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Apr 6, 2015
04/15
Apr 6, 2015
by
Peter Dourmashkin
texts
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Topics: Physics, Classical Mechanics, Physics
Source: http://www.flooved.com/reader/3310
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In this chapter, we will focus our attention on simple graphs where the relationship denoted by an edge is symmetric. Afterward, in Chapter 6, we consider the situation where the edge denotes a oneway relationship, for example, where one web page points to the other.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1744
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A graph with directed edges is called a directed graph or digraph
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1735
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The approach to �nding exact solutions that has been taken so far has involved initially solving the �eld equations in the interaction region IV and then investigating the conditions under which these solutions can be considered as the result of collisions of plane waves. In this way, resulting solutions are found �rst and the initial conditions are obtained subsequently. In this chapter it is appropriate to return to the original problem of specifying the initial data and then attempting...
Topics: Physics, Special Relativity, General Relativity and Gravitation, Classical General Relativity,...
Source: http://www.flooved.com/reader/2788
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by
Daniel Kleitman;Peter Shor
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As we have also noted, the form of the equations here is one in which the sj variables are solved for in the sense that each appears with coefficient 1 in exactly one equation. This form is convenient for evaluating these s variables when all the x variables are set to 0, so we associate the origin in the x variables with this form for the equations.
Topics: Maths, Optimization and Control, Optimization, Linear Programming, Network Problems, Slack...
Source: http://www.flooved.com/reader/1867
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by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Another Proof of the Superposition Principle: The superposition principle is so important a concept that it is worth reviewing yet again. Here we will use the integrating factors formula for the solution to �rst order linear ODE�s to give another simple proof of this principle.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1441
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by
Vera Mikyoung Hur
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Ordinary differential equations are differential equations whose unknowns are functions of a single variable. They commonly arise in dynamical systems and electrical engineering. Partial differential equations are differential equations whose unknown depend two or more independent variables. In this course, we focus only on ordinary differential equations. The order of a differential equation is the largest integer n, for which an nth derivative occurs in the equation.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1536
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In this lecture we will continue to study the category S_p of ppro�nite spaces, where p is a prime number. Our main goal is to connect S_p with the category of E�algebras over the �eld Fp, following the ideas of Dwyer, Hopkins, and Mandell.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2220
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So far we�ve focussed on a statistical mechanics, studying systems in terms of their microscopic constituents. In this section, we�re going to take a step back and look at classical thermodynamics. This is a theory that cares nothing for atoms and microscopics. Instead it describes relationships between the observable macroscopic phenomena that we see directly.
Topic: Maths
Source: http://www.flooved.com/reader/3131
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Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/988
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by
Tomasz S. Mrowka
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There are a bunch of basic formulas in dealing with forms, the exterior derivative and contraction and the Lie derivative.
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2144
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Recall from last time the statement of the following lemma: given L a holomorphic line bundle with curvature...
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2115
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Following on from our investigation of the phonon and interacting electron system, we now turn to another example involving bosonic degrees of freedom � the problem of quantum magnetism.
Topic: Maths
Source: http://www.flooved.com/reader/3067
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Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Vectors and Matrices, Differential...
Source: http://www.flooved.com/reader/1065
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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Mathematics
Source: http://www.flooved.com/reader/1101
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This material closely follows selections from Chapter 3 of Enumerative Combinatorics 1 by Richard Stanley
Topics: Maths, Algebra, Statistics and Probability, Probability, Combinatorics, Mathematics
Source: http://www.flooved.com/reader/1266
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by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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To start we will de�ne �rst order linear equations by their form. Soon, we will understand them by their properties. In particular, you should be on the lookout for the statement of the superposition principle and in later sessions a conceptual de�nition of linearity.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1389
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by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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De�nition 1. Solutions x1(t), . . . , xn(t) to (3) are called linearly dependent if there are constants ci, not all of which are 0, such that c1x1(t) + . . . + cnxn(t) = 0, for all t.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1396
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Remember Riesz representation theorem in Hilbert Space.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Methods,...
