Topology - people.math.harvard.edu

A Mathematical Theory of Communication

J. W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit, can store one bit of information. N such devices can store N bits, since the total numberof possible states is 2N and log 2 2 N = N. If the base 10 is used the units may be called decimal digits. Since log2 M = log10 M log10 2 = 3: 32log10 M;

  Devices, Early

Group Theory and the Rubik's Cube

Again, we’re going to rewrite this using new symbols. Let mean multiplication, and let e= 1, a= 2, b= 4, and c= 3. Then, the multiplication table for (Z=5Z) looks like e a b c e e a b c a a b c e b b c e a c c e a b Notice that this is exactly the same as the table for addition on Z=4Z!

  Table, Multiplication, Multiplication table

Theory of functions of a real variable.

In Chapter II I do the basics of Hilbert space theory, i.e. what I can do without measure theory or the Lebesgue integral. The hero here (and perhaps for the first half of the course) is the Riesz representation theorem. Included is the spectral theorem for …

  Basics

Sequences and Series: An Introduction to Mathematical Analysis

he walked half of the remaining distance, so now he was 3/4 of the way to the grocery. In the following ten minutes he walked half of the remaining ... of the nautilus shell, the number of seeds in consecutive rows of a sunflower, and many natural …

  Shell, Half

SOME FUNDAMENTAL THEOREMS IN MATHEMATICS

FUNDAMENTAL THEOREMS Theorem: I(V(J)) = p J. The theorem is due to Hilbert. A simple example is when J= hpi= hx2 2xy+ y2iis the idealJgeneratedbypinR[x;y];thenV(J) = fx= ygandI(V(J)) istheidealgeneratedby x y. Forliterature,see[294]. 13. Cryptology An integer p>1 is primeif 1 and pare the only factors of p. The number kmod pis the ...

  Fundamentals, Theorem

Lie algebras - people.math.harvard.edu

8 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA A+B+ 1 2 A2 +AB+ 1 2 B2 − 1 2 (A+B+···)2 = A+B+ 1 2 [A,B]+··· where [A,B] := AB−BA (1.1) is the commutator of Aand B, also known as the Lie bracket of Aand B.

  Algebra

Unit 5: Change of Coordinates

B= S 1v. Theorem: If T(x) = Ax is a linear map and S is the matrix from a basis change, then B = S 1AS is the matrix of T in the new basis B. Proof. Let y = Ax. The statement [y] B= B[x] Bcan be written using the last theorem as S 1y = BS 1x so that y = SBS 1x. Combining with y = Ax, this gives B = S 1AS. 5.4. If two matrices A;B satisfy B = S ...

Higher Algebra - people.math.harvard.edu

to Yoneda’s lemma, this property determines the space Zup to homotopy equivalence. Moreover, since the functor X7!K(X) takes values in the category of commutative rings, the topological space Z is automatically a commutative ring object in the homotopy category H of topological spaces.

  Yoneda

Lines and Planes in R3

Lines and Planes in R3 A line in R3 is determined by a point (a;b;c) on the line and a direction ~v that is parallel(1) to the line. The set of points on this line is given by fhx;y;zi= ha;b;ci+ t~v;t 2Rg This represents that we start at the point (a;b;c) and add all scalar multiples of the vector ~v.

  Line, Points, Point of

Lecture 9: Partial derivatives

Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then ∂ ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y), where y is considered a constant. It is called partial derivative of f with respect to x. The partial derivative with respect to y is defined similarly. We also use the short hand notation ...

  Derivatives

Spectral and Algebraic Graph Theory

\Algebraic Graph Theory" by Chris Godsil and Gordon Royle. Other books that I nd very helpful and that contain related material include \Modern Graph Theory" by Bela Bollobas, \Probability on Trees and Networks" by Russell Llyons and Yuval Peres, \Spectra of Graphs" by Dragos Cvetkovic, Michael Doob, and Horst Sachs, and ...

  Theory, Graph, Algebraic, Graph theory, Algebraic graph theory

Spectral Graph Theory and its Applications

What I’m Skipping Matrix-tree theorem. Most of algebraic graph theory. Special graphs (e.g. Cayley graphs). Connections to codes and designs. Lots of work by theorists.

  Applications, Theory, Graph, Algebraic, Spectral, Algebraic graph theory, Spectral graph theory and its applications

A Short Tutorial on Graph Laplacians, Laplacian Embedding ...

The spectral graph theory studies the properties of graphs via the eigenvalues and eigenvectors of their associated graph matrices: the adjacency matrix and the graph Laplacian and its variants. Both matrices have been extremely well studied from an algebraic point of view. The Laplacian allows a natural link between discrete

  Theory, Tutorials, Graph, Embedding, Algebraic, Graph theory, Laplacian, Tutorial on graph laplacians, Laplacian embedding

EmilyRiehl - Mathematics

It is difficult to preview the main theorems in category theory before developing ... The complete graph on n vertices is characterized by the property that graphhomomorphismsG !K ... Manyfamiliarvarietiesof“algebraic”objects—suchasgroups,rings,modules,pointed

  Theory, Graph, Algebraic

1.10 Matrix Representation of Graphs

The matrix representation of a graph is often convenient if one intends to use a computer to obtain some information or solve a problem concerning the graph. This kind of representation of a graph is conducive to study properties of the graph by means of algebraic methods. Let σ= 1 2 ··· n i1 i2 ··· in be a permutation of the set {1,2 ...

  Graph, Algebraic

500 - OCLC

Including elementary number theory; analytic number theory; algebraic number theory; geometry of numbers; probabilistic number theory; specific fields of numbers (e.g., rational numbers, algebraic numbers, real numbers, complex numbers) Class rational functions in 512.9; class real functions, complex functions in 515; class numerical methods in 518

  Theory, Algebraic

LINEAR ALGEBRA METHODS IN COMBINATORICS

theory have been the winners. In this volume, an explicit Ramsey graph construction (Sec-tions 4.2, 5.7) serves as simple illustration of the phenomenon. Some of the much more complex examples known to be directly relevant to the theory of computing are mentioned brie y, along with a number of open problems in this area (Section 10.2).

  Theory, Graph

Introduction to Modern Algebra - Clark University

CONTENTS v 3.9 Real and complex polynomial rings R[x] and C[x]. . . . . . . . . . . . . . . .87 3.9.1 C[x] and the Fundamental Theorem of Algebra ...

  Algebra