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To get around the ambiguity of English, mathematicians have devised a special minilanguage for talking about logical relationships. This language mostly uses ordinary English words and phrases such as �or�, �implies�, and �for all�. But mathematicians endow these words with de�nitions more precise than those found in an ordinary dictionary.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1720
76
76
texts
eye 76
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Our goal in this lecture is to prove the following result: Theorem 1. Let n and k be nonnegative integers. Then the tensor product K(n) _ J(k) is an injective object in the category of unstable Amodules.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2216
86
86
2002
2002
by
Chiang C. Mei
texts
eye 86
favorite 0
comment 0
Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2622
81
81
2004
2004
by
Richard Stanley
texts
eye 81
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The main goal of this section is to give a formula in terms of _A(t) for r(A) and b(A) when K = R (Theorem 2.5). We �rst establish recurrences for these two quantities.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/982
136
136
texts
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Theorem (Schensted): Let � be a permutation of {1, 2, . . . , n} written in oneline notation. Let P and Q be the Standard Young Tableaux (SYT) in the image of the (RobinsonSchenstedKnuth) RSK algorithm, i.e. RSK(�) = (P, Q), with shapes sh(P) = sh(Q) = _. Then the length of the longest increasing subsequence in � equals the length of the �rst row of _ and the length of the longest decreasing subsequence in � equals the length of the �rst column of _.
Topics: Maths, Algebra, Statistics and Probability, Probability, Combinatorics, Mathematics
Source: http://www.flooved.com/reader/1269
116
116


by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
texts
eye 116
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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
93
93
texts
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In this lecture and in the next, we�ll briefly review secondorder PDEs. We�ll begin with one of the simplest of such PDEs: the Laplace equation.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Partial Differential Equations (PDEs),...
Source: http://www.flooved.com/reader/1340
145
145
texts
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We have seen that a central question about representations of quivers is whether a certain connected quiver has only �nitely many indecomposable representations. In the previous subsection it is shown that only those quivers whose underlying undirected graph is a Dynkin diagram may have this property. To see if they actually do have this property, we �rst explicitly decompose representations of certain easy quivers.
Topics: Maths, Algebra, Representation Theory, Mathematics
Source: http://www.flooved.com/reader/1658
135
135
texts
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We�ll see now how Schur�s lemma allows us to classify subrepresentations in �nite dimensional semisimple representations.
Topics: Maths, Algebra, Representation Theory, Mathematics
Source: http://www.flooved.com/reader/1655
87
87
Sep 16, 2004
09/04
by
Emma Carberry
texts
eye 87
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comment 0
Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Inverse Function Theorem,...
Source: http://www.flooved.com/reader/1105
86
86
Apr 13, 2004
04/04
by
Jeff Viaclovsky
texts
eye 86
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Elliptic regularity: Hitherto we have always assumed our solutions already lie in the appropriate C^(k,_) space and then showed estimates on their norms in those spaces. Now we will avoid this a priori assumption and show that they do hold a posteriori. This is important for the consistency of our discussion
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1332
73
73
texts
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In this lecture, we introduce the complementary slackness conditions and use them to obtain a primaldual method for solving linear programming.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1321
92
92


by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts
eye 92
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comment 0
In session on Phase Portraits, we described how to sketch the trajectories of a linear system x' = ax + by, y' = cx + dy where a, b, c, d constants.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1431
151
151


by
Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
texts
eye 151
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This is meant as a followup 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
144
144


