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Flooved is an online education platform founded in 2011, that seeks to provide free education to a global audience by providing lecture notes, handouts, and study guides online, beginning with undergraduate mathematics and physics content.



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Nov 14, 2013 Santosh Vempala
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Theorem 1. If G is bipartite, then P=M.
Topics: Maths, Graph Theory, Optimization and Control, Basics, Connectivity and Matchings, Optimization,...
Source: http://www.flooved.com/reader/1319
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Nov 14, 2013 Victor Flynn
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These notes are modi�ed from previous versions (due to Neil Dummigan,Alan Lauder and Roger Heath-Brown) and have been recently revised by me. They draw mainly upon �A Classical Introduction to Modern Number Theory�, by Ireland and Rosen, and �Algebraic Number Theory�, by Stewart and Tall. While I take full responsibility for their current contents, considerable thanks are clearly due to Neil, Alan and Roger.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1060
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Nov 14, 2013 Vladan Vuletic
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Today � Finish proof: inverse Fourier transform � Exact de�nition of expectation values, uncertainties � Proof (see notes): _x_k � 1, _x_p � h�, ___t � 1, _E_t � �h 2 2 2 2 � Compton scattering Photoelectric e_ect
Topics: Physics, Particle Physics and Fields, Quantum Physics, Specific Reactions and Phenomenology, Photon...
Source: http://www.flooved.com/reader/3099
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Nov 14, 2013 Richard M. Dudley
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2163
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Nov 14, 2013 Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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This is meant as a follow-up on the review of vectors and matrices in the previous session.
Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Linear Algebra, Linear Independence,...
Source: http://www.flooved.com/reader/1458
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Nov 14, 2013 Fredrik Jonsson
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This course is intended as an introduction to the wide �eld of phenomena encountered in nonlinear optics. The course covers:� The theoretical foundation of nonlinear interaction between light and matter.� Perturbation analysis of nonlinear interaction between light and matter.� The Bloch equation and its interpretation.� Basics of soliton theory and the inverse scattering transform. It should be emphasized that the course does not cover state-of-the-art material constants of nonlinear...
Topics: Physics, Acoustics, Optics and Waves, Optics�, Nonlinear Optics, Wave Optics, Polarization,...
Source: http://www.flooved.com/reader/2971
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This handout provides the essential elements needed to understand �nite random matrix theory. A cursory observation should reveal that the tools for in�nite random matrix theory are quite di_erent from the tools for �nite random matrix theory. Nonetheless, there are signi�cantly more published applications that use �nite random matrix theory as opposed to in�nite random matrix theory. Our belief is that many of the results that have been historically derived using �nite random...
Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics
Source: http://www.flooved.com/reader/975
Flooved
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In the previous chapters we have considered the collision between two electromagnetic waves, permitting them also to be coupled to gravitational waves. In this chapter a situation is considered in which a plane gravitational wave approaches and collides with a plane electromagnetic wave. In this situation, the �eld equations are identical to those of previous chapters but the initial conditions are di_erent.
Topic: Maths
Source: http://www.flooved.com/reader/2793
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Nov 14, 2013 Richard Stanley
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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Posets and Zorn�s Lemma, Groups, Hasse Diagrams...
Source: http://www.flooved.com/reader/1045
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Nov 14, 2013 Vladan Vuletic
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Can we �derive� Newton�s F = ma from the SE?
Topics: Physics, Quantum Physics, Quantum Mechanics, Quantum Systems with Finite Hilbert Space,...
Source: http://www.flooved.com/reader/3084
Flooved
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In this lecture and the next two lectures, we�ll briefly review partial differential equations (PDEs). As PDEs are much more difficult to solve than ODEs, we shall start with the simplest of PDEs, those of the first order.
