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Flooved
by Jeff Viaclovsky
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Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1005
Flooved
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Topic: Maths
Source: http://www.flooved.com/reader/1484
Flooved
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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Mathematics
Source: http://www.flooved.com/reader/1043
Flooved
texts

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Topic: Maths
Source: http://www.flooved.com/reader/1029
Flooved
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Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1021
Flooved
by Dmitry Panchenko
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Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1031
Flooved
texts

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Number of primitive roots- suppose that m is an integer such that there is a primitive root g mod m
Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory,...
Source: http://www.flooved.com/reader/1139
Flooved
texts

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This is an elementary introduction to simplicial sets, which are generalizations of �-complexes from algebraic topology. The theory of simplicial sets provides a way to express homotopy and homology without requiring topology. This paper is meant to be accessible to anyone who has had experience with algebraic topology and has at least basic knowledge of category theory.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/1112
Flooved
by Emma Carberry
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We�ll assume that the curves are in R3 unless otherwise noted. We start o_ by quoting the following useful theorem about self adjoint linear maps over R2:
Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Mathematics
Source: http://www.flooved.com/reader/1111
Flooved
by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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In our discussion of the unit step function u(t) we saw that it was an idealized model of a quantity that goes from 0 to 1 very quickly. In the idealization we assumed it jumped directly from 0 to 1 in no time. In this note we will have an idealized model of a large input that acts over a short time. We will call this model the delta function or Dirac delta function or unit impulse.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1253
Flooved
by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
texts

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As we have seen with the applet, Euler�s method is rarely exact. In this section we try to understand potential sources of error, and �nd ways to estimate or bound it.
Topics: Maths, Numerical Analysis, Error, Computation of Ordinary Differential Equations (ODEs) and Partial...
Source: http://www.flooved.com/reader/1256
Flooved
by Michel X. Goemans
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Many optimizations problems arising in practice are NP hard. Under the widely accepted conjecture that P=/ NP, we cannot compute e_ciently and exactly an optimal solution for all possible instances = of these problems. Several approaches have been used to deal with this intractability. On one hand, dynamic programming, branch and bound, and implicit enumeration algorithms always �nd an optimal solution by navigating the space of feasible solutions in a more e_cient way than an exhaustive...
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1259
Flooved
by Katrin Wehrheim
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The aim of this handout is to provide a detailed proof of the equivalence between the two de�nitions of compactness: existence of a �nite subcover of any open cover, and existence of a limit point of any in�nite subset.
Topics: Maths, Analysis and Calculus, Topology and Metric Spaces, Analysis, Compact Spaces, Mathematics
Source: http://www.flooved.com/reader/1271
Flooved
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The purpose of these notes is to give some examples illustrating how naive numerical approximations to PDE's may not work at all as expected. In addition, the following two important notions are introduced: (I) von Neumann stability analysis - helps identify when (and if ) numerical schemes behave properly. (II) Artificial viscosity - a tool in stabilizing numerical schemes. These notes should be read in conjunction with the use of the MatLab scripts (in the Athena 18311-Toolkit at MIT) whose...
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1179
Flooved
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This lecture is an intruduction to quantum computation. Quantum computation is motivated in part by the field of Physics, and experiments.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1154
Flooved
texts

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The level set �nite di_erence method is properly developed in the books by its origina_tors: Sethian [ ] and Osher and Fedkiw [ ]. Here we concentrate on an essential point: upwind di_erencing.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1694
Flooved
texts

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The colliding wave problem has been described in terms of two approaching waves in regions II and III, in a background region I, which is here taken to be �at. According to the work of Penrose (1980), the initial data are well set, so that a unique solution exists in the interaction region IV at least in the neighbourhood of the boundaries of regions II and III. It is therefore necessary �rst to state the relevant �eld equations in the interaction region, and then to attempt to solve them...
Topic: Maths
Source: http://www.flooved.com/reader/2801
Flooved
texts

