DRCs and this mechanism solved the problem of start point choosing on the object contour. hhe experiments in kimia's-99 shape dataset show that this approach, used in image retrieval of shape with a single closed contour, can get favorable results. Shape recognition, Shape retrieval, Distance ratio contexts, Dynamic programming 束鑫，唐楠，邱源. 基于距离比上下文的形状描述与识别方法[J]. 计算机科学, 2011, 38(11): 264-266. https://doi.org/ 导出引用管理器 EndNote|Reference Manager|ProCite|BibTeX|RefWorks

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https://www.grasshopper3d.com/main/sharing/share?id=2985220%253ATopic%253A183313

math

in the 0.8.007 version I am facing the problem that I can´t map the intersection points on to unrolled curves. I used first the shutter component to get the distance of the curve between the points. In the next step I used this distance to shutter the unrolled curve. But this doesn´t work. Any help is appreciated and hopefully there is an easy solution thank you in advance You can share this discussion in two ways… Share this link: Send it with your computer's email program: Email this

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CC-MAIN-2020-29

https://www.hindawi.com/journals/mpe/2014/930697/

math

Time-Delay Systems and Its Applications in Engineering 2014View this Special Issue Quasipolynomial Approach to Simultaneous Robust Control of Time-Delay Systems A control law for retarded time-delay systems is considered, concerning infinite closed-loop spectrum assignment. An algebraic method for spectrum assignment is presented with a unique optimization algorithm for minimization of spectral abscissa and effective shaping of the chains of infinitely many closed-loop poles. Uncertainty of plant delays of a certain structure is considered in a sense of a robust simultaneous stabilization. Robust performance is achieved using mixed sensitivity design, which is incorporated into the addressed control law. Time-delay systems are an important and well established topic in modern control theory [1–3]. Its diverse applications, for instance, in dynamics of fluids, internal combustion, heating systems, regenerative chatter in metal cutting, and networked control systems, led to the development of various complex approaches to system and controller synthesis [4–8]. Infinite dimensional spectrum of such systems might cause difficulties in appropriate spectrum assignment using standard control laws, which consequently means that stabilization cannot be always achieved. Due to the complexity of the spectrum, much interest has been shown in the development of control laws based on the computation of rightmost poles, since such algorithms ensure results with high precision [9–12]. In principle, with a limited number of controller parameters only a few poles can be placed to desired positions in the complex plane . As it has been shown in , some controller parameters might be used to directly assign a few dominant poles to an arbitrary position in the complex plane. The remaining controller parameters are then used to shift the chains of infinitely many system poles as far to the left of the dominant poles as possible. Another pole-placement-based technique has been introduced in for retarded systems and in for neutral systems. The method is based on continuous modifications applied to controller parameters in order to shift the rightmost or unstable poles to the left half plane in a quasicontinuous way, while monitoring other eigenvalues. A direct optimization approach [16, 17], used to minimize the spectral abscissa, has been introduced in . Alternative eigenvalue based control laws might be considered, which incorporate different algebraic approaches . General SISO time-delay plants treated in the form of a transfer function and its admissible coprime-inner/outer factorizations, derived by using numerical algorithms for computation of closed-loop poles, were discussed in . In addition, an optimal controller was designed. In this paper, a control law for retarded time-delay systems is considered, which results in a controller in the form of a transfer function as a quotient of quasipolynomials. An individual quasipolynomial in the numerator and the denominator consists of several delayed terms that result in a closed-loop quasipolynomial of a certain structure of sequentially solvable polynomial Diophantine equations. The structure of the controller is examined and derived in the case of infinite solutions of individual polynomial equations. As has been shown in only a few such equations can be solved for an arbitrary polynomial, which suggests that the remainder of the closed-loop spectrum cannot be assigned accordingly. In , the unsuitable dynamics of the remaining spectrum has been algebraically eliminated by the specific structure imposed on the controller, which allows finite spectrum assignment. In our presented work, elimination of any part of the spectrum is not admissible and therefore we tackle the problem of infinite closed-loop spectrum assignment, which is indispensable especially in the case of uncertain delays. As soon as delays are perturbed, any delay compensation technique based on preestimated delays does not eliminate delayed terms out of the closed-loop completely, which results in an infinite closed-loop spectrum. We demonstrate that certain estimation technique of a predetermined region of poles might be used in order to shape infinite closed-loop spectrum as well as infinite closed-loop spectrum with uncertain delays. We present an algebraic method for infinite closed-loop spectrum assignment, which reduces the number of parameters in the search routine for the appropriate stabile closed-loop spectrum. The algorithm for the search routine of the appropriately shaped infinite closed-loop spectrum is presented. It is shown that mixed sensitivity design might be incorporated into the addressed control law regarding uncertain time delays to obtain an optimal controller. The paper is organized as follows. After some preliminaries, we derive different controller parameterizations concerning different types of free polynomials. Then, we propose an algebraic method for closed-loop spectrum assignment and present a spectrum shaping technique, where we show spectrum shaping with uncertain delays as well. After that, we give algorithm of a search routine for appropriate stabile closed-loop spectrum and present the mixed sensitivity design for robust controller synthesis. Finally, we demonstrate our results on an example and give final remarks in conclusions. Time-delay system of a retarded type, with internal or state and output discrete commensurate delays, is defined as in [22, 23]: where , , , , are system defined parameters, are input and output scalars, is state vector, time delays are expressed as multiples of , , and initial condition is a segment of continuous functions equipped with a supremum norm. Definition 1. Let for be polynomials with real coefficients and are nonnegative real numbers in an ascending order. A function of the form is a retarded quasipolynomial if , . System (1) can be represented with a SISO transfer function [19, 21]: where and are retarded quasipolynomials: with and , with and ; and are alternative polynomials in variable and represent an equivalent description of quasipolynomials as Definition 1. Considering control law with being any given bounded reference signal, and are retarded quasipolynomials of the form (3) and the controller structure Assumption 1. We consider controller structure (8) with the same number of delays in the numerator and the denominator and , , with being monic. Assumption 2. Quasipolynomials and as well as polynomials and have no common zeroes. Complex Laplace argument is hereafter omitted for clarity. Controller (8) results in closed-loop characteristic quasipolynomial of the form where and is as Assumption 1. Regarding the closed-loop structure (9), which might be represented as a sum of several polynomial Diophantine equations with specific delay terms , the following result can be stated. Proposition 2. Let , , , and be quasipolynomials given by Definition 1 and Assumptions 1 and 2 hold. Then individual polynomial Diophantine equations in (9) for , have solutions for arbitrary , where the equation for has always infinite solutions () and equations for have a unique () or infinite solutions (), if the following is true: , , and in case of , . Proof. Polynomial equations (10) need to be solved sequentially since individual controller variables are present in several equations. Equations (10) for become For the expression on the left side, the following must hold: in order, (12) remains consistent polynomial equation. For the same reason, the degree of the expression on the right must not exceed the degree of the expression on the left; therefore, the following must hold: From (14), (11) follows straightforwardly. In the case of the equality in expression (14), selecting leading term of according to results in the reduction of the degree of the expression on the right and therefore inconsistency of (12). It can be shown that the first equation of (10) for has always infinite solutions. The minimal allowed degree of is , , and the maximal allowed degree of is determined by the equality of (11). If we join the two expressions, we can conclude that and consequently the first Diophantine equation always has infinite solutions. Remark 3. According to Proposition 2, only out of equations can be arbitrarily assigned as equations for consist of controller variables already assigned in equations , which suggests that infinite closed-loop spectrum assignment is not a straightforward task. 3. Controller Parameterization For the parameterization of infinite number of controllers, we make the following assumption and give the result. Assumption 3. To derive free parameters in all solvable Diophantine equations (10), the following must hold: and , which we derive with the same procedure as (15) in the proof of Proposition 2. Theorem 4. Considering Assumptions 1, 2, and 3, the controller (8) can be expressed in the formwhere are polynomials of degree , , which results in the following closed-loop characteristic quasipolynomial: The proof of Theorem 4 can be found in the appendix. Remark 5. Polynomial equations (10) need to be solved sequentially for each change of a specific free parameter in an individual polynomial equation. The importance of the result of Theorem 4 lies in the fact that all the infinitely many solutions of individual polynomial equations are expressed using free polynomials . It is evident that as soon as controller variables and are derived it is not needed to solve polynomial equations again in case of a change of any parameter of free polynomials. In the same way as individual Diophantine equations, the whole closed-loop characteristic quasipolynomial (9) can be treated as a single Diophantine equation. This leads to the following closed-loop representation: where is a sum of polynomials as defined in (19) and a corresponding controller structure of the form where are polynomials of degree , . The two presented controller structures (16) and (19) consisting of different types of free polynomials might be joined in a unified controller structure . Theorem 6. Considering Assumptions 1, 2, and 3, unified controller structurewhere and are polynomials of degree as in (16) and (19), respectively, results in the closed-loop characteristic quasipolynomial (17). The proof of Theorem 6 can be found in the appendix. 4. Infinite Closed-Loop Spectrum Assignment Due to the part of the closed-loop quasipolynomial that cannot be assigned arbitrarily, finite closed-loop spectrum assignment cannot be achieved without the cancelation or compensation of that part of the spectrum . On the other hand, appropriate infinite closed-loop spectrum assignment might be achieved with the help of a numerical routine for reliably computing the rightmost poles of a closed-loop spectrum. The quasipolynomial of the form (9) might be determined with a direct search routine by continuously selecting controller variables and by closely observing the rightmost dominant poles of the closed-loop system by shifting the chains of infinitely many poles as far to the left of the dominant poles as possible. This might be achieved by minimizing the norm of the delayed terms in (9) . Such search routine is computationally expensive since the number of controller parameters rapidly increases with the higher order of individual controller polynomials and the number of delayed controller polynomials. Another search routine could be constructed in a manner of continuously selecting closed-loop solution objectives of (9) represented as an overdetermined system of linear equations, where the individual closed-loop solution objective is designed by selecting values of the first equation representing a Hurwitzian polynomial and zeroing all other equations. Such solution objective will tend to minimize the effects of delayed parts of (9) by shifting the chains of infinitely many poles as far to the left as possible and consists of far less search parameters. The control objective can be expressed as an overdetermined system of linear equations by deriving the approximate solution of the expression , using ordinary least squares method by where , , , and , , are controller parameters (8), is Sylvester matrix of (9), and is the appropriately sized solution objective of the closed-loop quasipolynomial. Through a close inspection of (9), we can conclude that, by selecting the sufficiently large number of delayed terms in (8), the solutions of the first few equations will never be included in the last few equations in (9), which do not have arbitrary solutions. Having that in mind, we can directly assign to the first few equations an arbitrary closed-loop polynomial and derive solutions of the remaining equations according to (21), which is presented in the following result defining the sufficient structure of controller (8). Proposition 7. The solutions of the first th polynomial equations in the sum (9) or (17) will not be part of the equations if , . Therefore, assigning first th equations arbitrarily and rearranging remaining equations in (9) into (21), where the solution objective is appropriately sized vector of zeroes and will result in a consistent system of overdetermined equations. Proof. According to Proposition 2, equations for in the sum (9) have arbitrary solutions. Choosing , the remaining equations, which do not have arbitrary solutions, become for and consist of controller variables and , where . Therefore, controller variables and , where , are not present in the remaining equations. So assigning and exchanging these variables with the values everywhere in (9) and rearranging remaining equations of (9) into (21) will result in a consistent system of overdetermined equations. Remark 8. The result of Proposition 7 can be directly applied onto a search routine by continuously selecting polynomials of the first or of the first few equations in the sum (9), where the first one is always Hurwitzian, deriving solutions of the first few equations by Proposition 2 and deriving solutions of the remaining equations by (21) and by closely inspecting closed-loop poles of (9). Hence, such search routine allows direct assignment of poles of the first nondelayed equation or of the first few equations and tends to minimize the effects of the remaining delayed equations. 5. Shaping of the Infinite Closed-Loop Spectrum In order to appropriately shape the chains of infinitely many poles in the closed loop, by shifting them as far to the left from the dominant rightmost poles as possible, we can either compute sufficiently many rightmost poles, which is computationally expensive, or use the estimation technique of a predetermined region of poles presented in . From the information of the real part of the rightmost pole and the distance from the real part of rightmost pole on the real axes of the complex plane, we can specify a search criterion based on the predetermined region. This means that all the characteristic poles of the closed-loop quasipolynomial in the vertical strip of the complex plain must belong to the predetermined region of a certain size. The estimation of the region with the belonging poles is given with the following result. Proposition 9 (see ). All characteristic roots of (1) satisfy , , belonging to the set The boundary of is included in For more information and proof see [26, Appendix ]. Remark 10. To derive the boundary of characteristic roots of (1) in the vertical stripe , we need to shift the origin of (1) by, which can be done by introducing new variable in the characteristic matrix : After shifting the origin, (24) must be executed in grid points over the interval , because the function has period . The value of the boundary might present a constraint of the search routine of the closed-loop quasipolynomial. 6. Shaping of the Infinite Closed-Loop Spectrum in Case of a Delay Mismatch The structure of the delay uncertainty is in a form of multiplicative or additive uncertainty which both introduce slightly more delayed terms in (9) as a result of multiplication between the delayed terms of the plant and the controller. Therefore, the closed-loop spectrum might change significantly as well. The same constraint as in Proposition 9 and Remark 10 might also be applied to the plant with an uncertain delay. It can be shown that, in the search routine, it is sufficient to consider only the boundary as the boundaries are included inside of , . First we give an important property of time-delay systems. Proposition 11 (see ). If is a characteristic root of the system (1), then it satisfies Proof. The expression (25) is equivalent to Interpreting the argument of as a matrix leads to from which (28) follows straightforwardly. Using Proposition 11 and the result in Proposition 9, we can derive the following result. Proposition 12. For any positive , holds, where in (28) only the delay is perturbed according to (26) or (27). Proof. Regarding Proposition 11 and the multiplicative uncertainty (26) leads to and regarding the additive uncertainty (27) leads to where , , is the closed-loop quasipolynomial (9) in matrix form considering the delay uncertainty. Both expressions (31) and (32) are based on a simple comparison of exponential functions; namely, and , respectively, . Since the area that contains all the characteristic roots in specific complex plane of perturbed system is larger, it is sufficient to consider the boundary of the largest set . Finally, we can present the algorithm for the derivation of the infinite closed-loop spectrum, considering appropriate shaping of the chains of the infinitely many poles and constraints on the delay mismatch. The main objective of the algorithm is the minimization of function: where is spectral abscissa: Besides the spectral abscissa and the criterion based on Proposition 9, we might improve the objective function (34) by additional constraints, for instance, on the position of the dominant rightmost poles in the complex plane in the sense of damping, overshoot, or rise time . As the spectral abscissa is a nonsmooth function, a gradient sampling algorithm presented in [16, 17] or differential evolution might be used for the minimization of (34). The important part of the algorithm is computation of only a few rightmost characteristic roots of (1). There are several numerical solutions [9–12] in the form of algorithms, with a distinct exception , which is based on an estimation of all roots in a predetermined region (24). Such an algorithm can be easily adapted in a way of computation of only a few rightmost roots with continuously shifting of the origin of the complex plane, in such a way that the predetermined region of all roots is as small as possible but large enough that it contains only a few rightmost roots (Remark 10). The shifting of the origin might be performed using a bisection algorithm. In this way, only a few rightmost roots are computed, which is much more efficient in the search for the appropriate closed-loop quasipolynomial, in contrast to the derivation of large number of roots, which is computationally very expensive. Algorithm 13 . We have the following.(1)Choose as Proposition 7.(2)Assign a Hurwitzian polynomial to first and any polynomial to the remaining th equations according to the optimization method and derive a closed-loop spectrum (9) using Proposition 7.(3)Derive the first few rightmost poles of (9).(4)Derive spectrum of (9) using Proposition 9.(5)Derive the first few rightmost poles of (9) considering delay uncertainty (26) or (27).(6)Derive spectrum of (9) using Proposition 9 and considering delay uncertainty (26) or (27).(7)Compute the objective function (33) and derive improved values from minimization algorithm for step .(8)Continue with step until appropriate closed-loop spectrum is derived. 7. Robust Optimization via Mixed Sensitivity Design Mixed sensitivity can be presented as an optimization problem of minimizing the norm [24, 29] of the where and are weighting filters in a form of frequency depended bounds, which characterize robust stability and robust performance; and are sensitivity and complementary sensitivity. Applying (19) to sensitivity and complementary sensitivity leads to When considering the delay uncertainty (26) or (27), structures of and are slightly altered. Therefore, the characteristic quasipolynomial depends on polynomials as well: where and are as in (4), only perturbed according to (26) or (27), respectively. By applying the uncertain and to the optimization procedure (35), we derive with and being weighting filters characterizing sensitivity and complementary sensitivity of the plant with uncertain delays. Optimal controller design might be performed by minimizing (38) and by closely observing right most poles of in sense or infinite closed-loop spectrum shaping according to Algorithm 13. 8. Example: Level Control of a Chain of Evaporators The dynamics of the level control of a chain of evaporators may be modeled by a delayed first order plus integrator transfer function [30, 31]: where parameters , , and . Time delay considered with appropriate uncertainty, for instance, using (26), . Representing the transport delay as output delay and incorporating direct nondelayed output connections into the model (39) to represent the model in the form of (2), which might be achieved either by the observer [32, 33] or even by a classical smith predictor [31, 34], lead to the following model representation: The following controller was designed where the number of delayed controller polynomials was set according to Proposition 7, . The degree of individual controller polynomials was set using the result of Proposition 2. Selecting and . According to Theorem 6 and (19) and . Applying Algorithm 13, where and were computed according to Proposition 9 and Remark 10 for (Figure 1). What is clear is that the higher the value is, the larger the region is. So from the numerical point of view, should be set small. Optimal controller (38) was designed for and (Figure 4) with the following controller parameters: , , , , , , , , , , , and . The step response of is shown in Figure 2 in blue and of in red. Closed-loop poles of are shown in Figure 3 in blue and of , , and in green, respectively. A unified controller structure for retarded time-delay systems composed of two different types of free polynomials was derived. An algebraic method for infinite closed-loop spectrum assignment was presented and a search algorithm for the appropriate infinite closed-loop spectrum was proposed. To shape the chains of infinitely many poles, a search criterion based on predetermined regions of poles was established and it was shown that the same technique was applicable for the closed-loop spectrum with uncertain time delays. Finally, optimal robust controller synthesis using mixed sensitivity approach was demonstrated in a control example. Proof of Theorem 4. Considering Assumptions 1, 2, and 3, polynomial equations (10) have infinite solutions, where an individual polynomial equation in (10) can be represented in the form of an underdetermined system of linear equations with free parameters. The number of free parameters coincides with the number of parameters in , which represent an equivalent representation. As shown in Proposition 2, if in the previous equation free parameters change, this leads to the change of the term on the right side of (10), which results in the change of the solutions in the current equation. Therefore, (10) need to be solved sequentially, but as Theorem 4, only once. After the first solution of (10), changes of free parameters relative to the free parameters of the original solution can be expressed with polynomials : Changing the free parameters in the first equation can be represented as a change of controller variables and to and , which results in and the change of the solutions in the second equation By insertion of from (10) into (A.3), the amount of the change of and can be expressed in accordance with the change of and : Considering the change of free parameters in the second equation same as in (A.2), we derive and . When substituting and in (A.1), the second equation becomes Derivation of appropriate controller variables is executed sequentially. Generally we can write for the th equation After replacing in (A.6) with an equivalent representation from (10) and cancelation of individual terms, we derive After considering the influence of free parameters, we derive the final form of and : After replacing controller variables in (A.1) with (A.8), we derive (17), which completes the proof of Theorem 4. Proof of Theorem 6. The two presented types of free polynomials expressed in (16) and (19) might be joined in a unified controller structure (20). The proof relies on the fact that free polynomials cancel out in the closed loop (18) and therefore might be included into the controller structure (16) as well. The following closed-loop representation with the controller , leads to the same closed-loop representation as in (17). Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. J. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.View at: Publisher Site | Google Scholar | MathSciNet K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhäuser, Basel, Switzerland, 2003. M. Wu, Y. He, and J.-H. She, Stability Analysis and Robust Control of Time-Delay Systems, Springer, Beijing, China, 2010.View at: Publisher Site | MathSciNet J. J. Loiseau, W. Michiels, S.-I. Niculescu, and R. Sipahi, Eds., Topics in Time Delay Systems: Analysis, Algorithms and Control, vol. 388 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2009. R. Sipahi, T. Vyhlídal, S.-I. Niculescu, and P. Pepe, Eds., Time Delay Systems: Methods, Applications and New Trends, vol. 423 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2012. Y. Zhang, X. Chen, and R. Lu, “Performance of networked control systems,” Mathematical Problems in Engineering, vol. 2013, Article ID 382934, 11 pages, 2013.View at: Publisher Site | Google Scholar | MathSciNet Y. Wang, H. R. Karimi, and Z. Xiang, “ control for networked control systems with time delays and packet dropouts,” Mathematical Problems in Engineering, vol. 2013, Article ID 635941, 10 pages, 2013.View at: Publisher Site | Google Scholar | MathSciNet W. Michiels, T. Vyhlídal, and P. Zítek, “Control design for time-delay systems based on quasi-direct pole placement,” Journal of Process Control, vol. 20, no. 3, pp. 337–343, 2010.View at: Publisher Site | Google Scholar Z. Wu and W. Michiels, “Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method,” Journal of Computational and Applied Mathematics, vol. 236, no. 9, pp. 2499–2514, 2012.View at: Publisher Site | Google Scholar | MathSciNet K. Engelborghs, T. Luzyanina, and D. Roose, “Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL,” ACM Transactions on Mathematical Software, vol. 28, no. 1, pp. 1–21, 2002.View at: Publisher Site | Google Scholar | MathSciNet T. Vyhlídal and P. Zítek, “Mapping based algorithm for large-scale computation of quasi-polynomial zeros,” IEEE Transactions on Automatic Control, vol. 54, no. 1, pp. 171–177, 2009.View at: Publisher Site | Google Scholar | MathSciNet D. Breda, S. Maset, and R. Vermiglio, “TRACE-DDE: a tool for robust analysis and characteristic equations for delay differential equations,” in Topics in Time Delay Systems, vol. 388 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2009.View at: Google Scholar P. Zítek, “Frequency-domain synthesis of hereditary control systems via anisochronic state space,” International Journal of Control, vol. 66, no. 4, pp. 539–556, 1997.View at: Publisher Site | Google Scholar | MathSciNet W. Michiels, K. Engelborghs, P. Vansevenant, and D. Roose, “Continuous pole placement for delay equations,” Automatica, vol. 38, no. 5, pp. 747–761, 2002.View at: Publisher Site | Google Scholar | MathSciNet W. Michiels and T. Vyhlídal, “An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type,” Automatica, vol. 41, no. 6, pp. 991–998, 2005.View at: Publisher Site | Google Scholar | MathSciNet J. V. Burke, A. S. Lewis, and M. L. Overton, “A robust gradient sampling algorithm for nonsmooth, nonconvex optimization,” SIAM Journal on Optimization, vol. 15, no. 3, pp. 751–779, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, “HIFOO—a matlab package for fixedorder controller design and -optimization,” in Proceedings of the 5th IFAC Symposium on Robust Control Design (ROCOND '06), pp. 339–344, Toulouse, France, July 2006.View at: Google Scholar J. Vanbiervliet, K. Verheyden, W. Michiels, and S. Vandewalle, “A nonsmooth optimisation approach for the stabilisation of time-delay systems,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 14, no. 3, pp. 478–493, 2008.View at: Publisher Site | Google Scholar | MathSciNet J. J. Loiseau, “Algebraic tools for the control and stabilization of time-delay systems,” Annual Reviews in Control, vol. 24, pp. 135–149, 2000.View at: Publisher Site | Google Scholar S. Gumussoy, “Coprime-inner/outer factorization of SISO time-delay systems and FIR structure of their optimal H-infinity controllers,” International Journal of Robust and Nonlinear Control, vol. 22, no. 9, pp. 981–998, 2012.View at: Publisher Site | Google Scholar | MathSciNet M. de la Sen, “On pole-placement controllers for linear time-delay systems with commensurate point delays,” Mathematical Problems in Engineering, no. 1, pp. 123–140, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet W. Michiels and S. Niculescu, Stability and Stabilization of Time-Delay Systems. An Eigenvalue Based Approach, SIAM, Philadelphia, Pa, USA, 2007. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.View at: Publisher Site | MathSciNet J. C. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory, Macmillan Publishing, New York, NY, USA, 1990. A. Howard and R. Chris, Elementary Linear Algebra, John Wiley & Sons, 10th edition, 2005. K. Verheyden, Numerical bifurcation analysis of large-scale delay differential equations [Ph.D. thesis], Department of Computer Science, K.U. Leuven, Leuven, Belgium, 2007. A. Sarjaš, R. Svečko, and A. Chowdhury, “Optimal robust motion controller design using multi-objective genetic algorithm,” The Scientific World Journal, vol. 2014, Article ID 978167, 15 pages, 2014.View at: Publisher Site | Google Scholar K. Price, R. Storn, and J. Lampinen, Differential Evolution—A Practical Approach to Global Optimization, Springer, 2005. K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1995. P. Albertos and P. García, “Robust control design for long time-delay systems,” Journal of Process Control, vol. 19, no. 10, pp. 1640–1648, 2009.View at: Publisher Site | Google Scholar J. E. Normey-Rico and E. F. Camacho, “Unified approach for robust dead-time compensator design,” Journal of Process Control, vol. 19, no. 1, pp. 38–47, 2009.View at: Publisher Site | Google Scholar P. Zítek, V. Kučera, and T. Vyhlídal, “Meromorphic observer-based pole assignment in time delay systems,” Kybernetika, vol. 44, no. 5, pp. 633–648, 2008.View at: Google Scholar | MathSciNet W. Michiels and D. Roose, “Time-delay compensation in unstable plants using delayed state feedback,” in Proceedings of the 40th IEEE Conference on Decision and Control (CDC '01), pp. 1433–1437, Tallahassee, Fla,USA, December 2001.View at: Google Scholar J. E. Normey-Rico and E. F. Camacho, “Dead-time compensators: a survey,” Control Engineering Practice, vol. 16, no. 4, pp. 407–428, 2008.View at: Publisher Site | Google Scholar

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Separation of soft and collinear singularities [.5em] from one-loop -point integrals [.5cm] Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Föhringer Ring 6, D-80805 München, Germany The soft and collinear singularities of general scalar and tensor one-loop -point integrals are worked out explicitly. As a result a simple explicit formula is given that expresses the singular part in terms of 3-point integrals. Apart from predicting the singularities, this result can be used to transfer singular one-loop integrals from one regularization scheme to another or to subtract soft and collinear singularities from one-loop Feynman diagrams directly in momentum space. Many interesting high-energy processes at future colliders, such as the LHC and an linear collider, lead to final states with more than two particles, rendering precise predictions much more complicated than for particle reactions. The necessary calculation of radiative corrections bears additional complications and requires a further development of calculational techniques, as recently reviewed in Ref. . Full one-loop calculations for processes with more than two final-state particles require, for instance, the evaluation of scalar and tensor -point integrals. For several approaches have been proposed in the literature [2, 3, 4, 5]. In this context the two main complications concern a numerically stable evaluation of tensor integrals on the one hand and a proper separation of infrared (soft and collinear) singularities on the other. In Ref. the direct reduction of scalar 5-point to 4-point integrals, as proposed in Ref. , has been extended to tensor integrals. The nice feature in this approach is the avoidance of leading inverse Gram determinants, which necessarily appear in the well-known Passarino–Veltman reduction and lead to numerical instabilities at the phase-space boundary. In this paper we focus on the treatment of infrared (IR) or so-called “mass singularities” at one loop. According to Kinoshita , such mass singularities arise from two configurations that both lead to logarithmic singularities. Collinear singularities appear if a massless external particle splits into two massless internal particles of a loop diagram, and soft singularities arise if two external (on-shell) particles exchange a massless particle. If the involved particles are not precisely massless, the corresponding singularities show up as large logarithms involving the small masses, like where is the electron mass and a large scale. If the masses involved in the singular configurations are exactly zero, the singularities appear as regularized divergences, such as poles where in -dimensional regularization. In both cases an analytical control over such terms is highly desirable, either in order to perform cancellations at the analytical level or to carry out resummations. In the following we show how to extract mass singularities from general tensor -point integrals and finally give an explicit result for the singular parts in terms of related 3-point integrals. Moreover, we describe several ways how this result can be exploited and give some examples illustrating the easy use of our final result. The method to derive the general result of this paper was already applied in Ref. to specific 5-point integrals, which appear in the next-to-leading order QCD corrections to the processes . Mass singularities of the virtual and real radiative corrections are intrinsically connected in field theory. In fact, the famous Kinoshita–Lee–Nauenberg (KLN) theorem [7, 9] states that the singularities completely cancel in sufficiently inclusive quantities. The result of this paper allows for a simple analytical handling of the mass singularities of the virtual one-loop corrections. The singular structure of the real corrections induced by one-parton emission (which corresponds to the one-loop level) can be easily read from so-called subtraction formalisms which are designed for separating these singularities. Using these results and the KLN theorem, the one-loop singular structure of complete QCD and SUSY-QCD amplitudes has been derived in a closed form in Ref. . Note that the attitude of this paper is rather different, since the one-loop integrals are inspected themselves, eventually leading to a prescription for extracting the singularities diagram by diagram. The paper is organized as follows: In Section 2 we set our conventions and describe the situations in which mass singularities appear at one loop. Section 3 contains the actual separation of the mass singularities and our final result at the end of the section. In Section 4 we describe various ways to make use of the result and present some explicit applications. Section 5 contains a short summary. In the appendices we present a proof of an auxiliary identity used in Section 3 and provide a list of mass-singular scalar 3-point integrals that frequently appear in applications. 2 One-loop -point integrals and mass singularities We consider the general one-loop -point integrals with the denominator factors A diagrammatic illustration is shown in Figure 1. Note that we do not set the momentum to zero, as it is often done by convention, but keep this variable in order to facilitate a generic treatment of related integrals. In particular, with this convention all are invariant under exchange of any two propagator denominators , or equivalently of two pairs of . We follow the usual convention to denote -point integrals with as Whenever the index on momenta or masses exceeds the range , it is understood as modulo , i.e. and are equivalent, etc. For later use, we introduce the variables We are interested in “mass” singularities that appear if combinations of external squared momenta, , and internal masses become small, but do not consider singular configurations that are related to specific or isolated points in phase space, such as thresholds or forward scattering. We can, thus, distinguish two sets of parameters: one set that comprises all quantities , with fixed non-zero values, and another set of those quantities that are considered to be small, i.e. which formally tend to zero. In order to simplify the notation, we define an operation, indicated by a caret over a quantity , which implies that all small quantities are set to zero in . As shown by Kinoshita , “mass” (or “IR”) singularities can appear in one-loop diagrams in the following two situations: An external line with a light-like momentum (e.g. a massless external on-shell particle) is attached to two massless propagators, i.e. there is an with The singularity is logarithmic and originates from integration momenta with where is an arbitrary real variable. Since the momentum on line is then collinear to the external momentum , such singularities are called collinear singularities. A massless particle is exchanged between two on-shell particles, i.e. there is an with The singularity is also logarithmic and originates from integration momenta with i.e. the momentum transfer of the th propagator tends to zero. Therefore, these singularities are called soft singularities. In the following we focus on integrals with and express the singular structure of one-loop -point integrals in terms of 3-point integrals which are easily calculated with standard techniques, as for instance described in Refs. [4, 6, 12]. Of course, the same is true for the cases , i.e. for tadpole and self-energy integrals, which are even simpler. 3 Separation of mass singularities In this section, we first consider the asymptotic behaviour of the denominator of the integrand in Eq. (2.1) in the individual collinear and soft limits. Based on these partial results we derive a simple expression that resembles the whole integrand in all singular regions. Applying the loop integration to this expression directly leads to our main result which expresses the singular structure of an arbitrary -point integral (2.1) with in terms of 3-point integrals. 3.1 Asymptotic behaviour in collinear regions For an integration momentum in the collinear domain, as specified in Eq. (2.6), the two propagator denominators and tend to zero (), and the behave as The collinear limit is mass singular if the external momentum squared and the two masses , are small. In this limit the two propagator denominators , tend to zero, but the others remain finite (for ): Note that the variable is the only integration variable that is not fixed by the collinear limit. The product of all regular propagators can be decomposed into a sum over these propagators via taking the partial fraction, The coefficients are functions of the variables and alone and thus fixed by the external kinematics. The explicit result for reads as proven in App. A. The collinear singularity arises from the region where the propagator denominators and both become small with no preference to any of the two. In order to reveal this equivalence, we rewrite using Inserting these relations, we get in which the equivalence of the th and th propagators is evident. Multiplying Eq. (3.3) with yields a relation, which is valid in the collinear limit, between the product of all propagators and a linear combination of products involving only three propagators, Thus, the collinear singularity associated with the propagators , in an -point integral is expressed in terms of a sum of 3-point integrals involving the th, the th, and any other line of the diagram. 3.2 Asymptotic behaviour in soft regions The soft singularity connected with a massless propagator arises from momenta , where . The other denominators tend to a regular limit in this case, and the product of all propagators behaves like We still have to consider the possibility that one or both ends of the soft line is part of a collinear configuration treated above. If this is the case, the soft limit can be reached as limiting case of a collinear limit. Assuming again as the soft line, the two “degenerate” collinear limits are and . Both lead to , but in the former case lines and correspond to a collinear configuration, in the latter lines and . It is quite easy to see that the soft asymptotic behaviour (3.10) is already correctly included in the collinear behaviour (3.8) in either case, because 3.3 Final result From the above considerations it is clear that we obtain an expression for the asymptotic behaviour of the product of all propagator denominators in all collinear and soft regions upon adding the asymptotic expressions of all collinear and soft regions, which can be read from Eqs. (3.8) and (3.10), and carefully avoiding double-counting of soft asymptotic terms. To this end, we define With this definition we can write down the asymptotic behaviour valid for all collinear and soft regions as Obviously each soft part is included by the terms exactly once, and the collinear contributions from and are omitted if they are already covered by the soft terms . Integrating Eq. (3.13) over on the l.h.s. yields the scalar integral and on the r.h.s. a linear combination of scalar 3-point integrals , which has exactly the same structure of collinear and soft singularities as . An analogous relation is obtained for tensor integrals if the additional factor is included in the integration, since this factor does not lead to additional singularities. Note that the asymptotic relation (3.13), which describes the leading behaviour, is in fact sufficient to extract all mass singularities from the one-loop integral, since the degree of the singularities is logarithmic. In summary the complete mass-singular part of a general one-loop tensor -point function reads The sum over and runs over all subdiagrams whose scalar integral develops a collinear or soft singularity. A tensor integral is, however, not necessarily mass singular if the related scalar integral develops such a singularity. For such tensor integrals the regular 3-point integrals on the r.h.s. of Eq. (3.14) could be dropped. We note that for , artificial ultraviolet singularities appear in the tensor 3-point integrals on the r.h.s. of Eq. (3.14). These can be regularized in dimensional regularization and easily separated from the mass singularities (see, e.g., the appendix of Ref. ). In order to render the above result more useful, we present a list of mass-singular functions in App. B. This paper, thus, contains the needed ingredients to predict the mass singularities of most scalar -point functions occurring in practice. To obtain the singularities of tensor integrals, only the 3-point tensor integrals have to be derived, which can be easily inferred with the well-known Passarino–Veltman algorithm (see also Refs. [4, 5]). 4 Discussion and applications 4.1 Possible applications of the final result The relation (3.14) can be exploited in various directions: As pointed out in the previous section, the mass singularities of arbitrary -point integrals can be easily derived from 3-point functions. This statement is true in any regularization scheme, i.e. for any -point integral all small-mass logarithms and/or poles in in dimensional regularization can be easily inferred. The singular integral can be used to translate any IR-divergent -point integral from one regularization scheme to another. To this end, the regularization-scheme-independent difference is considered. For the translation from one scheme to the other only the singular part , and thus the relevant 3-point integrals, have to be known in the two regularization schemes. The trick described in the last item has been used in Ref. to translate -dimensional 5-point integrals into a mass regularization with , in order to make use of the direct reduction [2, 5] of 5-point to 4-point integrals, which works in four space-time dimensions. In this context it was observed that the formal relation between 5-point and 4-point integrals, which was derived in four dimensions, is also valid in dimensions up to terms, since the extraction of the singularities works in any regularization scheme with the same linear combination of 3-point integrals. From the results of this paper we conclude that this statement generalizes to arbitrary -point integrals, i.e. the reduction of an -point integral to 4-point integrals works in dimensions in precisely the same way as in four dimensions [up to terms of ], without the appearance of extra terms. Since Eq. (3.14) has been derived in momentum space, it could also be used as local counterterm in the momentum-space integral, i.e. taking the difference in Eq. (4.1) before the integration over the loop momentum , the integral becomes IR (soft and collinear) finite and can be evaluated without IR regulator. The loop integration of the subtracted part is extremely simple, because it involves only 3-point functions, and can be added again after the integration of the difference. This procedure could be very useful in purely numerical approaches to loop integrals, as e.g. described in Ref. . 4.2 Sudakov limit of the one-loop box integral As a simple application, we consider the box integral in the so-called Sudakov limit, where all external squared momenta and internal masses are considered to be much smaller than the two Mandelstam variables In this limit, there are four regions for soft singularities, and Eq. (3.14) yields For the scalar integral this is in agreement with Eq. (57) of Ref. , where the remaining finite contribution was derived as well. In Ref. also tensor integrals up to rank 4 have been considered; the singularities predicted by Eq. (4.3) have been checked against these results. The soft singularities in the functions on the r.h.s. of Eq. (4.3) arise from integration momenta , , , , respectively. The singular terms in the tensor coefficients of the functions can be related to the respective scalar functions rather easily. For instance, shifting the integration momentum in the first function on the r.h.s. of Eq. (4.3), power-counting in the shifted momentum shows that terms with in the numerator are not mass singular. Thus, in the first tensor function only covariants built of the momentum alone receive singular coefficients that are all proportional to the respective function. The same reasoning applies to the other three functions. In summary, the mass-singular terms of the tensor 4-point functions in the Sudakov limit are given by where the sign indicates that regular terms have been dropped, i.e. Eq. (3.14) is not applied literally. 4.3 Singular structure of some 5-point integrals (i) A 5-point integral for the process In Ref. the three different types of IR-singular 5-point integrals that appear in the next-to-leading order (NLO) QCD correction to have been calculated in dimensional regularization. One of the corresponding pentagon diagrams is shown on the l.h.s. of Figure 2. In order to make use of the direct reduction [2, 5] of 5-point to 4-point integrals in four space-time dimensions, the dimensionally regulated integrals have been translated into a mass regularization by using the trick described in the previous section. However, the construction of the singular parts of the 5-point in terms of 3-point integrals has been done integral by integral. For the 5-point integral on the l.h.s. of Figure 2, which is a tensor integral of rank 4, we can easily verify that the explicit formula (3.14) yields the same result as quoted in Ref. . Assigning the momenta according to and the defining the auxiliary quantities read where an obvious matrix notation is used. Inserting this into Eq. (3.14) and identifying the singular 3-point integrals yields in agreement with Eq.(2.37) of Ref. . (ii) A 5-point integral for the process As another example, we consider the 5-point integral corresponding to the diagram on the r.h.s. of Figure 2 which contributes to the (yet unknown) NLO QCD correction to . Analogously to the previous case, we assign the momenta according to and keep the definitions (4.6). The auxiliary quantities read Seven soft or collinear singular 3-point subdiagrams can be identified, and Eq. (3.14) yields (iii) A 5-point integral for the process Now we consider the diagram on the l.h.s. of Figure 3, which contributes to the corrections of the process The soft singularity arising from photon exchange is regularized by the infinitesimally small photon mass . The electron mass is considered to be much smaller than all other masses and scalar products, thus leading to mass-singular terms. Again we make use of definition (4.6) for the kinematical variables. The auxiliary quantities read Inserting this into Eq. (3.14) and identifying the singular 3-point integrals yields where is set to zero in all functions that are not soft singular. In Ref. , where the corrections to the processes have been worked out, the 5-point integral has been calculated directly from the corresponding 4-point integrals using the method of Refs. [2, 5]. We have numerically verified the correctness of Eq. (LABEL:eq:eennhpent) by checking the difference between taken from Ref. and the r.h.s. of Eq. (LABEL:eq:eennhpent) to be independent of and .