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1336
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Nov 14, 2013 John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
Source: http://www.flooved.com/reader/1450
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Nov 14, 2013 Abhinav Kumar
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1130
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Nov 14, 2013 Daniel Kleitman;Peter Shor
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Seeking the Best Expected Rank: This problem can also be handled by the same general approach. If there are N possible choices all together, we can compute a threshold for each stage, where we accept the first candidate which is less than or equal to the threshold. Call this threshold T(s), where s is the number of candidates still unseen.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1886
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Nov 14, 2013 Zoltan Spakovszky
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In this section, we examine the issue of how we obtain the heat needed for work production. For the most part, this is from converting chemical energy into heat, so the discussion will be on reacting mixtures of gas which are involved in chemical combustion processes. The topic addressed is �thermochemistry�, which is the combining of thermodynamics with chemistry to predict such items as how much heat is released from a chemical reaction. This is the �Q� or �q� that we have used in...
Topic: Maths
Source: http://www.flooved.com/reader/3204
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Nov 14, 2013 Richard Fitzpatrick
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Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism, Physics
Source: http://www.flooved.com/reader/2723
Flooved
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Let V be a �nite dimensional vector space over F2, and let TV denote Lannes�s T-functor. In previous lectures we have established two very important properties of TV : � The functor TV is exact. � For every 2-pro�nite space X, there is a canonical isomorphism... Our goal in this lecture is to deduce a conceptual consequence of these facts, which makes no mention of modules over the Steenrod algebra.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2213
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Nov 14, 2013 Jin Au Kong
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Topics: Physics, Electromagnetism and Electromagnetic Radiation, Mathematical Methods in Physics, Calculus...
Source: http://www.flooved.com/reader/2569
Flooved
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In this chapter we will consider only vacuum solutions. This will enable us to continue to concentrate on the collision of gravitational waves. Various techniques will be discussed here, and the main solutions that have been obtained using them will be described in the next chapter. Many techniques are also known by which solutions of the Einstein�Maxwell equations, and other non-vacuum solutions, can be generated from known vacuum solutions. These will be discussed later in Chap-ter 15.
Topic: Maths
Source: http://www.flooved.com/reader/2786
Flooved
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We now introduce the Fourier Transform and show how it is related to the solution of the Heat Problem on an in�nite domain.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1670
Flooved
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2611
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Nov 14, 2013 Alan Guth;Barton Zwiebach
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Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Source: http://www.flooved.com/reader/3161
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Nov 14, 2013 Alan Guth;Barton Zwiebach
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Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Source: http://www.flooved.com/reader/3155
Flooved
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Generating Functions- for a sequence a0. . .we can de�neA(x) =ann�0a xn
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1125
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Nov 14, 2013 Vladan Vuletic
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Photoelectric e_ect Observed 1888; explanation, Einstein 1905. A negatively charged metal plate loses charge slowly if illuminated with light, while a positive charge remains. Light of su_ciently short wavelength releases electrons, the electron current is proportional to the intensity of the light. Light below a certain cut o_ frequency _0 does not release any electrons, no matter how high the intensity. Energy of released electrons can be measured by determining voltage V that prevents them...
Topic: Maths
Source: http://www.flooved.com/reader/3100
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Nov 14, 2013 Erik Demaine
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Topics: Maths, Statistics and Probability, Probability, Mathematics
Source: http://www.flooved.com/reader/1568
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Nov 14, 2013 Predrag Cvitanovi?;R. Mainieri
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So far we have concentrated on describing the trajectory of a single initial point. Our next task is to de�ne and determine the size of a neighborhood of x(t). We shall do this by assuming that the �ow is locally smooth and by describing the local geometry of the neighborhood by studying the �ow linearized around x(t). Nearby points aligned along the stable (contracting) directions remainin the neighborhood of the trajectory x(t) = ft(x0); the ones to keep an eye on are the points which...
Topics: Physics, Quantum Physics, Quantum Mechanics, Physics
Source: http://www.flooved.com/reader/2743
Flooved
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Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�, Physics
Source: http://www.flooved.com/reader/2858
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Nov 14, 2013 Eduardo Fradkin
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The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section that a consistent description of this theory cannot be done outside the framework of (local) relativistic Quantum Field Theory.
Topics: Physics, Particle Physics and Fields, Quantum Physics, General Theory of Fields and Particles,...
Source: http://www.flooved.com/reader/2878
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Nov 14, 2013 Daniel H. Rothman
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Summary Finally, we summarize the characteristics of dissipative systems: � Energy not conserved. � Irreversible. � Contraction of areas (volumes) in phase space.