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The subject to be discussed in this book is the collision and interaction of gravitational and electromagnetic waves. This is a particularly importanttopic in general relativity since the theory predicts that there will be a non-linear interaction between such waves. The e_ect of the non-linearity, however, is unclear. It is appropriate therefore to look in some detail at the simplest possible situation in which the e_ect of the non-linearity will be manifest: namely the interaction between...
Topic: Maths
Source: http://www.flooved.com/reader/2783
Flooved
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We start this course by exploring the framework of Newtonian mechanics, understanding the axioms and what they have to tell us about the way the Universe works. We then move on to look at a number of forces that are at play in the world. Nature is kind and the list is surprisingly short. Moreover, many of forces that arise have special properties, from which we will see new concepts emerging such as energy and conservation principles. Finally, for each of these forces, we turn the mathematical...
Topic: Maths
Source: http://www.flooved.com/reader/2816
Flooved
texts

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It is appropriate at the conclusion of this work to make some attempt to summarize the main results that have been obtained, and to point out some areas that still need further clari�cation.
Topic: Maths
Source: http://www.flooved.com/reader/2797
This chapter is a continuation of the review of all presently known exact solutions which describe the collision of plane electromagnetic waves, or a combination of both gravitational and electromagnetic waves. Attention is concentrated here only on diagonal solutions. These solutions may be considered as a generalization of the solutions representing the collision of gravitational waves with colinear aligned polarization that have been described in Chapter 10. It may be mentioned that the...
Topic: Maths
Source: http://www.flooved.com/reader/2792
Flooved
by Lawrence Evans;Mr. J. Edward Ladenburger
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Conservative forces. In general the work done by a force depends on the path taken from the initial position to the �nal one, as well as on the location of those points. But, as we have seen, there are some forces for which the work depends only on the endpoints, not on the path. These are called conservative forces.
Topic: Maths
Source: http://www.flooved.com/reader/2885
Flooved
texts

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These notes cover the development of the current scienti�c concepts of space and time through history, emphasizing the newest developments and ideas.The presentation will be non-mathematical: the concepts will be introduced and explained, but no real calculations will be performed. The various concepts will be introduced in a historical order (whenever possible), this provides a measure of understanding as to how the ideas on which the modern theory of space and time is based were developed....
Topic: Maths
Source: http://www.flooved.com/reader/2869
Flooved
texts

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These notes cover the development of the current scienti�c concepts of space and time through history, emphasizing the newest developments and ideas.The presentation will be non-mathematical: the concepts will be introduced and explained, but no real calculations will be performed. The various concepts will be introduced in a historical order (whenever possible), this provides a measure of understanding as to how the ideas on which the modern theory of space and time is based were developed....
Topic: Maths
Source: http://www.flooved.com/reader/2866
Flooved
by Ron Shamir
texts

