85
85

Oct 22, 2009
10/09

Oct 22, 2009
by
Jonathan Kelner

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Today we�ll go over some of the details from last class and make precise many details that were skipped. We�ll then go on to prove Fritz John�s theorem. Finally, we will start discussing the Brunn-Minkowski inequality.

Topics: Maths, Linear Algebra and Geometry, Algebra, Geometry, Number Fields, Minkowski�s Theorem on...

Source: http://www.flooved.com/reader/2033

174
174

Sep 17, 2009
09/09

Sep 17, 2009
by
Jonathan Kelner

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Today�s lecture covers three main parts: � Courant-Fischer formula and Rayleigh quotients � The connection of _2 to graph cutting � Cheeger�s Inequality

Topics: Maths, Linear Algebra and Geometry, Graph Theory, Optimization and Control, Vectors and Matrices,...

Source: http://www.flooved.com/reader/2019

360
360

Sep 17, 2009
09/09

Sep 17, 2009
by
Louis Scharf

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This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual.

Topics: Maths, Linear Algebra and Geometry, Algebra, Vectors and Matrices, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1669

139
139

Apr 23, 2009
04/09

Apr 23, 2009
by
Richard Melrose

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Topics: Maths, Linear Algebra and Geometry, Differential Equations (ODEs & PDEs), Vectors and Matrices,...

Source: http://www.flooved.com/reader/1580

108
108

2009
2009

2009
by
Kiran S. Kedlaya

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eye 108

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2065

102
102

2009
2009

2009
by
Vera Mikyoung Hur

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An important result of mechanics is that a system of masses attached in (damped or undamped) springs is stable. A similar result is in network theory. In these notes, we study the differential equation of the form y" + py' + qy = f(t), where p, q are constants and f(t) represents the external forces.

Topics: Maths, Linear Algebra and Geometry, Differential Equations (ODEs & PDEs), Linear Algebra,...

Source: http://www.flooved.com/reader/1540

175
175

2009
2009

2009
by
Kiran S. Kedlaya

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eye 175

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Projective Spaces and Varieties and...

Source: http://www.flooved.com/reader/2069

144
144

2009
2009

2009
by
Kiran S. Kedlaya

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eye 144

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Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Elementary Number Theory,...

Source: http://www.flooved.com/reader/2070

117
117

2009
2009

2009
by
Kiran S. Kedlaya

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eye 117

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2074

189
189

2009
2009

2009
by
Kiran S. Kedlaya

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We�re going to use the language of category theory freely. Fortunately, it�s easy to learn because it corresponds naturally to the way you (hopefully) already think about mathemat_ical objects. (I could give a reference, but in fact you should be �ne just looking these things up in Wikipedia.)

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2052

111
111

2009
2009

2009
by
Kiran S. Kedlaya

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eye 111

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2066

123
123

2009
2009

2009
by
Kiran S. Kedlaya

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eye 123

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2056

134
134

2009
2009

2009
by
Kiran S. Kedlaya

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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

119
119

2009
2009

2009
by
Kiran S. Kedlaya

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In this lecture, we specialize to the case of the global sections functor for sheaves on a locally ringed space, and thus obtain the de�nition of sheaf cohomology.

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2072

365
365

Mar 6, 2008
03/08

Mar 6, 2008
by
Sanjoy Mahajan

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Do you ever walk through a proof, understand each step, yet not believe the theorem, not say �Yes, of course it�s true�? The analytic, logical, sequential approach often does not convince one as well as does a carefully crafted picture. This di_erence is no coincidence. The analytic, sequential portions of our brain evolved with our capacity for language, which is perhaps 105 years old. Our pictorial, Gestalt hardware results from millions of years of evolution of the visual system and...

Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Analysis and Calculus,...

Source: http://www.flooved.com/reader/1919

92
92

2008
2008

2008
by
Abhinav Kumar

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eye 92

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2191

108
108

2008
2008

2008
by
Abhinav Kumar

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eye 108

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comment 0

Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Algebraic Geometry, Minimal...

Source: http://www.flooved.com/reader/2192

85
85

2008
2008

2008
by
Abhinav Kumar

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eye 85

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2199

120
120

2008
2008

2008
by
Abhinav Kumar

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eye 120

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Riemann-Roch Theorem, Mathematics

Source: http://www.flooved.com/reader/2189

98
98

2008
2008

2008
by
Abhinav Kumar

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eye 98

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comment 0

Topics: Maths, Linear Algebra and Geometry, Algebra, Geometry, Differential Geometry, Algebraic Geometry,...