Topics: Maths, Dynamics and Relativity, Dynamical Systems, Mathematics
Source: http://www.flooved.com/reader/1830
Flooved
Nov 14, 2013 Alan Guth;Barton Zwiebach
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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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Nov 14, 2013 Eduardo Fradkin
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We will now turn to the problem of the quantization of the simplest gauge theory, the free electromagnetic �eld. This is an abelian gauge theory. In Physics 583 we will discuss at length the quantization of non-abelian gauge �elds. Unlike abelian theories, such as the free electromagnetic �eld, even in the absence of matter �elds, non-abelian gauge theories are not free �elds and have highly non-trivial dynamics. In Physics 582 we will discuss canonical quantization and path-integral...
Topics: Physics, Particle Physics and Fields, Quantum Physics, General Theory of Fields and Particles,...
Source: http://www.flooved.com/reader/2880
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1957
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Nov 14, 2013 Jeff Viaclovsky
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Approximations. Let ... Then h _ C� (in�nitely di_erentiable with continuous derivatives).
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1792
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Nov 14, 2013 Rodolfo R. Rosales
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In two dimensions a Hopf bifurcation occurs as a Spiral Point switches from stable to unstable (or vice versa) and a periodic solution appears. There are, however, more details to the story than this: The fact that a critical point switches from stable to unstable spiral (or vice versa) alone does not guarantee that a periodic solution will arise, though one almost always does. Here we will explore these questions in some detail, using the method of multiple scales to find precise conditions...
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1176
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Nov 14, 2013 Thomas Ward;Alan Camina
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Topics: Maths, Analysis and Calculus, Complex Analysis, Contour Integration and Cauchy�s Theorem,...
Source: http://www.flooved.com/reader/3421
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Nov 14, 2013 Anirban Kundu;Jyoti Prasad Saha
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Topic: Maths
Source: http://www.flooved.com/reader/3380
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Nov 14, 2013 Daniel Kleitman;Peter Shor
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A binary tree is a tree in which each vertex has either 2 or 0 descendants. We can assign a code word to each leaf of a binary tree by starting at the top and assigning 0 to one descendant and 1 to the other, adding the bit onto the bits already possessed by the parent. If we do this the words at depth d in the tree will have a code word of length d.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1871
Flooved
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Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�, Physics
Source: http://www.flooved.com/reader/2846
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Nov 14, 2013 Ben Simons
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Previously, we have seen that, when treated in a mean-�eld or saddle-point approximation,the �eld theory of the weakly interacting Bose gas shows a transition to a Bose-Einsteincondensed phase when _ =0where the order parameter, the complex condensate wave-function _0acquires a non-zero expectation value, |_0| = _ ___Ld/g. The spontaneousbreaking of the continuous symmetry associated with the phase of the order parameter isaccompanied by the appearance of massless collective phase...
Topics: Physics, Condensed Matter, Particle Physics and Fields, Quantum Physics, General Theory of Fields...
Source: http://www.flooved.com/reader/3060
Flooved
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Today we will brie�y survey matroid representation and then discuss some problems in matroid optimization and the corresponding applications. The tools we develop will help us answer the following puzzle: Puzzle: A game is played on a graph G(V, E) and has two players, George and Ari. Ari�s moves consist of ��xing� edges e _ E. George�s moves consist of deleting any un�xed edge. The game ends when every edge has been either �xed or deleted. Ari wins if the graph at the end of...
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1923
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Nov 14, 2013 David Tong
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Our goal in this section is to use the techniques of statistical mechanics to describe the dynamics of the simplest system: a gas. This means a bunch of particles, �ying around in a box. Although much of the last section was formulated in the language of quantum mechanics, here we will revert back to classical mechanics. Nonetheless, a recurrent theme will be that the quantum world is never far behind: we�ll see several puzzles, both theoretical and experimental, which can only truly be...
Topic: Maths
Source: http://www.flooved.com/reader/3130
Flooved
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Topics: Maths, Statistics and Probability, Probability, Statistics, Hypothesis testing, Uniformly Most...