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This lecture introduces the problem of locating a given pattern in a string and the standard solutions to that problem.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2951
Flooved
by David Tong
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The purpose of this section is to lay down the foundations of kinetic theory, starting from the Hamiltonian description of 10^23 particles, and ending with the Navier-Stokes equation of �uid dynamics. Our main tool in this task will be the Boltzmann equation.This will allow us to provide derivations of the transport properties that we sketched in the previous section, but without the more egregious inconsistencies that crept into our previous derivaion. But, perhaps more importantly, the...
Topic: Maths
Source: http://www.flooved.com/reader/2943
Flooved
by Mr. Travis Byington;Lawrence Evans;Mr. Ryan Magee;Hao Zhang
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Overview - The discovery of radioactivity in 1896 caused a shock among chemists, because it suggested that the atom is not the eternal, immutable object they assumed it to be. This became even clearer within a few years, when radioactive process were studied carefully and it was shown that, in some of them, existing elements in the sample under study disappeared and were replaced by others that had not previously been there. ...
Topic: Maths
Source: http://www.flooved.com/reader/2913
Flooved
by John W. Norbury
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Chapter 1. Lagrangian Field Theory 1.1 Units. We start with the most basic thing of all, namely units and concentrate on the units most widely used in particle physics and quantum �eld theory (natural units). We also mention the units used in General Relativity, because these days it is likely that students will study this subject as well.
Topic: Maths
Source: http://www.flooved.com/reader/3070
Flooved
by J. D. Huba
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Topic: Maths
Source: http://www.flooved.com/reader/2984
Our objective here is to formulate a general approach to the subject of wave propagation in anisotropic dielectrics which makes use of ideas familiar from other branches of mathematical physics -- viz. the �eigenvalue problem.�.18 For reasons that will soon become abundantly clear, treatments of �crystal optics� focus on the behavior of the dielectric displacement vector, r D (r r , _) rather than on the electric field vector.19 For non-magnetic dielectrics the components of the...
Topic: Maths
Source: http://www.flooved.com/reader/3001
Flooved
by Thomas Ward
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Course objectives: In order to reach the more interesting and useful ideas, we shall adopt a fairly brutal approach to some early material. Lengthy proofs will sometimes be left out, though full versions will be made available. By the end of the course, you should have a good understanding of normed vector spaces, Hilbert and Banachspaces, �xed point theorems and examples of function spaces. These ideas will be illustrated with applications to di_erential equations.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/3363
Flooved
by Peter Ouwehand
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These notes are for a short course in set theory at the undergraduate level at Stellenbosch University. No pretense at orignality is claimed. Though ampli�ed by material from a number of additional sources, the debt to the �rst few chapters of the book Set Theory, by Thomas Jech, Springer 2003, should be easily discernible.
Topics: Maths, Logic, Numbers and Set Theory, Set Theory, Mathematics
Source: http://www.flooved.com/reader/3368
Flooved
by Charlie Zender
texts

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This document describes mathematical and computational considerations pertaining to radiative transfer processes and radiative transfer models of the Earth system. Our approach is to presenta detailed derivation of the tools of radiative transfer needed to predict the radiative quantities (irradiance, mean intensity, and heating rates) which drive climate. In so doing we begin with discussion of the intensity �eld which is the quantity most often measured by satellite remote sensing...
Topic: Maths
Source: http://www.flooved.com/reader/3325
Flooved
texts

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So far we have introduced the concepts of kinematics to describe motion in one dimension; however we live in a multidimensional universe. In order to explore and describe motion in this universe, we begin by looking at examples of two-dimensional motion, of which there are many; planets orbiting a star in elliptical orbits or a projectile moving under the action of uniform gravitation are two common examples. We will now extend our definitions of position, velocity, and acceleration for an...
Topic: Maths
Source: http://www.flooved.com/reader/3308
Flooved
by Joachim Lambek
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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Real Numbers, Propositional Logic, Set Theory,...
Source: http://www.flooved.com/reader/3381
Flooved
by James S. Milne
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This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/3434
Flooved
by Prof. Joseph Fields
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f(x). However, many of the functions of importance both within mathematics itself as
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/3477
Flooved
by James S. Milne
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These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings�s proof of certain �niteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/3397
Flooved
by James S. Milne
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An algebraic number �eld is a �nite extension of Q; an algebraic number is an element of an algebraic number �eld. Algebraic number theory studies the arithmetic of algebraic number �elds � the ring of integers in the number �eld, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a �eld is a Galois extension of the �eld with abelian Galois group. Class �eld theory describes the abelian extensions of...
Topic: Maths
Source: http://www.flooved.com/reader/3402
Flooved
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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds intersecting transversely.
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1965
Flooved
texts

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0.1. Coherent Sheaves on a Complex Manifold (contd.) - Let X be a com_plex manifold, OX the sheaf of holomorphic functions on X.
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1968
Flooved
texts

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0.1. Lagrangian Floer Homology (contd). Let (M, _) be a symplectic man_ifold, L0, L1 compact Lagrangian submanifolds intersecting transversely.
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1966
Flooved
texts

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De�nition 1: A matroid M = (S, I) is a �nite ground set S together with a collection of sets...
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1944
Flooved
texts

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Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1949
Flooved
texts