Source: http://www.flooved.com/reader/2201

97
97

2008
2008

2008
by
Abhinav Kumar

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eye 97

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comment 0

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2208

95
95

2008
2008

2008
by
Abhinav Kumar

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eye 95

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2195

83
83

2008
2008

2008
by
Abhinav Kumar

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eye 83

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2198

85
85

2008
2008

2008
by
Abhinav Kumar

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eye 85

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2200

98
98

2008
2008

2008
by
Abhinav Kumar

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eye 98

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comment 0

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2205

79
79

2008
2008

2008
by
Abhinav Kumar

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eye 79

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2202

91
91

2008
2008

2008
by
Abhinav Kumar

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eye 91

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2197

110
110

2008
2008

2008
by
Abhinav Kumar

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eye 110

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2190

77
77

2008
2008

2008
by
Abhinav Kumar

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eye 77

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2210

94
94

2008
2008

2008
by
Abhinav Kumar

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eye 94

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2206

102
102

2008
2008

2008
by
Abhinav Kumar

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eye 102

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comment 0

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2204

133
133

2008
2008

2008
by
Abhinav Kumar

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eye 133

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2196

98
98

2008
2008

2008
by
Abhinav Kumar

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eye 98

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comment 0

Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2203

108
108

2008
2008

2008
by
Abhinav Kumar

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eye 108

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2207

140
140

2008
2008

2008
by
Abhinav Kumar

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eye 140

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2194

82
82

2008
2008

2008
by
Abhinav Kumar

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eye 82

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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics

Source: http://www.flooved.com/reader/2193

232
232

Sep 28, 2004
09/04

Sep 28, 2004
by
Alan Edelman

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This handout provides the essential elements needed to understand �nite random matrix theory. A cursory observation should reveal that the tools for in�nite random matrix theory are quite di_erent from the tools for �nite random matrix theory. Nonetheless, there are signi�cantly more published applications that use �nite random matrix theory as opposed to in�nite random matrix theory. Our belief is that many of the results that have been historically derived using �nite random...

Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics

Source: http://www.flooved.com/reader/975

82
82

Sep 28, 2004
09/04

Sep 28, 2004
by
Per-Olof Persson

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In this section, the distributions of the largest eigenvalue of matrices in the _-ensembles are studied. Histograms are created �rst by simulation, then by solving the Painlev�e II nonlinear di_erential equation.

Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics

Source: http://www.flooved.com/reader/977

123
123

Sep 23, 2004
09/04

Sep 23, 2004
by
Brian Sutton

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In class, we saw the connection between the so-called Hermite matrix and the semi-circular law. There is actually a deeper story that connects the classical random matrix ensembles to the classical orthogonal polynomials studied in classical texts such as [1] and more recent monographs such as [2]. We illuminate part of this story here. The website www.mathworld.com is an excellent reference for these polynomials and will prove handy when completing the exercises. In any computational...

Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Orthogonal, Hermitian and...

Source: http://www.flooved.com/reader/978

133
133

Sep 16, 2004
09/04

Sep 16, 2004
by
Raj Rao

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eye 133

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Topics: Maths, Linear Algebra and Geometry, Vectors and Matrices, Matrices, Mathematics

Source: http://www.flooved.com/reader/974

90
90

Sep 16, 2004
09/04

Sep 16, 2004
by
Emma Carberry

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eye 90

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Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Inverse Function Theorem,...

Source: http://www.flooved.com/reader/1105

146
146

Jul 30, 2004
07/04

Jul 30, 2004
by
Thomas R. Covert

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Because any enumeration of a set of simple roots is related to any other enumeration by some �nite number of permutations, the relationship between any two Cartan matrices for a root system is an isomorphism by the conjugate product of permutation matrices.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1667

90
90

Jul 18, 2004
07/04

Jul 18, 2004
by
Andrew Brooke-Taylor

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We shall associate a �Cartan matrix� to the system _ _ _ and derive some properties of this matrix. An abstract Cartan matrix will be any square matrix with this list of properties. It turns out that an abstract Cartan matrix essentially determines the root system. In this paper we will work toward making this statement precise and proving it. The problem of classi�cation of root systems is reduced then to the classi�cation of the Cartan matrices.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1662

101
101

Jun 2, 2004
06/04

Jun 2, 2004
by
Juha Valkama

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Let V be a Euclidean space, i.e. a real �nite dimensional linear space with a symmetric positive de�nite inner product _,_.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1666

102
102

May 10, 2004
05/04

May 10, 2004
by
Eric Broder

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Let V be a Euclidean space, that is a �nite dimensional real linear space with a symmetric positive de�nite inner product _,_.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1664

182
182

May 10, 2004
05/04

May 10, 2004
by
Matthew Herman

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We can now define an abstract root system in a Eucledian space.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1661

120
120

May 10, 2004
05/04

May 10, 2004
by
Philip Brocoum

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Now we are ready to introduce the notion of a re�ection in a Eu_clidean space. A re�ection in V is a linear mapping s : V_ V which sends some nonzero vector _ _ V to its negative and �xes pointwise the hyperplane H_ orthogonal to _. To indicate this vector, we will write s = s_. The use of Greek letters for vectors is traditional in this context.

Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics

Source: http://www.flooved.com/reader/1665

117
117

2004
2004

2004
by
Emma Carberry

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eye 117

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Topics: Maths, Linear Algebra and Geometry, Analysis and Calculus, Geometry, Differential Geometry,...

Source: http://www.flooved.com/reader/1107

132
132

2004
2004

2004
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

110
110

2004
2004

2004
by
Emma Carberry

texts

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eye 110

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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Mathematics

Source: http://www.flooved.com/reader/1104