Source: http://www.flooved.com/reader/1483
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119
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1486
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Topics: Maths, Algebra, Coding and Cryptography, Mathematics
Source: http://www.flooved.com/reader/1478
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by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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Open up the applet Linear phase portraits: Cursor Entry. This ap_plet is similar to Linear phase portraits: Matrix Entry.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1407
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by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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De�nitions: A sinusoidal function (or sinusoidal oscillation or sinusoidal signal) is one that can be wrtten in the form f (t) = A cos(_t _ _).
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1429
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Fourier series. Let us now try applying our knowledge of Hilbert space to a concrete Hilbert space such as L^2(a, b) for a �nite interval (a, b) _ R. You showed that this is indeed a Hilbert space. One of the reasons for developing Hilbert space techniques originally was precisely the following result.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1574
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One may consider systems of classical particles or �elds some of which are bosonic and some fermionic. In this case, the space of states will be a supervector space, i.e. the direct sum of an even and an odd space (or, more generally, a supermanifold � a notion we will de�ne below). When such a theory is quantized using the path integral approach, one has to integrate functions over supermanifolds. Thus, we should learn to integrate over supermanifolds and then generalize to this case our...
Topics: Physics, Quantum Physics, Quantum Field Theory, Physics
Source: http://www.flooved.com/reader/1532
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Now, what I want to show today, and not much more, is that the ej form an orthonormal basis of L2(R), meaning they are complete as an orthonormal sequence. There are various proofs of this, but the only �simple� ones I know involve the Fourier inversion formula and I want to use the completeness to prove the Fourier inversion formula, so that will not do. Instead I want to use a version of Mehler�s formula.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1583
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We will now study some important properties of solutions to the heat equation �tu_D�u = 0. For simplicity, we sometimes only study the case of 1 + 1 spacetime dimensions, even though analogous properties are veri�ed in higher dimensions.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1617
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I talked about step functions, then the covering lemmas which are the basis of the de�nition of Lebesgue measure � which we will do after the integral � then properties of monotone sequences of step functions.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1586
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In this �nal lecture, we will study the behaviour of the Bloch equations in di_erent regimes of resonance and relaxation. The Bloch equations are formulated as a vector model, and numerical solutions to the equations are discussed.
Topics: Physics, Acoustics, Optics and Waves, Optics�, Nonlinear Optics, Wave Optics, Polarization,...
Source: http://www.flooved.com/reader/2974
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273
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Since ancient times, the notion of ray or beam propagation has been one of the mosten during and fundamental concepts in optical physics. As a zeroth order approximation we might consider a plane wave to be a model of a beam and its propagation vector to be a model of a ray. This is a reasonable start, but it is a much too restricted view and we can do much better. What we need is a solution to Maxwell's equations which is like a plane wave, but limited in spatial extent. One approach, the...
Topics: Physics, Acoustics, Optics and Waves, Electromagnetism and Electromagnetic Radiation, Optics�,...
Source: http://www.flooved.com/reader/3004
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by
Peter Dourmashkin;Kate Scholberg
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In this example we will consider what happens if you bend your knees when you hit the ground if you are jumping from a height.
Topics: Physics, Classical Mechanics, Physics
Source: http://www.flooved.com/reader/3026
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Our �nal lecture is on domain walls (also known as kinks). We do all the usual stu_: solutions, moduli spaces, brane constructions. We focus in the applications on the relationship between kinks and monopoles, leading to a quantitative correspondence between 2d sigma models and 4d gauge theories. Further applications to the 2d black hole, and 3d ChernSimonsHiggs theories are also described.
Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Gauge Field Theories,...
Source: http://www.flooved.com/reader/3118
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by
Alan Guth;Barton Zwiebach
texts
eye 130
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Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes, D...
Source: http://www.flooved.com/reader/3171
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Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/981
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We now introduce a very important object called the energymomentum tensor. As we will see, it encodes some very important conservation laws associated to solutions of (1.0.2).
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1603
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De�nition. A domain D in 3space is simplyconnected if each closed curve in it can be shrunk to a point without ever getting outside of D during the shrinking.
Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Topology and Metric Spaces, Geometry,...