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1. Deformations of Complex Structures An (almost) complex structure (X, J) splits the complexi�ed tangent and (wedge powers of) cotangent bundles as...
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1972
Flooved
texts

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Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1953
Topics: Maths, Linear Algebra and Geometry, Geometry, Mathematics
Source: http://www.flooved.com/reader/1958
Flooved
texts

eye 88

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Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/983
Flooved
texts

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Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics
Source: http://www.flooved.com/reader/974
Flooved
by Elizabeth Stanway
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Objectives: (1) To recap some basic aspects of SR (2) To introduce important notation.
Topic: Maths
Source: http://www.flooved.com/reader/3123
Flooved
by David Tong
texts

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In this section we�ll explore a couple of phase transitions in some detail and extract some lessons that are common to all transitions.
Topic: Maths
Source: http://www.flooved.com/reader/3132
Flooved
by Max Tegmark
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Topics: Key concept summary � Summary of useful metrics � Special relativity concept summary: Space and time uni�ed into 4D spacetime. � Analogous uni�cation for other 4-vectors (momentum+energy, etc.). � Lorentz transform relates 4-vectors in di_erent inertial frames. Ex_ample: fast moving clocks are slower, shorter and heavier. E = mc2 . Example: nuclear power. � General relativity concept summary: Spacetime is not static but dynamic, globally expanding and locally � curving...
Topic: Maths
Source: http://www.flooved.com/reader/3107
Flooved
by Eric Poisson
texts

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Topic: Maths
Source: http://www.flooved.com/reader/3127
Flooved
by Richard Earl
texts

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In this �rst course in abstract algebra we concentrate on groups. Two other important algebraic structures are rings and �elds � you will likely have met the �eld axioms in LinearAlgebra I and Analysis I.
Topics: Maths, Algebra, Groups, Geometry and Groups, Groups, Group Actions, Permutations, Group Actions,...
Source: http://www.flooved.com/reader/1069
Solutions expanded around an irregular singular point are distinctive in one aspect: they are usually in the form of an exponential function times a Frobenius series. Due to the factor of the exponential function, a solution near an irregular singular point behaves very differently from that near a regular singular point. It may blow up exponentially, or vanish exponentially, or oscillate wildly.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1146
Flooved
by Santosh Vempala
texts

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A directed graph is a graph in which every edge has a direction. A capacity is the maximum flow allowed on an edge, and is represented by ci,j, where the edge connects the vertices i and j in the direction from i to j.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1311
Flooved
texts

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What we really need is a machine that can not only move backwards and forwards, but also write to the tape and halt at any time of its choosing. And that�s what a Turing machine is. The ability to write essentially gives Turing machines an unlimited memory, since any information that can�t �t in the machine�s internal state can always be written to the tape. The ability to halt at discretion means that Turing machines aren�t �tied to the input� the way �nite automata are, but...
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1310
Flooved
texts

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Regular Perturbation: In some physical problems, the solution is dependent on a parameter K. When the parameter K is very small, it is natural to expect that the solution not be very different from the one with K set to zero
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1334
Flooved
by Jeff Viaclovsky
texts

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Remark: Since many of our results rely on the regularity of the Newtonian Potential, and hence use Proposition 2 of Lecture 9, we will assume througout that the H�older constant _ ranges in the open interval (0, 1).
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1328
Flooved
by Jeff Viaclovsky
texts

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Now we would like to extend our estimates to general domains. Note this is a global estimate, i.e. upto the boundary.We will use the remark from last time concerning domains with a portion of a hyperplane on the boundary which will provide us with a needed estimate
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1329
Flooved
by Jeff Viaclovsky
texts

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Extending interior Schauder estimates to �at boundary part, Global Schauder estimates, Banach Spaces and Contraction Mapping Theorem
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1330
Flooved
texts