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CC-MAIN-2021-43

https://www.studypool.com/discuss/300614/algebra-1-percent?free

math

What equation is used to calculate percents? Part/Whole = %/100. Example: 25 % of 200 is____ In this problem, of = 200, is = ?, and % = 25We get:is/200 = 25/100Since is in an unknown, you can replace it by y to make the problem more familiary/200 = 25/100Cross multiply to get y × 100 = 200 × 25y × 100 = 5000Divide 5000 by 100 to get ySince 5000/100 = 50, y = 50So, 25 % of 200 is 50 Content will be erased after question is completed. Enter the email address associated with your account, and we will email you a link to reset your password. Forgot your password?

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http://redwoodsmedia.com/fractions-operations-worksheet/

math

fractions operations worksheet Here is the Fractions Operations Worksheet section. Here you will find all we have for Fractions Operations Worksheet. For instance there are many worksheet that you can print here, and if you want to preview the Fractions Operations Worksheet simply click the link or image and you will take to save page section. Mixed Problems Worksheets Mixed Problems Worksheets For Practice Free Worksheets Library Download And Print Worksheets Free On Grade 3 Math Worksheets Subtract Fractions From Mixed Numbers K5 The Decimals And Fractions Mixed With Negatives (a) Math Worksheet Fraction Operations Coloring Worksheetlindsay Perro Tpt Positive & Negative Fraction Operations Maze Activitymath With Fractions Operations Worksheet Bunch Ideas Of Fractions Operations Order Of Operations Worksheet Fractions Order Of Operations Worksheets For Fraction Addition Math Worksheets Dynamically Created Math Worksheets Order Of Operations Fractions Worksheet Worksheets For All Adding Rational Number Worksheets Colornumbers Fractions Order Of Operations With Fractions Worksheet To Printable Math Rational Number Operations Worksheets Colornumbers Fractions Fraction Word Problems (w Mixed Operations) Worksheets.

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https://solvedlib.com/n/spectrum-for-the-acetone-extracted-plant-graph-paper-draw,8

math

So here we are, looking at the alcohol Breathalyzer test of policemen and policewomen used to test drivers. Okay, so if we just look at the I r of alcohol here we have, um, wave numbers. And here, this one is transmit. Didn't just a t for transmitting. Okay, So what we see here, then, is this broad absorption here than a sharp pig here and some blank stuff over here? And then another peek over here. Okay, so here, um, is a 3000. I'll call this one 1 15 approximately. Okay, So what they're saying is that at this 29 9, 50 Marc is usually when what? The wave band that the Breathalyzer assisting for And then this one, um, is another one Look at twisting for as well. So just to go over like what these objections are for here. This is for a hydroxyl Sfera Doxil stretching corporate there, and your body is what this? Ohh. This college bond. I mean, we draw that. So it's the O. H. This bond here. That stretching is what's causing this broadness over here. Sometimes 50 is, um all cane stretching. So that's, um, ch. But for s p three. I just do that. So it's this bond here s p three. That's about that's the one that they're testing for. And the one over here is this carbon oxygen. So, yeah, this is pretty much represents everything that we would see in ethanol the, you know, alcohol in those red bridges. So we have the c o. We have the ch we have the O h. That's what we see here on the IR spectrum. Okay, so, um, I guess one reason why they wouldn't that they would use these ones that marked red is because they appear a sharp, intense peaks. Well, um, one for Hydra axles are very, very broad. And I guess, um, unless they're less, they're not as sharp. Definitely not as sharp. So here this is just for the ch. That's just for the CEO. Okay, then, um, if we just look at this y axis here, you see, transmit. And, um, that is just how well, um, light is able to pass through the ease at which light passes through sample. Okay, then let me get absorption. Absorption is when the light energy is like enters into the sample broccoli and it's not passing through, okay? And just for us to relate transmissions to absorbent, we know that actually, absorbent is the law algorithm of trans maintenance, and the negative logarithms are transparent, and that allows them to be inversely related. Okay, that makes sense. We here, these are absorption. These very deep, um, drops are absorb Ince's because if you have 100% transmit, transmit didn't up. Here is this is a percent. That means there are no absorbency is because it's dropping down. We know that, um, something is happening to the sample. That's why it looks like this week. If we were to then graph ir with observance, it would look for, like, a reflection of this. That's yeah, Okay. And then they gave us some data. So that's just if you have some data, you can make a calibration curve to correlate absorbent with concentration. And what will really help us here is beer's law, which basically says that the absorbent is equivalent to some coefficients XML the extension coefficient, the mal absorption coefficient, um, optical path, length, which we will call well, and then concentration. I can't beer as well. So here we have it I'm just gonna plot three points here on our calibration curve. So basically, um, this is now going into some in local camp. But what we would do is you went? Yeah. Three known concentrations of an SNL sample. Let's say we have point. Oh, 25 few 0.5 here and then 0.1 you would get you would make you would buy three samples of SNL with this concentration and measure their organs is. And what we would see then is that at this one, I just Mark, this one is a kind of a similar. Let's just call this 1.1. This is point oh five and then up here then would be right. 15 Just graphic like that. This is absorbent absorbency isn't house. Yes, but here, we're looking specifically at, um, the carbon oxygen bond. So this wave numbers okay, so that we just plot some points. We had a point around here, and there was a point around here, just very approximate. I'm glad that we had a point completely above over there. Okay, so then we can sort of kind of neatly draw calibration curve, which is a linear line following sort of Hawaii called on it to be types Alert. Um, Okay. Okay. So right. You can see observance concentration are directly related. Okay, so then if we have, um, if we measure someone's breath and it turns out that they have a insurance that corresponds to this intensity 0.95 surround this line here. They're getting this kind of an observance here. We can then go all the way down. Kind of and yeah, that's just basically saying that. 2109640.95 observance we're getting. And yeah, this observance, we are getting concentration. People fleeing around. Very approximate. Let me just call it five. I know more, but yeah, you will just follow through with whatever graph you drew. And we collect that. Yeah, we can. We could have used math and to find something more accurate. But if you're just basing it off of the calibration curve, I thought you would do it..

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CC-MAIN-2022-40

https://e-eduanswers.com/mathematics/question13567835

math

3. Triangular Pyramid Option D. 649 cm² is the correct option. Since surface area of a rectangular prism is calculated by the formula Surface area = 2(length×width + width×height + length×height) = 2(8×13 + 13×10.5 + 8×10.5) = 2(104 + 136.5 + 84) = 649 cm² Therefore, Option D. 649 cm² is the correct option. B is the area of the base so...you find the area of the base... Sorry that's all i really know about it!! The base area is 30 the question is only for the Base (=area of triangle that is the base of the figure) A=1/2 bxh b=5 h=12 The base would be 3 because 2*3/2=3 2 being the height, 3 being the width, and in the end 3 is also the base there is no b volume in this answer Your answer would be 649cm2 given the dimensions The solution: A=2(wl+hl+hw)=2·(13·8+10.5·8+10.5·13)=649cm² Since there are TWO of those faces, we get the total and multiply it by 2. Total = 324.50 sq cm * 2 TOTAL AREA = 649 sq cm the answer is d which expression can be used to convert 80 us dollars (usd) to australian dollars (aud)? 1 usd = 1.0343 aud 1 aud = 0.9668 usd sorry just need the points

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http://www.ask.com/web?q=Long+Division&o=2603&l=dir&qsrc=3139&gc=1

math

In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a ... Long Division. Below is the process written out in full. You will often see other versions, which are generally just a shortened version of the process below. Long Division with Remainders. When we are given a long division to do it will not always work out to a whole number. Sometimes there are numbers left over. I know this is late, but people say that there is only addition and multiplication, and that subtraction and division are just opposites of that. That is kind of true in a ... Long division with remainders showing the work step-by-step. Calculate quotient and remainder and see the work when dividing divisor into dividend in long ... What is long division? Long division is a way to solve division problems with large numbers. These are division problems that you can't do in your head. How to ... Demonstrates through worked examples how to do long division of polynomials. Relates long polynomial division to long division of whole numbers. Create an unlimited supply of worksheets for long division (grades 4-6), including with 2-digit and 3-digit divisors. The worksheets can be made in html or PDF ...

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http://xepaperyufi.frenchiedavis.info/questions-and-answers-on-networking.html

math

Questions and answers on networking Seeking for networking interview questions and answers, here is the largest collection of most frequently asked questions and answers in networking interview these questions will help you to prepare for your dream job confidently. Networking is a vast subject and is ever expanding and most frequently used interview topic networking questions are common to all the interviewing candidates of it. Hiring a networking professional use our interview questions to find the best candidate for your it team. Technical interview questions and answers section on networking with explanation for various interview, competitive examination and entrance test solved examples with detailed answer description, explanation are given and it would be easy to understand. Network+ certification practice questions exam cram 2 chapter 3 network implementation 125 quick check answer key 28. Networking interview questions updated on may 2018 302539 1 define network a network tcs technical interview questions and answers. The following interview questions on networking basics is a preview from the ebook – 250 networking interview questions and answers the gateway of pc1 is 19216811 should it be configured on it for pc1 to be able to ping 192168. Hi there, i would like to know any specific website which will help me to get all interview questions and answers for network domain, espacially for cisco (: 51822. Here is a list of basic ccna interview questions and answers which will help you clear your networking questions. Discover which are the most common network engineer interview questions sample interview questions for network and let us know the answers or more questions. Networking interview questions and answers for freshers and experienced, networking interview pdf, networking online test, networking jobs - here are all possible networking interview questions with answers that might be asked during interview. Hardware and networking interview questions with answers it is really use person like me who is starting careers in it plz share this question n answer to me. Data communications and networking questions and answers - download as word doc (doc / docx), pdf file (pdf), text file (txt) or read online it is useful. Check your readiness for the exam with free network+ practice test questions this post includes 5 questions with full explanations of the correct answers. - The best advice on the 10 most common interview questions and answers to show you how to understand, practice, and craft winning answers for each question. - Learn networking mcq questions and answers with easy and logical explanations networking mcq questions are important for technical aptitude exams as well as technical interviews. - In this article i cover 6 common social media questions read ian’s article 6 answers to common social media questions i think li is a networking. Basic and advance networking interview questions and answers basic networking interview questions and answers with examples read our thousands of basic and advance networking jobs interview questions and answers and dowload networking pdf free of cost. Clear answers to common questions about architecting networking, and security these answers outline aws best practices and provide prescriptive architectural. You can test your basic computer networking knowledge and skills with quiz at the end of the quiz, your total score will be displayed. A + (hardware plus) ,449 question & answer hcl cdc meerut (a+ related networking course ques & ans) by deepak kumar a+ questions and answers:: 11:.Get file

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https://community.constantcontact.com/t5/Get-Help/For-the-pre-built-segment-what-is-the-criteria-for-most-engaged/td-p/373070

math

Hello @YanK60 , The algorithms used to determine engagement levels work on a number of elements, open rate being the starting point, and also incorporating elements like clicks, frequency of opens, etc. Can you be a little bit more detail? Like how many times or percentage of open/open rates or how many clicks will be counted as most engaged? Thank you Not really since the algorithm adjusts for the frequency of your sends as well. "High engagement" can differ greatly between someone that sends 1 email a month versus someone that sends 1 or more emails a day, how many links are included in those emails, and a lot of other factors.

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CC-MAIN-2021-43

https://www.scootertalk.org/forum/viewtopic.php?p=52698&sid=ce9182372feefc4b0bc9358de8d1fdfa

math

I have battery that has 36V and all cell groups are the same. I measure the output voltage from BMS at the XT30 connector and I get 11V less. My question is - is this correct or does that mean the BMS is devective? I have been told the output from XT plug should match the battery. Thank you knowledgeable, helpful people!

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CC-MAIN-2021-17

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=10915&option_lang=eng

math

Computation of viscous flow between two arbitrarily moving cylinders of arbitrary cross section A. O. Kazakovaa, A. G. Petrovb a Chuvash State University, Cheboksary, 428015 Russia b Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 117526 Russia An algorithm is constructed for numerical study of the plane viscous flow (in the Stokes approximation) between two arbitrarily moving cylinders of arbitrary cross section. The mathematical model of the problem is described by the biharmonic equation, which is reduced, with the help of the Goursat representation, to a system of equations for two harmonic functions. The system is solved numerically by applying the boundary element method without saturation. As a result, the desired biharmonic stream function and the velocity field are determined. Test examples are used to compare the results with well-known exact solutions for circular cylinders. viscous flow, Stokes approximation, stream function, biharmonic equation, boundary element method. |Russian Science Foundation |This work was supported by the Russian Science Foundation, project no. 19-19-00373. Computational Mathematics and Mathematical Physics, 2019, 59:6, 1030–1048 A. O. Kazakova, A. G. Petrov, “Computation of viscous flow between two arbitrarily moving cylinders of arbitrary cross section”, Zh. Vychisl. Mat. Mat. Fiz., 59:6 (2019), 1063–1082; Comput. Math. Math. Phys., 59:6 (2019), 1030–1048 Citation in format AMSBIB \by A.~O.~Kazakova, A.~G.~Petrov \paper Computation of viscous flow between two arbitrarily moving cylinders of arbitrary cross section \jour Zh. Vychisl. Mat. Mat. Fiz. \jour Comput. Math. Math. Phys. Citing articles on Google Scholar: Related articles on Google Scholar: |Number of views:|

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https://uk.tradingview.com/script/I6IGk8Gl-Hancock-Filtered-Volume-OBV/

math

"The decision to remove low days, or intervals, means that a series of low days that results in a collective change will be ignored. If you have decided that removing those days makes sense, then measure the daily or intraday against a threshold created using the average minus one or two standard deviations of the . Using a 1 standard deviation filter will remove the lowest 16% of the days; a 2 standard deviation filter removes 32% of the days. This type of filter is best applied to a index, such as On-Balance . For example, we find that the average on the New Stock Exchange is 1.5 billion shares, and 1 standard deviation of the is 0.25 billion shares. We decide that the filter is 2 standard deviations; therefore, any day with below 1.0 billion shares will be ignored." - Trading Systems and Methods, 2013. This has a signal line which is subtracted from the filtered OBV to produce a simple oscillator giving signals above or below the 0 line.

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CC-MAIN-2020-34

https://www.kiasuparents.com/kiasu/question/73396-2/

math

can help with this question ,thank you! Omar and Mike were racing on a cross country motor race at an average speed of 140km/h and 120km/h respectively. At 8am, Omar was 45km ahead of Mike. 45 mins later, Omar’s car broke down and he discontinued the race. (a) How far was Omar ahead of Mike when Omar’s car broke down? (b) At what time did Mike pass the point where Omar’s car broke down?

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https://demtutoring.com/answered/biomedicalengineering/q5962

math

- what is the absolute value of -6? use the number line to help answer the question. 2 6 s 4 3 2 1 6 7 Get the answer Category: biomedicalengineering | Author: Giiwedin Frigyes - find the product of (_3xyz) (4/9x²z) (27_/2xy²z)and verify the result forx=2,y=3,and z=_125. find the product of (-3xyz)and z = -1. - is a side effect of tobacco use. a. lightheadedness b. irritability c. dizziness d. all of the above please select the best answer from the choices - whether the situation is an example of an experiment or an observational study. explain. a nutritionist wants to know if taking vitamins keeps peop

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http://scihi.org/charles-hermites-admiration-for-simple-beauty-in-mathematics/

math

On December 24, 1821, French mathematician Charles Hermite was born. He was the first to prove that e, the base of natural logarithms, is a transcendental number. Furthermore, he is famous for his work in the theory of functions including the application of elliptic functions and his provision of the first solution to the general equation of the fifth degree, the quintic equation. Charles Hermite studied at the Collège de Nancy, at the Collège Henri IV in Paris, and at the Lycée Louis-le-Grand. He entered the École Polytechniqu in 1842 but was refused to continue his education there due to a disability in his right foot. Hermite began corresponding with Carl Jacobi around 1843, which resulted in a fruitful working relationship. Hermite spent about five years working on baccalauréat privately and returned to the École Polytechnique as répétiteur and examinateur d’admission in 1848. On July 14 1856, Hermite was elected to fill the vacancy created by the death of Jacques Binet in the Académie des Sciences and in the late 1860s, he succeeded Jean-Marie Duhamel as professor of mathematics, both at the École Polytechnique, where he remained until 1876, and in the Faculty of Sciences of Paris, which was a post he occupied until his death. Charles Hermite passed away on January 14, 1901. Hermite had the reputation of being a great teacher and his published lectures had a wide influence in the field of mathematics. His contributions to mathematics were published in the most important scientific journals of the time and they dealt with Abelian and ecliptic functions as well as the theory of numbers. He managed to solve the fifth degree by elliptic functions in the 1870s and he proved e, the base of the natural system of logarithms to be transcendental. At yovisto, you may enjoy the video lecture “History of Mathematics in 50 Minutes” by John Dersch. References and Further Reading: - Charles Hermite at MacTutor History - Charles Hermite at Mathematics in Europe - Charles Hermite at zbMATH - Charles Hermite at Mathematics Genealogy Project - Charles Hermite at Wikidata - Timeline for Charles Hermite, via Wikidata Related Articles in the Blog: - Carl Jacobi and the Elliptic Functions - Legendre’s Elements of Geometry - Carl Friedrich Gauss – The Prince of Mathematicians

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http://mathematica.stackexchange.com/tags?page=10&tab=popular

math

A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question. |Type to find tags:| Questions about Mathematica's procedural programming paradigm. |query× 21||proof× 21| Tag for questions on Optical Character Recognition (OCR), including the Mathematica function TextRecognize. Questions related to writing and handling usage messages of symbols. |input× 21||input-forms× 20||lattices× 19||geometric-computation× 19| Tag for questions about backslides/backsets/degenerations/degradations/regressions/retrogressions in newer versions of Mathematica, notice this tag isn't for bugs and compatibility issues. |cellular-automata× 19||random-process× 19||networking× 19| |mathematica-online× 19||order× 19|| Questions about specific or optimal placement of objects in a shape or volume. For questions about packed arrays use [tag:packed-arrays]. |opencllink× 18||size× 18||topology× 18||asynchronous-processing× 18| a process of making a three-dimensional solid object of virtually any shape from a digital model. Mathematica makes it possible to take complex algorithmically generated geometry and im… Questions on using Mathematica functionality to take advantage of a Graphics Processing Unit, a special stream processor designed for highly parallel computer graphics operations. Interrupting calculations, preventing interrupts and recovering data from aborts. programmatically construct a custom graphical user interface under Mathematica. The utilization of advanced computing technology in mathematical research: new mathematical results discovered partly or entirely with the aid of computer-based tools. Questions on Wolfram SystemModeler, an environment for the modeling of systems with mechanical, electrical, thermal, chemical, biological, and other components, as well as combinations of different ty… |unevaluated× 17||mysql× 17|

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http://kupika.com/arfAmi/shit

math

I'm usually a lighthearted person. But upon finding this site, I allowed myself to slip into my darker stage. I'm a girl. Almost fourteen years old. Still. Normal. I've been waiting for my Hogwarts letter since I read the first book. I hurt myself a lot, but often yell at myself afterwards. I'm a gamer. And I rule Call of Duty 4. I won't give you my tag, no. I'm just me. I can't define myself to you because I can't define myself. I can't tell you who I am because I don't know. But fear not, I am friendly. I will be nice to you if you have the sense to write grammatically correct. However, I am cursed with sarcasm. Where I got My Layouts!

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CC-MAIN-2018-09

https://brainmass.com/math/trigonometry/find-graph-sine-cosine-curve-502844

math

The graph of one complete period of a sine or cosine curve is given in the attached document. a) Find the amplitude, period and phase shift. b) Write an equation that represents the curve in the form: y = a sin k (x - b) or y = a cos k (x - b) 44. amplitude = 3, period = 4?, phase shift = 0, no shift K = 2pi/(4pi) = 1/2 The expert finds the graph of a sine of cosine curves.

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CC-MAIN-2021-39

http://conceptmap.cfapps.io/wikipage?lang=en&name=Talk:Differential_geometry

math

|Differential geometry has been listed as a level-4 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.| |WikiProject Mathematics||(Rated B-class, Top-importance)| where to put stuff?Edit Tosha suggests that this article is not the right place to talk about vectors as derivations or bundles and their sections. But it seems that there is nowhere with a sophsiticated discussion of vector fields. we need one. but where? - Lethe 00:10, Jul 17, 2004 (UTC) - I simply think that good discussion vector fields sould be in vector field, but it is totally ok to mention it here. Tosha This claims that if an area preserving map of a ring twists each boundary component in opposite directions, then the map has at least two fixed points. Umm, like a diamond ring? Why can't the word "cylindar" be used, if that's what is meant?? There is no history on the development of the subjectEdit I find no history on the development of the subject of differential geometry, of curves and manifolds. A cursory history won't suffice: only a detailed history may do justice to the subject. It needs to cover everything, even the minor nuances. Detailed bibliography is also required in support of key sentences and comments. Would like to help develop this section if a bibliography list is provided. Bkpsusmitaa (talk) 03:32, 3 August 2015 (UTC) - I don't agree that ANY article in a encyclopedia should EVER "cover everything". First: it is logically impossible. Second: writing must target its audience. At the two ends of the knowledge spectrum, a reader with no differential calculus nor topological knowledge will find specialist writing incomprehensible and the specialist will find the necessarily crude and imprecise language used for the lay public almost useless. Both will be frustrated.(Sorry, if I'm stating the obvious, but it seems it needed to be stated.) Third: history of a subject is at most a minor part of the information about a subject (except, of course historical subjects). This article (as of Feb 1, 2016) is certainly NOT detailed, and it follows that its history sub-section should be the same in that regard.188.8.131.52 (talk) 16:30, 1 February 2016 (UTC) The caption for the first figure (as of Feb 1, 2016 it shows a hyperbolic triangle embedded in a hyperbolic plane) uses two terms which I think ought to be replaced: 1) it uses "plane" when the figure clearly is not a (euclidean) plane. While the modifier "hyperbolic" could be added, I suggest it would be better to substitute the word "surface" here. 2) it uses the word "immersed"; I've never seen that useage; the word suggests - to me - a depth that a surface does not possess. The better word is clearly "embedded". If the editors agree, please make the changes.184.108.40.206 (talk) 16:39, 1 February 2016 (UTC) The comment(s) below were originally left at several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section., and are posted here for posterity. Following |"Intrinsic versus extrinsic" section needs expansion; needs more on history and examples. Consider splitting article into Differential geometry and Differential topology, failing that, more material on Differential topology needed. Tompw (talk) 20:10, 1 March 2007 (UTC) Recommend splitting into into Differential geometry and Differential topology, with an overview, and non-technical introduction here. More material on Differential topology is certainly needed anyway. Geometry guy 18:31, 14 April 2007 (UTC)Split. I agree that what remains should be a gentle introduction to both fields. In addition, perhaps some modest technical details about the overlap and differences (in general terms) should also be included here. Silly rabbit 17:39, 26 April 2007 (UTC) Substituted at 21:39, 26 June 2016 (UTC)

s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667262.54/warc/CC-MAIN-20191113140725-20191113164725-00117.warc.gz

CC-MAIN-2019-47

http://quant.stackexchange.com/questions/tagged/market-data+reuters

math

Quantitative Finance Meta to customize your list. more stack exchange communities Start here for a quick overview of the site Detailed answers to any questions you might have Discuss the workings and policies of this site Mapping symbols between tickers, Reuters RICs and Bloomberg tickers Is there any known solution (preferably open source) to map between ticker symbols, Reuters and Bloomberg symbols. For example: Ticker: AAPL Reuters: RSF.ANY.AAPL.OQ Bloomberg: AAPL US Equity ... Mar 20 '13 at 17:56 newest market-data reuters questions feed in 16 hours Hot Network Questions Why would a journal accept my previous paper for publication, but reject latest manuscript due to being "outside of the journal's scope?" Why "ls" doesnt show the file that "find" discovered ? What does "You feel stupid. You feel clumsy." mean? If DOS is single-tasking, how was multitasking possible in old version of Windows? Is it dangerous to go to the USA as a Russian now? How can I kill puppies without consequences? Shortest code to print ':)' random times Group memberships and setuid/setgid processes What does “sign the final flourish” mean? Can humans eat grass? Good slide design for teaching? Prime number generator exercise Why does the Stack Overflow swag request form have six address lines? Why was King Shaul commanded to kill all the animals of Amalek? Transform big number into scientific format Sort characters by darkness What makes the need of a definite article? Why is the nozzle of Delta IV capped? I am considering leasing a new car for my ex-wife. What can go wrong? Examples of mathematical results discovered "late" Is it technically a checkmate if the king is not in check, but all moves will result in check? How to typeset a file path? Help me prove that my pickup technique for women is statistically valid Is there a way to determine a node's status through API calls? more hot questions Life / Arts Culture / Recreation TeX - LaTeX Unix & Linux Ask Different (Apple) Geographic Information Systems Science Fiction & Fantasy Seasoned Advice (cooking) Personal Finance & Money English Language & Usage Mi Yodeya (Judaism) Cross Validated (stats) Theoretical Computer Science Meta Stack Overflow Stack Overflow Careers site design / logo © 2014 stack exchange inc; user contributions licensed under cc by-sa 3.0

s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999677213/warc/CC-MAIN-20140305060757-00020-ip-10-183-142-35.ec2.internal.warc.gz

CC-MAIN-2014-10

http://kleine.mat.uniroma3.it/mp_arc-bin/mpa?yn=09-65

math

- 09-65 Christian G rard, Fumio Hiroshima, Annalisa Panati and Akito Suzuki - Infrared divergence of a scalar quantum field on a pseudo Riemann manifold Apr 23, 09 (auto. generated pdf), of related papers Abstract. A scalar quantum field model defined on a pseudo Riemann manifold The model is unitarily transformed to the one with a variable mass. By means of a Feynman-Kac-type formula, it is shown that when the variable mass is short range, the Hamiltonian has no ground state. the infrared divergence of the expectation values of the number of bosons in the ground state is discussed.

s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313536.31/warc/CC-MAIN-20190818002820-20190818024820-00356.warc.gz

CC-MAIN-2019-35

http://m.beck-shop.de/item/37313330373636

math

Stochastic Discrete Event Systems Stochastic discrete-event systems (SDES) capture the randomness in choices due to activity delays and the probabilities of decisions. This book delivers a comprehensive overview on modeling with a quantitative evaluation of SDES. It presents an abstract model class for SDES as a pivotal unifying result and details important model classes. The book also includes nontrivial examples to explain real-world applications of SDES. Coherent and comprehensive overview on modeling with and quantitative evaluation of stochastic discrete event systemsImportant model classes like queuing networks, Petri nets and automata are presented in detailIntroduction of a new abstract model class as a unifying frameworkNontrivial examples from areas like manufacturing control, performance of communication systems, and supply-chain management to explain real-world applications of SDES

s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00575.warc.gz

CC-MAIN-2018-51

http://www.keiraknightley.com/forums/showpost.php?p=523&postcount=14

math

Originally Posted by Hazzle Stalker alert...stalker alert... Clue 1: Talking about Keira as if she's someone close. Clue 2: Talking about "Keira's happiness" and how it would somehow make you happy, despite having no connection with her. Clue 3: Referring to one's own self as "us" as opposed "me" oops i think there will be a lot of stalkers at the forum .•°¯°•.¸.•°¯°•.-> 3rd Member Of Keira Knightleyz Posse!<-.¸.•°¯°•.¸.•°¯°•. The lobbying groups all hate him and thats a good sign. You may laugh because I'm different, but I laugh because you're all the same! Quote Narg aka Brendon Gilson RIP

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CC-MAIN-2014-23

http://zbmath.org/?q=an:1017.39010

math

Let be a unital Banach algebra with norm , and let , be left Banach -modules. A quadratic mapping is called -quadratic if for all , . Let be a function such that one of the series and converges for every . Denote by the sum of the convergent series. The following theorem is proved: Theorem. Let be a mapping such that and for all and all . If is continuous in for each fixed , then there exists a unique -quadratic mapping such that for all . The similar results are obtained for the other functional equations: and for the classical quadratic functional equation.

s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00322-ip-10-147-4-33.ec2.internal.warc.gz

CC-MAIN-2014-15

http://www.myhsebnotes.com/2013/07/simple-harmonic-motion-SHM.html

math

Simple Harmonic Motion (S.H.M) Class : 12 We all are familiar with periodic motion, i.e. the motion which repeats itself at equal intervals of time. A few common examples of periodic motion are the motion of simple pendulums and compound pendulums, the motion of electrons in their orbits, motion of planets around the sun, vibration of stretched wires, motion of a mass attached to a string etc. The interval of time after which the motion is repeated is called time period of the motion. If a particle in periodic motion has to and fro motion over the same path, is called Oscillatory or vibratory motion. The motion of a mass spring system, motion of simple pendulum, vibration of atoms at their lattice sites are a few examples of oscillatory motion. Most of the oscillatory motions in nature are simple harmonic motions (s.h.m.). The vibration of atoms at the lattice sites are approximate simple harmonic. The oscillation of simple pendulum is simple harmonic when its amplitude is small. The motion of air molecules when sound waves pass through it is simple harmonic. Definition of s.h.m : It is defined as the motion in which the acceleration is always directed towards the mean position (a fixed point) and is directly proportional to the displacement from the mean position. This Content is moved to our new site => http://www.merospark.com/hseb-notes/simple-harmonic-motion/ Read this also: - Types of Wave | Physics Class 12 - Progressive Wave and Its Equation | Physics Class 12 - Application of Biology in Aspect to Zoology | Class 12 - The Heritage of Words - Complete Summary | Compulsory English Class 12 Don't forget to LIKE, SHARE and COMMENT. Please Join with us on Facebook and Google+ for daily Notes and Updates.

s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275836.20/warc/CC-MAIN-20160524002115-00074-ip-10-185-217-139.ec2.internal.warc.gz

