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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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Apr 6, 2015 Peter Dourmashkin
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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 one-way 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
Flooved
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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
Flooved
- 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
Flooved
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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 n-th derivative occurs in the equation.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1536
In this lecture we will continue to study the category S_p of p-pro�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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Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/988
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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
Flooved
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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
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, 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
Flooved
- 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
Flooved
- 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
Flooved
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1486
Flooved
- W.T. Tutte
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Topics: Maths, Algebra, Coding and Cryptography, Mathematics
Source: http://www.flooved.com/reader/1478
Flooved
- 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
Flooved
- 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
Flooved
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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
Flooved
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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
Flooved
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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
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
Flooved
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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
Flooved
- Fredrik Jonsson
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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
Flooved
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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
Flooved
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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
Flooved
- David Tong
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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 Chern-Simons-Higgs 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
Flooved
- 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
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/981
We now introduce a very important object called the energy-momentum 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
Flooved
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De�nition. A domain D in 3-space is simply-connected 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
Linear algebra is essentially about solving systems of linear equations, an important application of mathematics to real-world 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
Flooved
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Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1789
Flooved
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Inequalities: Generalized Minkowski inequality.
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1795
Flooved
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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
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
Flooved
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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
Flooved
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This lecture covers the proof of the Bessy-Thomasse 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
Flooved
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2101
Flooved
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Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2081
Flooved
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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Geometry of Manifolds, Smooth Map,...
Source: http://www.flooved.com/reader/2136
Flooved
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2131
Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2584
Flooved
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2208
Flooved
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Topics: Maths, Statistics and Probability, Statistics, Estimation Techniques, Maximum Likelihood...
Source: http://www.flooved.com/reader/2171
Flooved
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Content: Operators of the scalar Klein Gordon �eld ; The scalar charge-current density; Klein Gordon charge-current 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
Flooved
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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
Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Non-Newtonian...
Source: http://www.flooved.com/reader/2591
Flooved
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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
Flooved
- Michael Fowler
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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
Flooved
- John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1455
Flooved
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Induction is by far the most powerful and commonly-used 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
Flooved
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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
Flooved
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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
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1951
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 zero-sum 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
Flooved
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2134
Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2167
Flooved
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Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2164
Flooved
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Topics: Physics, Physics
Source: http://www.flooved.com/reader/2676
Flooved
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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
Flooved
- 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
Flooved
- 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 L-C 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
Flooved
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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
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 non-interacting many-particle 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
Flooved
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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
Flooved
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We shall describe the kinematics of circular motion, the position, velocity, and acceleration, as a special case of two-dimensional 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
One of the �rst exact solutions of Einstein�s equations describing the mutual scattering of gravitational waves was that of Khan and Penrose (1971). This describes the collision and subsequent interaction of two plane impulsive gravitational waves and their subsequent interaction. Following this pioneering work, many more exact solutions have been obtained. However, the general character of almost all solutions that have subsequently been found is basically the same as that of the original...
Topics: Physics, Special Relativity, General Relativity and Gravitation, Classical General Relativity,...
Source: http://www.flooved.com/reader/2798
Flooved
- Alan Guth;Barton Zwiebach
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Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Properties of...
Source: http://www.flooved.com/reader/3165
Flooved
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We will discuss one of the classes of problems that have been suggested and deployed for communications of this public key type. It was actually developed here at M.I.T., a number of years ago. The hard problem that is more or less equivalent to breaking this code is that of factoring a rather large number. (There are variants of the coding scheme in which the inversion problem is exactly equivalent to factoring such a number.)
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1854
Flooved
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Having established that the low energy properties of the atomic chain are represented by a free scalar classical �eld theory, we now turn to the formulation of the quantum system.
Topics: Physics, Condensed Matter, Particle Physics and Fields, Quantum Physics, General Theory of Fields...
Source: http://www.flooved.com/reader/3063
Flooved
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Klein-Gordon: The Schrodinger equation...contains �rst derivatives with respect to time, t, and second derivatives with respect to (xyz). We now wish to handle all four coordinates (x1x2x3x4) on equal footing.
Topic: Maths
Source: http://www.flooved.com/reader/2935
Flooved
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All lecture courses on string theory start with a discussion of the point particle. Ours is no exception. We�ll take a �ying tour through the physics of the relativistic point particle and extract a couple of important lessons that we�ll take with us as we move onto string theory.
Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Source: http://www.flooved.com/reader/3188