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RSA: RSA (together with its variants) is probably the most widely-used cryptographic protocol in modern electronic commerce. Much like Di_e-Hellman, it is built on modular arithmetic.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1306
Flooved
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Quantum states of n qubits: If you have an object that can be in two perfectly distinguishable states |0> or |1>, then it can also be in a superposition of the |0> and |1> states: _ |0> + _ |1> where _ and _ are complex numbers such that: |_| 2 + |_| 2 = 1
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1308
Flooved
by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Our work with these differential operators will be based on several rules they satisfy. In stating these rules, we will always assume that the functions involved are suf�ciently differentiable, so that the operators can be applied to them.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1416
Flooved
by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
texts

eye 106

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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1410
Flooved
texts

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Poisson Summation: This is really a correlation between fourier transforms and fourier series
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1499
Flooved
texts

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While integrals like _ f(z) dz and _ M dx+ N dy have been de�ned in the text (p.101), di_erential forms like dx, dy and dz = dx + i dy have not been de�ned (and the de�nition is more subtle), we shall develop the theory of harmonic functions (p.162-170) without di_erential forms.
Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1511
Flooved
by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts

eye 95

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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
Source: http://www.flooved.com/reader/1450
Flooved
by Catherine Wilkins
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The course will be in two parts: Part I in Michaelmas Term Weeks 3-8 and Hilary Term Weeks 1-2, and Part II in Hilary Term Weeks 3-8. For Part I you will be allocated a computer to yourself during scheduled sessions in the Statistics Department, and you may work collaboratively with others in these sessions. None of the work in Part I will be assessed, but instead will act as a foundation enabling you to work individually during Part II. This individual work will be assessed and will count...
Topics: Maths, Algebra, Numerical Analysis, Study Guides, Study Skills and Assignment Guidelines, Number...
Source: http://www.flooved.com/reader/1477
Flooved
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Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1485
Flooved
texts

eye 106

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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1505
Flooved
by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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eye 210

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If the ODE system has constant coef�cients, and its eigenvalues are real and distinct, then a natural choice for the fundamental matrix would be the one whose columns are the normal modes. There is another choice however which is suggested by (2) and which is particularly useful in showing how the solution depends on the initial conditions.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1449
Flooved
by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts

eye 135

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The work in the preceding note with fundamental matrices was valid for any linear homogeneous square system of ODE�s, x' = A(t) x . However, if the system has constant coef�cients, i.e., the matrix A is a constant matrix, the results are usually expressed by using the exponential matrix, which we now de�ne.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1447
Flooved
texts

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Topics: Maths, Analysis and Calculus, Complex Analysis, Riemann Surfaces, The Complex Logarithm, Mathematics
Source: http://www.flooved.com/reader/1515
Flooved
texts

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Differential inequality and uniqueness. We prove the uniqueness theorem for linear second-order differential equations with variable coef�cients.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1556
Flooved
texts

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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1519
Flooved
texts

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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1541
Flooved
texts