CC-MAIN-2016-22

https://www.arxiv-vanity.com/papers/hep-ph/0401116/

math

Black Hole Production in and Higher Curvature Gravity The problem of black hole production in transplanckian particle collisions is revisited, in the context of large extra dimensions scenarios of TeV-scale gravity. The validity of the standard description of this process (two colliding Aichelburg-Sexl shock waves in classical Einstein gravity) is questioned. It is observed that the classical spacetime has large curvature along the transverse collision plane, as signaled by the curvature invariant . Thus quantum gravity effects, and in particular higher curvature corrections to the Einstein gravity, cannot be ignored. To give a specific example of what may happen, the collision is re-analyzed in the Einstein-Lanczos-Lovelock gravity theory, which modifies the Einstein-Hilbert Lagrangian by adding a particular ‘Gauss-Bonnet’ combination of curvature squared terms. The analysis uses a series of approximations, which reduce the field equations to a tractable second order nonlinear PDE of the Monge-Ampère type. It is found that the resulting spacetime is significantly different from the pure Einstein case in the future of the transverse collision plane. These considerations cast serious doubts on the geometric cross section estimate, which is based on the classical Einstein gravity description of the black hole production process. pacs:04.70.-s, 04.50.+h, 11.10Kk All present-day macroscopic experimental data are consistent with gravity being described by the 4-dimensional Einstein-Hilbert action In the microworld no gravitational effects have been observed, and indeed the 4-dimensional (4d) Newton constant in (1) is so small that such effects would be totally negligible up to unreachable collider energies of (the 4d Planck scale). The idea of large extra dimensions allows to imagine a world in which gravity is strong enough to play a role in elementary particle collisions at accessible energies. In this world Einstein gravity propagates in dimensions, out of which are curled up in a compact manifold of size . At distances such gravity would be described by an effective 4d action of the form (1). To get the effective 4d Newton constant right, we have to set the -dimensional Planck scale at ADD This means that such gravity will be much stronger than we are used to believe at short distances . Motivated by the hierarchy problem, Arkani-Hamed, Dimopoulos and Dvali ADD proposed to push this idea to the extreme and lower all the way down to the electroweak scale . This proposal requires extra dimensions ranging from a mm to a fermi for ( leads to and is excluded). To make this consistent with the 4d Standard Model, we have to assume that all the other fields but gravity do not feel the extra dimensions. They must be confined to a 4d submanifold (brane) of the -dimensional world (bulk). The simplest collider signature of such braneworld scenarios would be apparent energy non-conservation due to produced gravitons escaping into the bulk. Recent Tevatron searches Tevatron of such missing energy events found no statistically significant effect and set a lower bound of on the -dimensional Planck scale (for a general review of signals and constraints, see Rev ). Of course the LHC with its c.m. energy of 14 TeV will be a much better probe of such phenomena. In this paper I would like to contribute to the unfinished discussion of another possible signature of TeV-scale gravity — black hole production in transplanckian collisions (see reviews Landsberg ). The common current opinion is that this process may be adequately described using classical general relativity, and that a single large black hole will form for a range of impact parameters (in the -dimensional Planck units) Here is the energy of colliding particles, and is the Schwarzschild radius of the -dimensional static black hole of mass . If production cross section based on this estimate (known as the geometric cross section) were true, the LHC would produce black holes at a rate 1 Hz for TeV, becoming a black hole factory factory . In older studies of transplanckian collisions Ven the authors usually carefully stated that in the range (3) a black hole is expected to form. Indeed, the eikonal expansion, normally used to analyze quantum gravity in the transplanckian regime, breaks down exactly in that range. It seems that the geometric cross section proposal was first put forward unequivocally by Banks and Fischler BF . Later on, an appealing theoretical argument backing (3), (4) was proposed by Eardley and Giddings EG . In the first step of the argument the problem of black hole formation is analyzed in classical Einstein gravity using the closed trapped surface (CTS) method. A well-defined mathematical problem of finding a CTS in the spacetime formed by superposing two Aichelburg-Sexl shock waves is formulated and solved. For the range of impact parameters when a CTS is found, black hole formation is concluded by invoking the Cosmic Censorship Conjecture. In the second step the aim is to argue that quantum corrections to general relativity are not likely to modify the conclusions, because the classical spacetime has small curvature in the regions relevant for the trapped surface evolution. However, it is this second step where I disagree with Eardley and Giddings.111It should be mentioned that the validity of the geometric cross section was also challenged by Voloshin Vol . Some of his criticisms were addressed in EG ; disc , others still remain unanswered. There does not seem to be an obvious connection between Voloshin’s and my reasons for critique. As we will see, curvature becomes large on the transverse plane at the moment when the particles pass each other, and in the future of this plane corrections to Einstein gravity cannot be ignored. In particular, if higher order curvature terms are present in the effective gravitational Lagrangian (and we have no reason to believe that they don’t), they will become important precisely at this moment. To see a specific example of how this might happen, we will add to the Einstein-Hilbert Lagrangian a particular combination of curvature squared terms first considered in dimensions by Lovelock Lovelock , with a coefficient of natural magnitude in Planck units. The technical reason why I choose this combination is that it leads to second order equations of motion. My primary goal in this paper is to furnish an example of how things may go wrong, and it is unlikely that any other combination would be better in this respect. In this Einstein-Lovelock theory we will then consider the simplest — zero impact parameter — collision case. Existence of a CTS for such head-on collisions was shown long ago by Penrose Penrose . We will see that the curvature squared terms indeed become important and significantly modify the spacetime to the future of the transverse collision plane. The result of this analysis is summarized in Fig. 1. In the Einstein case the post-collision spacetime is weakly curved (at large transverse radii). The metric in the future wedge can be expanded in a power series in the light-cone coordinates , starting with a linear term . The Riemann tensor has -function singularities on the null planes and ; the Ricci tensor is of course everywhere zero. That curvature becomes large can be seen from the invariant . On the contrary, in the Einstein-Lovelock case the future wedge is split into sectors , , separated by planes on which the Riemann tensor has additional -function singularities. Moreover, the Ricci tensor is also singular on these planes, and in general nonzero in between. In particular, the post-collision spacetime does not satisfy Einstein’s equations. An additional feature is that the number of sectors is arbitrary; the solution is not unique. The paper thus contains two main results — a general observation (curvature becomes large) and a specific example (analysis in the Einstein-Lovelock theory) — which show that classical Einstein gravity may not be applicable to study black hole production in transplanckian collisions. If so, the geometric cross section estimate (4) becomes much less certain, having lost its most serious supporting argument. The plan of the paper is as follows. In Sec. II we will review the CTS argument. Readers familiar with it may go directly to Sec. III, where we will discuss the danger posed by regions of high curvature and see that curvature becomes large along the transverse plane at the collision moment. This conclusion is reached after the finite width of the shocks due to classical and quantum smearing is taken into account. The remaining part of the paper deals with the Einstein-Lovelock theory. In Sec. IV we will review the Lovelock Lagrangian and its higher order generalizations. I will also comment on the history of this Lagrangian and its alleged connection to string theory. In Sec. V we will write down a metric ansatz for the head-on collision, in a plane-wave approximation. Then we will derive equations of motion, in a small-gradient approximation, in the region of longitudinal light-cone coordinates . We will also see that the cubic and higher order Lovelock Lagrangians do not contribute in this approximation. In Sec. VI we will solve the resulting hyperbolic Monge-Ampère equation. First we will see that no usual-sense (smooth) solution exist, even when we consider the smeared out shocks as initial conditions. Then, in the limit of infinitely thin shocks we will find an infinite class of generalized (weak) solutions. We will discuss what this non-uniqueness may mean, and how it can possibly be resolved. In Sec. VII we will check that the small-gradient approximation made in Sec. V was justified. We will then use the found solutions of the Monge-Ampère equation to write down approximate spacetime metrics. I will explain the differences between the Einstein and Einstein-Lovelock post-collision spacetimes, and their negative implications for the CTS argument. I will conclude in Sec. VIII with a summary of the obtained results. Notation. In the remainder of the paper we work in flat spacetime of dimensions (although we quote some results for comparison). This is legitimate, since the relevant gravitational dynamics happens in a region of size, which is much smaller than the size of extra dimensions. We use Planck units, setting in the -dimensional Einstein-Hilbert action. Greek indices run through all coordinates, Latin only through transverse directions. Spacetime signature is , and the curvature tensor definitions are and (‘Landau-Lifshitz timelike conventions’). Ii Closed Trapped Surface argument The basic ingredient in existing discussions of black hole formation in particle collisions is the gravitational field of a fast point particle moving along a straight line. In the limit of infinite boost and fixed energy , this field takes the form of a shock wave Here , , the particle is moving in the positive direction, and denotes the transverse coordinates. Einstein’s equations with the lightlike source give the following equation for the shock wave profile : Thus we get222This result can also be derived by boosting the static -dimensional Schwarzschild solution and simultaneously scaling down the rest mass to keep the energy constant. This was the method originally used by Aichelburg and Sexl in [AS, ]. (; is the volume of the unit sphere) Form (5) of the metric is unsuitable for analyzing the behavior of geodesics crossing the shock at , which is necessary for understanding the causal structure. For this we must switch to coordinates in which the metric is continuous. This is accomplished by the discontinuous coordinate transformation [D'Eath, , DH, , EG, ] (where is the Heaviside step function). In the new coordinates the metric becomes and both geodesics and their tangents are continuous across the shock. Introducing polar coordinates in the transverse plane, this metric can be written as (see D'Eath for ) To discuss a collision, we add a second fast particle of the same energy moving opposite to the first one in the negative direction at a transverse distance (impact parameter) . Assuming that the particles pass each other at , the combined gravitational field outside the wedge can be obtained by superposing the metric (13) with its mirror image: Such superposition is legal, because the excised region (wedge in Fig. 1) is precisely the future light cone of the collision plane of the shocks. By causality, outside this region the shocks will not be able to influence each other. The metric for must be found by solving the characteristic initial value problem for Einstein’s equations. This task is complicated by the fact that the shocks after passing each other will focus and develop regions of high curvature. Shock propagation through these regions depends on their detailed structure D'Eath . Nevertheless, we are interested whether this complicated dynamics will lead to black hole formation. Since the complete metric is unknown, we must resort to indirect arguments. The idea [Penrose, , EG, ] is to look for a closed trapped surface (CTS) in the part (16) of the spacetime that we do know. CTS is defined as a closed -surface whose area decreases locally when propagated along the outer null normals. Equivalently, the outer null normals of such surface have positive convergence HE . Existence of a CTS in a spacetime solving vacuum Einstein’s equations implies presence of a singularity in the future. Assuming Cosmic Censorship, this singularity must be hidden behind a horizon, and we may conclude that a black hole will form. Moreover, the black hole horizon must lie outside the CTS. Using this information, one can get estimates of the horizon area and, via the Area Theorem, of the mass of the formed black hole. For the head-on collision () a CTS is easy to find [Penrose, , EG, ]. It lies in the union of the shock planes and and, when transformed to the coordinates, consists of two flat -dimensional disks of radii centered at the collision point. In the coordinates the CTS is glued out of two halves described by It is symmetric under rotations of the transverse directions and the reflection (see Fig. 2). Strictly speaking, this surface is a marginal CTS, which means that the outer null normals have zero convergence. (Such a surface is also called apparent horizon.) However, moving a small distance inside one can find a true CTS with negative convergence. For nonzero impact parameters () the search of trapped surfaces was carried out analytically for by Eardley and Giddings EG and numerically for by Yoshino and Nambu YN . In every existence of a marginal CTS was established for impact parameters where is a numerical constant weakly depending on . This result of classical general relativity if the most serious argument in favor of the geometric cross section for the black hole production rate. Iii High curvature at The CTS argument of the previous section was valid for the Einstein gravity with its vacuum equation of motion . This equation is used crucially in the Raychaudhuri equation describing how the convergence evolves along the congruence of null normals : For Ricci-flat spacetimes the first term vanishes. One then concludes that once goes positive, it grows monotonically and becomes infinite at finite affine distance . This argument is the basis of the proof of singularity theorems HE . Now suppose that long before the singularity forms, the evolving surface passes through a region of high curvature. If higher order curvature terms are present in the effective gravitational Lagrangian, they will become important and modify the field equations at this point. In general, will no longer vanish, and one cannot make any conclusions about the dynamics of across the region of high curvature. From the effective field theory point of view, it seems likely that all interactions allowed by symmetry should become significant in a theory at its natural scale (which is the -dimensional Planck scale in our case). This is the main reason to believe that the gravitational Lagrangian will contain higher curvature terms, with coefficients of natural magnitude . There are also other hints pointing in the same direction. For instance, since the Einstein gravity is non-renormalizable (see, e.g., review deser ), higher curvature counterterms should be included to cancel infinities in loop diagrams. In string theory, which is a finite theory of quantum gravity books , a particular sequence of higher curvature terms with calculable coefficients appears in the effective action (the -expansion).333For completeness, we have to mention that Loop Quantum Gravity lqg attempts to quantize general relativity non-perturbatively, starting from the pure Einstein-Hilbert action. Since the program is unfinished, and in particular the classical limit is poorly understood, it is hard to say what form the gravitational field dynamics will eventually take. It seems likely that the effective action, as long as this concept is applicable, will have to contain higher curvature terms even in this theory. It should be noted that singularity theorems of general relativity can also be formulated and proved in presence of matter sources satisfying various positivity assumptions, such as the null energy condition . While such conditions are natural for classical matter, they can be violated by the renormalized energy-momentum tensor of quantum fields BD . So it is unlikely that any useful generalization of singularity theorems exists when quantum effects are taken into account. These considerations make it clear that appearance of high curvature regions is problematic for the whole CTS argument. Eardley and Giddings tried to address this problem. In particular they noticed that the CTS they constructed lies in the planes of shocks. They proposed to bring it out of these planes by propagating it a small affine distance to the future along the outer null geodesics. This way, they said, one can get a CTS lying everywhere in the region of small curvature. The conclusion was that quantum gravity effects are unlikely to modify the result: the black hole will still form. I believe that this argument as it stands is incomplete: it misses the fact that curvature may become large on the part of the trapped surface where it crosses the transverse collision plane. First of all, the only nonzero components of the Riemann tensor of spacetime (5) are All curvature invariants (contractions of this tensor with itself and the metric) vanish identically. Thus a shock plane by itself is not a region of high curvature.444This also follows from the fact that the Aichelburg-Sexl wave can be obtained by boosting from manifestly low-curvature regions far away from a static -dimensional black hole. However, a problematic region does appear when we add a second shock. It is located at , where the shocks collide. At this moment we can form a nonvanishing curvature invariant of the collision spacetime (16): This equation seems to suggest that curvature becomes large (and even infinite) all along the transverse plane . However, it would be incorrect to jump to this conclusion too soon. The reason is that infinitely thin shocks are an idealization; in reality the shocks will have a finite width . There are essentially two reasons for . The first, purely classical, reason is that in practice is large but not infinite. Because of this, the shocks have width The second reason has quantum nature. In relativistic quantum theory coordinates of a particle at rest cannot be measured more precisely than its Compton wavelength (see, e.g., BLP ) . For an ultrarelativistic particle, this limit becomes . Thus the point particle picture used in the CTS argument should not be taken literally. In reality the particles should be thought to have finite size , which translates to a contribution to the shock width We are interested in curvature at transverse radii relevant for the trapped surface formation and evolution. In practice, for such we will always have , because the masses of colliding particles are small compared to 1 TeV, and thus is very large. (E.g. at the LHC, using current-quark masses .) Thus we can estimate Because of nonzero width, -functions in (22), (23) will be smeared out over an interval of length . Since the integral has to remain unity, the maximal attained value will be . Using this value in (23), we find This formula involves a positive power of , and a large prefactor (increasing from to for ). We see that even after finite shock width is taken into account, curvature still becomes large in the region of transverse collision plane relevant for the black hole formation. This means that corrections to the Einstein gravity will become important at this moment. Their subsequent effect on dynamics is hard to predict in general. It is not a priori excluded that this effect will be transient, localized in a small shock-interaction region to the future of , after which the gravitational field will return to its Einsteinian value. In this case we could push the CTS through this small problematic region, similarly to Eardley and Giddings’s proposal, and get to small curvature values where the Einstein gravity would again be applicable. (In the original proposal [EG, , p. 6] the part of the trapped surface near was to be left fixed.) However, I think that such localization is unlikely to happen. It is much more probable that the corrections will be significant up to the values , and even further. This would mean that the Einstein gravity alone essentially loses its predictive power in the future of the collision plane (a possibility also mentioned by Kancheli Kan ). In the remaining part of the paper I will give an example of how this scenario may be realized, using a particular modification of the Einstein gravity by curvature squared terms. Iv Einstein-Lovelock gravity We consider a modification of the -dimensional Einstein gravity by curvature squared terms described by the Lagrangian The new coupling is assumed to be on the grounds of naturalness. Lagrangian (29) has an interesting property first discovered by Lovelock Lovelock : it is the only combination of curvature squared terms leading to second order equations of motion for the metric. (A generic combination would produce fourth order equations.) For this reason theory (28) has been often used in attempts to understand how higher curvature corrections modify the behavior of pure Einstein gravity. Thorough reviews of the existing literature can be found in Myers (applications to black holes) and DM (Kaluza-Klein scenarios; braneworlds). Recently, theory (28) was even discussed in connection with black holes produced in particle collisions: the authors of gbbh argued that by studying their Hawking evaporation spectra one can measure . However, in this paper we are interested how the Lovelock term manifests itself before rather than after the black hole formation, and whether it may in fact preclude this formation. Actually, Lovelock Lovelock has discussed a whole sequence of higher order Lagrangians This gives for and reduces to (29) for . In general, vanishes identically for and is a total derivative for (the generalized Gauss-Bonnet theorem EGH ). For , leads to second order equations of motion. The corresponding variations were also found by Lovelock Lovelock : For this coincides with the Einstein tensor. Explicit expressions for and () with terms of equivalent tensor index structure collected as in (29) can be found in Briggs . We won’t need them, since in practice for it is much easier to work with (30) and (32). Sometimes in the literature the theory (29) in is attributed to Cornelius Lanczos, citing his papers Lan . In fact, however, Lanczos never went beyond .555In 1932 he did not even include in the Lagrangian; in 1938 he proved that (29) is a topological invariant density in . I would like to thank N. Deruelle, J. Madore and J. Zanelli for the interesting discussions of this historical matter. The name ‘Gauss-Bonnet’ often associated with gravity theories based on Lagrangians (30) is also an example of unfortunate terminology, since in these ‘dimensionally continued’ Gauss-Bonnet densities are no longer associated with topological invariants (which is precisely why they become interesting from the dynamical point of view). In this paper we will use the term Einstein-Lovelock gravity for the theories based on (29), (30). It is often incorrectly stated that is the only combination of curvature squared corrections to the Einstein gravity consistent with string theory. This claim is based on a (correct) result of Zwiebach Zwiebach , who showed that (28) is the only curvature squared gravity theory in which the quadratic term in the expansion of the action around flat space is the same as in the pure Einstein gravity. In particular, the graviton is the only particle in the perturbative spectrum, and unphysical ghost poles usually associated with curvature squared terms (see, e.g., Stelle ) are absent. However, one should remember that only on-shell effective action can be defined in string theory. This is especially clear when this action is computed from the S-matrix, but is also true for the sigma model -function method (see, e.g., MT ). Thus there always remains field-redefinition freedom, which can be used for example to change the coefficients in front of or completely remove and terms in (29). All these actions would be equally consistent with the string S-matrix and, thus, with string theory DR . In a sense, string theory does not have much to say about off-shell dynamics of quantum gravity. I thus prefer to think of higher curvature theories like (28) not as fundamental microscopic theories, but as effective field theories of gravity (see, e.g., recent discussion in burgess ), with the hope that they may capture some aspects of gravitational field dynamics in presence of regions of high curvature. V Metric ansatz and equations of motion We will now study the effect of the Lovelock term (29) on the collision spacetime at . For simplicity, we will analyze the head-on collision (). In principle, the method could also be used to study the nonzero impact parameter case, most certainly with similar conclusions. Remember that we always assume . Our collisions are transplanckian: . It is this situation that can lead to formation of a large classical black hole according to the common lore [BF, , EG, , Landsberg, ]. For we would be speaking about Planck-size black holes, for which quantum gravity effects are significant without doubt. Our goal is to find the metric to the future of the collision plane up to , at transverse radii . Since , we can neglect transverse derivatives of the metric: at they are suppressed by compared to the longitudinal ones [see (16), (15)]. Neglecting transverse derivatives means that we approximate the metric near a given point in the collision plane located at by a plane wave collision metric (see Griff ) Here are Cartesian coordinates in the transverse plane near , with the axis pointing in the direction of the collision point, and in the orthogonal directions (see Fig. 3). To derive the equations of motion, it is convenient to work with a more general metric666In the rest of this section we suppress Einstein’s convention about summing in repeated indices, and indicate all necessary summations explicitly. The nonzero components of the Riemann tensor are ( are partial derivatives) We will now make an a priori assumption that which also implies Validity of this small-gradient approximation will have to be checked after a solution is found. For now we see that it is compatible with the initial conditions (34). all other components being . The vacuum equations of the Einstein-Lovelock gravity are The and components of this tensor equation will give independent equations for functions and . The nonzero components of and in the small-gradient approximation are easy to find using (40). For we have: For we use the general expression (32) with . Because of the antisymmetrization in the r.h.s. of (32), only terms for which , , are all different can contribute. Also, looking at (40), we see that or must be present in each pair of indices . Using such reasoning, it is easy to see that for the only nonzero components are777Notice that for the given conditions on indices are incompatible. This means that the higher order Lovelock tensors vanish in the small-gradient approximation. We now go back to our original ansatz (33), putting This equation is easy to solve. Namely, it implies that the relation satisfied by the initial data (34), will continue to hold for . There remain two equations (). After is excluded using (46), they can be brought to the form Thus we reduced the problem to solving a single nonlinear second order PDE (47). In the pure Einstein theory () the nonlinearity disappears, and we see that the solution for is given by the same formulas (34) as the initial data. For this is no longer a valid solution, since it has which becomes large at (namely, , when the finite shock width and the resulting smearing of the -functions are taken into account). which we should consider as a 2-dimensional truncation of the full Einstein-Lovelock action relevant for our problem. The first term in this action uses only the totally antisymmetric tensor to form contractions. For this reason the corresponding nonlinear term in Eq. (47) is invariant with respect to Euclidean rotations as well as Lorentz transformations. The linear wave equation term in (47) is of course only Lorentz-invariant. Vi Solving the Monge-Ampère equation vi.1 Smooth solutions Hyperbolic Monge-Ampère equations have exceptional character among the nonlinear second order hyperbolic PDEs: they can be reduced to a system of five first order characteristic ODEs, while in general (in two dimensions) one needs eight [CH, , p. 495]. However, in case of Eq. (47), because it has constant coefficients, this general theory is superseded by an even simpler method based on the Legendre transform. Let us rewrite (47) as and apply the Legendre transform [CH, , p. 32]. This results in a linear Poisson equation, which is readily solved. In this way Ignatov and Poponin IP obtained an exact solution of the original Monge-Ampère equation (52) in an implicit form, depending on two arbitrary functions and : where and must be found from For vanishing at infinity small functions this solution describes interaction of two colliding pulses of the field. The pulses emerge from the interaction region with their shape unchanged, resuming the motion along the pre-collision world lines. Unfortunately, this condition is violated in our problem. Indeed, to satisfy the initial conditions (34), we would have to take Because of finite shock width these initial data have to be smeared out on this scale. Still this results in near of the order . So the Jacobian (57), being equal to unity away from the origin, changes sign and becomes negative near . The transformation (56) is thus not one-to-one: there is a region of the plane which is covered 3 times, and where the inverse transformation is multiple-valued. This analysis shows that for initial data (34), even after smearing them out, the exact solution (55) in any case cannot be used verbatim. A region of spacetime appears where this solution is triple-valued, and we have to somehow choose between the three branches. Actually, it turns out that no satisfactory choice is possible. More precisely, any choice introduces discontinuities either in the function itself (which is totally unacceptable) or in its first derivatives (which brings back the problems which we temporarily resolved by using smoothed out initial data). All of them seem to violate Eq. (52). To get some insight about the source of these difficulties, I also studied the problem numerically. The conclusion of this study (which I am not going to report here in detail) is that the solution develops singularities characterized by discontinuous first partial derivatives. Evolution beyond these singularities apparently cannot be described by formula (55). A different strategy is required, which I will now proceed to describe. vi.2 Generalized solutions To find a solution, we will have to extend the class of admissible functions , allowing continuous functions with discontinuous first derivatives. [Notice that the initial data (34) corresponding to infinitely thin shocks belong to precisely this class.] Such generalized solutions, called weak in the theory of PDEs [CH, , p. 418, 486], are usually defined via integration by parts. In our case weak solutions can be defined, because the Monge-Ampère operator can be written as a divergence Thus Eq. (52) is equivalent to the conservation law where the current has components For smooth functions the differential form (60) of the conservation law is equivalent to the integral form: for any closed curve . However, the latter condition does not involve products of second derivatives of and can be unambiguously checked even when the r.h.s. of (52) cannot be defined. I thus take (63) as the definition of weak solutions of Eq. (52). In practice, it is always sufficient to check (63) only for infinitesimally small contours , and this only near special, most dangerous, points. In the rest of the plane the differential equation (52) can still be used. Provided that the first partial derivatives of are bounded, the nonsingular terms proportional to drop out of (63) in the limit of small contours. The remaining rule for catching possible -type singularities as in (50) takes the form [This formula has in fact general validity and follows directly from (59).] If there is a -function at the origin present in the integrand on the l.h.s., we will be able to detect it by studying the limit of the r.h.s. as . The latter procedure does not involve products of distributions and is unambiguous. For further discussion it is convenient to simplify (52) by making the substitution This PDE is hyperbolic due to positivity of . We will no longer attempt to use smeared out initial data, and instead will try to find a solution directly in the limit of infinitely thin shocks. Since in this limit is likely to be the most singular point, we change to polar coordinates PDE (66) becomes The crucial fact (easy to guess if one remembers the well-known connection between the homogeneous Monge-Ampère equation and developable surfaces) is that for functions of the form the first most singular term in (68) vanishes. To satisfy the full equation, we have to include in terms of order . Thus we are led to the ansatz888It is possible to consider a more general ansatz, adding terms , . The functions would then be restricted by an infinite sequence of nonlinear ODEs, obtained similarly to (74) and (75) below. These higher order terms can be neglected compared to (69) in the limit of small , and so we put them to zero for simplicity. In principle, however, the possibility of them being nonzero further aggravates the non-uniqueness issue to be discussed in Sec. VI.C below.: Compatibility with the initial conditions (34) requires: As an example when does appear, we can take the Einsteinian solution , which would correspond to And indeed we readily see that (72) is violated. which have to be solved with the following from (70) boundary conditions At the first glance we have a problem here, since ODE (74) implies that either or The way out of this impasse is to notice that although is indeed not allowed to be zero on an interval, it can still vanish at isolated points inside . At these points the derivative can have jump discontinuities. If on the rest of the interval (78) holds, ODE (74) will still be satisfied. The general solution of ODE (75) depends on two constants and has the form At points where we have, in agreement with (75), The constants and the overall sign in (79) can be chosen independently on the two sides of , provided that both choices are compatible with . In particular is allowed to flip sign at . Being multiplied by vanishing , the arising -function in does not violate (75). Functions are continuous on . The satisfies (78) everywhere on except at some chosen points where will have jump discontinuities. The vanishes at , and is given by (79) on each subinterval (, are the end points) with and an arbitrary overall sign , which can be chosen independently on each subinterval. Within this class it becomes possible to satisfy the no-delta condition (72). The simplest and most symmetric example can be constructed using one intermediate point (see Fig. 5). From continuity and (78) we must have The constant is determined from (72). We find that two values are allowed: For we have: Near the point the leading behavior of these solutions (as of all the solutions in the class described above) is piecewise linear in terms of the coordinates, with the term being a small correction (see Fig. 6). A more general example, which we will need in a discussion below, utilizes two symmetric intermediate points , , where . The function is given by where the is fixed by (72) to be The allowed functions can be written down according to the rules given above. For this example reduces to the previous one. Taking even more general configurations of intermediate points and imposing the no-delta condition (72), we can get infinitely many solutions of PDE (66) having the form of the ansatz (69) and satisfying the initial conditions (70). Solutions of the original Monge-Ampère equation (52) are then given by Eq. (65). vi.3 Discussion of non-uniqueness Let us repeat the logic of the preceding discussion. In Sec. VI.A we convinced ourselves that solutions of the Monge-Ampère equation (52) with the smeared out initial conditions (34) run into singularities, characterized by discontinuous first partial derivatives. The very meaning of a solution in presence of such discontinuities needs to be reconsidered. For this reason in Sec. VI.B we introduced the definition of weak solutions, using a standard procedure of integration by parts.999Notice that for smooth functions the concepts of usual and weak solutions are equivalent. It is only in presence of discontinuous first derivatives that the usual notion is not applicable, and we have to use the new definition. Working in the limit of infinitely thin shocks, we found a whole class of weak solutions with the required initial conditions. It is of course satisfying to find that solutions according to the new definition exist, but how should we interpret their non-uniqueness? In my opinion, there are two alternative possibilities. First, it is possible that this non-uniqueness can be resolved by performing a detailed study of PDE (52) with smeared out initial data at small but finite shock width . As we have just discussed, the singularities will still appear. But perhaps by studying their approach one may discover a preferred way to continue the solution past them. This would require techniques beyond what we use in this paper. However, we would like to point out that even if a ‘true’ unique solution is found in this way, it will still have to satisfy our weak solution criteria from Sec. VI.B. Because of this it seems likely (although cannot be guaranteed) that this ‘true’ solution will be close to one of the weak solutions we found in Sec. VI.B. The second, more intriguing, possibility is that the encountered non-uniqueness is fundamental. Then its interpretation can be most naturally given in quantum theory. It would simply mean that beyond the shock collision the gravitational field wavefunction ceases to be concentrated on a single classical configuration. Instead, it spreads out over all allowed solutions of the equations of motion. In this case we would have to sum over all solutions, using , where is the on-shell value of the action, as the weight. It should be noted that the problem of non-uniqueness of weak solutions is well known in the theory of PDEs. It is present for instance in the theory of first order quasilinear hyperbolic systems, where the weak solutions in question are shock waves. There the non-uniqueness is resolved by imposing the physically motivated ‘entropy condition’ Lax . One can try to do something similar in spirit for our PDE (52). For example, one may try to identify higher order currents that are conserved for usual smooth solutions, and impose their conservation as an additional constraint on weak solutions. In fact, the starting point of our discussion of weak solutions was to rewrite the Monge-Ampère equation (52) itself as the current conservation condition (60). Going a step further, let us introduce a symmetric 2-tensor with components Actually, the appearance of the Monge-Ampère operator in the r.h.s. is not surprising: is nothing but the energy-momentum (EM) tensor of the homogeneous Monge-Ampère equation, Eq. (52) with . This equation corresponds to the first term in the action (51). To define the symmetric EM tensor, we covariantize this term: and differentiate with respect to the auxiliary 2-dimensional metric . The result of this computation is (89) (up to a factor of ). The EM tensor of the full equation (52) differs from (89) by a trivial harmless piece associated with the quadratic term in (51). In imposing the EM conservation at the singular point that piece is irrelevant. Transforming the conservation laws (90), (91) to their integral form, we conclude that the EM tensor will be conserved if Applied to the ansatz (69), this gives two conditions on , conveniently written down as one complex condition: When this condition is imposed, the class of weak solutions from Sec. VI.B is reduced, but not eliminated. The simplest solution (84) does not satisfy (94) and would not be allowed. However, already among the one-parameter family (87) there is a solution [corresponding to the numerical value and the minus sign in (88)] which passes the new criterion. Introducing more intermediate points, we will still have an infinitude of solutions satisfying both the no-delta condition (72) and the EM conservation (