Source: http://www.flooved.com/reader/1639
Flooved
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We introduce another powerful method of solving PDEs. First, we need to consider some preliminary de�nitions and ideas.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1671
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Nov 14, 2013 Jeff Viaclovsky
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Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/993
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Nov 14, 2013 Prof. Richard Hammack
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such a function f, a single real number input x determines a unique single output value
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/3480
Flooved
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Matsubara Summations Basic idea: by introducing auxiliary function g(z) that has simple poles of strength unity at z = i_n, Cauchy�s theorem implies
Topics: Physics, Condensed Matter, Particle Physics and Fields, Statistical Physics and Thermodynamics,...
Source: http://www.flooved.com/reader/3057
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Nov 14, 2013 R. Victor Jones
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Consider a totally coherent wave propagating in the positive direction....For later reference, we note that the full (non-normalized) Jones vector representation of this field is given by...We can be easily shown that...
Topics: Physics, Acoustics, Optics and Waves, Optics�, Quantum Optics�, Wave Optics, Polarization,...
Source: http://www.flooved.com/reader/2991
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Nov 14, 2013 Emma Carberry
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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Mathematics
Source: http://www.flooved.com/reader/1098
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Nov 14, 2013 Albert R. Meyer
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In particular, when we make an estimate by repeated sampling, we need to know how much con�dence we should have that our estimate is OK. Technically, this reduces to �nding the probability that an estimate deviates a lot from its expected value. This topic of deviation from the mean is the focus of this �nal chapter.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1719
Today�s lecture covers three main parts: � Courant-Fischer formula and Rayleigh quotients � The connection of _2 to graph cutting � Cheeger�s Inequality
Topics: Maths, Linear Algebra and Geometry, Graph Theory, Optimization and Control, Vectors and Matrices,...
Source: http://www.flooved.com/reader/2019
Flooved
Nov 14, 2013 Predrag Cvitanovi?;Gregor Tanner;G�bor Vattay
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Semiclassical approximations to quantum mechanics are valid in the regime where the de Broglie wavelength _ _ _/p of a particle with momentump is much shorter than the length scales across which the potential of the system changes signi�cantly. In the short wavelength approximation the particleis a point-like object bouncing o_ potential walls, the same way it does in the classical mechanics. The novelty of quantum mechanics is the interference of the point-like particle with other versions...
Topics: Physics, Quantum Physics, Quantum Mechanics, Physics
Source: http://www.flooved.com/reader/2752
Flooved
Nov 14, 2013 Denis Auroux
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The three theorems we have studied: the divergence theorem and Stokes� theorem in space, and Green�s theorem in the plane (which is really just a special case of Stokes� theo_rem) are widely used in physics and continuum mechanics, in the study of �elds, potentials, heat �ow, wave motion in liquids, gases, and solids, and thermodynamics, to name some of the uses. Often partial di_erential equations which model some physical situation are de_rived using the vector integral calculus...
Topics: Maths, Analysis and Calculus, Calculus, Mathematics
Source: http://www.flooved.com/reader/1821
Flooved
Nov 14, 2013 Katrin Wehrheim
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Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some collection of isolated points. In fact, it turns out that �most� continuous functions are non-differentiable at all points. (To understand what this statement could mean, you should take courses in topology and measure theory.) However, Weierstrass was not, in fact, the �rst to construct...
Topics: Maths, Analysis and Calculus, Mathematics
Source: http://www.flooved.com/reader/1197
Flooved
Nov 14, 2013 Prof. Peter J. Cameron
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Topic: Mathematics
Source: http://www.flooved.com/reader/3518
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Nov 14, 2013 Mr. Travis Byington;Lawrence Evans;Mr. Ryan Magee;Hao Zhang
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...This course take up those loose ends and discusses the discoveries about them by scientists of the 18th and 19th centuries. It turns out that the questions are all related to one another. Like gravity, electric and magnetic interactions also require acting �at a distance� without any apparent intervening medium. In the mid-19th century the description of these phenomena turned away from the mechanical point of view, and centered instead on the idea of �elds. In the last half of that...
Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�,...
Source: http://www.flooved.com/reader/2907
Flooved
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2598
Flooved
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Our �rst example of a public-key cryptosystem, in which two people exchanging messages did not have to meet beforehand, was Di_e-Hellman. We then talked about the RSA cryptosystem, which is probably the most widely used today. Here are the basics of how it works:
Topic: Maths
Source: http://www.flooved.com/reader/1309
Flooved
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In this lecture we will introduce Multiplicative Weights, a simple technique with many applications. We start with an example.