by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts
eye 144
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comment 0
Separable Equations We will now learn our �rst technique for solving differential equation. An equation is called separable when you can use algebra to separate the two variables, so that each is completely on one side of the equation. We illustrate with some examples.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Nonlinear...
Source: http://www.flooved.com/reader/1426
115
115
texts
eye 115
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Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1513
135
135
texts
eye 135
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we show that the spherical representation z _ Z is conformal. This means that if l and m are two lines in the plane intersecting in z at an angle _, then the corresponding circles C and D through N and Z intersect Z at the same angle _
Topics: Maths, Analysis and Calculus, Complex Analysis, Mathematics
Source: http://www.flooved.com/reader/1504
117
117
texts
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In this course, we will mainly consider the case of free particles, in which V = 0 (i.e., the homogeneous Schr�odinger equation). In the case of free particles, there is an important family of solutions to (1.0.1), namely the free waves. The free wave solutions provide some important intuition about how solutions to the homogeneous Schr�odinger equation behave.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1607
100
100
texts
eye 100
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We will now study the Laplace and Poisson equations on a domain (i.e. open connected subset) _ _ R^n
Topic: Maths
Source: http://www.flooved.com/reader/1613
108
108
texts
eye 108
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Let�s discuss how the wave equation arises as an approximation to the equations of �uid mechanics. For simplicity, let�s only discuss the case of 1 spatial dimension.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
Source: http://www.flooved.com/reader/1600
302
302
texts
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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
190
190
texts
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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
104
104
2007
2007
by
Daniel Kleitman;Peter Shor
texts
eye 104
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The definition of the variance of f is equivalent to its interpretation as the mean square of f's deviation from its mean. However, it can be used to put an upper limit on the probability that a random variable takes on values with deviations greater than x� for any x greater than 1. The simplest and worst such bound is called Tchebyshev's inequality.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1873
87
87
2007
2007
by
Daniel Kleitman;Peter Shor
texts
eye 87
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We will describe an algorithm (discovered by V.Strassen) and usually called �Strassen�s Algorithm) that allows us to multiply two n by n matrices A and B, with a number of multiplications (and additions) which is a small multiple of ... , when n is of the form 2^k.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1865
97
97
2007
2007
by
Daniel Kleitman;Peter Shor
texts
eye 97
favorite 0
comment 0
To do a 2^k FFT mod a prime p you need to choose a prime p whose remainders include 2^kth roots of unity, and you need to find one such root that is not a 2^(k1)th root of unity
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1888
78
78
May 11, 2004
05/04
by
Alantha Newman
texts
eye 78
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1 Multi�ows and Disjoint Paths  Let G = (V,E) be a graph and let...
Topics: Maths, Optimization and Control, Optimization, Mathematics
Source: http://www.flooved.com/reader/1936
130
130
2009
2009
by
Kiran S. Kedlaya
texts
eye 130
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Hartshorne only treats �atness after cohomology (so see III.9) and doesn�t talk about descent at all. The EGA reference for �atness is EGA IV, part 2, �2. I�m not sure if descent is discussed at all in EGA, so I gave references to SGA 1 instead.
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2060
80
80
Sep 10, 2009
09/09
by
Jonathan Kelner
texts
eye 80
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This course requires linear algebra, so here is a quick review of the facts we will use frequently.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2023
167
167
2006
2006
by
Chiang C. Mei
texts
eye 167
favorite 0
comment 0
Topics: Physics, Special Relativity, General Relativity and Gravitation, Gravitational Waves, Wave...
Source: http://www.flooved.com/reader/2043
80
80
May 1, 2002
05/02
by
Michael Mitzenmacher
texts
eye 80
favorite 0
comment 0
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2008
101
101
May 15, 2002
05/02
by
G. Plaxton
texts
eye 101
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This section of the lecture examines some of the contemporary methods used to visualize computer networks. More information can be found at www.cybergeography.org which contains links to numberous internetrelated visualization projects.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/2009