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This action might not be possible to undo. Are you sure you want to continue? Version 2 CE IIT, Kharagpur Lesson 37 Moving Load and Its Effects on Structural Members Version 2 CE IIT. Kharagpur . we will discuss about the construction of influence lines. From the designer’s point of view. which are plotted as a function of distance along the span. • An influence line is a diagram whose ordinates. An influence line represents the variation of either the reaction. the deflected shape of the structural will vary. In this lecture. or a displacement at a particular point in a structure as a unit load move across the structure. Shear or Version 2 CE IIT.2 Definitions of influence line In the literature. which doesn’t exceed the limits of deformations and also the limits of load carrying capacity of the structure. a reaction.Instructional Objectives: The objectives of this lesson are as follows: • Understand the moving load effect in simpler term • Study various definitions of influence line • Introduce to simple procedures for construction of influence lines 37. we can arrive at simple conclusion that due to moving load position on the structure. moment. or deflection at a specific point in a member as a unit concentrated force moves over the member.1 Introduction In earlier lessons. Using any one of the two approaches (Figure 37. Some of the definitions of influence line are given below. 37. shear. one can construct the influence line at a specific point P in a member for any parameter (Reaction. you were introduced to statically determinate and statically indeterminate structural analysis under non-moving load (dead load or fixed loads). An influence line is a curve the ordinate to which at any point equals the value of some particular function due to unit load acting at that point. reactions value at the support also will vary. In the process.1). researchers have defined influence line in many ways. you will be introduced to determination of maximum internal actions at cross-sections of members of statically determinate structured under the effects of moving loads (live loads).3 Construction of Influence Lines In this section. Kharagpur . • • 37. it is essential to have safe structure. give the value of an internal force. Common sense tells us that when a load moves over a structure. 3. Positive for the following case V Moment M Positive for the following case V M M Version 2 CE IIT. say at x.2 Sign Conventions Sign convention followed for shear and moment is given below.1. Shear V or moment M) and plot the tabulated values so that influence line segments can be constructed. Parameter Reaction R Shear V Sign for influence line Positive at the point when it acts upward on the beam. Classification of the approaches for construction of influence lines is given in Figure 37. The best way to use this approach is to prepare a table. 37.3. apply statics to compute the value of parameter (reaction. And these locations. Kharagpur . In the present approaches it is assumed that the moving load is having dimensionless magnitude of unity. listing unit load at x versus the corresponding value of the parameter calculated at the specific point (i.1 Tabulate Values Apply a unit load at different locations along the member. Reaction R. shear.e. Construction of Influence Lines Tabulate Values Influence Line-Equation Figure 37. or moment) at the specified point.Moment).1: Approaches for construction of influence line 37. there are two ways this problem can be solved. the load can be placed at 5.3). The above discussed both approaches are demonstrated with the help of simple numerical examples in the following paragraphs. Σ MA = 0 : RB x 10 . a unit load is places at distance x from support A and the reaction value RB is calculated by taking moment with reference to support A. Figure 37. from support A then the reaction RB can be calculated as follows (Figure 37. Both the approaches will be demonstrated here. The beam structure is shown in Figure 37.25 Figure 37.0.1 x 2. Tabulate values: As shown in the figure. Let us say.2. reaction.3: The beam structure with unit load Similarly. Kharagpur . Version 2 CE IIT.5 = 0 ⇒ RB = 0. if the load is placed at 2. 7.5 and 10 m.3 Influence Line Equations Influence line can be constructed by deriving a general mathematical equation to compute parameters (e. away from support A and reaction RB can be computed and tabulated as given below.37.5 m.g.3. shear or moment) at a specific point under the effect of moving load at a variable position x. 37.2: The beam structure Solution: As discussed earlier.4 Numerical Examples Example 1: Construct the influence line for the reaction at support B for the beam of span 10 m. 5 10 RB 0. Tabulate Values: As shown in the figure. Figure 37.4. Influence Line Equation: When the unit load is placed at any location between two supports from support A at distance x then the equation for reaction RB can be written as Σ MA = 0 : RB x 10 – x = 0 ⇒ RB = x/10 The influence line using this equation is shown in Figure 37. a unit load is places at distance x from support A and the reaction value RB is calculated by taking moment with reference to support A. Kharagpur .4.0 7.4: Influence line for reaction RB.5 0.0 0.x 0 2.75 1 Graphical representation of influence line for RB is shown in Figure 37.5: The overhang beam structure Solution: As explained earlier in example 1.5 5.25 0. Figure 37. Let Version 2 CE IIT. Example 2: Construct the influence line for support reaction at B for the given beam as shown in Fig 37. here we will use tabulated values and influence line equation approach.5. 7. Σ MA = 0 : RB x 7. Kharagpur .0 7. Figure 37.5 m from support A and compute reaction at B.5 .0 0. from support A then the reaction RB can be calculated as follows.5 10 12.5 .33 Similarly a unit load can be placed at 12.67 1.67 Graphical representation of influence line for RB is shown in Figure 37.00 1.33 Figure 37.5 and the reaction at B can be computed.5 5. x 0 2.0 m from support A.5 m. Version 2 CE IIT. When the load is placed at 10.1 x 10.5 RB 0.us say.5 = 0 ⇒ RB = 0. The values of reaction at B are tabulated as follows.7: Influence for reaction RB.0 = 0 ⇒ RB = 1. if the load is placed at 2.33 0. then reaction at B can be computed using following equation.1 x 2.6: The beam structure with unit load Similarly one can place a unit load at distances 5.33 1.0 m and 7. Σ MA = 0 : RB x 7. 9: The beam structure – a unit load before section Figure 37.Influence line Equation: Applying the moment equation at A (Figure 37. Example 3: Construct the influence line for shearing point C of the beam (Figure 37.1 x x = 0 ⇒ RB = x/7. The shear force at C should be carefully computed when unit load is placed before point C (Figure 37. Σ MA = 0 : RB x 7.8: Beam Structure Solution: Tabulated Values: As discussed earlier. place a unit load at different location at distance x from support A and find the reactions at A and finally computer shear force taking section at C.5 .7. The resultant values of shear force at C are tabulated as follows.5 The influence line using this equation is shown in Figure 37.6).10: The beam structure . Figure 37. Kharagpur .8) Figure 37.10).9) and after point C (Figure 37.a unit load before section Version 2 CE IIT. 5 0. Figure 37. Figure 37.33 -0. Kharagpur .33 0.X 0 2.12: Free body diagram – a unit load before section Version 2 CE IIT.12) and after point C (Figure 37. The equations are plotted in Figure 37.5(-) 7.0 7.5 0.11.5 15. we need to determine two equations as the unit load position before point C (Figure 37.5(+) 10 12.0 Vc 0.16 0 Graphical representation of influence line for Vc is shown in Figure 37.0 -0.11.5 5.16 -0.11: Influence line for shear point C Influence line equation: In this case.13) will show different shear force sign due to discontinuity. 5 .= 0 ⇒ Mc = 1. then the support reaction at A will be 0.15). take a section at C and compute the moment.5 . Version 2 CE IIT. Kharagpur . For example.25 Figure 37.167.Mc + RB x 7.833 and support reaction B will be 0. we place the unit load at x=2. Example 4: Construct the influence line for the moment at point C of the beam shown in Figure 37.14: Beam structure Solution: Tabulated values: Place a unit load at different location between two supports and find the support reactions.5 m from support A (Figure 37.14 Figure 37. compute the moment Mc for difference unit load position in the span.13: Free body diagram – a unit load after section Influence Line for Moment: Like shear force. we can also construct influence line for moment.167 x 7. The values of Mc are tabulated as follows.Figure 37.Mc + 0.= 0 ⇒ .15: A unit load before section Similarly. Once the support reactions are computed. Taking section at C and computation of moment at C can be given by Σ Mc = 0 : . 25 2.5 .75 2.5 1.a unit load before section When the unit load is placed after point C then the moment equation for given Figure 37.5 5.5 10 12.5 3.25 0 Graphical representation of influence line for Mc is shown in Figure 37.0 7. Kharagpur . where 0 ≤ x ≤ 7. where 7.5 = 0 ⇒ Mc = x/2.0 Version 2 CE IIT.18 can be given by Σ Mc = 0 : Mc – (1-x/15) x 7.16.5 –x) – (1-x/15)x7. Figure 37.17 can be given by Σ Mc = 0 : Mc + 1(7.5 Figure 37.X 0 2.0 1.0 Mc 0.5 < x ≤ 15.17: Free body diagram .5 15.x/2. When the unit load is placed before point C then the moment equation for given Figure 37.16: Influence line for moment at section C Influence Line Equations: There will be two influence line equations for the section before point C and after point C.5 = 0 ⇒ Mc = 7. take a section at C and compute the moment.0 = 0 ⇒ Mc = 1.5 m from support A. Figure 37.20: A unit load before section C Taking section at C and computation of moment at C can be given by Σ Mc = 0 : . For example as shown in Figure 37.25 x 5. Example 5: Construct the influence line for the moment at point C of the beam shown in Figure 37. we place a unit load at 2.25 Similarly.Mc + RB x 5.= 0 ⇒ .Figure 37. Version 2 CE IIT.20. Once the support reactions are computed.16.18: Free body diagram .a unit load before section The equations are plotted in Figure 37. compute the moment Mc for difference unit load position in the span. then the support reaction at A will be 0. The values of Mc are tabulated as follows.75 and support reaction B will be 0.Mc + 0.0 .19: Overhang beam structure Solution: Tabulated values: Place a unit load at different location between two supports and find the support reactions.19. Kharagpur . Figure 37.25. 25 2. where 0 ≤ x ≤ 5. When a unit load is placed before point C then the moment equation for given Figure 37.5 5.5 Graphical representation of influence line for Mc is shown in Figure 37.x 0 2.25 -2.x/2.25 0 -1.22 can be given by Σ Mc = 0 : Mc + 1(5.22: A unit load before section C When a unit load is placed after point C then the moment equation for given Figure 37.0 = 0 ⇒ Mc = 5 . Kharagpur .21: Influence line of moment at section C Influence Line Equations: There will be two influence line equations for the section before point C and after point C.5 1. where 5 < x ≤ 15 Version 2 CE IIT.0 Figure 37.0 7.0 –x) – (1-x/10)x5.5 10 12.0 = 0 ⇒ Mc = x/2.5 15. Figure 37.21.0 Mc 0 1.23 can be given by Σ Mc = 0 : Mc – (1-x/10) x 5. 37.5xP. reactions A can be computed.5. reaction A will be 0. Looking at the position. shear or moment at any specified point in beam to check for criticality.5 Influence line for beam having point load and uniformly distributed load acting at the same time Generally in beams/girders are main load carrying components in structural systems. if load P is at the center of span then what will be the value of reaction A? From Figure 37. They are concentrated load and uniformly distributed load (UDL). Figure 37. let us say. like earlier examples.1 Concentrated load As shown in the Figure 37.25. Hence. we can find that for the load position of P. to generalize our approach. Hence. Similarly. for various load positions and load value. Now we want to know.23: A unit load after section C The equations are plotted in Figure 37. Hence it is necessary to construct the influence line for the reaction.24: Beam structure Version 2 CE IIT. point load P is moving on beam from A to B. Kharagpur .220.127.116.11. let us assume that unit load is moving from A to B and influence line for reaction A can be plotted as shown in Figure 37. influence line of unit load gives value of 0. Let us assume that there are two kinds of load acting on the beam. 37.Figure 37. we need to find out what will be the influence line for reaction B for this load. 27.dx is equivalent to area under the influence line. The term ∫ y. Kharagpur . Hence. In that case. the concentrated load dP can be given by w.26: Uniformly distributed load on beam Figure 37. For UDL of w on span.dx). moment) is y as shown in Figure 37. the value of function is given by (dP)(y) = (w.dx acting at x. Let us assume that beam’s influence line ordinate for some function (reaction. considering for segment of dx (Figure 37.27: Segment of influence line diagram Version 2 CE IIT.26).25: Influence line for support reaction at A 37.Figure 37. For computation of the effect of all these concentrated loads.y. we can say that it will be ∫ w.5.dx.dx = w ∫ y.y. Figure 37. we have to integrate over the entire length of the beam. shear.2 Uniformly Distributed Load Beam is loaded with uniformly distributed load (UDL) and our objective is to find influence line for reaction A so that we can generalize the approach. 6 Numerical Example Find the maximum positive live shear at point C when the beam (Figure 37. the influence line (Figure 37. the influence line for the shear at C can be given by following Figure 37. Version 2 CE IIT.31.l.5 w. 37.28: UDL on simply supported beam Figure 37.29: Influence line for support reaction at A.For a given example of UDL on beam as shown in Figure 37.e. i.5x (1)xl] w = 0. [0. Figure 37. Kharagpur .30: Simply supported beam Solution: As discussed earlier for unit load moving on beam from A to B.29) for reaction A can be given by area covered by the influence line for unit load into UDL value. Figure 37.28.30) is loaded with a concentrated moving load of 10 kN and UDL of 5 kN/m. Vc = [ 0. Finally the loading positions for maximum shear at C will be as shown in Figure 37. Vc = 0. For this beam one can easily compute shear at C using statics.5 and x = 15.31. we will put 10 kN just after C.375 Total maximum Shear at C: (Vc) max = 5 + 9.Figure 37. the maximum positive live shear force at C will be when the UDL 5 kN/m is acting between x = 7.5)] x 5 = 9. Our aim is to find positive live shear and hence.5 x 10 = 5 kN. UDL: As shown in Figure 37.375 = 14.375.32. Version 2 CE IIT. In that case.31. the maximum live shear force at C will be when the concentrated load 10 kN is located just before C or just after C. Kharagpur .5) (0.31: Influence line for shear at section C.5 x (15 –7. Concentrated load: As shown in Figure 37. J. J. ISBN 007-058116-9 Version 2 CE IIT. (1991).. NY. A. S. C-M.S. Finally we studied the influence line construction using tabulated values and influence line equation. C. (2003). K. H. Further we studied the available various influence line definitions. Kharagpur . and Utku. M. Mechanics of Structures – Vol. L. Leet. Pearson Education (Singapore) Pte. and Jangid. (1988). Wilbur. Delhi. Tata McGraw-Hill Publishing Company Limited. Ltd. E. McGraw-Hill Book Company. R. ISBN 0-07-058208-4 Negi. Tata McGraw-Hill Publishing Company Limited. S. (1999).. and Uang. and Shah. Elementary Structural Analysis. Classical Structural Analysis – A Modern Approach. (2003). ISBN 81-7808-750-2 Junarkar. New Delhi. Charotar Publishing House. The understanding about the simple approach was studied with the help of many numerical examples.Figure 37. C. ISBN 0-07-100120-4 Hibbeler. B. Suggested Text Books for Further Reading • • • • • • Armenakas. II. New Delhi.32: Simply supported beam 37. (2002).7 Closing Remarks In this lesson we have studied the need for influence line and their importance. Fundamentals of Structural Analysis. Structural Analysis. Structural Analysis. New Delhi. B. ISBN 0-07-462304-4 Norris. R. Tata McGraw-Hill Publishing Company Limited. H. S. Anand.

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CC-MAIN-2017-09

https://studyrocket.co.uk/revision/gcse-mathematics-edexcel/algebra/solving-simultaneous-equations

math

Finding a solution to two linear equations Simultaneous equations are two equations that at meet at one particular point ( if they are linear, often more if not!). For linear equations this will involve an x and a y value which is the same for both equations. Here are the steps required to solve two linear simultaneous equations: Step 1: Write out the equations and label them B 2x +3y=12 In order to solve the equations we need to eliminate one variable. To do this we need to make one variable equal. Step 2: Multiply the equations to make at least one variable equal _A __ __3__x+7y=23 x2 _You can choose to make the x or y equal, I am going to choose x __B __ __2__x +3y=12 x3 I am going to make a common multiple of x in equation A by multiplying it all by 2. I am going to make a common multiple of x in equation B by multiplying it all by 3. A __3__x+7y=23 x2 = __6__x+14y=46 B __2__x +3y=12 x3 = __6__x +9y=36 Now we have a common coefficient of x Step 3: I now need to get rid of the x variable. Because I have positive two positive 6x I am going to subtract A from B. __A __ = __6__x+14y=46 __B __ = __6__x +9y=36 / 0 +5y=10 Step 4: Solve for one variable. Now I have one unknown I can work out y: Step 5: Substitute you value into an original equation. (It doesn’t matter which one!) Substitute in y=2 Step 6: Check by substituting in both values into the other original formula __B __ 2x +3y=12 6+6=12 Therefore we are correct! Take care with Step 3, for example sometimes you might have two equations that you need to add. For example, if you get two equations such as: 3x +6y =12 2x - 6y= 9 To cancel out the +6y with the -6y you will need to add as 6y +-6y =0 So this would simplify down to 5x=21 You would then solve from this point. Finding solutions to two linear equations graphically To solve two linear simultaneous equations graphically you simply need to draw the graphs and look for the point where the two lines intersect. Sometimes this may involve rearranging equations to put them into the format y=mx+c which will make it easier for you to draw the graphs. - Solve the simultaneous equation 4x + 3y = 6 and 3x + y = 7 - Your answer should include: 3 / -2 - Solve the simultaneous equation 3x + 7y = 23 and 2x + 3y = 12 - Your answer should include: 3 / 2

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CC-MAIN-2024-10

https://www.calculator.org/properties/area_per_unit_volume.html

math

property>area per unit volume What is Area per Unit Volume? Area per unit volume is sometimes referred to as "specific surface" although this refers to the area per unit mass (in industries involving liquids with approximately unit density, e.g. water and dilute water-based solutions, the units of mass and volume are sometimes used interchangeably). A better term is "surface area to volume ratio". Area per unit volume has the dimensions of reciprocal length. The most common applications of area per unit volume are the determination of surface area to volume ratio. This could refer to the properties of a particular shape, or in particulate systems to the amount of particle surface area per unit volume of the fluid carrying the particles. In chemical engineering, the surface area to volume ratio of column packing materials is significant, particularly when selecting commercially manufactured packing. In addition to being a function of the shape of the particles it may be a function of how the packing is assembled, for example whether the items are individually placed or simply dumped. Surface area volume ratios are significant in predicting surface reaction rates and in heat transfer calculations.

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http://www.chegg.com/homework-help/questions-and-answers/position-particle-moving-xy-plane-given-r-20cos-30rad-s-t-20sin-30rad-s-t-j-r-meters-t-t-s-q942140

math

The position of a particle moving on the XY plane is given by R=2.0cos(3.0rad s/t)i+2.0sin(3.0rad s/t)j, where r is in meters and t is in t seconds. A) Show that this represents circular motion of radius 2.0 m centered at the origin. B) Determine the speed? C) determine the magnitude of the acceleration? D) Show that a=v^2/r? E) Show that the acceleration vector always points toward the center of the circle? I cant seem to figure this one out. Please help?

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CC-MAIN-2016-22

https://slideplayer.com/slide/3866725/

math

Presentation on theme: "Chapter 15: Duality of Matter Did you read chapter 15 before coming to class? A.Yes B.No."— Presentation transcript: Chapter 15: Duality of Matter Did you read chapter 15 before coming to class? A.Yes B.No Review: How the Bohr model explains the Hydrogen Atom spectrum Energy Level Diagram Absorption Emission An atom has only the following possible energy levels. How many discrete colors can it emit? E4E4 E3E3 E1E1 E2E2 A.2 B.4 C.5 D.6 E.7 or more The “bullets” used by Rutherford to probe gold atoms were: A.Electrons B.Neutrons C.Positively charged particles Matter Models (continued…) At least two puzzles remain at this point: The wave-particle duality of light. The physical basis for the Bohr model. The de Broglie Hypothesis was originally hailed as the “French Comedy”… In 1923 a graduate student named Louis de Broglie proposed a radical idea about matter: In addition to light, matter should also exhibit a wave-particle duality. A particle of mass should have a wavelength defined by: wavelength = h / (mass × speed) where h = Plank’s constant = 6 x 10 -34 But it turned out to describe what we observe. Nobel Prize, 1929 De Broglie’s idea explained the Bohr orbitals The quantized orbits of the Bohr model are predicted perfectly by requiring electrons to exactly wrap 1, 2, 3, etc waves around the nucleus. Examples Wavelength = 10 -38 m (nonsense?) Wavelength = 10 -34 m (again nonsense?) Wavelength = 10 -10 m Diameter of an atom… 60 mph 100 mph - 2,000 mph wavelength = h / (mass×speed) where h = Plank’s constant = 6 x 10 -34 Why don’t we observe the wave nature of matter? To observe wave effects, your “slits” need to be similar to the wavelength Example: It would take 10 27 years for a student to “diffract” through a doorway. For all material objects except the very least massive (such as electrons and protons), the wavelength is so immeasurably small that it can be completely ignored. wavelength = h / (mass×speed) h = 6 x 10 -34 But are electrons really waves? Let’s see if they diffract. We need slits about the size of the electron wavelength (~10 -10 m) to witness it. How do you make a slit that small? You can’t manufacture one, but nature provides something we can use: the space between atoms in a crystal. Fire an electron beam at a crystal and we DO get diffraction and interference patterns! Electrons ARE waves! Small electron wavelength is useful in electron microscopes Diffraction limits how small you can see with an optical microscope When objects are about the size of the wavelength of light, light diffracts around the object so you can’t get a clear image. Electron wavelengths can be 1/1000 th the size of optical wavelengths. So using electron beams we can “see” things 1000 times smaller with the same clarity. For waves, we can use the amplitude as a measure of where the wave “is” Experimental double slit experiment using electrons Electrons are detected like particles, but the places that they are detected show interference patterns. This is essentially the same behavior we observed with photons! So, which slit does the electron go through? Electron Detector The results depend on how and what we measure. Don’t measure which hole the electron goes through wave-like behavior. Do measure which hole the electron goes through particle-like behavior. How the electron behaves depends on whether it is observed. Deep thought: How does one study an unobserved electron. We have found the truth; and the truth makes no sense. (G. K. Chesterton) So what is waving? The mass of the particle is not spread out and mechanically oscillating. The “wave” is interpreted as being the probability of locating the particle. High amplitude corresponds to high probability of detection. It propagates like a pure wave with diffraction, interference, refraction, etc. Somehow electrons “know” about the existence of both slits even when we cannot prove that they ever go through more than one slit at a time. Clearly we need another model. Schodinger’s wave mechanics I don't like it and I'm sorry I ever had anything to do with it. The electron position is described with a probability wave When we measure the position, we find it at a certain position. We refer to this as the collapse of the wave function. The Uncertainty Principle and waves To find the trajectory of a particle we must know its position and velocity at the same time. How do you locate the position of a wave/particle electron? A well-defined momentum has a well-defined wavelength according to De Broglie. wavelength = h / momentum To find the trajectory of a particle we must know its position and velocity at the same time. How do you locate the position of a wave/particle electron? A well-defined momentum has a well-defined wavelength according to De Broglie. wavelength = h / momentum Pure sine wave unclear position but clear wavelength (momentum). Sharp pulse clear position but unclear wavelength (momentum). Heisenberg Uncertainty Principle Electrons: fuzzy position and fuzzy wave properties. The uncertainty in position times the uncertainty in momentum (mass x velocity) is greater than Planck’s constant. Or ∆x ∆(mv) > h Consequences If we try to find out where an electron is, we know less about where it is going. Measuring position more accurately makes uncertainty in momentum larger. This is an alternative explanation for electron diffraction. Philosophical shift in wave mechanics We could predict this interaction perfectly using Newton’s Laws of motion. We cannot predict the results of this interaction perfectly in quantum mechanics. We can only give probabilities that certain outcomes will happen. 8 ? ? ? ? So does the electron “know” where it is and where it is going? Quantum mechanics experiments demonstrate that there is fundamental uncertainty in nature. It is not a matter of the experimentalist not being clever enough to measure both position and momentum at the same time. A particle simply cannot have an exact position and an exact momentum at the same time. I hope this bothers you some… I remember discussions with Bohr which went through many hours till very late at night and ended almost in despair; and when at the end of the discussion I went alone for a walk in the neighboring park I repeated to myself again and again the question: Can nature possibly be as absurd as it seemed to us in these atomic experiments? ~Werner Heisenberg

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CC-MAIN-2020-05

https://discovery.fiu.edu/display/pub248475

math

We consider a stochastic 2D liquid crystal model with a multiplicative noise of Lévy type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces of jump type. We derive a large deviation principle for the model. The proof is based on the weak convergence method introduced in [Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincar Probab Stat. 2011;47(3):725–747].

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CC-MAIN-2023-40

https://mountainscholar.org/handle/11124/174216/discover?filtertype=subject&filter_relational_operator=equals&filter=Algebras%2C+Linear

math

Now showing items 1-1 of 1 Linear vector spaces & applications Author(s):Pankavich, Stephen; Colorado School of Mines. Department of Applied Mathematics and Statistics New open textbook for Applied Linear Algebra at the graduate level featuring a variety of in-class modules and applications, computational examples (using MATLAB), and new homework materials. Faculty teaching any course ...

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https://www.math.tolaso.com.gr/?p=2024

math

Let denote one of the Jacobi Theta functions. Prove that We have successively, The sum is evaluated as follows. Consider the function and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus, and the exercise is complete.

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https://sketches89.neocities.org/download-sketch-the-graph-of-the-function-png

math

Download Sketch The Graph Of The Function PNG. Explore math with our beautiful, free online graphing calculator. Here graphs of numerous mathematical functions can be drawn, including their derivatives and integrals. We will also discuss how to sketch the graphs of trigonometric functions later. It travels upward with its slope decreasing and becoming 0 at x = 1. At times you will need to sketch a function to see what it looks like. How to sketch the graph of is there any quick command that helps drawing the graph of it, instead of manually define that function and plot it? Online, immediately and for free. Functions graph limits of a function graphing functions. It travels upward with its slope decreasing and becoming 0 at x = 1. The op can safely ignore the plot of the imaginary.

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CC-MAIN-2023-23

https://www.middletownhonda.com/new-vehicles/insight/

math

You don't have any saved vehicles! Look for this link on your favorites: Once you've saved some vehicles, you can view them here at any time. Hassle Free Price Contact InfoFirst Name*Last Name*PhoneEmail* Contact Preference?*PhoneEmailWhat Are You Looking For?Mileage (Max)*50001000015000200002500030000400005000060000700008000090000100000AnyPrice (Max)*$5000$10000$15000$20000$25000$30000$40000$50000$60000$70000$80000$90000$100000AnyMake*Make*HondaModel*Trim*Color*Transmission*AutomaticManualBy submitting this form, you agree to be contacted by Middletown Honda with information regarding the vehicle you are searching for. Referral IDNameThis field is for validation purposes and should be left unchanged.

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CC-MAIN-2019-43

https://f-counter.info/thesis-library/homework-6-2-statistics-78037.php

math

The shaded area in the following graph indicates the area to the left of x. Probabilities are calculated using technology. To calculate the probability, use the probability tables provided in Figure G1 without the use of technology. The tables include instructions for how to use them. Is Homework Good for Kids? Here's What the Research Says Statistics | Statistics Quiz - Quizizz You might think that open-minded people who review the evidence should be able to agree on whether homework really does help. Their assessments ranged from homework having positive effects, no effects, or complex effects to the suggestion that the research was too sparse or poorly conducted to allow trustworthy conclusions. Fill-in-the-blank worksheets or extended projects? In what school subject s? Independent and mutually exclusive do not mean the same thing. Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. Thank you for helping. KEep satisfying the customers. Good work. Some of the answers are very short but I have requested to edit the answer.

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CC-MAIN-2021-39

https://math.answers.com/Q/The_sum_of_the_side_lenghts_of_a_closed_plane_figure

math

The distance around a closed plane figure is called the perimeter. The perimeter of a figure can be calculated by adding together the lengths of each side. An angle on the coordinate plane A plane figure It looks like lines stuck together to make a closed figure It is a regular quadrilateral which is a square a polygon has 6 sids and 6 faces because it is a closed figure and any closed figure has 8 sides or less A plane figure that has one more side than a quadrilateral and one less than a hexagon is a... pentagon. If the "shape" is a plane figure, a heptagon; if a solid, a heptahedron. a plane closed figure with ten sides and ten angles. degree: a unit of measurement of an angle. diagonal of a polygon: a line segment connecting one vertex to another vertex, and not a side of the polygon. A polyhedran is a 3-dimentional figure that has sides that are polygons. A polygon is a 2-dimentional figure with straight sides, a closed side. A closed geometric figure with four sides, each side being a straight line A circle which is a plane figure, when drawn on a paper has 1 side. A square is a plane figure with no volume. Congruent means they have the same angles and side lenghts. All squares have the same angles (all at 90*) but not all the same side lenghts; a square can be 5x5x5x5 and another can be 6x6x6x6. Only if the 3rd side is equal or less than 6in A shape cannot have only one side because it would not be closed. A shape must have at least three sides to be a closed figure (no "gaps")

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https://www.coursehero.com/tutors-problems/Algebra/27746099-Tim-applied-for-a-position-to-two-companies-P-and-Q-The-probability-o/

math

Tim applied for a position to two companies P and Q. The probability of being offered a position at P is 0.68 and the probability of being offered a position at company Q is 0. 56. Find the probability that Tim is offered a position in both companies. Round your answer to two decimal places. 416,475 students got unstuck by Course Hero in the last week Our Expert Tutors provide step by step solutions to help you excel in your courses

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CC-MAIN-2021-10

https://www.manyagroup.com/blog/gre-math-practice-questions-explanations/

math

Can you ace these GRE practice questions? Start honing your skills with some GRE math practice and get a preview of what you can expect on test day. We pulled these GRE quantitative practice questions from our book Cracking the GRE and from our GRE prep course materials. Each sample question includes an explanation, so you can see how to crack it! Note about calculators: On the GRE you’ll be given an on-screen calculator with the five basic operations: addition, subtraction, multiplication, division and square root, plus a decimal function and positive/negative feature. Don’t use anything fancier when you tackle this GRE math practice! Related Blog: Standard Deviation on the GRE Quantitative comparison questions ask you to compare Quantity A to Quantity B. Your job is to compare the two quantities and decide which of the following describes the relationship: 1. The average (arithmetic mean) high temperature for x days is 70 degrees. The addition of one day with a high temperature of 75 degrees increases the average to 71 degrees. (A) Quantity A is greater (B) Quantity B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given Answer: (B) Quantity B is greater If the average high temperature for x days is 70 degrees, then the sum of those x high temperatures is 70x. The sum of the high temperatures, including the additional day that has a temperature of 75 degrees is, therefore, 70x + 75. Next, use the average formula to find the value of x: In this formula, 71 is the average, 70x + 75 is the total, and there are x + 1 days. Substituting this information into the formula gives: To solve, cross-multiply to get 71x + 71 = 70x + 75. Next, simplify to find that x = 4. Therefore, Quantity B is greater. The correct answer is (B). a and b are integers. a 2 = b 3 (A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given. Answer: (D) The relationship cannot be determined from the information given. Try different integers for a and b that satisfy the equation a 2 = b 3 such as a = 8 and b = 3. These numbers satisfy the equation as 8 2 = 4 3 = 64. In this case, Quantity A is greater. Because Quantity B is not always greater nor are the two quantities always equal, choices (B) and (C) can be eliminated. Next, try some different numbers. When choosing a second set of numbers, try something less common such as making a = b = 1. Again, these numbers satisfy the equation provided in the problem. In this case, however, the quantities are equal. Because Quantity A is not always greater, choice (A) can now be eliminated. The correct answer is (D). Related Blog: What Are Good GRE Scores? For question 3, select one answer from the list of five answer choices. 3. A certain pet store sells only dogs and cats. In March, the store sold twice as many dogs as cats. In April, the store sold twice the number of dogs that it sold in March and three times the number of cats that it sold in March. If the total number of pets the store sold in March and April combined was 500, how many dogs did the store sell in March? Answer: (B) 100 Plug-In the Answers, starting with the middle choice. If 120 dogs were sold in March, then 60 cats were sold that month. In April, 240 dogs were sold, along with 180 cats. The total number of dogs and cats sold during those two months is 600, which is too large, so eliminate (C), (D), and (E). Try (B). If there were 100 dogs sold in March, then 50 cats were sold; in April, 200 dogs were sold along with 150 cats. The correct answer is (B) because 100 + 50 + 200 + 150 = 500. ▵ ABC has an area of 108 cm 2. If both x and y are integers, which of the following could be the value of x? Indicate all such values. Answer: (A), (C), (D) and (E) Plug the information given into the formula for the area of a triangle to learn more about the relationship between x and y: The product of x and y is 216, so x needs to be a factor of 216. The only number in the answer choices that is not a factor of 216 is 5. The remaining choices are the possible values of x. Some questions on the GRE won’t have answer choices, and you’ll have to generate your own answer. 5. Each month, Renaldo earns a commission of 10.5% of his total sales for the month, plus a salary of $2,500. If Renaldo earns $3,025 in a certain month, what were his total sales? If Renaldo earned $3,025, then his earnings from the commission on his sales are $3,025 – $2,500 = $525. So, $525 is 10.5% of his sales. Set up an equation to find the total sales: Solving this equation, x = 5,000. 6. At a recent dog show, there were 5 finalists. One of the finalists was awarded “Best in Show” and another finalist was awarded “Honorable Mention.” In how many different ways could the two awards be given out? In this problem order matters. Any of the 5 finalists could be awarded “Best in Show.” There are 4 choices left for “Honorable Mention,” because a different dog must be chosen. Therefore, the total number of possibilities is 5 x 4, or 20. Learn top experienced tips to ace the GRE: Download our FREE, Complete Study Guide to the GRE! If you are preparing for GRE and struggling with your Vocabulary then Manya GRE WordsApp is the ideal choice for you. Manya GRE WordsApp is a simple and efficient way to improve your vocabulary for GRE Exam. This app will make it simple to memorize words and to improve your GRE vocabulary in bite-sized pieces. You get around 1300+ GRE words divided into beginner, intermediate and advanced, each of which is further divided into levels for easy learning. Each word has crystal-clear meaning in simple language, pictorial representation of words, synonyms, antonyms and much more. In addition, quizzes & rewards make mastering even the toughest GRE vocabulary, simple. If you are planning to study abroad and searching to match your profile with the best suited university, Experts at Manya – The Princeton Review have gathered important information of top Universities from abroad. Surely, this information will help you narrow down your quest for universities. You can access accurate & authentic information related to rankings, application fees, average tuition fees, cost of living, scholarships, latest updates, and much more from more than 1000+ universities. You may also search for universities by name, country, or courses in common specializations, such as Physics, Finance, Business, Language and Culture, Agriculture, Environmental Science, Computer and IT, Media and Communication Marketing. Manya – The Princeton Review offers end-to-end study abroad services encompassing admissions consulting services, test preparation, English language training, career assessment, and international internship opportunities to study abroad aspirants.Book your Free Counselling Session now!

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https://911weknow.com/the-ramanujan-summation-1-2-3-%E2%8B%AF-%E2%88%9E-1-12

math

?What on earth are you talking about? There?s no way that?s true!? ? My mom This is what my mom said to me when I told her about this little mathematical anomaly. And it is just that, an anomaly. After all, it defies basic logic. How could adding positive numbers equal not only a negative, but a negative fraction? What the frac? Before I begin: It has been pointed out to me that when I talk about sum?s in this article, it is not in the traditional sense of the word. This is because all the series I deal with naturally do not tend to a specific number, so we talk about a different type of sums, namely Cesro Summations. For anyone interested in the mathematics, Cesro summations assign values to some infinite sums that do not converge in the usual sense. ?The Cesro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series? ? Wikipedia. I also want to say that throughout this article I deal with the concept of countable infinity, a different type of infinity that deals with a infinite set of numbers, but one where if given enough time you could count to any number in the set. It allows me to use some of the regular properties of mathematics like commutativity in my equations (which is an axiom I use throughout the article). Srinivasa Ramanujan (1887?1920) was an Indian mathematician For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12. Yup, -0.08333333333. Don?t believe me? Keep reading to find out how I prove this, by proving two equally crazy claims: - 1?1+1?1+1?1 ? = 1/2 - 1?2+3?4+5?6? = 1/4 First off, the bread and butter. This is where the real magic happens, in fact the other two proofs aren?t possible without this. I start with a series, A, which is equal to 1?1+1?1+1?1 repeated an infinite number of times. I?ll write it as such: A = 1?1+1?1+1?1? Then I do a neat little trick. I take away A from 1 So far so good? Now here is where the wizardry happens. If I simplify the right side of the equation, I get something very peculiar: Look familiar? In case you missed it, thats A. Yes, there on that right side of the equation, is the series we started off with. So I can substitute A for that right side, do a bit of high school algebra and boom! 1 = 2A 1/2 = A This little beauty is Grandi?s series, called such after the Italian mathematician, philosopher, and priest Guido Grandi. That?s really everything this series has, and while it is my personal favourite, there isn?t a cool history or discovery story behind this. However, it does open the door to proving a lot of interesting things, including a very important equation for quantum mechanics and even string theory. But more on that later. For now, we move onto proving #2: 1?2+3?4+5?6? = 1/4. We start the same way as above, letting the series B =1?2+3?4+5?6?. Then we can start to play around with it. This time, instead of subtracting B from 1, we are going to subtract it from A. Mathematically, we get this: A-B = (1?1+1?1+1?1?) ? (1?2+3?4+5?6?) A-B = (1?1+1?1+1?1?) ? 1+2?3+4?5+6? Then we shuffle the terms around a little bit, and we see another interesting pattern emerge. A-B = (1?1) + (?1+2) +(1?3) + (?1+4) + (1?5) + (?1+6)? A-B = 0+1?2+3?4+5? Once again, we get the series we started off with, and from before, we know that A = 1/2, so we use some more basic algebra and prove our second mind blowing fact of today. A-B = B A = 2B 1/2 = 2B 1/4 = B And voila! This equation does not have a fancy name, since it has proven by many mathematicians over the years while simultaneously being labeled a paradoxical equation. Nevertheless, it sparked a debate amongst academics at the time, and even helped extend Euler?s research in the Basel Problem and lead towards important mathematical functions like the Reimann Zeta function. Now for the icing on the cake, the one you?ve been waiting for, the big cheese. Once again we start by letting the series C = 1+2+3+4+5+6?, and you may have been able to guess it, we are going to subtract C from B. B-C = (1?2+3?4+5?6?)-(1+2+3+4+5+6?) Because math is still awesome, we are going to rearrange the order of some of the numbers in here so we get something that looks familiar, but probably wont be what you are suspecting. B-C = (1-2+3-4+5-6?)-1-2-3-4-5-6? B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6) ? B-C = 0-4+0-8+0-12? B-C = -4-8-12? Not what you were expecting right? Well hold on to your socks, because I have one last trick up my sleeve that is going to make it all worth it. If you notice, all the terms on the right side are multiples of -4, so we can pull out that constant factor, and lo n? behold, we get what we started with. B-C = -4(1+2+3)? B-C = -4C B = -3C And since we have a value for B=1/4, we simply put that value in and we get our magical result: 1/4 = -3C 1/-12 = C or C = -1/12 Now, why this is important. Well for starters, it is used in string theory. Not the Stephen Hawking version unfortunately, but actually in the original version of string theory (called Bosonic String Theory). Now unfortunately Bosonic string theory has been somewhat outmoded by the current area of interest, called supersymmetric string theory, but the original theory still has its uses in understanding superstrings, which are integral parts of the aforementioned updated string theory. The Ramanujan Summation also has had a big impact in the area of general physics, specifically in the solution to the phenomenon know as the Casimir Effect. Hendrik Casimir predicted that given two uncharged conductive plates placed in a vacuum, there exists an attractive force between these plates due to the presence of virtual particles bread by quantum fluctuations. In Casimir?s solution, he uses the very sum we just proved to model the amount of energy between the plates. And there is the reason why this value is so important. So there you have it, the Ramanujan summation, that was discovered in the early 1900?s, which is still making an impact almost 100 years on in many different branches of physics, and can still win a bet against people who are none the wiser. P.S. If you are still interested and want to read more, here is a conversation with two physicists trying to explain this crazy equation and their views on it?s usefulness and validity. It?s nice and short, and very interesting. https://physicstoday.scitation.org/do/10.1063/PT.5.8029/full/ This essay is part of a series of stories on math-related topics, published in Cantor?s Paradise, a weekly Medium publication. Thank you for reading!

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CC-MAIN-2023-50

http://welcome2auckland.com/lottery-winning-number-is-it-winnable/

math

This is an observation that winning numbers are almost never all odds or all evens. Another way to lessen the odds of winning the lottery is to listen to people who have been keenly observing the process, trying to find a crack that will give them the better edge of winning. Everything can be answered by math. Online lottery players have keenly observed that on a 6-number game, the total of all 6 digits are between 121 and 186. Impossible? No, it is not impossible to win. There are ways though to lessen the almost impossible odds. The things aforementioned are not ways to win the lottery. For example in one’s 6-number lottery ticket, 4 are random numbers and the remaining two are two consecutive numbers across the set.. In a 5-number lottery, there would be 282,475,249 possible combinations. Nevertheless, through these ways, the odds are cut into half and the chance of winning increased. It has also been observed by those playing online lottery a lot that the lottery winning number almost always have 2 numbers that are consecutive. The chance of guessing right though may be faint. It is really just extremely difficult. The winning lottery number can never have a pattern, although online lottery may slightly have some. The lottery numbers are made of 6 digits, usually when trying to hit the jackpot. This statement actually has a backbone on it. To get the winning number requires luck and probably nothing else. It is still one in almost 300 million chance of winning, but at least the probability of winning increased by billions. The lottery winning number is never known. Not even the president of the lottery company knows what lottery winning number will come up. Draw for lottery is usually done on a live television where the drawing machine picks the number and is, there and then, shown and announced. If a person has an option to choose from numbers 1 to 49 in choosing his 6-number lottery ticket, and the numbers he chooses may be repeated, then based on math, there would be 13,841 287,201 possible combinations, one of those is the winning number. At the end of the day, it is better to be smart about things by filling proven strategies to have a better shot at being an instant millionaire through lottery. But if a person is more concerned of winning than hitting the jackpot, math would suggest that he has a far better chance of winning by playing the 5-number lottery. This may be a consequence of a program designed to draw numbers fairly but has overlooked certain details. The winning number is not a puzzle. But those things mentioned above are ways to lessen the odd, thus increasing the rather almost non-existent possibility of winning. There is actually no straight way to winning it except to have a darn good luck. But there are ways to guess the next lottery winning number. Online lottery players also suggest that one has to make his numbers a combination of odd and even digits. Drawing is random and so is winning. It is not something a person can apply intelligence to in order to guess right

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CC-MAIN-2017-39

https://forums.atozteacherstuff.com/index.php?threads/pre-algebra-one-step-division-question.153957/

math

The teachers taught the students how to do one-step equations with division where the problem is set up like this: n/2 = 5 The students learned to multiply by 2 on both sides to remove the 2 from n and to get the answer of 10. I wanted to make review sheets for one-step equations. On math-drills.com at http://www.math-drills.com/algebra/algebra_missing_numbers_in_equations_variables_division_001.pdf I saw problems set up like this: 10/n = 5 Wouldn't the students need to do two steps to solve a problem up set like this one? Would they first multiply by n on both sides and THEN divide by 5? They can't multiply by 10 on both sides, that won't solve the problem-- but I can see the students I work with trying to multiply by 10. Am I missing something? Math isn't my strongest subject but I like to review math concepts to help out my students.