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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1545
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We study linear systems of n �rst-order differential equations. They are related to �rst-order matrix differential equations. When the corresponding matrix is constant, then the eigenvalues and the eigenfunctions of the matrix provide a useful framework to construct the general solution. The fundamental matrix is constructed as the exponential matrix.
Topic: Maths
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I am heading towards the spectral theory of self-adjoint compact operators. This is rather similar to the spectral theory of self-adjoint matrices and has many useful applications. There is a very e_ective spectral theory of general bounded but self-adjoint operators but I do not expect to have time to do this. There is also a pretty satisfactory spectral theory of non-self-adjoint compact operators, which it is more likely I will get to.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1579
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De�nition 8. An element K _ B(H), the bounded operators on a separable Hilbert space, is said to be compact (the old terminology was �totally bounded� and you might still see this) if the image of the unit ball is precompact, i.e. has compact closure � that is if the closure of ... is compact in H.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1578
In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is �transported� in a certain direction. Some important physical examples include the mass density �ow for an incompressible �uid, and the Boltzmann equation of kinetic theory. We discuss both linear transport equations and a famous nonlinear transport equation known as Burger�s equation. One of our major goals is to show that in contrast...
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1610
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The results from the previous lecture produced one solution to the Dirichlet problem... But how do we know that this is the only one? In other words, we need to answer the uniqueness question (6) from the previous lecture. The next theorem addresses this question. We �rst need to introduce some important spacetime domains that will play a role in the analysis
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1611
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The Fourier transform is a �continuous� version of the formula (1.0.1) for functions de�ned on the whole space R^n. Our goal is to write functions f de�ned on R^n as a superposition of di_erent frequencies. However, instead of discrete frequencies m, we will need to use �continuous frequencies�
Topic: Maths
Source: http://www.flooved.com/reader/1606
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We now switch to the simplest methods for integrating or di_erentiating a function from its function samples. A careful study of Taylor expansions reveals how accurate the constructions are.
Topics: Maths, Numerical Analysis, Mathematics
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The advantage of the bisection method is that it is guaranteed to co_verge to a root, by construction. On the other hand, convergence is rather show compared to the next 2 methods we now present. If there are several roots, the bisection method will converge toward one of them (we may not have no control over which root the method chooses.)
Topics: Maths, Numerical Analysis, Mathematics
Source: http://www.flooved.com/reader/1595
As we will soon see, the PDE (1.0.1) has a unique solution verifying (1.0.2) as long as f(x) is su_ciently di_erentiable and decays su_ciently rapidly as |x|_ �. Much like in the case of the heat equation, we will be able to construct the solution using an object called the fundamental solution.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1614
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Our goal in this section is to derive an integral representation formula for the solution to Poisson�s equation on domains _ _ R^n. Speci�cally, we will study the boundary value Poisson PDE
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1615
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I want to do a couple of �serious� applications of what we have done so far. There are many to choose from, and I will mention some more, but let me �rst consider the Diriclet problem on an interval. I will choose the interval [0, 2�] because we looked at it before.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
Source: http://www.flooved.com/reader/1581
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We will now prove our �rst result � Schur�s lemma. Although it is very easy to prove, it is fundamental in the whole subject of representation theory.
Topics: Maths, Algebra, Representation Theory, Mathematics
Source: http://www.flooved.com/reader/1654
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by Zdzislaw (Gustav) Meglicki
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In this module we are also going to take another look at quantum mechanics, asking if quantum mechanics can be simulated by jump Markov processes. There was a vigorous discussion about this in the literature.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1660
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Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1647
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Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R
Topics: Maths, Statistics and Probability, Mathematics
Source: http://www.flooved.com/reader/1646
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Discrete Random Variable: - de�ned by probability function (p.f.)
Topics: Maths, Statistics and Probability, Mathematics
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by Albert R. Meyer
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Mathematicians have developed a variety of measures and methods to help usunderstand how a random variable performs in comparison to its mean. The simplest and most widely used measure is called the variance of the random variable. The variance is a single value associated with the random variable that is large for random variables that are likely to deviate signi�cantly from the mean and that is small otherwise.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1733
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by Albert R. Meyer
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In this text, we will not devote much attention to the median. Rather, we will focus on the expected value, which is much more interesting and useful.
Topics: Maths, Mathematics
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In this section, we introduce a four step approach to questions of the form, �What is the probability that. . . ?� In this approach, we build a probabilistic model step-by-step, formalizing the original question in terms of that model.
Topics: Maths, Statistics and Probability, Probability, Statistics, Definitions and Theorems, Sampling...
Source: http://www.flooved.com/reader/1736
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Polar Coordinates Polar coordinates are a di_erent way of describing points in the plane. The polar coordinates (r, _) are related to the usual rectangular coordinates (x, y) by by x = r cos _, y = r sin _
Topics: Maths, Analysis and Calculus, Calculus, Mathematics
Source: http://www.flooved.com/reader/1805
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In this section, we consider the most common case � �nding a line which goes approximately through a set of data points.
Topics: Maths, Analysis and Calculus, Calculus, Mathematics
Source: http://www.flooved.com/reader/1806
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Normed space and Banach spaces. A normed space is a vector space V together with a function ||�|| : V_R such that ...
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
Source: http://www.flooved.com/reader/1786