Source: http://www.flooved.com/reader/1820
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Linear algebra is essentially about solving systems of linear equations, an important application of mathematics to realworld problems in engineering, business, and science, especially the social sciences. Here we will just stick to the most important case, where the system is square, i.e., there are as many variables as there are equations.
Topics: Maths, Analysis and Calculus, Calculus, Mathematics
Source: http://www.flooved.com/reader/1808
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Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1789
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Inequalities: Generalized Minkowski inequality.
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1795
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by
Nizameddin H. Ordulu
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eye 153
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Introduction  The �Plateau�s Problem� is the problem of �nding a surface with minimal area among all surfaces which have the same prescribed boundary. Let x be a solution to Plateau�s problem t for a closed curve...
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1910
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by
Michel X. Goemans
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Our �rst topic of study is matchings in graphs which are not necessarily bipartite. We begin with some relevant terminology and de�nitions.
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1922
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by
Daniel Kleitman;Peter Shor
texts
eye 98
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We will first describe in detail how a spreadsheet can be set up to perform the Fast Fourier Transform algorithm. We will then apply it to the task of multiplying large numbers.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1864
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by
Michel X. Goemans
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This lecture covers the proof of the BessyThomasse Theorem, formerly known as the Gallai Conjecture. Also, we discuss the cyclic stable set polytope, and show that it is totally dual integral (TDI) (see lecture 5 for more on TDI systems of inequalities).
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1943
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Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2016
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by
Kiran S. Kedlaya
texts
eye 124
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2101
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by
Kiran S. Kedlaya
texts
eye 165
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2081
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by
Tomasz S. Mrowka
texts
eye 86
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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Geometry of Manifolds, Smooth Map,...
Source: http://www.flooved.com/reader/2136
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by
Tomasz S. Mrowka
texts
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2131
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2584
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94
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2208
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by
Richard M. Dudley
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Topics: Maths, Statistics and Probability, Statistics, Estimation Techniques, Maximum Likelihood...
Source: http://www.flooved.com/reader/2171
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Content: Operators of the scalar Klein Gordon �eld ; The scalar chargecurrent density; Klein Gordon chargecurrent density operators; General form for the local operator application; The time derivative Hamiltonian; The position operator; The velocity operator; The acceleration operator; Acceleration operator with EM interaction; The angular momentum operator
Topic: Maths
Source: http://www.flooved.com/reader/2668
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Plasmas are complicated because motions of electrons and ions are determined by the electric and magnetic �elds but also change the �elds by the currents they carry. For now we shall ignore the second part of the problem and assume that Fields are Prescribed. Even so, calculating the motion of a charged particle can be quite hard.
Topics: Physics, Physics
Source: http://www.flooved.com/reader/2681
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, NonNewtonian...
Source: http://www.flooved.com/reader/2591
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The purpose of this chapter is to describe in some detail the character of the plane waves whose interactions we will be considering in the subsequent chapters.
Topic: Maths
Source: http://www.flooved.com/reader/2799
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The course falls rather naturally into three parts, which will take approximately equal times. Part I covers the needed developments in mathematics and science from the earliest times until Galileo: the Babylonians and Egyptians, the Greeks, the Arabs, and early Western Europe. Part II will be devoted to the works of Galileo and Newton, culminating in Newton�s Laws of Motion and his Law of Universal Gravitation: the revolution revealing that the heavens obeyed the same laws as earthly...
Topic: Maths
Source: http://www.flooved.com/reader/3256
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Topics: Maths, Analysis and Calculus, Topology and Metric Spaces, Analysis, Calculus, Cauchy Sequences,...
Source: http://www.flooved.com/reader/1275
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Euler�s Totient Function:The number of elements in a reduced residue system mod m is called Euler�s totient function: _(m)(ie., the number of positive integers � m and coprime to m)
Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Algebra, Elementary Number...
Source: http://www.flooved.com/reader/1135
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by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts
eye 111
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comment 0
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1455
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Induction is by far the most powerful and commonlyused proof technique in discrete mathematics and computer science. In fact, the use of induction is a de�ning characteristic of discrete �as opposed to continuous �mathematics.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1723
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The theorem below relates the greatest common divisor to linear combinations. This theorem is very useful; take the time to understand it and then remember it!