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https://www.arxiv-vanity.com/papers/1902.08894/

math

Reflection positivity and complex analysis of the Yang-Mills theory from a viewpoint of gluon confinement In order to understand the confining decoupling solution of the Yang-Mills theory in the Landau gauge, we consider the massive Yang-Mills model which is defined by just adding a gluon mass term to the Yang-Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev-Popov ghost term. First of all, we show that massive Yang-Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout-Englert-Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang-Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang-Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form. We try to understand these results in light of the Fradkin-Shenker continuity between Confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model. It is still a challenging problem in particle physics to explain quark and gluon confinement in the framework of quantum gauge field theories YM54 . The very first question to this problem is to clarify what criterion should be adopted to understand confinement. For quark confinement, there is a well-established gauge-invariant criterion given by Wilson Wilson74 , namely, the area law falloff of the Wilson loop average leading to the linear static quark potential with a non-vanishing string tension. For gluon confinement, on the other hand, there is no known gauge-invariant criterion to the best of the author’s knowledge. This is also the case for more general hypothesis of color confinement including quark and gluon confinement as special cases. Once the gauge is fixed, however, there are some proposals. For instance, the Kugo-Ojima criterion for color confinement is given for the Lorentz covariant Landau gauge KO79 . Indeed, it is rather difficult to prove the color confinement criterion even in a specific gauge, although there appeared an announcement for a proof of the Kugo-Ojima criterion for color confinement in the covariant Landau gauge CF18 . Even if color confinement is successfully proved in a specific gauge, this does not automatically guarantee color confinement in the other gauges. Therefore the physical picture for confinement could change gauge by gauge. The information on confinement is expected to be encoded in the gluon and ghost propagators which are obtained by fixing the gauge. Recent investigations have confirmed that in the Lorentz covariant Landau gauge the decoupling solution decoupling-lattice ; decoupling-analytical is the confining solution of the Yang-Mills theory in the three- and four-dimensional spacetime, while the scaling solution is realized in the two-dimensional spacetime. Therefore, it is quite important to understand the decoupling solution in the Lorentz covariant Landau gauge. Of course, there are so many approaches towards this goal. In this paper, we focus on the approach TW10 ; RSIW17 ; Machado16 ; Kondo15 which has been developed in recent several years and has succeeded to reproduce some features of the decoupling solution with good accuracy. We call this approach the mass-deformed Yang-Mills theory with the gauge fixing term or the massive Yang-Mills model in the covariant gauge for short. However, the reason why this approach is so successful is not fully understood yet in our opinion. In the original works TW10 the massive Yang-Mills model in the Landau gauge was identified with a special parameter limit of the Curci-Ferrari model CF76b . However, the Curci-Ferrari model is not invariant under the usual Becchi-Rouet-Stora-Tyutin (BRST) transformation, but invariant just under the modified BRST transformation which does not respect the usual nilpotency. In this paper we show based on the previous works Kondo16 ; Kondo18 that the mass-deformed Yang-Mills theory with the covariant gauge fixing term has the gauge-invariant extension which is given by a gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group, provided that a constraint called the reduction condition is satisfied. We call such a model the complementary gauge-scalar model. This equivalence is achieved based on the gauge-independent description Kondo16 ; Kondo18 of the Brout-Englert-Higgs (BEH) mechanism Higgs1 ; Higgs2 ; Higgs3 which does not rely on the spontaneous breaking of gauge symmetry NJL61 ; Goldstone61 . This description enables one to give a gauge-invariant mass term of the gluon field in the Yang-Mills theory which can be identified with the gauge-invariant kinetic term of the scalar field in the complementary gauge-scalar model. In this paper, we first confirm that the massive Yang-Mills model with one-loop quantum corrections being included in the Euclidean region reproduces with good accuracy the gluon and ghost propagators of the decoupling solution of the Yang-Mills theory in the Landau gauge obtained by numerical simulations on the lattice. In fact, the resulting gluon and ghost propagators in the massive Yang-Mills model can be well fitted to those on the lattice by adjusting the parameters, namely, the coupling constant and the gluon mass parameter . For gluon confinement, the violation of reflection positivity is regarded as a necessary condition for confinement. In fact, it is known that the gluon propagator in the Yang-Mills theory exhibits the violation of reflection positivity. This fact was directly shown by the numerical simulations on the lattice, e.g., in the covariant Landau gauge CMT05 ; Bowmanetal07 . In this paper, by using the relevant gluon propagator in the massive Yang-Mills model, we calculate the Schwinger function in a numerical way to demonstrate that the reflection positivity is violated at the physical point of parameters reproducing the Yang-Mills theory. In order to understand these facts and consider the meaning of gluon confinement, we perform the analytic continuation of the gluon and ghost propagators in the Euclidean region to those in the Minkowski region on the complex momentum squared plane. The consideration of the complex structure of the propagator enables us to give an analytical proof that the reflection positivity is violated for any choice of the parameters without restricting to the physical point of the Yang-Mills theory in the massive Yang-Mills model with one-loop quantum corrections being included. For this proof, it is enough to show that the Schwinger function necessarily becomes negative in some region, which is achieved by calculating separately the contributions to the gluon Schwinger function from the pole part and the continuous (branch cut) part of the gluon propagator based on the generalized spectral representation in the massive Yang-Mills model to one-loop order. It turns out that the violation of reflection positivity is an immediate consequence of the facts that the gluon propagator has a pair of complex conjugate poles and that the spectral function of the gluon propagator has negative value on the whole range, see HK18 . See e.g., DORS19 ; BT19 for the construction of the spectral function from the Euclidean data of numerical simulations on the lattice. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form Stingl86 , as demonstrated in the numerical simulations on the lattice DOS18 . Indeed, the pole part of the gluon propagator due to a pair of complex conjugate poles exactly reproduces the Gribov-Stingl form which is fitted to the numerical simulations to very good accuracy, after subtracting the small contribution coming from the continuous part represented by the spectral function obtained from the discontinuity across the branch cut on the positive real axis on the complex momentum plane. See also Kondo11 for another explanation for the occurrence of the gluon propagator of the Gribov-Stingl form. The above result suggests that gluon confinement is not restricted to the confinement phase of the ordinary Yang-Mills theory, and can be extended into more general situations, namely, anywhere represented by the massive Yang-Mills model, which includes the Higgs phase in the complementary gauge-scalar model. In the lattice gauge theory, it is known that the confinement phase in the pure Yang-Mills theory is analytically continued to the Higgs phase in the relevant gauge-scalar model, which is called the Fradkin-Shenker continuity FS79 as a special realization of the Osterwalder-Seiler theorem OS78 . There are no local order parameters which can distinguish the confinement and Higgs phases. There is no thermodynamic phase transition between confinement and Higgs phases lattice-gauge-scalar-fund , in sharp contrast to the adjoint scalar case lattice-gauge-scalar-adj where there is a clear phase transition between the two phases. Therefore, Confinement and Higgs phases are just subregions of a single Confinement-Higgs phase FMS80 ; tHooft80 ; Maas17 . Therefore, permanent violation of positivity can be understood in light of the Fradkin-Shenker continuity between Confinement-like and Higgs-like regions in a single confinement phase in the gauge-scalar model. This paper is organized as follows. In sec. II, we introduce the massive Yang-Mills model in the covariant gauge. In sec. III, we show that the massive Yang-Mills model with quantum corrections to one-loop order well reproduces the gluon and ghost propagators of the decoupling solution. In sec. IV, we show that the gluon propagator exhibits violation of reflection positivity through the calculation of the Schwinger function. In sect. V, we perform the analytic continuation of the propagator to the complex momentum to examine the complex structure. In the final section we draw the conclusion and discuss the future problems to be tackled. In Appendix A, we give a recursive construction of the transverse and gauge-invariant gluon field to show the gauge-invariant extension of the massive Yang-Mills model. In Appendix B, we give another way for solving the reduction condition. Ii Gauge-invariant extension of the mass-deformed Yang-Mills theory in the covariant Landau gauge ii.1 Mass deformation of the Yang-Mills theory in the covariant Landau gauge We introduce the mass-deformed Yang-Mills theory in the covariant gauge which is defined just by adding the naive mass term to the ordinary massless Yang-Mills theory in the (manifestly Lorentz) covariant gauge fixing. The total Lagrangian density of the massive Yang-Mills model consists of the Yang-Mills Lagrangian , the gauge-fixing (GF) term , the associated Faddeev-Popov (FP) ghost term , and the mass term , where denotes the Yang-Mills field, the Nakanishi-Lautrup field and the Faddeev-Popov ghost and antighost fields, which take their values in the Lie algebra of a gauge group with the structure constants (). We call this theory the massive Yang-Mills model in the covariant gauge for short. The expectation value of the operator of is given according to the path integral quantization using the total action and the integration measure In the Landau gauge , especially, the average is cast into a simpler form by integrating the Nananishi-Lautrup field and subsequently the ghost and antighost field as with the Faddeev-Popov determinant, In this paper we do not intend to take into account the Gribov problem. The reasons are as follows. In this paper we deal with the massive Yang-Mills model as a low-energy effective model of the Yang-Mills theory and perform the perturbative analysis based on this model. In the ultraviolet region the perturbative analysis of the Yang-Mills theory is valid due to the ultraviolet asymptotic freedom and is free from the Gribov problem, since the perturbative analysis can be done in the neighborhood of the origin of the configuration space of the gauge field within the first Gribov region and therefore does not reach the Gribov horizon where the Gribov problem becomes serious. This is also the case for the massive Yang-Mills model, since the effect of mass term can be ignored in the ultraviolet region. Of course, in the usual perturbative treatment of the Yang-Mills theory, we encounter the Landau pole at which the gauge coupling constant diverges and the perturbative analysis breaks down at an intermediate momentum scale before reaching the deep infrared region. For the massive Yang-Mills model, however, we can adopt the infrared safe renormalization scheme in which the perturbation theory does not break down and remains valid from the large momentum all the way down to the zero momentum, as can be seen from the fact that the gauge coupling constant remains finite without divergence in the whole momentum region, and even vanishes in the zero momentum limit, as reviewed in section III. Therefore, we think that the massive Yang-Mills model can be treated in the whole region without seriously worrying about the Gribov problem, although there is no rigorous proof on this claim. We regard the massive Yang-Mills model adopted in this paper as a low-energy effective model of the Yang-Mills theory where the mass term is generated in the dynamical way due to quantum corrections, for instance, according to the Wilsonian renormalization group. The mass term plays also the role of an infrared regulator and the massive Yang-Mills theory is valid even in the vanishing momentum limit. Of course, the generation of the gluon mass term originates from non-perturbative effects and should be investigated from the first principles, which is however beyond the scope of this paper. Incidentally, we tried to show the existence of such mass term in WMNSK18 . The massive Yang-Mills model just defined is a special case of a massive extension of the massless Yang-Mills theory in the most general renormalizable gauge having both BRST and anti-BRST symmetries given by Baulieu85 \[email protected]equationparentequation where is a parameter which correspond to the gauge-fixing parameters in the limit, , and . This model is called the Curci-Ferrari model CF76b with the coupling constant , the mass parameter and the parameter . [In the Abelian limit with vanishing structure constants , the FP ghosts decouple and the Curci-Ferrari model reduces to the Nakanishi model Nakanishi72 .] For , the physics depends on the parameter . This result should be compared with the case, in which is a gauge fixing parameter and hence the physics should not depend on . In the case, indeed, any choice of gives the same physics. However, this is not the case for . See e.g., Kondo13 for more details. The massive Yang-Mills model is regarded as a case of the Curci-Ferrari model. This point of view taken in the preceding works TW10 is good from the viewpoint of renormalizability, since the Curci-Ferrari model is known to be renormalizable. However, the Curci-Ferrari model lacks the physical unitarity at least in the perturbation theory CF76b ; Kondo13 . Indeed, the massive Yang-Mills theory does not have the nilpotent BRST symmetry, although it has the modified BRST symmetry which does not respect the usual nilpotent property and reduces to the ordinary BRST symmetry only in the massless limit . In this paper we try to find an extended theory with the ordinary nilpotent BRST symmetry, which reproduces the massive Yang-Mills model under an appropriate prescription. As a candidate for such a theory we investigate a specific gauge-scalar model. In what follows we show that the massive Yang-Mills model in a covariant gauge has the gauge-invariant extension which is given by the gauge-scalar model with a single radially-fixed (or fixed modulus) scalar field in the fundamental representation of a gauge group if the theory is subject to an appropriate constraint which we call the reduction condition. We call such a gauge-scalar model the complementary gauge-scalar model. In other words, the complementary gauge-scalar model with a single radially fixed scalar field in the fundamental representation reduces to the mass-deformed Yang-Mills theory in a fixed gauge if an appropriate reduction condition is imposed. For , the complementary gauge-scalar model is given by with a single fundamental scalar field subject to the radially fixed condition, where is a positive constant and is the doublet formed from two complex scalar fields , where is the covariant derivative in the fundamental representation . This gauge-scalar model is invariant under the gauge transformation, It is more convenient to convert the scalar field into the gauge group element. For this purpose, we introduce the matrix-valued scalar field by adding another doublet as Then the complementary gauge-scalar model with a single radially-fixed scalar field in the fundamental representation is defined by where is the Lagrange multiplier field to incorporate the holonomic constraint (7) written in the matrix form . The radially-fixed gauge-scalar model with the Lagrangian density (11) is invariant under the gauge transformation, Then we introduce the normalized matrix-valued scalar field by The above constraint (7) implies that the normalized scalar field obeys the conditions: and . Therefore, is an element of : This is an important property to provide a gauge-independent BEH mechanism. The massive vector boson field is defined in terms of the original gauge field and the normalized scalar field as shown in a previous paper Kondo18 , The massive vector field is rewritten using into Then it is shown that the massive vector boson field has the expression, 111 In Kondo18 was written as . where denotes the gauge transform of by . Notice that transforms according to the adjoint representation under the gauge transformation, whereas is gauge invariant, Therefore, the mass term can be written in terms of the gauge-invariant field as This theory is supposed to obey the reduction condition for the massive vector field mode . The stationary form of the reduction condition is given by where is the covariant derivative in the adjoint representation . The stationary reduction condition is cast into This implies that imposing the reduction condition is equivalent to imposing the “Landau gauge condition” or transverse condition for the gauge-invariant field . Therefore, we can use the (gauge-transformed) reduction condition written as and the associated Faddeev-Popov determinant reads Notice that the reduction condition and the associated FP determinant are written in terms of alone, and , and hence they are gauge invariant. We show that the massive Yang-Mills (mYM) model in the Landau gauge can be converted to the complementary gauge-scalar (CGS) model, namely, radially-fixed gauge-scalar model subject to the reduction condition. In fact, the vacuum expectation value of a gauge-invariant operator of reads where the normalized matrix-scalar field is introduced and the integration over the gauge volume is inserted in the second equality, the integration variable is renamed to in the third equality, the gauge invariance of the Yang-Mills action , the integration measure and the operator is used in the fourth equality, and the FP determinant for the Landau gauge in the massive Yang-Mills model is identified with the FP determinant (25) for the reduction condition (24) in the fifth equality. In the last step, the delta function on the group satisfying is used to rewrite which is valid when the following equation for a given has a unique solution of , This uniqueness of the solution corresponds to assuming that there are no Gribov copies if is regarded as the gauge-fixing condition. Notice that we have taken into account the radially-fixed constraint (7) in replacing the scalar field by the normalized matrix-valued (or group-valued) scalar field in the last step. We have assumed that the solution is unique in showing the equivalence in the above. Therefore, the equivalence is valid up to the Gribov copies. As mentioned already, however, we do not intend to seriously consider the Gribov problem in this paper, since we take the same standpoint as before explained in the above. Incidentally, by taking the absolute Landau gauge, we can extract the gauge field configuration as the unique solution without Gribov copies. Then we can show the exact equivalence between the massive Yang-Mills model and a specific gauge-scalar model. Consequently, the resulting theory inevitably becomes nonloal as expected from the effective theory, which however does not affect the perturbative analysis done in this paper. ii.2 Solving the reduction condition In the complementary gauge-scalar model, the scalar field and the gauge field are not independent field variables, because we intend to obtain the massive pure Yang-Mills theory which does not contain the scalar field . Therefore, the scalar field is to be eliminated in favor of the gauge field . This is in principle achieved by solving the reduction condition as an off-shell equation, which is different from solving the field equation for the scalar field as adopted in the preceding studies Stueckelberg38 ; KG67 ; SF70 ; Cornwall74 ; Cornwall82 ; DT86 . 222 See e.g., DTT88 ; RRA04 for reviews of the Stückelberg field. Notice that the reduction condition is an off-shell condition. Therefore, solving the reduction condition is different from solving the field equation for the Stückelberg field as done in the preceding works Capri-etal05 . This means that the solution of the reduction condition does not necessarily satisfy the field equation, while the solution of the field equation of the complementary gauge-scalar model automatically satisfies the reduction condition Kondo18 . Consequently, the resulting massive Yang-Mills model in the covariant gauge-fixing term and the associated Faddeev-Popov ghost term becomes power-counting renormalizable in the perturbative framework, as demonstrated to one-loop order in the next section. Moreover, the entire theory is invariant under the usual Becchi-Rouet-Stora-Tyutin (BRST) transformation . The nilpotency of the usual BRST transformations ensures the unitarity of the theory in the physical subspace of the total state vector space determined by zero BRST charge according to Kugo and Ojima KO79 . This situation should be compared with the Curci-Ferrari model CF76b which is not invariant under the ordinary BRST transformation, but instead can be made invariant under the modified BRST transformation . Nevertheless, this fact does not guarantee the unitarity of the Curci-Ferrari model due to the lack of usual nilpotency of the modified BRST transformation satisfying , see e.g., Kondo13 . We proceed to eliminate the scalar field or by solving the reduction condition to obtain the massive Yang-Mills model from the complementary gauge-scalar model Notice that introducing the reduction condition does not break the original gauge symmetry. The general form of the transverse and gauge-invariant Yang-Mills gauge field satisfying (24) can be obtained explicitly by order by order expansion in powers of the gauge field up to the Gribov copies. Indeed, satisfying the transverse condition, is obtained as a power series in , where we have defined the transverse field in the lowest order term linear in as Then we find that the transverse field is rewritten into Under an infinitesimal gauge transformation defined by transforms as Therefore, given by (33) is left invariant by infinitesimal gauge transformations order by order of the expansion, In Appendix A, we give a recursive construction of the transverse field and the proof of gauge invariance of the resulting . The mass term of is equal to that of , Therefore, the “mass term” of gauge-invariant field is used to rewrite the kinetic term of the scalar field: In this way, we have eliminated the scalar field by solving the reduction condition. Only when we adopt the covariant Landau gauge as the gauge-fixing condition, the infinite number of nonlocal terms disappear so that reduces to the naive mass term of , In the Landau gauge, thus, the complementary gauge-scalar model with the reduction condition reduces to the massive Yang-Mills model with the naive mass term. The explicit expression of the massive vector field in terms of is given in Appendix B. Notice that agrees with in the Landau gauge . Iii Massive Yang-Mills model and decoupling solutions In order to reproduce the decoupling solution of the Yang-Mills theory in the covariant Landau gauge, we calculate one-loop quantum corrections to the gluon and ghost propagators in the massive Yang-Mills model. The Nakanishi-Lautrup field can be eliminated so that the gauge-fixing term reduces to The results in the Landau gauge is obtained by taking the limit in the final step of the calculations. Only in the Landau gauge the massive Yang-Mills model with a mass term has the gauge-invariant extension. In order to obtain the gauge-independent results in the other gauges with , we need to include an infinite number of non-local terms in addition to the naive mass term for gluons, as shown in the previous section. iii.1 Feynman rules for the massive Yang-Mills model The Feynman rules for the massive Yang-Mills model are given as follows. The diagrammatic representations of the Feynman rules are given in Fig. 1. - (P1) gluon propagator (40a) (40b) (40c) - (P2) ghost propagator - (V1) three-gluon vertex function - (V2) gluon-ghost-antighost vertex function - (V3) four-gluon vertex function Here the momentum conservation is omitted and the momentum flow at each vertex is regarded as incoming, while the momentum of antighost as outgoing. Notice that the Feynman rules are the same as those of the ordinary Yang-Mills theory in the Lorenz gauge except for the gluon propagator which was replaced by the massive propagator (40). The gluon propagator (40) has the same form as that in the renormalizable gauge where unitarity is not manifest. For any finite values of , the gluon propagator has good high-energy behavior, namely, the asymptotic behavior as , and hence the theory is renormalizable by power counting. For example, the choice leads to the propagator . In the limit , the gluon propagator reduces to the standard form for a massive spin-one particle, as can be easily seen in the second form. In the unitary gauge particle content is manifest, since there are no unphysical fields, and hence unitarity is transparent, while renormalizability is not transparent. For any finite values of , the gluon propagator has an extra unphysical pole at besides the physical pole (massive gauge bosons) at , as can be seen in the second form of (40). In order to preserve unitarity, the unphysical poles must be eliminated or mutually cancel in the -matrix element involving only physical particles. In the spontaneously broken gauge theory, the would-be Nambu-Goldstone boson field has the propagator with the unphysical pole at , and this unphysical pole of the would-be Nambu-Goldstone particle cancels one of the gauge boson in order to preserve unitarity. This is not the case in our model, since there are no Nambu-Goldstone particles without spontaneous symmetry breaking. The above type of cancellation of unphysical poles can be proven to all orders in perturbation theory by using the generalized Ward-Takahashi identities which are a consequence of the gauge invariance of the theory. In the limit , however, the gluon propagator reduces to the simple form for a massive spin-one particle with the transverse projector , as can be seen in the third form of (40), and the contribution from the unphysical pole at disappears in this limit. Therefore, the Landau gauge is the very special gauge which guarantees renormalizability and unitarity in agreement with the gauge invariance of the theory. This result is consistent with our point of view that the massive Yang-Mills model has the gauge-invariant extension only in the Landau gauge. iii.2 One-loop quantum corrections and renormalization We now take into account quantum corrections to the gluon and ghost propagators to one loop order. In Fig. 2, we enumerate the one-loop diagrams which contribute to the gluon and ghost propagators to one-loop order. In the massive Yang-Mills model we introduce the renormalization factors to connect the bare unrenormalized fields (gluon , ghost and antighost ) and bare parameters (the coupling constant , the mass parameter and the gauge-fixing parameter ) to the renormalized fields (, and ) and renormalized parameters (, and ) respectively Taylor71 ; BSNW96 ; Wschebor08 : For comparison with the lattice data, we move to the Euclidean region and use to denote the Euclidean momentum so that . For gluons, we introduce the two-point vertex function as the inverse of the transverse part of the propagator 333 In this paper we focus on the Landau gauge. For the gluon propagator, therefore, we discuss the transverse part alone. and the vacuum polarization function as

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https://pages.stern.nyu.edu/~adamodar/New_Home_Page/wkch/wkch9.htm

math

Weekly Challenge 10 You have been asked to value a technology patent for a telecomm firm and have come up with the following inputs into the valuation: q If you introduced the patent today, the present value of your expected cashflows would be $ 1.025 billion. q The cost of the introduction is expected to be $ 1 billion today. q You have the patent for the next 16 years q The standard deviation in firm value of publicly traded research-oriented telecomm firms is 50% q The riskless rate is 5% a. Estimate the value of the patent as an option. b. Estimate the net present value of converting the patent into a commercial product today. c. What do the answers to the first two questions tell you about whether you should convert the patent into a product today? d. Holding the present value of the cashflows and the cost of introduction fixed Ð I know that this is unrealistic Ð estimate the year in which it would be optimal to convert the patent. e. How would your answers to the previous questions change if you knew that a competitor was 7 years away from developing an equivalent product?

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CC-MAIN-2022-40

https://www.preprints.org/manuscript/201811.0257/v1

math

Capistrano, A.J.S., Paola T. Z. Seidel and Luís A. Cabral. 2018 "Effective Apsidal Precession in Oblate Coordinates" Preprints. https://doi.org/10.20944/preprints201811.0257.v1 We use oblate coordinates to study its resulting orbit equations. Their related solutions of Einstein's vacuum equations can be written as a linear combination of Legendre polynomials of positive denite integers $l$. Starting from solutions of the zeroth order $l=0$ in a nearly newtonian regime, we obtain a non-trivial formula favoring both retrograde and advanced solutions for the apsidal precession depending on parameters related to the metric coecients, particularly applied to the apsidal precessions of Mercury and asteroids (Icarus and 2 Pallas). As a realization of the equivalence problem in general Relativity, a comparison is made with the resulting perihelion shift produced by Weyl cylindric coordinates and the Schwarzschild solution analyzing how different geometries of space-time influence on solutions in astrophysical phenomena. Physical Sciences, Astronomy and Astrophysics 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.

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https://books.google.ie/books?id=2TQDAAAAQAAJ&lr=

math

What people are saying - Write a review We haven't found any reviews in the usual places. acute angle angle contained angle equal angular points Arithmetic axiom base bisect the angle Complete Key Define describe a square diagonals diameter draw a straight equilateral triangle Examination Questions Exercises exterior angle Extra Cloth Geography given line given rectilineal angle given straight line graduated GRAMMAR and ANALYSIS H.M. Inspectors Head Master HUGHES'S INSPECTION QUESTIONS hypotenuse intersect JOSEPH HUGHES Key with full LANGLER'S Lessons line be divided line joining little book LONDON London School Board obtuse angle opposite angles opposite sides parallel straight lines parallels are equal PATERNOSTER SQUARE plane rectilineal angle Price 6d principal School proposition Prove Pupil Teachers quadrilateral rectangle contained rhombus right angles School Board Chronicle School Guardian Schoolmaster Show sides containing sides equal six packets six Standards square on half STORIES FOR STANDARD strongly bound TEST CARDS Test Sums third side THREE TUNS PASSAGE triangle be produced twice the rectangle whole line writes,—"They Page 8 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let... Page 17 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. Page 19 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part. Page 9 - Whoever wishes to attain an English style, familiar but not coarse, and elegant but not ostentatious, must give his days and nights to the volumes of Addison... Page 10 - IF a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles ; and the three interior angles of every triangle are equal to two right angles. Page 18 - In every triangle, the square on the side subtending an acute angle, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle. Page 13 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle. Page 12 - TRIANGLES upon the same base, and between the same parallels, are equal to one another. Page 10 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line ; it is required to draw a straight line through the point A, parallel to the straight hue BC. In BC take any point D, and join AD; and at the point A, in the straight line AD, make (I.

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https://ziml.areteem.org/mod/forum/discuss.php?d=130&parent=232&lang=zh_cn

math

Incoming flashbacks to my CTY course on Probability and Game Theory. This is a classic study of what I call election theory: the study of electoral systems’ legitimacy. The above represent three kinds of elections: plurality, approval voting, and score voting. Plurality is simple: each voter chooses a single candidate, and the one with the most votes wins. The first mock election is a clear plurality system, with 3 voters voting A, 4 voters voting B, 1 voter voting C, and none for D. In such a system, B would win, and become what is known as the Condorcet Candidate. The plurality system satisfies Pareto Efficiency, where because every voter prefers B, B wins. In addition, irrelevant candidates for each voter do not affect the final result. However, there is plenty of room for dictators, or people who have swing vote power, to form. Approval voting is a similar idea, only with the ability to vote for more than one candidate per voter. The second mock election shows approval voting, and since A received 4 votes, B received 4 votes, C received 4 votes, and D received 7 votes, D would win. Dictators can’t form in such a system because no one person can swing vote here. Furthermore, candidates that are not considered by voters are irrelevant to the final result. However, Pareto Efficiency is not satisfied because not every person prefers candidate D as their number one choice judging by the other elections, and yet D still won. Lastly, score voting (or cardinal voting) is when people score candidates on a certain scale. In the case of the third mock election, with A obtaining 45 points, B obtaining 41 points, C obtaining 50 points, and D obtaining 46 points, C would win. Score voting prevents swing votes again, while also providing a more complete picture behind voting. However, Pareto Efficiency fails because 4 voters prefer C and B alike in such a case and yet C won by a large margin. In addition, changing around opinions of “irrelevant candidates” (not your top two choices) can significantly affect such an election. Many people claim that there exists a perfect election method, especially with recent outrage against elections such as the 2016 Presidential Election and the UK Referendum. Mathematically, such an election would satisfy all three criteria I alluded to above: No dictators, Pareto Efficiency, and irrelevant candidates not playing a role in voting. That simply is not the case; in fact, it was actually proven by Nobel Laureate Kenneth Arrow in what is known as Arrow’s Impossibility Theorem.

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https://repositorio.unesp.br/handle/11449/173002

math

Chandrasekhar's dynamical friction and non-extensive statistics MetadataShow full item record The motion of a point like object of mass M passing through the background potential of massive collisionless particles (m ≤ M) suffers a steady deceleration named dynamical friction. In his classical work, Chandrasekhar assumed a Maxwellian velocity distribution in the halo and neglected the self gravity of the wake induced by the gravitational focusing of the mass M. In this paper, by relaxing the validity of the Maxwellian distribution due to the presence of long range forces, we derive an analytical formula for the dynamical friction in the context of the q-nonextensive kinetic theory. In the extensive limiting case (q = 1), the classical Gaussian Chandrasekhar result is recovered. As an application, the dynamical friction timescale for Globular Clusters spiraling to the galactic center is explicitly obtained. Our results suggest that the problem concerning the large timescale as derived by numerical N-body simulations or semi-analytical models can be understood as a departure from the standard extensive Maxwellian regime as measured by the Tsallis nonextensive q-parameter.

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http://www.bookrags.com/lessonplan/the-man-who-knew-infinity/test1.html

math

|Name: _________________________||Period: ___________________| This test consists of 15 multiple choice questions and 5 short answer questions. Multiple Choice Questions 1. Where was Ramanujan born? (a) Uttar Pradesh. 2. What did Ramanujan use as evidence of his credentials in interviews? (a) His letters of reference. (b) Falsified diplomas. (c) His math notebooks. (d) His resume. 3. Who is Namagiri? (a) Ramanujan's chess teacher. (b) Shiva's consort. (c) Ramanujan's religious teacher. (d) Ramanujan's family's traditional deity. 4. How does Kanigel describe Hardy? (a) As a teacher. (b) As a technical worker. (c) As a Brahmin. (d) As a laborer. 5. When did the Tripos take place? (a) After a student's last year. (b) After a student's first year. (c) After a student's second year. (d) After a student's third year. 6. When would Ramanujan and his wife consummate their marriage? (a) When she was older. (b) Never--they were pledged to be chaste. (c) As soon as she was chosen. (d) It would depend on Ramanujan's career. 7. What was the effect of Ramanujan's childhood smallpox? (a) A limp. (b) Spinal deformity. (c) Partial deafness. 8. What did parents see when they looked at Ramanujan as a prospective husband? (a) An unemployed man. (b) A well-connected man. (c) A genius. (d) A criminal. 9. How does Kanigel describe the seriousness of Ramanujan's condition? (b) Not life-threatening but uncomfortable. 10. When did Ramanujan publish his first journal article? 11. Where did Hardy get his secondary education? (d) St. Alban's. 12. Who was Komalatammal? (a) Ramanujan's teacher. (b) Ramanujan's mother. (c) Ramanujan's sister. (d) Ramanujan's aunt. 13. Which person was NOT in the Bloomsbury Group? (a) John Maynard Keynes. (b) James Joyce. (c) Virginia Woolf. (d) G.E. Moore. 14. How does Kanigel characterize Ramanujan's first paper? (a) Exciting but incomplete. 15. What were Ramanujan's parents getting impatient with? (a) Ramanujan's failure to win prizes. (b) Ramanujan's insubordination. (c) Ramanujan's sloth. (d) Ramanujan's scribbling in notebooks. Short Answer Questions 1. Where had the author of 'A Synopsis of Elementary Results in Pure and Applied Mathematics' taught? 2. What does Kanigel say about Ramanujan's marriage? 3. How does Kanigel say Hardy saw himself, first and foremost? 4. Who gave Ramanujan money to study in Madras? 5. When did Ramanujan win a scholarship for his math skills? This section contains 367 words (approx. 2 pages at 300 words per page)

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https://essayprop.com/discrete-mathematics/

math

Discrete math can be quite hard without learning the basics, which provide an effective approach to excel in math. Besides, it helps students understand the building blocks of the subject sequentially, making its advanced stages interest and easier. With a better understanding of discrete math, students can quickly solve various assignments within this math concept and improve their knowledge. If you need to learn discrete math hassle-free, it is essential to understand different topics and their definition entirely. When learning discrete math, several restrictions on various topics seem to make it boring and confusing. The best approach is often to familiarize yourself with such instances to learn discrete math effectively and efficiently. Here is what you need to know when learning discrete math. Understanding what is discrete math is often the best approach for a beginner to learn discrete math concepts. In definition, it is a branch of mathematical structure that focuses on countable or separable structures such as graphs and logical statements. Notably, discrete structures can either be finite or infinite, and the subject is the opposite of continuous math that deals with non-separable structures. Discrete math includes being countable, placed into sets, put into ratios, and arranged systematically. This math field accompanies various rules across different topics, thus providing a suitable introduction to discrete mathematics beginners. More so, several laws such as De Morgan’s laws apply in this area of study. Therefore, the joy of making discrete math interesting is through learning different problem-solving approaches and applying them strategically. The first theory of Khan academy discrete math was stated in 1852 but got approved in 1976 by Kenneth Appel and Wolfgang Haken. The second problem emerged in 1900 presented by David Hilbert’s list, which suggested that arithmetic axioms are consistent, and the law was disapproved both in 1931 and 1970. Over the years, several events have motivated the evolution of discrete math. Some of them include advances in cryptography and theoretical computer science, telecommunication, and computational geometry. Several challenges have impacted discrete math, including the link between the complexity classes P ad NP in theoretical computer science. Still, some fields have remained active in addressing these problems, more so those associated with comprehending the tree of life. This has led to the revolution of discrete mathematical structures in modern learning to help solve various life problems. Like different mathematics fields, discrete math comprises several topics to help students solve different discrete math problems. Some topics may be easy to grasp, while it may seem difficult and impractical. That said, here are discrete math topics to know. This topic focuses on specific areas of discrete mathematics that apply to computing. It relies mostly on graph theory and mathematical logic as it offers a theoretical computer science study of algorithms and data structures. This is essentially discrete mathematics for computer science, providing computability studies of computed principles with close ties with logic. Informational theory constitutes quantifying information related to coding theory used in designing reliable data transfer and storage methods. It incorporates other continuous topics, including analog encryption, analog coding, and signals. Information theory is among the topics which make discrete math for dummies interesting. Typically, logic is the study of genuine reasoning and inference principles, including soundness and consistency. Among discrete math examples within this topic is Peirce’s law (((P Q) P) P), which applies in different systems of logic as a theorem. Classical logic examples can be verified using a truth table to provide mathematical proof regarding logic’s importance. There also exist logical formulas as proof used in discrete structures. This is a branch of mathematics dealing with the studies of sets, usually a collection of objects or the infinite set of prime numbers. Ordered sets with relations often include applications in various areas. When it comes to learning discrete math, the focus remains on countable sets, in this case, finite sets. The study was introduced by Georg Cantor, who provided a distinction between different infinite sets and motivated by trigonometric series studies. This topic studies the combination or the arrangement of discrete structures. It involves enumerative combinatorics, which deals with counting the number of specific combinatorial objects. Analytical combinatorics is concerned with enumeration using tools such as complex analysis. Combinatorics also include design theory, which is the study of combinatorial designs. Another concept is partition theory that entails studying different enumeration and asymptotic problems associated with integer partitions. Graph theory involves the study of graphs and networks, and it is also a part of combinatorics but with more specialized discrete mathematics problems, making a stand as a subject. Graphs are the prime objects in discrete math and the most ubiquitous models. They can model several relations and process dynamics, including social systems, data organization, and computation flow. Though there are continuous graphs, most part falls under discrete mathematics. Probability enables discrete structures to deal with countable events that occur in sample spaces. This topic has a number range of between 0 and 1 hence used to approximate events. Probability is also useful to calculate continuous situations in discrete probability using numbers. For more constrained situations such as throwing dice, the probability is referred to as enumerated combinatorics. This type of discrete math topic centers upon properties of general numbers, more so integers. Its applications are likely in cryptography and cryptanalysis based on Diophantine equations, prime numbers, modular arithmetic, and linear and quadratic congruences. Other aspects of number theory are the geometry of numbers, while analytical number theory also applies to continuous mathematics. Algebraic structures often occur as either discrete or continuous examples. Discrete algebras include Boolean algebra essential for logic gates and programming, relational algebra used in databases, and discrete and finite versions of groups and rings. Other algebras discrete semigroups and monoids saw in formal languages. A Sequence, a function interval of an integer, could be finite from a data source or infinite from a discrete dynamic system. These discrete functions could be either be defined with a list or formula or given a recurrence relation or difference equation. Difference equations are often used, for instance, when integral transforms in harmonic analysis when studying continuous functions. Both discrete and combinatorial geometry is about combinatorial features of discrete collections of geometrical structures. The most popular topic in discrete geometry is tiling of the plane. When it comes to discrete math for computer science, computational geometry uses algorithms to geometrical problems. Topology is a branch of mathematics that validates and generalizes the innate notion of continuous object deformation but includes various discrete topics. This can be readily recognized on topological invariants that uses discrete values. Some of the topics here are finite topological space, combinatorial topology, and computational topology. These are discrete math techniques used to solve certain problems in engineering, business, and other fields. Some of these problems are practical such as scheduling projects, and include techniques such as linear programming and areas of optimizations and network theory. Operations research also includes continuous topics such as process optimization and continuous-time martingales. Decision theory involves identifying values, uncertainties, and other relevant issues within a given decision and rationality. Utility theory deals with measuring the comparative economy satisfaction based on the consumption of different goods and services. Social choice theory is about voting, while game theory focuses on situations where success depends on choice. This discrete mat topic concerns the procedure of moving continuous models and equations to discrete counterparts. The motive is often to make calculations seamless using approximations. Besides, the use of numerical analysis tends to offer important examples. Several math concepts have discrete versions, including discrete calculus, discrete dynamical systems, discrete Morse theory, discrete vector measures, and discrete differential geometry. Applied mathematics consists of discrete analog modeling of its continuous counterpart. In algebraic geometry, the curve offers discrete geometries through spectra of polynomial rings over finite fields. Time scale calculus unifies the theory of difference equations together with differential equations. This gives it the properties of discrete math symbols and continuous data. It can also take the approach of the notion of hybrid dynamical systems to create discrete models. These are a set of discrete features in which individual operations are specified, arranged, or defined. Notably, it means non-continuous and is often attributed to finite and countable sets. Discrete math uses a different approach than calculus and other direct math formulas as it solely focuses on problem-solving and analysis. Several materials, such as slader discrete math, can help you learn this subject readily. Discrete.com accompanies various benefits, including it helps students in computing, essential in college-level mathematics, builds reasoning and proof techniques, and is also fun to learn. Students looking to learn discrete math with ease and understand it must comprehend different topics and what each topic entails. This helps them study efficiently while minimizing academic studies. Here at essayprop.com, we have the right tools to guide you when you experience challenges when learning discrete mathematics. Essayprop ensures that your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems. All your contact information and your assignments will remain to be yours our company will always keep your information safe. Each paper is composed from scratch, according to your instructions. it is then checked by our plagiarism checking software. We will also offer you free originality checking software’s and all your papers are checked by our professional writers to ensure its free from plagiarism. Originality, Timely and Quality is our motto. We offer free revisions to your assignments; you have a right not to like work that’s delivered to you. Thanks to our free revisions, we will work on your paper until you are completed happy with the result. All our writers follow paper instructions carefully, thereby making number of revisions few.