Topics: Maths, Game Theory, Zero-Sum Games, Mathematics
Source: http://www.flooved.com/reader/2028
Flooved
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Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1630
Flooved
Nov 14, 2013 Richard Melrose
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The wave equation in one space and one time variable
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1501
Flooved
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2623
Flooved
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Covariance of X and Y is de�ned as: ... Positive when both high or low in deviation
Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1625
Flooved
Nov 14, 2013 James Nearing
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If the subject of ordinary di_erential equations is large, this is enormous. I am going to examine only one corner of it, and will develop only one tool to handle it: Separation of Variables. Another major tool is the method of characteristics and I�ll not go beyond mentioning the word. When I develop a technique to handle the heat equation or the potential equation, don�t think that it stops there. The same set of tools will work on the Schroedinger equation in quantum mechanics and on the...
Topics: Physics, Condensed Matter, Mathematical Methods in Physics, Structural, Mechanical, and Thermal...
Source: http://www.flooved.com/reader/3023
Flooved
Nov 14, 2013 Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2616
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Nov 14, 2013 Eduardo Fradkin
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We now turn our attention to interacting quantum �eld theories. All of the results that we will derive in this section apply equally to both relativistic and non-relativistic theories with only minor changes. Here we will use the path integrals approach we developed in previous chapters.
Topics: Physics, Particle Physics and Fields, Quantum Physics, General Theory of Fields and Particles,...
Source: http://www.flooved.com/reader/2872
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Nov 14, 2013 R. Artuso;Predrag Cvitanovi?;H.H. Rugh
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As we shall see, the trace formulas and spectral determinants work well, sometimes very well. The question is: Why? And it still is. The heuristic manipulations of chapter 18 were naive and reckless, as we are facing in�nite-dimensional vector spaces and singular integral kernels. We now outline the key ingredients of proofs that put the trace and determinant formulas on solid footing. This requires taking a closer look at the evolution operators from a mathematical point of view, since up to...
Topics: Physics, Quantum Physics, Quantum Mechanics, Physics
Source: http://www.flooved.com/reader/2779
Flooved
Nov 14, 2013 Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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In this note we state Green�s formula and look at some examples. We will prove it in the next note.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1397
Flooved
Nov 14, 2013 Prof. Peter J. Cameron
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This course introduces the basic notions of probability theory and develops them to the stage where one can begin to use probabilistic ideas in statistical inference and modelling, and the study of stochastic processes. Probability axioms. Conditional probability and independence. Discrete random variables and their distributions. Continuous distributions. Joint distributions. Independence. Expectations. Mean, variance, covariance, correlation. Limiting distributions
Topic: Mathematics
Source: http://www.flooved.com/reader/3509
Flooved
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We shall associate a �Cartan matrix� to the system _ _ _ and derive some properties of this matrix. An abstract Cartan matrix will be any square matrix with this list of properties. It turns out that an abstract Cartan matrix essentially determines the root system. In this paper we will work toward making this statement precise and proving it. The problem of classi�cation of root systems is reduced then to the classi�cation of the Cartan matrices.
Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics
Source: http://www.flooved.com/reader/1662
Flooved
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We now consider D�Alembert�s solution for a �nite string.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1676
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Nov 14, 2013 Daniel Kleitman;Peter Shor
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A sequence of zeroes and ones can represent a message, a sequence of data in a computer or in digital communications, but it also can represent all sorts of mathematical or real world objects. Thus a sequence of length n can represent the outcome of n coin tosses, or a binary valued vector of length n, or a polynomial of degree up to n-1 with binary coefficients, or a number. We will now discuss three other interpretations for such a sequence.
Topics: Maths, Mathematics
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Nov 14, 2013 Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
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Nov 14, 2013 Richard Melrose
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Now say that X is a set, R is a ring of subsets and we have a function � : R _ [0, �). This is the �measure�, what we�re looking for. One of the properties we need for this function is additivity : �(A _ B) = �(A) + �(B), if A, B _ R, A _ B = _ From just this property we can derive a number of properties about �....
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
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