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1711
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by
Daniel Kleitman;Peter Shor
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In this Chapter we will describe some other schemes for secret coding, consider some questions about how long it should take to find primes, and go into further detail about the steps needed to construct an RSA code.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1855
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by
J. Bernstein;A. Rita
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Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1951
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In this lecture, we will study various applications of the theory of Multiplicative Weights (MW). In this section, we brie�y review the general version of the MW algorithm that we studied in the previous lecture. The following sections then show how the theory can be applied to approximately solve zerosum games and linear programs, and how it connects with the theory of boosting and approximation algorithms.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2027
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by
Tomasz S. Mrowka
texts
eye 94
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2134
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by
Richard M. Dudley
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eye 100
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comment 0
Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2167
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by
Richard M. Dudley
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eye 80
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2164
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Topics: Physics, Physics
Source: http://www.flooved.com/reader/2676
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We will go here through an easy going, step by step, derivation of the Dirac equation (in the �chiral� representation), with the main focus on the actual physical meaning of all it�s properties.
Topics: Physics, Acoustics, Optics and Waves, Particle Physics and Fields, Waves, Specific Theories and...
Source: http://www.flooved.com/reader/2675
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by
Lawrence Evans;Mr. J. Edward Ladenburger
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Phases of matter. Until now we have analyzed mostly systems in which the particles have a �xed average spatial relation to each other, i.e., are bound. This kind of system is a solid. If the particles are not bound, but still are on average close enough together to interact continuously with nearest neighbors, we have a liquid. When the particles are on average far apart and only interact occasionally when they collide with each other, we have a gas. These situations de�ne the three...
Topics: Physics, Physics
Source: http://www.flooved.com/reader/2886
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by
Mr. Travis Byington;Lawrence Evans;Mr. Ryan Magee;Hao Zhang
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AC generators and rms values It is relatively easy to devise a source (a �generator�) which produces a sinusoidally varying emf. A rotating coil in a magnetic �eld gives an important example; another is the sinusoidally varying potential across the inductor in an LC oscillator, used in many situations to provide an emf for another circuit. This kind of oscillating source is called an AC (alternating current) generator. Its output is described by...
Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�,...
Source: http://www.flooved.com/reader/2904
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To prepare the ground for the quantization of the electromagnetic field, let us revisit the classical treatment of a simple harmonic oscillator with one degree of freedom. We start, with the assertion that the Hamiltonian of such an oscillator must have the form...
Topics: Physics, Acoustics, Optics and Waves, Quantum Physics, Optics�, Quantum Optics�, Physics
Source: http://www.flooved.com/reader/3003
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The aim of this course is to develop the machinery to explore the properties of quantum systems with very large or in�nite numbers of degrees of freedom. To represent such systems it is convenient to abandon the language of individual elementary particles and speak about quantum �elds. In this lecture, we will consider the simplest physical example of a free or noninteracting manyparticle theory theory which will exemplify the language of classical and quantum �elds. Our starting point...
Topics: Physics, Condensed Matter, Particle Physics and Fields, General Theory of Fields and Particles,...
Source: http://www.flooved.com/reader/3059
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In this section we �nally get to quantum electrodynamics (QED), the theory of light interacting with charged matter. Our path to quantization will be as before: we start with the free theory of the electromagnetic �eld and see how the quantum theory gives rise to a photon with two polarization states. We then describe how to couple the photon to fermions and to bosons.
Topics: Physics, Particle Physics and Fields, Quantum Physics, General Theory of Fields and Particles,...
Source: http://www.flooved.com/reader/3077
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by
Peter Dourmashkin
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We shall describe the kinematics of circular motion, the position, velocity, and acceleration, as a special case of twodimensional motion. We will see that unlike linear motion, where velocity and acceleration are directed along the line of motion, in circular motion the direction of velocity is always tangent to the circle. This means that as the object moves in a circle, the direction of the velocity is always changing. When we examine this motion, we shall see that the direction of change...
Topic: Maths
Source: http://www.flooved.com/reader/3309