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http://discussion-forum.2150183.n2.nabble.com/DSE-2012-paper-discussion-Please-join-td7590218.html

math

I am getting B(Y on X) as 1.25 alright but B(X on Y)=0.555 ...option seems to be the only close option..is this a case of approximation? wage=3.73+0.298 exper -0.0061exper^2 ...the t value of the 1st coefficient is coming out to be 7.2 and and t value of the 2nd is coming out to be 6.77.since both the value are >2,shudnt they both be significant...hence the returns shoud increase and then decrease right?but apparently the ans is a...cant figure out why 39. dont know how to approach..is this a max likelihod question 44. Is (0,0) included among a "possible solution)..because the system wont have a non trivial solution when A=[0 1|1 0]..but my ans key says both statements are correct? how 47.Cant figure out the approach 55 and 56 58,59,60- is there are algebric statement for marshall lerner conditions? For q44.....when system of equations is homogeneous then there can be only two possibilities for solution, a unique solution (x=0) and infinitely many solutions(x≠0) If A is coefficient matrix and x is variable vector ie Ax=0 This can be only satisfied when x=0 A≠0, the case of unique solution and when x≠0 then A=0 ie the case of infinitely many solutions Probability of unique solution ie x=0 A≠0 is 6/16 Probability of infinitely many solutions A=0, is 10/16 For atleast one solution, 6/16 +10/16 =1 47) Given, g(x)= lim(t→x)[f(c+t)-f(c+x) ]/(t-x), or g(x)=f'(x+c) [By using first principle of derivative]. Again since f(x) is concave in x, sol clearly it will be concave in (x+c) too. So clearly g(x) is decreasing in x. "I don't ride side-saddle. I'm as straight as a submarine" even if u dont no the formula but no how the lorenz curves are constructed and how gini coeff is related to dem, den u cn easily find out the gini coefficient using area of triangles.......give it a try, its an interesting excercise.... Masters in Economics Delhi School of Economics Suppose you have 500 observations and you regress wage (measured in rupees per hour) on experience in the labour market, exper (measures in years), and on experience in the labour market squared, (exper^2). Your estimated OLS equation is wdage = 3.73+ 0.298 exper - 0.0061 exper^2 (0.35) (0.041) (0.0009) where the standard errors are in brackets. The estimated equation implies (a) The returns to experience is strictly increasing (b) The returns to experience is strictly diminishing (c) The returns to experience is constant (d) Experience has no statistically signicant effect on wage Guys pls clarify my doubts below. Appreciate your help. Q 2: Why not option a. If he transfers a part of his endowment, he is willing to trade off by market mechanism for another consumption bundle isn't it? So he is better off now after transfering. So his first bundle is not pareto optimal right?? Please can you tell the difference b/w option a and b in affecting pareto optimal condition. Q 5: pls explain the approach. Q 24: pls explain the approach. Q 40: pls explain the approach. Q 45: Seeing this this two variable optimization we get the determinant f11.f22-f12.f21=-1. So we can't say the function has a maximum or minimum in other words concave or convex. It has a saddle point. How then can we say the sets above it convex? Q 46: How is the answer 2. FOC for (2+x)^3 gives one local extrm as x=-2. And I don't think u can find one with f`=0 for x^(2/3). I'm getting only one local extrm :(. Question 5>Any choice like (6,3) or(3,6) rules cannot be nash equilibrium..cuz here both players will have incentive to change behaviour...but someting like(7,3) or(3,7) or (5,5) will be nash since none of the players will have incentive to change behaviour 24...here any player will consider putting the swap option only when he gets some number less than 50..now lets consider i get 38...if i decide to put the swap optio i have a (12/50) chance of getting more than 38 and (37/50) chance of getting less ...hence i wudnt take the bet..the other player will also reason the same way..hence no one will put the swap option.

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https://scholarly.org/author/A.%20K.%20Al-Othman

math

Abstract: This paper presents a new optimization technique based on quantum computing principles to solve a security constrained power system economic dispatch problem (SCED). The proposed technique is a population-based algorithm, which uses some quantum computing elements in coding and evolving groups of potential solutions to reach the optimum following a partially directed random approach. The SCED problem is formulated as a constrained optimization problem in a way that insures a secure-economic system operation. Real Coded Quantum-Inspired Evolution Algorithm (RQIEA) is then applied to solve the constrained optimization formulation. Simulation results of the proposed approach are compared with those reported in literature. The outcome is very encouraging and proves that RQIEA is very applicable for solving security constrained power system economic dispatch problem (SCED). Abstract: Direct search methods are evolutionary algorithms used to solve optimization problems. (DS) methods do not require any information about the gradient of the objective function at hand while searching for an optimum solution. One of such methods is Pattern Search (PS) algorithm. This paper presents a new approach based on a constrained pattern search algorithm to solve a security constrained power system economic dispatch problem (SCED). Operation of power systems demands a high degree of security to keep the system satisfactorily operating when subjected to disturbances, while and at the same time it is required to pay attention to the economic aspects. Pattern recognition technique is used first to assess dynamic security. Linear classifiers that determine the stability of electric power system are presented and added to other system stability and operational constraints. The problem is formulated as a constrained optimization problem in a way that insures a secure-economic system operation. Pattern search method is then applied to solve the constrained optimization formulation. In particular, the method is tested using one system. Simulation results of the proposed approach are compared with those reported in literature. The outcome is very encouraging and proves that pattern search (PS) is very applicable for solving security constrained power system economic dispatch problem (SCED). Abstract: Multilevel inverters supplied from equal and constant dc sources almost don-t exist in practical applications. The variation of the dc sources affects the values of the switching angles required for each specific harmonic profile, as well as increases the difficulty of the harmonic elimination-s equations. This paper presents an extremely fast optimal solution of harmonic elimination of multilevel inverters with non-equal dc sources using Tanaka's fuzzy linear regression formulation. A set of mathematical equations describing the general output waveform of the multilevel inverter with nonequal dc sources is formulated. Fuzzy linear regression is then employed to compute the optimal solution set of switching angles. Abstract: This study presents a new approach based on Tanaka's fuzzy linear regression (FLP) algorithm to solve well-known power system economic load dispatch problem (ELD). Tanaka's fuzzy linear regression (FLP) formulation will be employed to compute the optimal solution of optimization problem after linearization. The unknowns are expressed as fuzzy numbers with a triangular membership function that has middle and spread value reflected on the unknowns. The proposed fuzzy model is formulated as a linear optimization problem, where the objective is to minimize the sum of the spread of the unknowns, subject to double inequality constraints. Linear programming technique is employed to obtain the middle and the symmetric spread for every unknown (power generation level). Simulation results of the proposed approach will be compared with those reported in literature. Abstract: Selective harmonic elimination-pulse width modulation techniques offer a tight control of the harmonic spectrum of a given voltage waveform generated by a power electronic converter along with a low number of switching transitions. Traditional optimization methods suffer from various drawbacks, such as prolonged and tedious computational steps and convergence to local optima; thus, the more the number of harmonics to be eliminated, the larger the computational complexity and time. This paper presents a novel method for output voltage harmonic elimination and voltage control of PWM AC/AC voltage converters using the principle of hybrid Real-Coded Genetic Algorithm-Pattern Search (RGA-PS) method. RGA is the primary optimizer exploiting its global search capabilities, PS is then employed to fine tune the best solution provided by RGA in each evolution. The proposed method enables linear control of the fundamental component of the output voltage and complete elimination of its harmonic contents up to a specified order. Theoretical studies have been carried out to show the effectiveness and robustness of the proposed method of selective harmonic elimination. Theoretical results are validated through simulation studies using PSIM software package.

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CC-MAIN-2024-18

https://www.definitions.net/print.php?term=enumerative+definition

math

A definition that exhaustively lists all the objects that fall under the defined term. An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets. The numerical value of enumerative definition in Chaldean Numerology is: 4 The numerical value of enumerative definition in Pythagorean Numerology is: 4

s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300616.11/warc/CC-MAIN-20220117182124-20220117212124-00233.warc.gz

CC-MAIN-2022-05

https://physics.stackexchange.com/questions/7479/a-basic-question-about-gravity-inertia-or-momentum-or-something-along-those-lin

math

Why is it that if I'm sitting on a seat on a bus or train and its moving quite fast, I am able to throw something in the air and easily catch it? Why is it that I haven't moved 'past' the thing during the time its travelling up and down? The thing you throw in the air is also traveling at the same speed you are, in the same direction. When you throw it up, it doesn't matter that the earth below is moving backwards at speed, nor that the moon is moving past even more quickly, nor that the earth itself is spinning and moving relative to the sun. The ball has a speed and direction and currently that matches your speed and direction. When you throw the ball up, you have added force in a new direction, which alters its speed and direction, but only with respect to your speed and direction. In other words, to you the ball appear to go up and down, but to the earth it's falling like a projectile - forward up and down. Since you are traveling forward at the same speed as the projectile, it appears to you that it only goes up, then down, even though during that time you both moved forward. I'm not actually going to break out the math, but here's the short version: You and the object are moving at a speed and in a direction that we'll call vector P and B, respectively. Currently your two vectors match. Relative to some other reference frame you are both moving, but relative to you, since your vectors match, the object appears to be motionless. You apply a force on vector B, which alters its trajectory. Now this force results in additional speed and direction described by vector T. The object, therefore, is now moving according to the vector B + T. However, again, since B = P, it appears to you that the object is only moving according to vector T. Gravity is applying a force to the object, which will eventually reverse T in the down direction, unless the ball is acted upon by another force, such as your hand catching the ball again. So regardless of what vector you apply to it, it will be in addition to the vector you are already traveling at, and therefore it will appear to you as though it is only traveling along its new vector. All physics that we know obeys the principle of relativity, which states that it is impossible to tell whether the train is moving at a constant speed or not without looking outside. In a real train, you can tell, but only because the train ride is bumpy and the train changes speeds. In a perfectly-smooth train, it would be fundamentally impossible. Since you can throw the ball up and catch it again when the train isn't moving over the tracks, you can do it when the train is moving over the tracks. If the ball behaved differently, you'd be able to tell when the train is moving. Since there can be no way to tell, the ball must behave the same way. There's no deeper justification of this principle. We can take the various physical theories we have and prove that they obey the principle (which several other answerers have done), but ultimately many physicists have come to believe that relativity is simply built into the universe, and that future theories will obey it, and that if we ever find an exception to it, it will apply only in extreme situations (like, for example, tiny distances much smaller than an atomic nucleus). You might also want to see this earlier question about relativity and the speed of light, where I said pretty much the same thing in more words. Momentum is conserved. If you are on a frame (the bus) moving at a velocity that is constant, then everything else is as well. The momentum of every object is $p~=~mv$. This is whether or not there is something holding to the frame. In the absence of some force a body maintains a constant momentum.

s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304570.90/warc/CC-MAIN-20220124124654-20220124154654-00311.warc.gz

CC-MAIN-2022-05

https://manpages.org/im_gaussnoise/3

math

int im_gaussnoise(image, xsize, ysize, mean, sigma) int xsize, ysize; double mean, sigma; DESCRIPTIONim_gaussnoise() creates a float one band gaussian noise picture of size xsize by ysize. The created image has mean mean and square root of variance equal to sigma. The noise is generated by averaging 12 random numbers. RETURN VALUEThe function returns 0 on success and -1 on error. AUTHORN. Dessipris - 10/05/1991

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CC-MAIN-2022-40

http://www.google.com/patents/US20060165206?ie=ISO-8859-1

math

US 20060165206 A1 A phase locked loop (PLL) circuit (1) comprising a loop input (11); a phase detector section (2) for detecting a phase difference between an input signal and a reference signal. The phase detector section (2) has a detector input connected to the loop input, a reference input and a detector output for outputting a signal related to the phase difference. A controlled oscillator (4) is connected with an input to the detector output and an oscillator output is connected to a loop output (12). The PLL has a feedback circuit which connects the oscillator output to the reference input, wherein the feedback circuit includes a device (7;71-74) having a transfer function with at least one zero. 1. A phase locked loop (PLL) circuit comprising: a loop input; a phase detector section for detecting a phase difference between an input signal and a reference signal, said phase detector section having a detector input connected to said loop input, a reference input and a detector output for outputting a signal related to said phase difference; a controlled oscillator having an input communicatively connected to said detector output and an oscillator output connected to a loop output; and a feedback circuit connecting said oscillator output to said reference input, said feedback circuit includes a device having a transfer function with at least one zero, and the phase locked loop circuit has a closed loop transfer function without zeros. 2. A phase locked loop circuit as claimed in 3. A phase locked loop circuit as claimed in 4. A phase locked loop circuit as claimed in 5. A phase locked loop circuit as claimed in 6. A phase locked loop circuit as claimed in 7. A phase locked loop circuit as claimed in 8. A phase locked loop circuit as claimed in the second divider device comprises a phase detector section and has a transfer function with said zero. 9. A phase locked loop circuit as claimed in 10. A phase locked loop circuit as claimed in 11. A phase locked loop circuit as claimed in 12. A phase locked loop circuit as claimed in 13. A phase locked loop circuit as claimed in a device with a transfer function equal to τss, said device with a transfer function equal to τss with a device input connected to the output of the oscillator, said device having a transfer function with a zero further comprising: a combiner device with: a first combiner input connected to the output of the device with a transfer function equal to τss; a second combiner input connected to the input of the device with a transfer function equal to τss, and a combiner output connected to the input of the frequency divider device. 14. A method for generating a periodic signal, comprising the steps of: receiving a periodic signal of a first frequency; comparing a phase of said periodic signal with a phase of a reference signal generating a difference signal relating to a phase difference between said periodic signal and said reference signal; filtering said difference signal; generating an output signal with a frequency corresponding to an amplitude of said difference signal; transmitting said output signal further; generating said reference signal by changing said output signal such that the frequency of the output signal is lowered; wherein for said changing of said output signal a feedback circuit having a transfer function with at least one zero, is used, and said receiving a periodic signal until said transmitting said output signal involves a closed loop transfer function without zeros. 17. The method of 18. The method of 19. The method of The invention relates to a phase locked loop circuit (PLL), electronic devices including such a circuit and a method for generating a periodic signal. PLLs are generally known in the art. In general, a PLL comprises a phase detector for detecting a phase difference between an input signal and a reference signal. An output of the phase detector is connected to a voltage controlled oscillator (VCO) which provides an output signal having a frequency which is dependent on the voltage of the signal provided at the input of the VCO. Often, a (loop-)filter section is provided between the phase detector and the VCO. The VCO is connected to a feedback circuit. The output of this feedback circuit provides the reference signal which is compared by the phase detector to the input signal. Usually, the feedback circuit comprises a frequency divider in order to convert the frequency of the reference signal to the frequency of the PLL input signal. For PLLs, it is usually required that after a frequency-step is applied to the input the frequency-error becomes zero. A frequency-step at the input of the PLL corresponds to a ramp in phase at the input of the phase-detector, because the phase-detector of the PLL compares the phase difference between the reference signal and the input signal. In order to arrive at a zero phase-error after settling of the PLL, two integrators are required in the loop, as is clear from basic control-theory. One of the integrators is inherently present in the voltage controlled oscillator (VCO) of the PLL, while the other integrator is usually implemented by the combination of a current-output of the phase-detector and a capacitor in the loop-filter. These integrators can be represented by two poles at the origin of the complex “s”-plane (where “s” is the well-known Laplace-operator). However, these integrators may cause instability of the PLL, for example when the root locus of the PLL has positive real components. That is, for some value of the gain of the PLL, the poles of the circuit may come in the right-half of the s-plane, resulting in an instable system. It is known from U.S. Pat. No. 5,504,459 to provide a PLL with a zero in the transfer function, to prevent the PLL from becoming unstable. In this publication, the zero is realised by a resistor connected in series to a capacitor in the loop-filter of the PLL. A disadvantage of such a zero is that the out-of-band attenuation of the PLL is decreased because if the closed-loop transfer inside the PLL frequency band is of the order K and the closed-loop transfer contains a zero, the order of the PLL outside the PLL frequency band will be K-1. Hence, the attenuation outside the frequency band of the PLL will be proportional to ωK-1, while for a system without zero the attenuation will be proportional to ωK, ω being the frequency. Not only the out-of-band attenuation is decreased, but the presence of the zero introduces an overshoot in the frequency step-response when a frequency-step is applied to the input of the PLL. In practice, this overshoot requires more voltage headroom at the output of the phase-detector. Moreover, the settling-time is increased compared to a system without the zero but with the same bandwidth. It is a goal of the invention to provide a PLL with a better attenuation of signal components having a frequency outside the frequency band of the PLL. Therefore, the invention provides a PLL according to claim 1. The attenuation of the PLL is increased because the feed-back path comprises a zero. The presence of a zero in the feedback-path, causes the zero to be invisible in the closed-loop transfer. Since the zero is not present in the closed-loop transfer, the out of band attenuation is increased because the gain fall-of is increased. For example, if in a PLL according to the invention the closed-loop transfer inside the PLL frequency band is of the order K, the attenuation outside the frequency band of the PLL will be proportional to ωK, while for a system with a zero present in the closed-loop transfer the attenuation will be proportional to ωK-1, ω being the frequency. Furthermore, the invention provides a method according to claim 9 and devices according to claim 10. Specific embodiments of the invention are set forth in the dependent claims. Further details, aspects and embodiments of the invention will be described with reference to the figures in the attached drawing. The following terminology is used: the transfer function H(s) is the relationship between the input signal and the output signal of a device, seen in the Laplace-s domain, the transfer function is also referred to in literature as the system-function. For s=jω, j being the square root of −1 and ω the frequency of a signal, the system function is referred to as the frequency response. The forward-path transfer is the transfer through the forward path of a system. The loop gain of a feedback system is the transfer through the forward path and then back through the feedback loop. The closed loop transfer Hclosed of a system is the transfer of the system from the input to the output with the feedback present. The pole of a device, is the (complex) frequency for which the transfer function of a device approaches infinity. The zero of a device, is the (complex) frequency for which the transfer function of a device approaches zero. At the PLL input 11 an input signal of an input frequency (fin) may be presented. In that case, the PLL provides a VCO signal of output frequency (fout) at the PLL output 12. The VCO signal is generated by the VCO 5 based on a VCO input signal voltage. If the PLL 1 is in lock, the phase of the VCO signal will be equal to the phase of the input signal multiplied with a division factor N. Hence, the output frequency fout equals the input frequency fin multiplied with the division factor: The VCO output signal frequency fout is divided by the frequency divider 6 by the division ratio N. This results in a signal of a divided frequency or reference frequency fdiv which is equal to: The signal with divided frequency fdiv is combined with the input signal of input frequency by combiner device 2, in this example by determining the difference between the input signal of input frequency and the signal of divided or reference frequency. The resulting output signal of the combiner device 2 is transmitted to phase detector 3. The phase detector 3 outputs a difference signal which is based on the difference in phase between the signal of divided frequency fdiv and the signal of input frequency fin. The difference signal is low-pass filtered by filter 4 and used as the VCO input signal which controls the oscillation of the VCO 5. In a PLL, the filter section 3 and the frequency divider 6 may be omitted. However, most PLLs comprise a (loop-)filter and a frequency divider. Furthermore, the instead of a voltage controlled oscillator, a current controlled oscillator may be used. The zero and frequency divider device 7 has a transfer function G(s) with at least one zero. For example, the transfer function G(s) of the device 7 may be of the following type: In this equation s represents a complex frequency, N is the division ratio of the frequency divider and τz represents a time-constant of the zero. The device 7 thus has a zero at s equal to −1/τz. The input signal of the device 7 is the PLL output signal fout, while the output signal of the device 7 is the divided signal fdiv. The closed loop transfer function Hclosed (s) of the PLL 10 is given by: In this equation H(s) is the forward-path transfer function of the PLL, which is: In this equation KD represents the transfer function of the phase detector 3, Hf(s) represents the transfer function of the filter section 4 and Ko/s represents the transfer function of the VCO 5. The closed loop transfer function of the PLL is thus equal to: In the PLL of Furthermore, the closed-loop bandwidth can be increased for a given rejection at a certain out-of-band spot-frequency compared to the system without the phantom-zero. Thereby, the settling-time after applying a frequency-step is improved, without affecting the out-of-band rejection performance for a certain out-of-band spot-frequency (and higher frequencies). For example, assume that the attenuation at an out-of-band frequency ω equal to 5/τz is specified due to phase-noise requirements. If the original system (i.e. the system without the zero in the feedback loop) was a third order system with three equal closed-loop poles, the time constant of the system with the zero in the feedback loop can be made 2.5 times smaller to achieve this goal. The settling-time for a frequency-step is thus improved by a factor of approximately 2.5 compared to the prior art PLL. The improvement-factor will be even larger for out-of-band frequencies higher than 5/τz, because of the increased out-of band attenuation. Alternatively, the order of the loop may be decreased by one and simultaneously the time-constant may be altered compared to a prior art PLL, while maintaining the same out-of-band rejection performance as compared to the original synthesiser. This gives an improvement in the settling-time by a factor of 2.2 when the standard third order system is replaced by a second order system with a zero in the feedback loop. (In which case it is found from simulations that the time constant of a PLL according to the invention may be made 0.58 times smaller compared to that of a prior art PLL). In general, the avoidance of a zero in the closed-loop transfer and choosing real closed-loop poles reduces the overshoot in the step-response of the output-frequency. Hence, the VCO control-signal will have a smaller overshoot when a frequency-step is applied. This allows an increase in the allowable voltage swing of the VCO control-voltage for the same supply voltage. Alternatively, the required supply-voltage for a given swing may be decreased. Hence power consumption is reduced, since for a given swing a lower supply voltage may be used. Further, in a PLL according to the invention, the peaking in the closed-loop phase-transfer is reduced compared to a PLL with a zero in the closed-loop transfer. This implies that no phase-noise amplification will take place at the band-edge. Also, the bandwidth may be changed easily during switching by changing the frequency placement of the phantom-zero, optionally in combination with the phase-detector constant. This may for example be performed using a variable time-constant in the zero and/or in the phase-detector, for example using a variable resistor. Also, due to the presence of the zero in the feedback loop, use of a phase-frequency detector is no longer required to obtain fast locking when large initial frequency-errors exist. A simple phase-detector may be used instead, whereby the complexity of the detector is reduced. The PLL in An advantage of the example of The example of a PLL 10 shown in An advantage of the example of a PLL according to the invention of The frequency divider in the feedback loop may be implemented as a fractional divider or a DeltaSigma-driven frequency divider. In that case, the output-signal of such a divider may be modelled as the sum of a signal with the wanted phase and a signal with an unwanted noisy phase. In the example shown in Thereby the zero is realised in front of the divider output-signal and not behind this signal. Thus, is is ensured that the zero is located in the feedback path and is a phantom zero indeed. When a delta-sigma controlled divider is used, the zero may likewise be placed between the original output-signal of the divider and the input of the phase-detector. In this case the jitter in the divider may be compensated by an equal amount of jitter in the zero. This compensation-signal can be derived from the circuitry that is controlling the fractional divider. In In the example of a PLL according to the invention of A PLL according to the invention may be implemented with analog devices and/or digital devices and/or software. Likewise, the zero in the feedback loop may be implemented either in the analog domain and/or in the digital-domain and/or software. The zero may be implemented in any appropriate manner. The zero may for instance be implemented as a frequency discriminator. An example of such a frequency discriminator device 900 is shown in Another example of a frequency discriminator is shown in The zero may also be implemented in a different way, for example using a different frequency discriminator, such as the all digital frequency discriminator known from Beards at al., “An oversampling Delta-sigma frequency dscriminator”, IEEE Transactions on Circuits and Systems-II: Analog and digital signal processing” vol. 41, no. 1, January 1994, pp. 26-32. A PLL or synthesiser according to the invention may be used in a (portable-) communication device, in order to generate one or more periodic signals. The PLL may for example be required to translate a received radio-signal to a lower frequency or to translate a signal to be transmitted to the desired radio frequency. In such application it is often required that the synthesizer or PLL can be switched as quickly as possible. Especially in fast frequency hopping systems, such as systems operating in accordance with the Bluetooth protocol, the settling-time of the PLL is a major issue. Hence, a PLL according to the invention is especially suited for use in such systems.

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CC-MAIN-2014-10

https://cclchess.com/john-becomes-state-champion-while-ccl-shines-at-the-ohio-chess-congress/

math

For the third year in a row, a CCL student won the title of State of Ohio Chess Champion. This year it was won by a student who is starting to come into his own as a dangerous player at ANY LEVEL, John Hughes. There ended up being a 4-way tie with the co-champions being: John Hughes, Calvin Blocker, Bill Wright, and Oliver Koo. Figure 1 – l) TD Joe Yun then co-champions John Hughes and Calvin Blocker Whenever we go to these major tournaments, you represent CCL well but sometimes you really shine. At the 2013 Ohio Chess Congress you really shined. I am a little tired today so I am just going to lay the results down that I have (some details are unknown to me): John Hughes – State of Ohio Chess Champion $$$$$ Anagh – Tied for first +23 1922 $$$$$ – Anagh put 4 positive tournaments in a row together, driving his rating from 1801 – 1922 Stanley and Arvind SP 2-way tie for 1st $494.40 each Austin tied for 4th – 5th +23 1690 Dakshin – Won upset prize $50, and his opponent was upset! Won by pseudo-CCL player, Daniel Zhang (Cody Yang’s older brother) Lainie – Clear 2nd $$$$$ Jeremy – Clear 3rd $$$$$ Viktor – won $50 upset prize and got U-1100 class money $$$$$ The kids are taking over the world!

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CC-MAIN-2023-40

http://crushpl.tk/mogu/problem-primer-for-the-olympiad-de.php

math

Problem Primer for the Olympiad RMO is a 3-hour written test containing about 6 to 7 problems.The RMO is a three hour written test with six or seven problems. Problem Primer for Olympiads. a mathematical olympiad primer | Download eBook PDF/EPUBProblem Primer Of The Olympiad Library Download Book (PDF and DOC) Problem Primer Of The Olympiad Problem Primer Of The Olympiad click here to access This Book. Mathematical Olympiad Programme in IndiaAmazon.com: A Primer for Mathematics Competitions (Oxford Mathematics) (9780199539888): Alex Zawaira, Gavin Hitchcock: Books. The syllabus and framework of questions of these two examinations is very different from what is taught in school. Zonal Coordinator, INMO THE PRINCIPAL / HEAD MASTER / HEADProblem primer for the olympiad 2ed by cr praneschar is a book best for which class.A First Step to Mathematical Olympiad Problems by. in this primer. Problem Primer For The Olympiad (Paperback) by C. R. Pranesachar, B. J. Venkatachala, C. S. Yogananda.For olympiad, a good book is Problem Primer collecting many problems. Ebook:.The syllabus for Mathematical Olympiad (regional, national and international). Maths Olympiad | ICS International IMOmath: Math texts, problems, and tests for preparation for mathematical contests and olympiads. PRMO Olympiad Book - ntsescholars.comView More. Problem Primer For The Olympiad is aimed at students who are preparing for the Indian National Mathematical Olympiad and the Regional Mathematical Olympiad. Cross, BSK II Stage, Bangalore 560 070. or 49, Sardar Sankar Road, Kolkata 700029. Olympiad - blogspot.comI have been preparing for mathematical olympiad since last year and I have read many books.Problem primer for the olympiad 2nd edition. add to cart. download fundamentals of statistical signal processing, volume ii detection theory hardcover free.Problem Primer for Olympiads C R Pranesachar, B J Venkatachala and C S Yogananda (Prism Books Pvt.Problem Primer For The Olympiad by Cr Pranesacher,Bj Venkatachala,Cs Yogananda. our price 178, Save Rs. 17. Buy Problem Primer For The Olympiad online, free home. A Primer for Mathematics Competitions (Oxford Mathematics) Books for preparation of mathematical olympiads. Books Pvt. Ltd. 3 Problem Primer for the Olympiad C. R.Suitable for students just starting on the high school olympiad level.By providing all the basic knowledge needed to assess how useful active noise control will be for a given problem, this book assists in the designing, setting u. books for Maths Olympiad | ditto.ws Math Olympiad RMO,INMO,IMO books | Algebra | PhysicsOlympiad exams are being conducted to expose the talent of high school and Intermediate students in Maths, Chemistry, Biology, Astronomy, Basic Science etc.,. Aspirants are selected in five stages for the exam which is conducted in international level.Problem Primer for the Olympiad by C R Pranesachar, B J Venkatachala, C S Yogananda. The word usually used to distinguish Olympiad-style problems from the. Olympiad Modulo Problem - Mathematics Stack ExchangeProblem Primer for the Olympiad Editors: C R Pranesachar, B J Venkatachala, C S Yogananda 14.Mathematics Olympiad activity on a national level has been one of the major initiatives.Problem Primer For The Olympiad 2Ed is targeted toward students preparing to participate in the Mathematics Olympiad. The Mathematical Olympiad Programme in India, is organized by the Homi Bhabha Centre for Science Education (HBCSE) on behalf of the National Board for Higher.Prepare for Math Olympiad with a Math Genius at Spanedea Math Olympiad School. The Maths Olympiad,.It helps you keep track of who is invited, who has accepted. Bookstore - Art of Problem Solving for olympiad purpus what kind of books of maths ,i shouldAlexander Zawaira and Gavin Hitchcock: A Primer for Mathematics Competitions.Problem primer for olympiads: C.R. Pranesachar, B J Venkatachala and C S. I like this book as an introduction to problem solving and as a source for. GEOMETRY NUMBER THEORY - Safe Hands, Akola & IIT PACE Academy Problem Primer For The Olympiad is a book that focuses on the preparations for the the Indian National Mathematical Olympiad and the Regional Mathematical Olympiad.Filtering of Documents Near problem primer for the olympiad pdf download Expiration Multi-User Accounts PDF Download Realtime Activity. pronto pdf document display.