108
108
2009
2009
by
Kiran S. Kedlaya
texts
eye 108
favorite 1
comment 0
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2066
246
246
texts
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1. K�hler Geometry
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2108
110
110
texts
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Sard�s Theorem: An extremely important notion in differential topology is that of general posi_tion or genercity. A particular map may have some horrible pathologies but often a nearby map has much nicer properties
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2129
84
84
texts
eye 84
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1. Symplectic Manifolds...
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2121
132
132
2007
2007
by
Kiran S. Kedlaya
texts
eye 132
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comment 0
In this unit, we describe a more intricate version of the sieve of Eratosthenes, introduced by Viggo Brun in order to study the Goldbach conjecture and the twin prime conjecture. It is most useful for providing lower bounds; for upper bounds, the Selberg sieve (to be introduced in the following unit) is much less painful.
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/2079
104
104
texts
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1. Homeomorphism Classification of Simply Connected Compact 4Manifolds...
Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Geometry, Geometry of Manifolds, Complex...
Source: http://www.flooved.com/reader/2119
108
108
texts
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In the early sixties Smales realized that many of the ideas of differential topology can be applied to aid in the study of PDEs and as part of this program he showed how to generalize Sard�s theorem to the in�nite dimensional case. First we need to introduce the correct kind of mappings of Banach manifolds.
Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
Source: http://www.flooved.com/reader/2135
90
90
May 8, 2003
05/03
by
Richard M. Dudley
texts
eye 90
favorite 0
comment 0
Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2155
147
147
texts
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In the previous class, we have introduced various concepts necessary for the study of EM waves in photonic crystal structures. We shall now use these concepts to explain various results such as: � Reconstruction of the permittivity pro�le. � The band diagrams for rectangular and triangular lattices. � ksurfaces for various eigenvalues. In particular, we will show an example of how a periodic structure can exhibit ksurfaces typicalof a negative refraction material (the concept of...
Topics: Physics, Acoustics, Optics and Waves, Electromagnetism and Electromagnetic Radiation, Waves,...
Source: http://www.flooved.com/reader/2572
94
94
2008
2008
by
Abhinav Kumar
texts
eye 94
favorite 0
comment 0
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2203
88
88
2008
2008
by
Abhinav Kumar
texts
eye 88
favorite 0
comment 0
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2206
101
101
2008
2008
by
Abhinav Kumar
texts
eye 101
favorite 0
comment 0
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2207
97
97
2008
2008
by
Abhinav Kumar
texts
eye 97
favorite 0
comment 0
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
Source: http://www.flooved.com/reader/2204
79
79
Apr 23, 2003
04/03
by
Richard M. Dudley
texts
eye 79
favorite 0
comment 0
Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2158
114
114
Apr 23, 2003
04/03
by
Richard M. Dudley
texts
eye 114
favorite 0
comment 0
Topics: Maths, Statistics and Probability, Statistics, Mathematics
Source: http://www.flooved.com/reader/2175
80
80
texts
eye 80
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In this lecture, we will revisit the relationship between unstable modules over the (mod 2) Steenrod algebra A and analytic functors from the category of F2 vector spaces to itself.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
Source: http://www.flooved.com/reader/2212
114
114
2002
2002
by
Chiang C. Mei
texts
eye 114
favorite 0
comment 0
Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2606
76
76
2002
2002
by
Chiang C. Mei
texts
eye 76
favorite 0
comment 0
Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
Source: http://www.flooved.com/reader/2600
116
116
texts
eye 116
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comment 0
A plasma is a gas in which an important fraction of the atoms is ionized, so that the electrons and ions are separately free. When does this ionization occur? When the temperature is hot enough. ...
Topics: Physics, Physics of Gases, Plasmas, and Electric Discharges, Physics of Plasmas and Electric...
Source: http://www.flooved.com/reader/2664
290
290
texts
eye 290
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Objectives: The GR �eld equations
Topics: Physics, Special Relativity, General Relativity and Gravitation, Classical General Relativity,...
Source: http://www.flooved.com/reader/2823
175
175
texts
eye 175
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comment 0
Topics: Physics, Electromagnetism and Electromagnetic Radiation, Classical Electromagnetism�, Physics
Source: http://www.flooved.com/reader/2846
126
126


by
Lawrence Evans;Mr. J. Edward Ladenburger
texts
eye 126
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Standing waves in a string. We consider again two harmonic waves with the same amplitude, wavelength and frequency, but now moving in opposite directions.
Topic: Maths
Source: http://www.flooved.com/reader/2903
252
252


by
Mr. Travis Byington;Lawrence Evans;Mr. Ryan Magee;Hao Zhang
texts
eye 252
favorite 1
comment 0
Superposition of Harmonic Waves The essential characteristic of energy transport by waves is that waves obey the superposition principle. This means that two waves in the same spatial region can interfere, rearranging the energy in space in a pattern often quite different from that of either wave alone. Since light propagates as a wave, we will analyze this phenomenon.We begin with a mathematical problem: How do we �nd the wave function for the combined wave resulting from interference of...
Topics: Physics, Acoustics, Optics and Waves, Optics�, Waves, Wave Optics, Interference and Coherence,...
Source: http://www.flooved.com/reader/2918
92
92
Nov 6, 1995
11/95
by
George Thompson
texts
eye 92
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These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the SeibergWitten invariants. Much of the necessary background material is given, including a crash course in topological �eld theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold theory. My main...
Topic: Maths
Source: http://www.flooved.com/reader/2970
108
108
texts
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With the foregoing preparation, we are now in a position to apply the classical analogy orcanonical quantization program to achieve the second quantization of theelectromagnetic field. As our starting point and for reference, we, once again, set forth the vacuum or microscopic Maxwell's equations in the time domain...
Topics: Physics, Acoustics, Optics and Waves, Quantum Physics, Optics�, Quantum Optics�, Physics
Source: http://www.flooved.com/reader/2990
123
123
Mar 2, 2000
03/00
by
R. Victor Jones
texts
eye 123
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We begin our discussion of optical pulse propagation 35 with a derivation of the nonlinear Schr�dinger (NLS) equation. To that end, we recall Equations [ VII23 ] and [ VII23 ] from the early lecture set entitled Nonlinear Optics I
Topics: Physics, Acoustics, Optics and Waves, Quantum Physics, Optics�, Quantum Optics�, Physics
Source: http://www.flooved.com/reader/2997
121
121
texts
eye 121
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Rotation Operators: Consider a scalar �eld f(r).Example 1: The temperature at point r After a rotation of the coordinate system, the same �eld is described by...
Topics: Physics, Mathematical Methods in Physics, Quantum Physics, Group Theory, Physics
Source: http://www.flooved.com/reader/2937
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267
texts
eye 267
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The purpose of this course is to answer these questions. We shall see that the second viewpoint above is the most useful: the �eld is primary and particles are derived concepts, appearing only after quantization. We will show how photons arise from the quantization of the electromagnetic �eld and how massive, charged particles such as electrons arise from the quantization of matter �elds. We will learn that in order to describe the fundamental laws of Nature, we must not only introduce...
Topic: Maths
Source: http://www.flooved.com/reader/3072
123
123
Oct 1, 2005
10/05
by
Ben Simons
texts
eye 123
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comment 0
The aim of this lecture is to explore the nature of the ground state and the character of the elementary excitation spectrum in the condensed phase.
Topics: Physics, Condensed Matter, Particle Physics and Fields, Quantum Physics, General Theory of Fields...
Source: http://www.flooved.com/reader/3069
289
289
Dec 28, 2010
12/10
by
Peter Dourmashkin
texts
eye 289
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We have already used Newton�s Second Law or Conservation of Energy to analyze systems like the blocspring system that oscillate. We shall now use torque and the rotational equation of motion to study oscillating systems like pendulums or torsional springs.
Topics: Physics, Classical Mechanics, Classical Mechanics of Discrete Systems, Fundamental Concepts, Simple...
Source: http://www.flooved.com/reader/3302
172
172
texts
eye 172
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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
110
110
2007
2007
by
Alan Guth;Barton Zwiebach
texts
eye 110
favorite 0
comment 0
Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Source: http://www.flooved.com/reader/3178
226
226
Sep 1, 2008
09/08
by
Peter Ouwehand
texts
eye 226
favorite 0
comment 0
Topic: Maths
Source: http://www.flooved.com/reader/3372
160
160
texts
eye 160
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comment 0
These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).
Topics: Maths, Algebra, Mathematics
Source: http://www.flooved.com/reader/3429
120
120
2004
2004
by
Emma Carberry
texts
eye 120
favorite 1
comment 0
Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Geometry, Differential Geometry,...
Source: http://www.flooved.com/reader/1106
119
119
texts
eye 119
favorite 0
comment 0
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1095
145
145
texts
eye 145
favorite 0
comment 0
Topics: Maths, Algebra, Number Theory, Mathematics
Source: http://www.flooved.com/reader/1122
114
114
Nov 1, 2003
11/03
by
Michael VaughanLee
texts
eye 114
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comment 0
In this course we are mainly concerned with �eld extensions, and �nding roots of polynomials, but before we get on to this we need to investigate the notion of characteristic.
Topics: Maths, Mathematics
Source: http://www.flooved.com/reader/1071
109
109
texts
eye 109
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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 nondifferentiable 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
121
121
texts
eye 121
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De�nition: An anitchain A of a poset P is a subset of elements of P such that for all x, y _ A, x _� y and y _� x. We denote the levels of a graded poset P as Pi where Pi = {x _ P : rank(x) = i}.
Topics: Maths, Algebra, Statistics and Probability, Probability, Combinatorics, Mathematics
Source: http://www.flooved.com/reader/1265
104
104
2004
2004
by
Richard Melrose
texts
eye 104
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comment 0
Lemma. MF is closed under �*(X _ X)�
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1494
83
83
2004
2004
by
Richard Melrose
texts
eye 83
favorite 0
comment 0
Density of the Schwartz Functions: To prove this we need to show the existence of a sequence of functions fn _ Sn for each f _ L2(R) such that fn f.
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1479
100
100


by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
texts
eye 100
favorite 0
comment 0
In summary, the procedure of sketching trajectories of the 2 _ 2 linear homogeneous system x_ = Ax, where A is a constant matrix
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
Source: http://www.flooved.com/reader/1439
115
115
2004
2004
by
Richard Melrose
texts
eye 115
favorite 0
comment 0
Topics: Maths, Differential Equations (ODEs & PDEs), Methods, Fourier Analysis, Mathematics
Source: http://www.flooved.com/reader/1489