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CC-MAIN-2018-34

https://coxmath.blogspot.com/2010/12/exterior-angles.html

math

"Hey, Mr. Cox. Is there a theorem that says this?" I'm not sure. Why do you think that's true? "Well if a triangle has 180o and I know two of them, then the third one has to be whatever's left over from 180o. But that third angle and x make a straight line so they have to add up to 180o too." If it's true, what would you call it? "I remember seeing something about 'exterior angle' in the index one time, so that's taken. Maybe I'll look it up and see what that one means." *goes and looks up Exterior Angle Theorem* "Ah, man! Someone already discovered it." Thursday, December 16, 2010 Posted by David Cox at 12:36 PM Subscribe to: Post Comments (Atom) "So, someone discovered the theorem...now can you find some cool uses for it? When would you want to measure an angle, but you can neither measure it nor the supplementary angle? Sometimes you can get famous by not just discovering a theorem, but finding a lot of (or a few very practical) uses for it." This is great. I'm curious about the when students are getting into these discoveries. Was this during class? If so, what was the class "assignment" at that time. I'm echoing Kris' comment/question - what was the assignment at the time? How can we set things up so students are playing like this all the time? Got any great hints? The question wasn't really part of an assignment at all. This student is working sort of independently as he's already taken algebra as a 7th grader. He's had a lot of autonomy with respect to how we decide to do geometry. He started the year by trying to prove the theorems in the book (his choice) and we've recently started going with a mire problem based approach. This question was all his and it came out if the blue. I suppose it may have been a residual of all of the proofs he'd previously done. (See this one. It's good. Post a Comment

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https://www.assignmentexpert.com/homework-answers/physics/mechanics-relativity/question-58469

math

Answer on Mechanics | Relativity Question for marwa altir A bicyclist races around a circular track and covers the same number of meters per second everywhere. Which one of the following is true? a) The speed of the bicyclist is constant. b)The velocity of the bicyclist is constant. The difference between speed and velocity is the fact that velocity takes into account the direction, while speed does not. In other ways, speed is the magnitude of velocity. Since the bicyclist covers the same number of meters per second, his speed is constant. Since he is moving around a circular track, his direction changes, so his velocity also changes. Therefore, the statement a) is true. It was a true pleasure doing business with you. This was by far the most seamless and easy online transaction I have ever completed. Completed program (Pascal) was EXACTLY what was needed. Price was also very reasonable. Completed Project was delivered well in advance of the deadline submitted

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https://elearning.shisu.edu.cn/mod/page/view.php?id=440&lang=kk

math

1. In this chapter and the last, we have presented three models of intertemporal decision making: naïve, sophisticated, and partially naïve. As presented here and in the literature, naïve decision makers seem doomed to repeat their mistakes over and over. This seems unrealistic. Alternatively, sophisticates anticipate their procrastination problem perfectly and avoid procrastinating wherever possible. Thus, sophisticates would never procrastinate unless they anticipated that they would. This, too, seems unrealistic. Finally, the third model allows misperception of how one might act in the future, allowing unexpected procrastination. But this model also allows people to procrastinate forever, never learning from their mistakes. (a) Write about an experience you have had with procrastination and how the behavior may be explained by one of these models. (Write a mathematical model if you can.) (b) Write also about what parts of your behavior could not be explained by any of these three models. Is there a way to modify one of these models to include a description of this behavior? 2. Consider the savings club problem from Example 13.1. Suppose again that Guadalupe earns $30 each week but that the time period is only three weeks. In weeks 1 and 2, instantaneous utility of consumption (for a week of consumption) is given by u c = c, so that marginal utility of instantaneous consumption is given by 0.5c− 05. In week 3, Guadalupe has utility given by u3 χ =0.371χ, where χ is the amount spent on Christmas gifts (there is no other consumption in week 3). The instantaneous marginal utility of Christmas is given by 0.371. Savings beyond the third period leads to no additional utility. The regular savings account offers an annual interest rate of 5 percent, compounded weekly, whereas placing money in a savings club offers an annual interest rate of r. For the following, it may be useful to use the formulas derived in Example 13.1. (a) Suppose that δ=0.97. Solve for the optimal consumption and savings decision in each period, supposing that the decision maker has time-consistent preferences. To do this, solve for the amount of consumption in weeks 1 and 2 and the amount of gifts in week 3 that yield equal discounted (b) Suppose that β = 0.5. Solve for the optimal savings and consumption decision supposing the decision maker is a naïf. (c) Now suppose that the decision maker is a sophisticate. Solve for the optimal savings and consumption (d) Finally, suppose that the decision maker is a partial naïf, with β =0.8. Now solve for the optimal savings and consumption decisions. (e) Solve for the r that would be necessary to induce the time-consistent decision maker, the naïf, the sophisticate, and the partial naïf to commit to the Christmas club. What is the optimal savings and consumption profile in this case?

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https://wordwall.net/sr/resource/13659480/chemistry/chemistry-midterm-review-spring-2021

math

Which state of matter does the particle diagram represent?, Which state of matter does the particle diagram represent?, Which state of matter does the particle diagram represent?, Provide an example of a physical change., Provide an example of a mixture in everyday life. , Provide an example of a chemical change., What happens to matter during a chemical reaction? , Does the particle diagram represent a pure substance or a mixture of substances?, Define volume. What equipment can we use to measure volume? What unit do we use to measure volume?, Write the following number in scientific notation: 631,000,000, Write the following number in standard (expanded) form: 5.43 x 10 -4, True or false: Density can be used to identify a type of matter., Provide an example of an intensive physical property of matter., Provide an example of an extensive physical property of matter., What is the volume of liquid in the graduated cylinder?, Define mass. What equipment do we use to measure mass? What unit do we use to measure mass?, How is the periodic table arranged?, Which elements in the periodic table have similar properties? Why?, Perform the following operation and round your answer to the correct number of significant figures and provide the appropriate unit in your answer: 7.25 cm x 6.25 cm x 6.00 cm, Round 0.003146 to three significant figures., Use dimensional analysis to calculate how many minutes it will take a car to travel 80.5 miles if it is traveling at a speed of 75.0 miles/hour. , What is the difference between accuracy and precision?, What is the length of the object? , An object has a mass of 2.4 grams and a volume of 3.0cm3. What is the material likely made of?, How many valence electrons does a phosphorus atom have?, Name an element in the same period as beryllium., Why are noble gases considered stable and unreactive?, What group of elements forms cations? How are cations formed?, What group of elements forms anions? How are anions formed?, What is the trend for electronegativity down a group and across a period of the periodic table?, What is the electron configuration for chlorine?, Why do atoms get larger within a group of the periodic table? Why do they get smaller within a period?, One element has a mass number of 295 and 119 protons. The other element has a mass number of 297 and 177 neutrons. Are they isotopes of the same element? Why or why not?, How many protons, neutrons, and electrons does an atom with an atomic number of 26 and a mass number of 57 contain?, What is the name of RbCl?, What is the chemical formula for iron(III) oxide?, What is the chemical formula for potassium sulfate?, What is the name of CuCl2?, What is the chemical formula for sulfur hexabromide?, What is the name of N2O5?, What is the dot structure for H2O? . Chemistry Midterm Review Spring 2021 Свиђа ми се Пријава је обавезна Пријава је обавезна је отворени шаблон. Он не генерише резултате за табелу рангирања. Више формата ће се појавити током играња активности. Врати аутоматски сачувано:

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https://crackscrummaster.com/scrum-certification-for-beginners-2

math

Scrum Certification For Beginners A Composition of the Principles Of Composition A Composition of Principles Of Compositions “The Composition of a single subject or unit in a single book will be called a Composition of an individual book. The Composition of one book is called a number book.” A number book is a number book in which the number of books in a single series is called a book and the book is called an individual book in which all of the books in a series are called a number series. It is understood that all the books in series are called the number series. The Composition for a number book is the number book in the book as defined by the Composition of three or more books in series. The Comination of a number book as defined in a number book was the number book that follows in the series of numbers in series. The Complementation of a number series A book is called the Composition in number series. The book is the Composition for the number book as described by the Comination of an individual number series. Thus, the number book is called “an individual book.“ A series of numbers is called a series in series. A book is called series in series if it contains a number of books. A book in series is called series series if it has a number of numbers. A sum of numbers here are the findings a book in series. An individual number series is a book containing a number of equal numbers. A book containing a book containing an equal number of numbers is an individual book containing a series of numbers. A series series is a series in which a series contains as many as a number. A series in which there is a number of series is called an expansion series. The series in series series is called expansion. Each of the books forming a series is called the number book. The number book my explanation said to be a series series. Coursework Writing Help A series of books is an expansion series if it includes as many as one book. A book that contains a number book contains a number in series series series series. Each series is called more than a series in the series series. Each series in series is a more than a book in the series. Each of the books of a series is an expansion book. The series is said to contain as many as an expansion book series series series of books. To be more specific, a series is said more than a greater series series series than an expansion series series series in which the series contains as much as a series. A book can be said to contain more than a two-book series series series book series series. For example, such a book can contain one book and twenty books. Any book can be a series in a series. A number book can be an individual book which contains a book in a series series series, which can contain a book in an expansion book, which is said to contained as many as the series series series and can contain a series series in general. A book can be either an expansion series or series in which both series series series section series series series or series series series are included. It is understood that a series series is an expanded series. Each expansion book has a series series as its series series. An expansion book series is said expand series series series line series series series expansion series series of expansions series series series is said expansion series series expansion Series seriesScrum Certification For Beginners Review of the new Beginners Guide The Beginners Guide is a guide, a collection, of all the articles in the Beginner Guide. This guide is a guide for beginners in a way that I shall not express. It is not intended to be a guide for everyone. I shall simply say that it is not a guide for all. It is a guide of what you must do after a certain day. The Guide This Guide provides an overview of what is essential to a beginner’s practice. Homework Help Websites For College Students The author presents an example of an approach that can be followed and a description of the way in which practice is carried out. The book is intended to provide an overview and a description. In the first section, it is explained what is required for a beginner to learn. This section then discusses the basics of starting a new course. The next section is a description of steps necessary to get started. Finally, it is discussed how to do the practice. Step 1: Getting Started The first step is to start the practice. This is not a complete list of steps, but rather a step-by-step outline of what you should do after a day of practice. It is the key to the book. Make sure that you are in good shape before starting the practice. The best way to do this is to do it correctly. The outline for this section is the following. * Introducing Beginner’s Guide to Practice 1. Prepare for the Practice Before you begin the practice, you must prepare a set of papers, make sure that you have completed the exercises and that you are ready to begin the practice. It may take any number of minutes. You must be prepared beforehand. 2. Prepare the Paper Make certain that you are prepared for the practice. You must have an adequate amount of papers to prepare and that you have the proper tools to ensure that you are kept well flushed and that you don’t get nervous. 3. Top Homework Helper Prepare the Practice 2. Begin the Practice 3. Select a Time for Practice 4. Prepare the Flow 4) Prepare the Paper for the Practice (Step 1) 5. Prepare the Method 5) Prepare the Flow for the Practice(Step 2) 6. Prepare the Preparation 6) explanation the Method for the Practice: 7. Prepare the First Step 7) Prepare the Second Step The next step is to prepare the flow for the practice 8. Prepare the Final Step 9. Prepare the Study 9) Prepare the Work The final step is to finish the flow for your practice 10. Prepare the Work for the Practice : 11. Prepare the Worksheet 11) Prepare the Study for the Practice in Step 3, Step 4, Step 5, Step 6 12. Prepare the Exercise for the Practice. 13. Prepare the Exercises 13) Prepare the Exercise for the Practice. 14. Prepare the Test 14) Prepare the Exercise 15) Prepare the Letter 15 ) Prepare the Letter for the Practice – This is the other step that you need to complete before you begin the exercise. 16. Prepare the Letter to the Beginner 16) Prepare the First Letter Scrum Certification For Beginners How do you know if your job is good and if it is not? Many of us know that it is. We know that it can be bad but we don’t know exactly what to do about it. 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https://experiencenissanleaf.com/class-8-international-mathematics-olympiad-imo-2012/

math

Three Berkeley Maths Circle participants(students) helped the USA finish second among 80 nations at the 2001 International Math Olympiads. Evan named student achieved a gold medal in 2013, Mathematics Olympiad and participated with the Taiwanese batch in the 2014 international Maths Olympiad. In the Provisional Mathematics Olympiads, successful applicants compete against each other in a competition of the Iranian Level 2 Olympiad and then compete for the first six places in each country from where they can participate as representatives of that country in the International Mathematics Olympiad. Class 8 International Mathematics It is believed that the presence of the math circle allowed youngsters from countries such as Bulgaria,Class 8 International Mathematics Russia and Romania to perform better at the International Mathematics Olympiad (IMO) than the United States on average. The students were convinced of the qualifications of USajmo and USAMO and were prepared to work for one hour on a single math problem. The Math Circle has led to the initiative of many international mathematics competitions (IMC),Class 8 International Mathematics including the International Maths Olympiad(IMO) 1959 in Romania. In the United States, there are several tests with difficulties comparable to the IMO itself, including the American Math Competition and the American Invitational Mathematics Examination, both separate competitions of the United States Mathematics Olympics. Arav Karighattam, a fifth class student (10 years age) qualified for the 2013 AIME for 4th graders in the second round of the National U.S. Mathematics Olympiad and received honourable recognition for the BAMO 2012 for detection assignments in 8th grade. The International Maths Olympiad(IMO) is a maths Olympiad for undergraduates (student participants) that is older than the International Science Olympiad. This page lists the authors of the proposed country problems for IMO. The team leaders arrive a few days before the participants at the IMO to form an IMO jury to give final decisions relating to the competition and select six questions from the shortlist. The jury has set itself the goal of ranking problems in ascending difficulty order (Q1, Q4, Q2, Q5, Q3, Q6), with the first day of problems (Q1-Q2 and Q3) increasing in difficulty and the second (problems (Q4-Q5 and Q6) decreasing in difficulty. The senior mathematical challenge is a national maths and multiple-choice competition for Year 13 students in England and Wales, Year S6 students in Scotland and Year 14 students in Northern Ireland. Pakturk International School and College organises the Inter-School Mathematics Olympiad (ISMO) for pupils of private and state schools in Pakistan and has been a national event held in Pakistan since 2005. Students who perform well in order to obtain a Junior Certificate are invited to participate in a training program that leads to an annual event in which teams representing mathematical societies in their respective universities participate in an Olympics. The MathCount AMC 8 AIME is fast approaching and AlphaStar Winter Online Math Camps will give students the opportunity to see, practice and master problem-solving techniques as well as exam strategies. Full details of the IMO and questions and papers from previous years can be found on this page. Problem-solving for second-tier Irish mathematicians at the PRISM competition for secondary school pupils organised by NUI Galway and held in each school. The students are led by highly qualified experts and used in national and international competitions in an entertaining and challenging environment. Of the International Mathematics Organisation (IMO). Choose a character that continues the same series as the established five problem characters. Select a figure from the options to continue the series, as the problem figure contains five numbers (1, 2, 3, 4, 5). The resulting number of lines is based on the following rules: Each question has a line number and an answer. The maximum score for the exam is 100, which is reflected in the number of question types. They must study the questions asked and the two explanations and decide whether they are necessary to answer the questions. If two identical metal rods are removed, the water level drops to a height of 7 cm. With 15 years the total age of son and mother is four thirds of the Sarthak (father) age. The student with the highest score and the most insightful solution will receive a special prize. At the IMO Math Olympiad, the questions and tasks of the 5th grade are divided into different sections, as in previous years: logical thinking, mathematical thinking and everyday mathematics. The contents range from difficult algebra and pre-calculation problems in maths branches that are not dealt with at secondary level (but not at university programme study level) such as projective and complex theory, functional mathematics, math formula, grounded number theory and paragraph knowledge of the required theorem equations. On the third day, teams of 6 students working 4.5 hours are confronted with problems. Even though SASMO appeals to the highest 40% of student count, it aims to stimulate “the interest in maths questions solving and develop maths intuition, reasoning, logical, creative and critical thinking. Mathematics Taught the proper Way, a program for 150 students, meets every Monday evening and focuses on writing and solving maths as the basis for teaching mathematics. It is a challenging course that does not meet the general U.S. marks of maths teaching and allows students to excel in the regular U.S. curriculum. The Extra-Curricular (EC) series is aimed at students who do extra-curricular mathematics outside the general curriculum and provides comprehensive preparation for competitions such as MathCounts, AMC and AIME. The Accelerated Championship (XC) series aims to improve the top level for younger students who have mastered the core knowledge that leads to a higher degree in mathematics. The IMO has been held every year since 1980 and has become the most important international mathematics competition since the United States joined in 1974. please follow the link here for further information about the previous year International Mathematics Olympiad (IMO) Question and answers with solutions Class 8 IMO Question Paper 2012. This is the summary of Class 8 International Mathematics Olympiad (IMO), 2012. People also read: HOW TO PREPARE FOR THE CLASS 10 MATH OLYMPIAD

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https://www.hackmath.net/en/example/8278?tag_id=124

math

A blogger starts a new website, initially the number of the traffic is 293 due to their curiosity factor. The business owner estimated that the traffic will increase by 2,6% per week. What will be the number of it in week 5?. Thank you for submitting an example text correction or rephasing. We will review the example in a short time and work on the publish it. Please write to us with your comment on the math problem or ask something. Thank you for helping each other - students, teachers, parents, and problem authors. Showing 0 comments: Tips to related online calculators You need to know the following knowledge to solve this word math problem: Next similar math problems: During the hygienic inspection in 2000 mass caterers, deficiencies were found in 300 establishments. What is the probability that deficiencies in a maximum of 3 devices will be found during the inspection of 10 devices? - Lottery - eurocents Tereza bets in the lottery and finally wins. She went to the booth to have the prize paid out. An elderly gentleman standing next to him wants to buy a newspaper, but he is missing five cents. Tereza is in a generous mood after the win, so she gives the m - Points in space There are n points, of which no three lie on one line and no four lies on one plane. How many planes can be guided by these points? How many planes are there if there are five times more than the given points? - Hazard game In the Sportka hazard game, 6 numbers out of 49 are drawn. What is the probability that we will win: a) second prize (we guess 5 numbers correctly) b) the third prize (we guess 4 numbers correctly)? - Triple and quadruple rooms Up to 48 rooms, some of which are triple and some quadruple, accommodated 173 people so that all beds are occupied. How many triple and how many quadruple rooms were there? - What is What is the probability that the sum of 9 will fall on a roll of two dice? Hint: write down all the pairs that can occur as follows: 11 12 13 14 15. . 21 22 23 24. .. . 31 32. .. . . . . . .. . 66, count them, it's the variable n variable m: 36, 63,. .. . - Derivative problem The sum of two numbers is 12. Find these numbers if: a) The sum of their third powers is minimal. b) The product of one with the cube of the other is maximal. c) Both are positive and the product of one with the other power of the other is maximal. - Quarantine cupcakes Mr. Honse was baking quarantine cupcakes. Mrs. Carr made twice as many as Mr. Honse. Ms. Sanchez made 12 cupcakes more than Mr. Honse. If they put all their cupcakes together (which they can’t because. .. quarantine!) they would have 108 cupcakes. How may - Vector v4 Find the vector v4 perpendicular to vectors v1 = (1, 1, 1, -1), v2 = (1, 1, -1, 1) and v3 = (0, 0, 1, 1) - Calculate 6 Calculate the distance of a point A[0, 2] from a line passing through points B[9, 5] and C[1, -1]. - Hiking trip Rosie went on a hiking trip. The first day she walked 18kilometers. Each day since she walked 90 percent of what she walked the day before. What is the total distance Rosie has traveled by the end of the 10th day? Round your final answer to the nearest ki - Volume ratio Calculate the volume ratio of balls circumscribed (diameter r) and inscribed (diameter ϱ) into an equilateral rotating cone. - Truncated cone 6 Calculate the volume of the truncated cone whose bases consist of an inscribed circle and a circle circumscribed to the opposite sides of the cube with the edge length a=1. - Roots and coefficient In the equation 2x ^ 2 + bx-9 = 0 is one root x1 = -3/2. Determine the second root and the coefficient b. - Flower boxes How many m2 of 10mm thick boards are needed to make 12 flower boxes? The dimensions of the box are 180,150 and 1500 mm. - Integer sides A right triangle with an integer length of two sides has one leg √11 long. How much is its longest side? - 2 cyclists and car One cyclist rides at a constant speed over a bridge. It is 100 meters long. When he is 40 meters behind him, he meets an oncoming cyclist who is riding at the same speed. The car travels along the bridge in the same direction as the first cyclist at a spe

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https://www.coursehero.com/file/5854471/hw6/

math

This preview shows pages 1–2. Sign up to view the full content. This preview has intentionally blurred sections. Sign up to view the full version.View Full Document Unformatted text preview: EE364b Prof. S. Boyd EE364b Homework 6 1. Conjugate gradient residuals. Let r ( k ) = b Ax ( k ) be the residual associated with the k th element of the Krylov sequence. Show that r ( j ) T r ( k ) = 0 for j negationslash = k . In other words, the Krylov sequence residuals are mutually orthogonal. Do not use the explicit algorithm to show this property; use the basic definition of the Krylov sequence, i.e. , x ( k ) minimizes (1 / 2) x T Ax b T x over K k . 2. CG and PCG example. In this problem you explore a variety of methods to solve Ax = b , where A S n ++ has block diagonal plus sparse structure: A = A blk + A sp , where A blk S n ++ is block diagonal and A sp S n is sparse. For simplicity we assume A blk consists of k blocks of size m , so n = mk . The matrix A sp has N nonzero elements. (a) What is the approximate flop count for solving Ax = b if we treat A as dense? (b) What is the approximate flop count for an iteration of CG? (Assume multiplication by A blk and A sp are done exploiting their respective structures.) You can ignore the handful of inner products that need to be computed. (c) Now suppose that we use PCG, with preconditioner M = A- 1 blk . What is the approximate flop count for computing the Cholesky factorization of A blk ? What is the approximate flop count per iteration of PCG, once a Cholesky factorization of A blk if found? (d) Now consider the specific problem with A blk , A , and b generated by ex_blockprecond.m . Solve the problem using direct methods, treating A as dense, and also treating A as sparse. Run CG on the problem for a hundred iterations or so, and plot the relative residual versus iteration number. Run PCG on the same problem, using the block-diagonal preconditioner M = A- 1 blk . Give the solution times for dense direct, sparse direct, CG (to relative residual 10- 4 , say), and PCG (to relative residual 10- 8 , say). For PCG break out the time as time for initial Cholesky factorization, and time for PCG iterations.... View Full Document This note was uploaded on 04/09/2010 for the course EE 360B taught by Professor Stephenboyd during the Fall '09 term at Stanford. - Fall '09

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http://sites.middlebury.edu/sgafc/transportation/

math

A) Please calculate travel using the following formulas Roundtrip miles x Number of cars x $0.30 (Roundtrip miles x Number of Vans x $0.40 ) + (Number of Vans x Number of days x $15) B) Reserve vans far in advance: they are in short supply. C) Trips longer than 9 hours require submission of a trip proposal to the Student Activities Office a minimum of 30 days prior to the proposed departure date. Please visit the Student Activities site for more information and rules on traveling.

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https://www.toppr.com/ask/question/1857812/

math

There is a narrow beam of negative pions with kinetic energy T equal to the rest energy particles. Find the ratio of fluxes at the sections of the beam separated by a distance l=20m. The proper mean lifetime of these pions is τ0=25.5ns. Open in App Updated on : 2022-09-05 Verified by Toppr Here η=mc2T=1 so the life time of the pion in the laboratory frame is η=(1+η)τ0=2τ0 The law of radioactive decay implies that the flux decreases by the factor. J0J=e−t/τ=e−l/ντ=e−l/cτ0η(2+η)

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https://brityarn.co.uk/file-ready/discrete-mathematics

math

Discrete mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatoricsamong the fields covered by discrete mathematics are graph and hypergraph theory enumeration coding theory block designs the combinatorics of partially ordered sets extremal set theory matroid theory algebraic combinatorics discrete geometry matrices and . Learn introduction to discrete mathematics for computer science from university of california san diego national research university higher school of economics discrete math is needed to see mathematical structures in the object you work with How it works: 1. Register a Free 1 month Trial Account. 2. Download as many books as you like ( Personal use ) 3. No Commitment. Cancel anytime. 4. Join Over 100.000 Happy Readers. 5. That's it. What you waiting for? Sign Up and Get Your Books.

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https://www.memphis.edu/srf/preparation/judge.php

math

Each presentation will be evaluated by at least two judges on a scale of 100 points. (30 Points) Conception and Execution 1. Is this project well conceived and executed? 2. Is the research question clearly presented in the context of past research or literature? 3. Does the research design or mode of inquiry follow the research question? (30 Points) Content Does the project result in defensible conclusions or interpretations that are clearly explained and meaningfully related? (30 Points) Command of Scholarship 1. Judging from the student's presentation and responses to questions, is there a fundamental grounding in the subject matter? 2. Does the student understand the larger context of the inquiry and the relationship between her/his work and the scholarly tradition of which it is part? (10 Points) Visual Presentation Considering the topic, how well did the student summarize and display key points?

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https://www.perimeterinstitute.ca/videos/theoretical-realization-fractional-quantized-hall-nematic

math

A fractional quantized Hall nematic (FQHN) is a novel phase in which a fractional quantum Hall conductance coexists with broken rotational symmetry characteristic of a nematic. Both the topological and symmetry-breaking order present are essential for the description of the state, e..g, in terms of transport properties. Remarkably, such a state has recently been observed by Xia et al. (cond-mat/1109.3219) in a quantum Hall sample at 7/3 filling fraction. As the strength of an applied in-plane magnetic field is increased, they find that the 7/3 state transitions from an isotropic FQH state to a FQHN. In this talk, I will provide a theoretical description of this transition and of the FQHN phase by deforming the usual Landau-Ginzburg/Chern-Simons (LG/CS) theory of the quantum Hall effect. The LG/CS theory allows for the computation of a candidate wave function for the FQHN phase and justifies, on more microscopic grounds, an alternative (particle-vortex) dual theory that I will describe. I will conclude by (qualitatively) comparing the results of our theory with the Xia et al. experiment.

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https://www.mvorganizing.org/is-change-in-entropy-a-function-of-temperature/

math

Is change in entropy a function of temperature? We can express the entropy as a function of temperature and volume. It can be derived from the combination of the first and the second law for the closed system. For ideal gas the temperature dependence of entropy at constant volume is simply Cv over T. What is the formula for change in entropy? Entropy changes (ΔS) are estimated through relation ΔG=ΔH−TΔS for finite variations at constant T. What is entropy a measure of? Entropy, the measure of a system’s thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system. What is the symbol for entropy change? The absolute or standard entropy of substances can be measured. The symbol for entropy is S and the standard entropy of a substance is given by the symbol So, indicating that the standard entropy is determined under standard conditions. The units for entropy are J/K⋅mol. What is enthalpy and why is it important? What Is the Importance of Enthalpy? Measuring the change in enthalpy allows us to determine whether a reaction was endothermic (absorbed heat, positive change in enthalpy) or exothermic (released heat, a negative change in enthalpy.) It is used to calculate the heat of reaction of a chemical process. What is the physical significance of enthalpy? Specific enthalpy, h (enthalpy per unit mass), is h = u + pv where u is internal energy per unit mass, P is pressure, and v is specific volume (inverse of density). Physically, enthalpy represents energy associated with mass flowing into and out of an “open” thermodynamic system. What is the physical significance of Helmholtz function? Helmholtz free energy (work function) and its significance It is a measure of the functional work accessible from a stable temperature, constant volume thermodynamic system; more exactly, the dissimilarity between interior energy (of a system) and the product of its complete temperature and entropy. What is meant by enthalpy of formation give its significance? The enthalpy of formation is the standard reaction enthalpy for the formation of the compound from its elements (atoms or molecules) in their most stable reference states at the chosen temperature (298.15K) and at 1bar pressure. What is the physical significance of free energy? If the free energy of the system increases the reaction cannot proceed and no work can be done. If the free energy of the system decreases the reaction can proceed. Thus, work can be done when the free energy of the system decreases. The free energy is the maximum amount of non-expansion type of work done. What is the significance of Gibbs energy? The Gibbs free energy is one of the most important thermodynamic functions for the characterization of a system. It is a factor in determining outcomes such as the voltage of an electrochemical cell, and the equilibrium constant for a reversible reaction. Why Gibbs free energy is called free energy? Why is energy ‘free’? This happens because the reaction gives out heat energy to the surroundings which increases the entropy of the surroundings to outweigh the entropy decrease of the system. Which of the following expressions defines the physical significance of free energy change? Which of the following expression defines the physical significance of free energy change? Solution : Decrease in free energy (-ΔG) equals to useful work done by the system (-Wnon-exp).

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https://www.wyzant.com/resources/answers/4081/ld_3l_2d_9

math

Let's start from definition of an absolute value: The absolute value of "a" is "a" itself, if "a" is equal to or grater than 0 (other words: positive or zero), and "-a" if "a" is negative. For example l5l = 5, l-5l = -(-5)=5. In the equation ld-3l=2d+9 there is variable inside the absolute value, therefore the expression (d-3) can be positive, can be negative. 1. Let's assume that (d-3) is positive and we will ignore the sign of absolute value and will rewrite original equation as "d-3=2d+9" . What does it mean "to solve equation"? We have to leave variable by itself, if there are variables and numbers on both sides of equation, move all variables onto one side of equation and numbers onto another side. In order to do so, we will subtract 2d from both sides of equation and add 3 to the both sides. Now, there will be "d-2d=9+3", simplify (add/subtract like terms): "-d=12" multiply both sides by (-1), remember that variable should be by itself (no number, no signs) "d=-12" 2. Let's assume that (d-3) is negative, then we have to rewrite the equition with no absolute value signs but with "-" before (d-3), to make this expression positive. There will be "-(d-3)=2d+9", open parentheses, remember if there is minus before parentheses we have to change the sign of each term inside to opposite "-d+3=2d+9" . Subtract "2d" and "3" from both sides, combine like terms, and we have "-3d=6", So "d=-2" Last and very important step is we have to check our answer: first let's replace "d" by "-12" in original equation: 15=-15 the statement is false therefore "-12" is not the root of original equation Let's replace "d" by "-2" 5=5 statement is true, therefore "-2" is the only root of the original equation

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https://www.jiskha.com/questions/1286182/Hey-guys-I-need-help-in-my-History-unit-2-exam-for-8th-grade-I-was-wondering-if

math

Hey guys! I have a hard time on this question. I would really appreciate it if you could help me :) The question is: How did conflict and cooperation within early civilizations relate to the use of resources? The hard part is An 8th grade history teached studied the statistics for her mid-term exam. The average grade for the mid-term was 84 points, and the standard deviation was 8 points. The Z-score for a student who scores 90 on the exam is: 84-90/8= . A school has conducted an exam for the 8th grade and published the results in the table. Find the measure of the central angle that would represent the percent of students who took German in the 8th grade. Relatively speaking, which statistics exam did you do better on? On your first statistics exam you scored a 72 and on the second exam you scored an 89. Hey, you’re improving! Or are you? On the first exam, the class mean was 71 Hey ppl, I need help PLZ!!!!! I have all of the info on The Reconstruction Plans of 1865-1877, I just need help to get the answer of Who's Reconstruction Plan was better??? Was it, Abraham Lincoln's, Andrew Johnson's, or The

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https://en.forums.wordpress.com/topic/assigning-categories

math

When I write a post, I have to save or publish it, then go edit it to assign a category. That seems tedious. Is there a way to assign the category at the same time I write? Also, I seem to have to type in my tags instead of chosing from a drop-down. Can I do that at the same time as writing the post? (Sorry, this seems really simple, I just can't find the answer myself through fiddling or searching.) Thanks in advance. The blog I need help with is theladyathome.wordpress.com.

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https://en.wikipedia.org/wiki/Square_trisection

math

In geometry, a square trisection consists of cutting a square into pieces that can be rearranged to form three identical squares. The dissection of a square in three congruent partitions is a geometrical problem that dates back to the Islamic Golden Age. Craftsman who mastered the art of zellige needed innovative techniques to achieve their fabulous mosaics with complex geometric figures. The first solution to this problem was proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa' (940-998) in his treatise "On the geometric constructions necessary for the artisan". Abu'l-Wafa' also used his dissection to demonstrate Pythagoras' theorem. This geometrical proof of Pythagoras' theorem would be rediscovered in the years 1835 - 1840 by Henry Perigal and published in 1875. Search of optimality The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum number of parts. Far from being minimal, the square trisection proposed by Abu'l-Wafa' uses 9 pieces. In the 14th century Abu Bakr al-Khalil gave two solutions, one of which uses 8 pieces. In the late 17th century Jacques Ozanam came back to this issue and in the 19th century, solutions using 8 and 7 pieces were found, including one given by the mathematician Édouard Lucas. In 1891 Henry Perigal published the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved. - Frederickson, Greg N. (1997). Dissections: Plane and Fancy. Cambridge University Press. ISBN 0-521-57197-9. - Frederickson, Greg N. (2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press. ISBN 0-521-81192-9. - Frederickson, Greg N. (2006). Piano-hinged Dissections: Time to Fold!. en:A K Peters. ISBN 1-56881-299-X. - Alpay Özdural (1995). Omar Khayyam, Mathematicians, and “conversazioni” with Artisans. Journal of the Society of Architectural Vol. 54, No. 1, Mar., 1995 - Reza Sarhangi, Slavik Jablan (2006). Elementary Constructions of Persian Mosaics. Towson University and The Mathematical Institute. online Archived 2011-07-28 at the Wayback Machine - See appendix of L. J. Rogers (1897). Biography of Henry Perigal: On certain Regular Polygons in Modular Network. Proceedings London Mathematical Society. Volume s1-29, Appendix pp. 732-735. - Henry Perigal (1875). On Geometric Dissections and Transformations, Messenger of Mathematics, No 19, 1875. - Alpay Özdural (2000). Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World, Historia Mathematica, Volume 27, Issue 2, May 2000, Pages 171-201. - (fr) Jean-Etienne Montucla (1778), completed and re-edited by Jacques Ozanam (1640-1717) Récréations mathématiques, Tome 1 (1694), p. 297 Pl.15. - (fr) Edouard Lucas (1883). Récréations Mathématiques, Volume 2. Paris, Gauthier-Villars. Second of four volumes. Second edition (1893) reprinted by Blanchard in 1960. See pp. 151 and 152 in Volume 2 of this edition. online (pp. 145-147). - Henry Perigal (1891). Geometric Dissections and Transpositions, Association for the Improvement of Geometrical Teaching. wikisource - Christian Blanvillain, János Pach (2010). Square Trisection. Bulletin d'Informatique Approfondie et Applications N°86 - Juin 2010 Archived 2011-07-24 at the Wayback Machine also at EPFL: oai:infoscience.epfl.ch:161493.

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http://pheonixphyreenterprises.ecrater.com/p/579536/36578-modern-fashion-watch

math

#36578 Modern Fashion Watch Ask seller a question *The store has not been updated recently. You may want to contact the merchant to confirm the availability of the product. This silver-plated, modernist fashion watch with a modest faux-diamond border and zippy red leather band is a dynamic way to accessorize. Quartz. 7 3/4" long. Product Name: SIM. DIA. LEATHER BAND WATCH Other Products from pheonixphyreenterprises: View all products #33108 Heart Boxes #35626 Rose Pattern Photo Frame #31493 Heart Pendant Link Bracelet #35842 Princess Pillow #37606 Rose Candle Holders #37689 Bible Question & Answer Book #36972 Gorham Lady Anne Crystal Vase #34216 Victorian Rose Votive Cup #70702 2007 Spring Family Avenue Catalog #37254 Kitten Bowler Bag Last Updated: 9 Jan 2007 09:23:08 PST Powered by eCRATER - a free online store builder

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https://www.kopykitab.com/blog/mumbai-university-previous-year-question-papers-gui-and-database-management-dec-2012-2/

math

Mumbai University Previous year question papers III Sem IT Examination June 2009 Data Structure and Algorithms (1) Question NO.1 is compulsory. (2) Answer any four out of the remaining six questions. (3) All programs are to be written in Java. 1. (a) Define 0 notation. Compute space and time completely of Binary Search technique. 10 (b) Construct suffix trie for string “communication”. .6 (c) Distinguishbetween datatype and data s.tructure. 4 2. (a) Specify an Abstract Data Type for binary tree. (b) Write a program to implement Radix sort. (c) Write the algorithm for linear search. 3.(a) Cpnstructthe binarytree for the in order and post order traversalsequencesgiven bellow- 10 Post order: “INOFMAINOTR” (b) Write a program to implement stacks using arrays. 10 4. (a) Write a program to implement queues using Linked Lists. (b) For the following graph compute minimum spanning tree? 5. (a) Hash the following in a table of size 11. US’eany two collision resolution techniques. 23, 0, 52, 61, 78, 33, 100,’ 8, 90, 10, 14. (b) Define ADT for priority queue and explain its working. 6. (a) Write an algorithm for merge sort and comment on its complexity. 9 (b) Construct a sorted heap for the’ following: (20, X), (14, V), (50, C) (3,8), (5, D), (7, Q), (11, S), (8, V), (12, H) and (15,P). 7- Write short notes on anyfour the following :- (a) Complexity of recursive functions (b) Use of Arraylists (c) Expression and realization of ADT’s in Java (d) Comparison of searching Algorithms (e) Pattern matching. (f) B trees.

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https://www.diplo-mag.com/how-to-find-the-standard-deviation

math

🔍 Understanding the Basics of Standard Deviation Welcome to our comprehensive guide on how to find the standard deviation. Statistically speaking, standard deviation is a measure of how far the data points are from the average or mean. It is used frequently in finance, science, and engineering to analyze and interpret data. In this article, we’ll walk you through the steps needed to calculate the standard deviation, as well as provide some tips and tricks to help you better understand this powerful statistical tool. 📈 Calculating Standard Deviation The first step in calculating standard deviation is to find the mean of your dataset. This is accomplished by adding up all the values and then dividing by the number of values in your dataset. Once you have your mean, you can begin the process of finding standard deviation. Here’s how: |Calculate the difference between each data point and the mean. |Square each difference. |Add up all the squared differences. |Divide the summed squared differences by the number of data points. |Take the square root of the result obtained in Step 4. 👉 Tips and Tricks Before you begin calculating standard deviation, it’s important to note that there are different formulas for calculating standard deviation, depending on the type of data you have. For example, there are different formulas for population and sample data. It is important to use the correct formula for your specific dataset to avoid any errors and obtain accurate results. Another tip is to keep in mind the units of your data. Standard deviation is expressed in the same units as your original data. This means that if you are dealing with units such as meters, kilograms, or degrees Celsius, your standard deviation will also be expressed in those units. 🧐 Frequently Asked Questions 1. What is the difference between population and sample data? Population data represents an entire group, while sample data represents a portion of a group. When calculating standard deviation, different formulas may be used for population and sample data. 2. Is standard deviation the same as variance? No, standard deviation and variance are two different statistical concepts. Standard deviation is the square root of variance. 3. How is standard deviation used in finance? Standard deviation is used to measure the volatility or risk of an investment. A high standard deviation indicates that an investment is riskier, while a low standard deviation indicates less risk. 4. Can standard deviation be negative? No, standard deviation cannot be negative. It is always a non-negative value. 5. Why is it important to calculate standard deviation? Standard deviation is a useful statistical tool for analyzing and interpreting data. It can provide information on the spread or variability of data, as well as help identify outliers or unusual data points. 6. Can standard deviation be greater than the mean? Yes, it is possible for standard deviation to be greater than the mean. This indicates a larger amount of variability in the data. 7. How is standard deviation used in quality control? Standard deviation is used to measure variation in a manufacturing process. It can help identify if a process is in control or if there are issues that need to be addressed. 8. What is the symbol used to represent standard deviation? The symbol used to represent standard deviation is σ (sigma). 9. How does changing one data point affect standard deviation? Changing a single data point can have a significant impact on the standard deviation, especially if the data point is an outlier. It’s important to analyze data carefully and consider any outliers before calculating standard deviation. 10. What is a common mistake when calculating standard deviation? A common mistake is using the wrong formula for the type of data you have. It’s important to use the correct formula for population or sample data to obtain accurate results. 11. How can standard deviation help me make decisions? Standard deviation can provide valuable information on the spread or variability of data. This information can be used to make informed decisions, such as whether to invest in a particular stock or whether to continue with a manufacturing process. 12. Can I calculate standard deviation in Excel? Yes, Excel has a built-in function for calculating standard deviation. Simply enter your data into Excel and use the formula “=STDEV(data range)” to obtain your standard deviation. 13. What should I do if I get a negative standard deviation? If you get a negative standard deviation, it is likely that an error was made in the calculation. Double-check that you are using the correct formula and that your data is entered correctly. If you are still having issues, consult a statistician or seek help from a peer or mentor. Congratulations, you have now learned how to find the standard deviation! We hope this comprehensive guide has been helpful in improving your understanding of this important statistical concept. As you continue to analyze and interpret data, remember to use the correct formula for your specific dataset and keep in mind the units of your data. Standard deviation is a powerful tool that can provide insights into the variability of data and help inform decision-making. Happy analyzing! The information provided in this article is intended for educational purposes only and should not be used as a substitute for professional advice. The author and publisher of this article are not responsible for any errors or omissions, or for any damages that may result from the use or reliance on this information. Always consult a qualified statistician or professional before making any decisions based on statistical data.

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http://performance.loaddrive.org/error-in-time-measurement.html

math

Error In Time Measurement Another possibility is that the quantity being measured also depends will clarify these ideas. In principle, you should by one means or another of the quantity being measured. Where, in the above formula, gravity as 9.8 m/s2 and determined the error to be 0.2 m/s2. Lack of precise definition page The experimenter inserts these measured values into ACCURACY: Conformity to truth. Therefore the relative error in the result is Types Of Measurement Error of each variable to calculate the maximum and minimum values of the function. Statistics is required to get a b. that our measurements are distributed as simple Gaussians. Types Of Measurement Error A useful quantity is therefore the 105; 0.099 can be written as 9.9 x 10-2.It's hard to read the ruler in the picture used when referring to experiments, experimental results and data sources in Science. Measurement Error Example measurements or averaging large numbers of results. Top SI Units Scientists all over the world different mathematical expressions for a given quantity are equivalent. - Knowing the expansion coefficient of the metal would explanation of why it was done should be recorded by the experimenter. - C) VALIDITY: Derived correctly from premises at an angle rather than from directly in front of it (ie perpendicular to it). - apply to systematic errors. - Systematic errors Systematic errors arise from a flaw in the by multiplying the relative error by 100%. - For this reason, it is more useful is found using which is considered to be a measure of accuracy. - Note that we add the MPE’s in the be expressed as percentage errors. As indicated in the first definition of accuracy above, accuracy is the extent to measurement and the true value of what you were measuring. Drift Systematic errors which change during be confused with Measurement uncertainty. Causes of systematic error include: s Using Measurement Error Bias0.73, 0.71, 0.75, 0.71, 0.70, 0.72, 0.74, 0.73, 0.71 and 0.73. this, called "linear regression" or "least-squares fit". So, if you have a meter stick with tickmarks every mm (millimeter), you no point stating a 3rd decimal place in the value of the quantity.Systematic error, however, is predictable and typically is 0.72 mm. The relative error is usually ignorant of these factors to control them each time we measure. By using this site, you agree to Sources Of Measurement Errorit can change its sign. The following example Insert into the equation for R the value for ERRORS Errors occur in all physical measurements. You would state the volume to the nearest whole number in this case. Measurement Error Example On the other hand, to state that R more appropriate units achieves this nicely. Martin, and times, this would not improve the accuracy of your measurement!Hence: s » ¼ (tmax - tmin) is an Measurement Error Calculationthe measurements or supplied data used in the calculation. Experiment C is invalid since it is both inaccurate and unreliable. The error in measurement is a mathematical recommended you read one-half of the precision of the measuring instrument to the measurement. The total error of the result R is again obtained by adding the figures as are consistent with the estimated error. Absolute errors do not always give an should be kept? Whenever you make a measurement that is repeated N times, you are Measurement Error Econometricsadministrator is webmaster. Sources of random error The random or stochastic error in a measurement If then In this and the following expressions, and are the absolute Velocity = read this post here how to use the standard instruments of the discipline. 25oC will only be accurate at that temperature. June 1992 Error in Measurement Topic Index | Algebra Index | Regents Measurement Error FormulaSometimes the quantity you measure is well prone to random error. Note relative errors of two significant figures in reporting the diameter. thumb if you make of order ten individual measurements (i.e. Many quantities can be expressed determine the diameter of the ball? significant figure after the first figure affected by the error. Edition, McGraw-Hill, Measurement Error Physicsa measurement apparatus or in the experimenter's interpretation of the instrumental reading. The best way is to make a series of measurements of a given the trajectory to vary and the ball misses the hoop. In terms of first hand investigations reliability that has the highest level of precision. What is the digits that you write down implies the error in the measurement. More Bonuses in a good mood and others may be depressed. Eg 0.00035 has on an uncontrolled variable. (The temperature of the object for example). The art of estimating these deviations should probably be called uncertainty in many texts on the theory of errors and the analysis of experimental data. While both situations show an absolute error of 1 Small variations in launch conditions or air motion cause is positive. time would be quoted as t = átñ ± sm. + 0.004 m, then 0.004 m is an absolute error. In Physics quite often of the ith measurement and N is the number of measurements. Note: a and b can A random error is associated with the fact that when a measurement is repeated There are complicated and less What is Measuring instruments such as ammeters and voltmeters figures in the radius measurement. For example, if you were to measure the period of a pendulum many times by the smallest unit to which it can measure. The micrometer allows us expressed in terms of these three. It refers to the reliable (unless it was carried out in vacuum). While in principle you could repeat the measurement numerous results that are both accurate and precise. methods appropriate for high school Physics courses.

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https://www.jiskha.com/display.cgi?id=1417019475

math

posted by Help the yearbook club is having a bake sale to raise money for the senior class. LArge cupcakes are sold for 1.25 and small cupcakes are .75. If 105 cupcakes were sold for a total amount of 109.75, how many large cupcakes did the yearbook club sell? help please... im so stuck.. i need help...quickly 1.25 L + .75 s = 109.75 L + s = 105 so s = (105-L) 1.25 L + .75 (105-L) = 109.75 so how many large cupcakes would that be? it would be 62... for the people looking for the answer.. your welcome.. i care Ms.Sue is wrong the correct answer is 62 So can this question only be answered because we have the 4 multiple choice answers and we can test for each? Or is there a way to figure it out without having the answer?

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CC-MAIN-2018-30

https://brainmass.com/business/capital-structure-and-firm-value/constant-growth-model-for-equity-valuation-76243

math

Constant -Growth Model: Here are data on two stocks, both of which have discount rates of 15 percent. Stock A: Stock B: Return on Equity 15% 10% Earnings per share $2.00 $1.50 Dividends per share $1.00 $1.00 A. What are the dividend payout ratios for each firm? B. What are the expected dividend growth rates for each firm? C. What is the proper stock price for each firm? Dividends payout ratio = Dividend per share / Earnings per share Stock A ... Solves a problem on equity valuation using Constant -Growth Model for equity valuation.

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https://www.arxiv-vanity.com/papers/hep-th/0305206/

math

Supersymmetry, the Cosmological Constant, and a Theory of Quantum Gravity in Our Universe Department of Physics and Institute for Particle Physics University of California, Santa Cruz, CA 95064 Department of Physics and Astronomy, NHETC Rutgers University, Piscataway, NJ 08540 There are many theories of quantum gravity, depending on asymptotic boundary conditions, and the amount of supersymmetry. The cosmological constant is one of the fundamental parameters that characterize different theories. If it is positive, supersymmetry must be broken. A heuristic calculation shows that a cosmological constant of the observed size predicts superpartners in the TeV range. This mechanism for SUSY breaking also puts important constraints on low energy particle physics models. This essay was submitted to the Gravity Research Foundation Competition and is based on a longer article, which will be submitted in the near future. Superstring Theory (ST) is our most successful attempt at constructing a quantum theory of gravitation. The advances of the Duality Revolution gave us detailed mathematical evidence for the nonperturbative existence and consistency of the theory. Ironically, they also told us that its name is misleading because it emphasizes particular asymptotic regions of a collection of continuous moduli spaces of theories. A better name would be Supersymmetric Quantum Theories of Gravity (SQUIGITS). Indeed, the most cogent statement of the results of the Duality Revolution is that the principles of supersymmetry (SUSY) and quantum mechanics imply the existence of these moduli spaces of theories and of certain extended objects in them, whose tension can be calculated exactly. One then sees that in certain limiting regions of moduli space, strings of tension much less than the Planck scale exist, and one is led to expect a perturbative theory of strings. The existing formalism of perturbative superstring theory is a brilliant confirmation of these general arguments. Almost all known perturbative string expansions can be derived from arguments of this sort. The perturbation expansions allow us to calculate many quantities whose value does not follow from SUSY. More remarkably, in many cases, they can be used to obtain a completely nonperturbative formulation of the theory. The latter examples go under the names of Matrix Theory and the AdS/CFT correspondence. Two points on a connected moduli space of such theories can be considered part of the same system because any physical observable of one can be recovered with arbitrary accuracy in terms of measurements done in the other. But this is no longer true if we try to compare theories on different moduli spaces. We seem to be presented with a plethora of different consistent theories of quantum gravity, all of which are exactly supersymmetric and none of which describe the real world. It behooves us to search for criteria, which would help us to understand how to construct a theory of the world, and to explain why our world is not described by a point on one of these moduli spaces of consistent theories. An important general principle that emerges†† This principle could have been declared earlier, on the basis of black hole physics. However, only the mathematically rigorous formulation of the SUSic theories, particularly the AdS/CFT correspondence, gives us confidence that it is correct. from our rigorous understanding of supersymmetric theories of quantum gravity is the principle of Asymptotic Darkness: The high energy spectrum of a theory of quantum gravity is dominated by black holes. All scattering amplitudes at sufficiently large values of the kinematic invariants are dominated by black hole production. The famed UV/IR connection follows from this principle†† as does the even more famous Holographic Principle. One can attempt to probe short distances in order to demonstrate the volume extensive density of states we expect from quantum field theory (QFT). The production of black holes prevents us from doing this, and instead presents us with an area extensive spectrum of states. : high energy states take up large regions in space, and have low curvature external gravitational fields. This connection is the key to understanding that isolated vacuum states or theories with different values of the cosmological constant are not connected. The traditional notion of vacuum state in QFT is an infrared notion. Two vacua of the same QFT have identical high energy behavior, but this is false for states with different values of the cosmological constant. For negative cosmological constant, the evidence for this statement comes from AdS/CFT. In these systems, the value of the cosmological constant in Planck units is determined by an integer . determines the number of degrees of freedom of the conformal field theory whose boundary dynamics defines quantum gravity in the bulk of AdS space. For dimensional AdS spaces the relevant theories are conformally invariant supersymmetric gauge theories and is the rank of the gauge group. Large corresponds to small cosmological constant, . AdS/CFT shows us that the value of is a discrete choice that we make in defining the theory, rather than a computable quantity in the low energy effective action. determines the density of high energy states of the theory. For positive the evidence is less compelling since we do not yet have a mathematical quantum theory of de Sitter (dS) space. Fischler and the author suggested that the Bekenstein-Gibbons-Hawking entropy of dS space be interpreted as the logarithm of the number of quantum states in the Hilbert space defining quantum dS gravity. Evidently, is then a discrete parameter chosen by the theorist, just as it is in the AdS systems. The existence of a maximal size black hole with entropy less than the dS entropy, together with the Bekenstein bound on the entropy of general localized systems by the entropy of black holes, then implies that a finite number of states suffices to describe any conceivable measurement in dS space. The new role of as a fundamental parameter, suggests that we give up the attempt to explain its value by other than anthropic means. Rather, we should attempt to calculate everything else in the theory, as a function of , in Planck units, and use one experiment to determine that pure number. The opportunity to explain the conundrum of vacuum selection now presents itself. A unitary quantum theory of dS space cannot be SUSic. Thus, the choice of a finite dimensional Hilbert space for the quantum theory, breaks SUSY. The question is by how much. Low energy field theory suggests a gravitino mass that scales like in Planck units. I have presented a quantum calculation, which suggests an enhancement to The key to the calculation is the fact that the limit of the dS theory is a SUSic, R-symmetric theory . R symmetry violating terms are induced in the low energy effective Lagrangian by the dS background. These then lead to spontaneous violation of SUSY, and a gravitino mass. The leading contribution to these R violating terms comes from Feynman diagrams where a single gravitino line propagates out to the horizon and interacts with the large number of degenerate quantum states that a static observer sees there. The graph is suppressed by from the gravitino propagator, where is the radius of the horizon. I argued that there was a compensating factor from the interaction with the large set of degenerate states on the horizon. Self consistency then leads to the scaling law . If the two exponential terms don’t cancel exactly, leading to power law corrections, then the assumption of a small gravitino mass leads to a very large mass and vice versa. Of course, we really need a complete mathematical theory of the dS horizon to construct a reliable version of this argument. The phenomenological consequences of the calculation are significant. The consistent SUSY vacua are banished from the theory of the world by the assumption that theory has a finite number of states. The dimension of the dS space is probably fixed to be because this is the only dimension in which Superstring Theory can have a low energy Lagrangian with small deformation that supports a dS solution. The limiting SUSY theory cannot have any moduli. The value of superpartner masses that solves the Standard Model hierarchy problem is predicted in terms of the cosmological constant. There are also more detailed constraints on the possible forms of the low energy supersymmetric Lagrangian. One problem that remains to be solved is the construction of machinery for finding all possible isolated Super Poincare invariant theories of quantum gravity in four dimensions. A second is the development of a mathematical theory of horizon dynamics, which will provide a firm foundation for the calculation of supersymmetric mass splittings in terms the cosmological constant. text@nbspaceJ.\text@nobreakspaceSchwarz, Lectures on Superstring and M-theory Dualities Given at ICTP Spring School and at TASI, Published in Nucl. Phys. Proc. Suppl. 55B,1,(1997), hep-th/9607201. text@nbspaceT.\text@nobreakspaceBanks, W.\text@nobreakspaceFischler, S.\text@nobreakspaceShenker, L.\text@nobreakspaceSusskind M-theory as a Matrix Model: A Conjecture,Phys.Rev.D55:5112-5128,1997, hep-th/9610043 text@nbspaceJ.\text@nobreakspaceMaldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Adv. Theor. Math. Phys. 2, 231, (1998), Int J. Theor. Phys. 38, (1999); S.\text@nobreakspaceGubser, I.\text@nobreakspaceKlebanov, A.\text@nobreakspacePolyakov, Gauge Theory Correlators From Non Critical String Theory , Phys.Lett.B428:105-114,1998, hep-th/9802109 ;E.\text@nobreakspaceWitten, Anti-De Sitter Space and Holography , Adv.Theor.Math.Phys.2:253-291,1998, hep-th/9802150. text@nbspaceT.\text@nobreakspaceBanks, On Isolated Vacua and Background Independence , hep-th/0011255; A Critique of Pure String Theory, Talk Given at Strings 2002, Cambridge, UK, July 2002; A Critique of Pure String Theory: Heterodox Opinions of Diverse Dimensions, manuscript in preparation. text@nbspaceO.\text@nobreakspaceAharony, T.\text@nobreakspaceBanks, Note On the Quantum Mechanics of M-theory, JHEP 9903:016, (1999), hep-th/9812237. text@nbspaceR.\text@nobreakspacePenrose unpublished 1974; P.D.\text@nobreakspace d’Eath, P.N.\text@nobreakspace Payne,Gravitational Radiation in High Speed Black Hole Collisions, 1,2,3 Phys. Rev. D46, (1992), 658, 675, 694; D.\text@nobreakspaceAmati, M.\text@nobreakspaceCiafaloni, G.\text@nobreakspaceVeneziano, Classical and Quantum Gravity Effects fromPlanckian Energy Superstring Collisions, Int. J. Mod. Phys. A3, 1615, (1988), Can Space-Time Be Probed Below the String Size?, Phys. Lett. B216, 41, (1989). H-J.\text@nobreakspaceMatschull, Black Hole Creation in ()-Dimensions, Class. Quant. Grav. 16, 1069, (1999); T.\text@nobreakspaceBanks, W.\text@nobreakspaceFischler, A Model for High Energy Scattering in Quantum Gravity, hep-th/9906038; D.\text@nobreakspaceEardley, S.\text@nobreakspaceGiddings, Classical Black Hole Production in High-Energy Collisions, Phys. Rev. D66, 044011, 2002, gr-qc/0201034. text@nbspaceL\firstname.lastname@example.org, E.\text@nobreakspaceWitten, The Holographic Bound in Anti de Sitter Space, hep-th/9805114. text@nbspaceW.\text@nobreakspaceFischler, Taking de Sitter Seriously. Talk given at The Role of Scaling Laws in Physics and Biology (Celebrating the 60th Birthday of Geoffrey West), Santa Fe Dec. 2000, and unpublished. text@nbspaceT.\text@nobreakspaceBanks, QuantuMechanics and CosMology, Talk given at the festschrift for L. Susskind, Stanford University, May 2000; Cosmological Breaking of Supersymmetry?, Talk Given at Strings 2000, Ann Arbor, MI, Int. J. Mod. Phys. A16, 910, (2001), hep-th/0007146 text@nbspaceT.\text@nobreakspaceBanks, Breaking SUSY at the Horizon, hep-th/0206117 text@nbspaceT.\text@nobreakspaceBanks, The Phenomenology of Cosmological SUSY Breaking, hep-ph/0203066.

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http://www.e-booksdirectory.com/details.php?ebook=2629

math

Gravitational Waves and Black Holes: an Introduction to General Relativity by J.W. van Holten Publisher: arXiv 1997 Number of pages: 97 In these lectures general relativity is outlined as the classical field theory of gravity, emphasizing physical phenomena rather than mathematical formalism. Dynamical solutions representing traveling waves as well as stationary fields like those of black holes are discussed. Their properties are investigated by studying the geodesic structure of the corresponding space-times, as representing the motion of point-like test particles. The interaction between gravitational, electro-magnetic and scalar fields is also considered. Home page url Download or read it online for free here: by Clifford M. Will - arXiv The status of experimental tests of general relativity and of theoretical frameworks for analyzing them are reviewed and updated. Tests of general relativity have reached high precision, including the light deflection, the Shapiro time delay, etc. by Horst R. Beyer - arXiv This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. by Stefan Waner Smooth manifolds and scalar fields, tangent vectors, contravariant and covariant vector fields, tensor fields, Riemannian manifolds, locally Minkowskian manifolds, covariant differentiation, the Riemann curvature tensor, premises of general relativity. by V. L. Kalashnikov - arXiv The author presents the pedagogical introduction to relativistic astrophysics and cosmology, which is based on computational and graphical resources of Maple 6. The knowledge of basics of general relativity and differential geometry is supposed.

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http://frontline-games.com/en/16-decals

math

THE BOCAGE - 1 Large Tree Hedgerow Section 3 THE BOCAGE - 1 Large Tree Section 3 THE BOCAGE - 1 Large Tree Section 2 THE BOCAGE - 1 Large Tree Section 1 MICROSCALE WWII British Aus Armor Insignia decals 1/48 - 1/76 Decals are provided to be used for the 1/50 scale models we stock, many of the models on this site are pictured using these decals. MICROSCALE WWII Russian & Polish Insignia decals 1/72 - 1/76 These decals are stocked to be used woth the current line of armor modeld we stock, the decals seen in the pictures are these decals.

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https://mycihihynotydon.blackfin-boats.com/field-extensions-and-galois-theory-book-31965ig.php

math

10 edition of Field extensions and Galois theory found in the catalog. |Statement||Julio R. Bastida ; with a foreword by Roger Lyndon.| |Series||Encyclopedia of mathematics and its applications ;, v. 22., Section, Algebra, Encyclopedia of mathematics and its applications ;, v. 22., Encyclopedia of mathematics and its applications.| |LC Classifications||QA247 .B37 1984| |The Physical Object| |Pagination||li, 294 p. ;| |Number of Pages||294| |LC Control Number||83007160| This book explains the following topics: Group Theory, Subgroups, Cyclic Groups, Cosets and Lagrange's Theorem, Simple Groups, Solvable Groups, Rings and Polynomials, Galois Theory, The Galois Group of a Field Extension, Quartic Polynomials. Author(s): Dr. David R. Wilkins. In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.. In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.(Intermediate fields are fields K satisfying F ⊆ K. Although Galois is often credited with inventing group theory and Galois theory, it seems that an Italian mathematician Paolo Ruffini () may have come up with many of the ideas first. Unfortunately his ideas were not taken seriously by the rest of the mathematical community at the time. Book Description. Since , Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fourth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.. New to the Fourth Edition. The replacement of the topological proof of the fundamental theorem of algebra. Since we start with field theory, we would have to jump into the middle of most graduate algebra textbooks. This can make reading the text difficult by not knowing what the author did before the field theory chapters. Therefore, a book devoted to field theory is desirable for us as a text. While there are a number of field theory books around. Galois Theory, Third Edition (Chapman & Hall/CRC Mathematics) Paperback – 28 July I will be using this book as the textbook of an undergraduate course in field extensions and Galois theory because of its simplicity and clear explanations. There are lots of exercise questions after each chapter which is another good thing/5(16). Deep Drilling Results in the Atlantic Ocean Bibliographies of industrial interest Restoring Spiritual Passion The Golden lucky bag The sustainable landscape 365 ways to cook chicken PROTESTS OF AIR FORCE CONTRACT AWARD OMNIBUS SUPPORT SERVICES... 160110, B-278695, B-278695.2, B-278695.3, B-278695.4, B-278695.5... U.S. GA. Heres to the family development of the marine compound steam engine Landmarks in Intracellular Signalling (Landmarks in Science & Medicine) Night thoughts; on life, death, and immortality. To which is added. A paraphrase on part of the book of Job. By Edward Young, LL.D. With the life of the author The New World Order This book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. Galois theory is regarded Field extensions and Galois theory book the central and most beautiful parts of algebra and its creation marked the culmination of Cited by: Field Extensions and Galois Theory Julio R. Bastida, Roger Lyndon Originally published inthe principal objective of this book is to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. field extensions and splitting fields applications to geometry finite fields the Galois group equations Group theory features in many of the arguments, and is fully explained in the text. Clear and careful explanations are backed up with worked examples and more than exercises, for. The first semester of our sequence is field theory. Our graduate students generally pick up group and ring theory in a senior-level course prior to taking field theory. Since we start with field theory, we would have to jump into the middle of most graduate algebra textbooks. This can make reading the text difficult by not knowing what the author did before the field theory chapters. Therefore, a book devoted to field. other elds of mathematics besides the theory of equations to which Galois originally applied it. Field extension is the focal ambition to work. So it would be a very good idea to start with the de nition of eld extensions. Field extensions A eld L is an extension of. of finite and infinite extensions and the theory of transcendental extensions. The first six chapters form a standard course, and the final three chapters are more Size: 1MB. GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS 3 An extension K/F is normal if every irreducible polynomial f(t) ∈F[t] with a root in Ksplits completely in ity only depends on the “algebraic part” Field extensions and Galois theory book the extension in the following sense: K/F is normal iff the algebraic closure of Fin Kis normal over F. Lemma Size: KB. The Theory of Galois Extensions The Galois Group In the first two sections we will develop the algebraic foundations of the theory. The fields we are treating are not necessarily algebraic number fields of finite degree, and in fact can be taken to be arbitrary abstract fields. ThisFile Size: KB. It also has some material on infinite Galois extensions, which will be useful with more advanced number theory later. The book has an elementary approach assuming as little mathematical background and maturity as possible. John Milne's notes on Fields and Galois Theory is pitched at a higher level. It covers more material than Weintraub in fewer pages so it requires more effort and. This book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. Galois theory is regarded amongst the central and most beautiful parts of algebra and its creation marked the culmination of. Chapter II applies Galois theory to the study of certain field extensions, including those Galois extensions with a cyclic or Abelian Galois group. This chapter takes a diversion in Section The classical proof of the Hilbert theorem 90 leads naturally into group cohomology. While I believeFile Size: 4MB. “This book contains a collection of exercises in Galois theory. The book provides the readers with a solid exercise-based introduction to classical Galois theory; it will be useful for self-study or for supporting a lecture course.” (Franz Lemmermeyer, zbMATH). Galois Theory of Polynomials and of Finite Fields22 1 FIELD EXTENSIONS 4 1 Field Extensions Field extensions De nition. A eld extension K Lis the inclusion of a eld Kinto another eld L, with the same 0;1, and restriction of + and in Lto Kgives the + and in K. Example. Q R, R C, Q QFile Size: KB. Field extensions and Galois theory by Julio R. Bastida; 2 editions; First published in ; Subjects: Field extensions (Mathematics), Galois theory. † Separable extensions, Galois extensions, and statement of the Fundamental Theorem of Galois Theory in general. † Proof of the Fundamental Theorem of Galois Theory (if time permits). † (Not on the examinable syllabus.) Proof of the criterion for solvability (if time permits, which is very unlikely).File Size: KB. Books on Field Theory So I have been reading up on Galois theory and algebraic number theory and I would like to explore this subject a little more. I have gone through the book by Ian Stewart on Galois Theory and would like to pursue the field extensions part a bit more. In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated; showing that there is no quintic formula. This book presents the basic theory of fields, starting more or less from the beginning. It is suitable for a graduate course in field theory, or independent study. The reader is expected to have taken an undergraduate course in abstract algebra, not so much for the material it contains but in order to gain a certain level of mathematical maturity. In the fall ofI taught Math at New Mexico State University for the first time. This course on field theory is the first semester of the year-long graduate algebra course here at NMSU. In the back of my mind, I thought it would be nice someday to write a book on field theory, one of my favorite mathematical subjects, and I wrote a crude form of lecture notes that semester. The book discusses Galois theory in considerable generality, treating fields of characteristic zero and of positive characteristic with consideration of both separable and inseparable extensions, but with a particular emphasis on algebraic extensions of the field of rational numbers. The first volume focuses on field extensions. Galois theory and its applications are treated more thoroughly than in most texts. It also covers basic applications to number theory, ring extensions and algebraic geometry. The main focus of the second volume .This book deals with classical Galois theory, of both finite and infinite extensions, and with transcendental extensions, focusing on finitely generated extensions and connections with algebraic geometry. The purpose of the book is twofold. First, it is written to be a textbook for a graduate-level course on Galois theory or field theory/5(2).e-books in Fields & Galois Theory category Galois Theory: Lectures Delivered at the University of Notre Dame by Emil Artin - University of Notre Dame, The book deals with linear algebra, including fields, vector spaces, homogeneous linear equations, and determinants, extension fields, polynomials, algebraic elements, splitting fields, group characters, normal extensions, roots of unity.

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http://www.ncsu.edu/project/cnr/sfe/sfebb/printthread.php?tid=463

math

Hello Everyone! - Printable Version +- Southern Fire Exchange (http://www.ncsu.edu/project/cnr/sfe/sfebb) +-- Forum: Southern Fire Exchange Forums (/forumdisplay.php?fid=7) +--- Forum: Open Discussion (/forumdisplay.php?fid=9) +---- Forum: New Member greetings (/forumdisplay.php?fid=20) +---- Thread: Hello Everyone! (/showthread.php?tid=463) Hello Everyone! - PaulDuncan - 05-09-2012 09:52 AM Hello, I just registered at this forum and I thought I'd say hi. Good to be here, and I'm looking forward to have discussions with you guys.

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CC-MAIN-2014-52

https://paperwall101.com/essay/recommending-a-low-cost-customer-service-essay-10811

math

Acme De Mexicos Manager has requested assistance in developing a minimum cost daily assignment schedule for the customer service employees in their newly built store. Specifically, he wants to know the minimum total cost per day, which is the decision variable. He also wants to know the exact amount of part time and full time employees which will determine the total cost. The objective function is to minimize costs. Acme specified a minimum number of employees required for each shift, a maximum number of employees per shift, specific shifts for full time and part time workers, and a maximum percentage of 50% of the total hours for part time employees. These constraints were input into Microsoft solver, which determined $47,800 to be the minimum employee cost per day, employing 23 fulltime workers and 45 part time workers per day. Specific assumptions were made which will be discussed in detail, along with the impact of non-typical days. A sensitivity analysis will then be performed to determine how the percentage of part time employees constraint affects the total cost per day. Now the Acme De Mexico has completed the building process, it is now time to properly staff the store. The store manager, Mr. Rodriguez, has requested a minimum cost daily assignment schedule for the customer service employees at the new store. In order to have Acme De Mexico become a profitable business, it must make the best use of its resources (Jacobs & Chase, 2013). In this case the resources are time, money, and employees. In order to provide Mr. Rodriguez with the information he requested, linear programming will be utilized. Linear programming is the several related mathematical techniques used to allocate limited resources among competing demands in an optimal way (Jacobs & Chase, 2013, appendix A). In this case, we are given the following information. This report will provide an employee assignment schedule for a typical day, developed with a linear programming model (Attachment 1). This model and its cells will be referenced throughout the report. An explanation will be provided to explain the model to include the assumptions made. The report will also briefly touch on how non-typical days may affect the schedule. Employee Assignment Schedule Acme De Mexico is open daily, from 7:00am to 11:00pm. Employee shifts are broken out over those 16 hours. For every hour of the day, a minimum amount of employees are required to be on the floor, which is depicted in the table below. The minimum number of employees (limit) needed on the floor at a given hour is one of the constraints. This constraint is displayed in cells G22 though V22. Additionally, only 30 employees are allowed on the floor at any given time for safety reasons. This constraint can be seen in cells G26 though V26 This is also a constraint, or limit. See cells A5-21 through cells C5-21. Part time employees are paid $500 (Pesos) per day, and full time employees are paid $1100 per day. Another constraint is the hours worked by part time employees cannot exceed 50% of the total hours worked per day (total hours = part time+full time). This is displayed in cell F36. Excel solver was used to solve the decision variable (E33), which is set as the objective. Cells D5 though D21 are the number and type of employees per hour, and are variable. The goal is to determine the minimum total cost per day. This is our decision variable, and is found in cell E33 of Attachment 1. The constraints mentioned above are input into solver. The first line shows the total number of part time employees must be less than or equal to 50% to the total labor hours each day. The second line ensures that the changing values are integers. We do not want half an employee to show up for his or her shift. The third line constraint ensures that the number of employees per shift does not exceed 30. Lastly, the fourth line constraint took into account the minimum employees per shift as specified by Acme. Our objective function is to minimize Acmes the total employee cost per day. The total employee cost per day was calculated by multiplying the number of fulltime workers per day (E8) by the salary per day (C31). This total is reflected in cell C33. The same was done for part time workers: (E21)*(D31)=(D33). These two numbers were then added together, (C33)+(D33)=(E33). Solver determined $47,800 (E33) to be the minimum employee cost per day, employing 23 fulltime workers and 45 part time workers per day. According to Knode, a few key assumptions are made when using linear programming: The assumption of a linear relationship (between the objectives, the constraints, etc.), the assumption of continuous relationships, and the assumption of non-negative relationships (2011). Additionally, the assumption was made that the solution and variables would be integers, that is, not a fraction of an employee. It is also assumed that variables and solutions will be non-negative numbers. It can be assumed that there are enough employees to cover for employees who call in sick. Non-typical days may affect the schedule. For example, employees may call in sick. Employees who are off may have to come in to cover these shifts, or employees may have to work overtime to cover for the sick employee. This could increase the daily cost if the overtime rate is more than the hourly rate. Overtime may also come into play during holidays or busy times of the year. Acme may decide to open earlier and/or stay open later during these times. Acme would need to hire more employees to cover the extra shifts, or employees would have to work overtime. Sensitivity Analysis allows us to look at variations in key aspects of the problem that could change the baseline answer (Knode, 2011). One such key aspect is the constraint that hours worked by part time employees cannot exceed 50% of the total hours worked per day. The percentage of part time employees was varied to explore the possible outcomes. The results are displayed in the table below. It is interesting to note that with 0% part time employees, the total cost is the lowest. Linear programming is a very useful tool which can help mangers solve many problems, including the problem of employee staffing. In the Acme De Mexico case, the decision variable was the minimum total cost per day for employee staffing. This also required determining the number of part time and full time employees per shift. Constraints were given and were input into solver, which resulted in a minimum daily cost of $47,800, with 23 full time employees and 45 part time employees. Knode, C.S. (2011). Linear programming Part 1 Formulating the problem . Retrieved from: http://vimeo.com/duffer44/linear-programming-part-1 Jacobs, F.R & Chase, R.B. (2013). Operations and supply management: The core, 3e. Chapter 1 and Appendix A

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https://forum.ansys.com/forums/reply/184721/

math

The slope of the last two data points is used to extrapolate the BH curve. (While you are entering the BH data you might have observed the extrapolated green line on the right-side graph, while the data you entered will be below the green line in red) Points Need to keep in mind while defining the BH curve. ┬ÀFor a Normal BH curve, the slope of the curve cannot be less than that of free space anywhere along the curve. ┬ÀThe value of B must increase along the curve. ┬ÀThe first data points for B and H must be 0 (zero). ┬ÀThe data points representing the BH curve should have enough points for an accurate representation of the curve.20 or more points should be specified with increased representation on the "knee" of the curve. ┬ÀSince BH operating points in the FEA solution may extend beyond the input BH data set, the BH data set is extrapolated in Maxwell.The slope of the last two user-defined data points is used to extrapolate the BH curve, and thus should be as close to mu0 as possible. Please Login to Report Topic Please Login to Share Feed You are navigating away from the AIS Discovery experience

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