This is the first semester of the basic graduate algebra course. It will include an introduction to groups, rings, and modules, culminating in the structure theorem for finitely generated modules over a principal ideal domain, with applications to the structure theory of linear operators on a finite dimensional vector space.
Discrete Mathematics With Graph Theory 3rd Edition.pdf
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This course is an introduction to algebraic number fields, that is, finite dimensional extensions of the rational numbers. We will cover all the standard topics like the finiteness theorems for the class number and the units. We will not only cover these topics from the classical abstract viewpoint, but students will also be introduced to the MAGMA computer package with which actual computations in, and about, number fields can be done. Note that this is the first of a two semester course. In the second semester we will move on to more advanced topics, in particular class field theory.
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
This course will survey application of the fundamental group in areas of current research interest in low dimensional topology within our department. The emphasis will be on geometric methods in the study of infinite groups with applications to theory of braids (and other arrangements), knots and links, three dimensional manifolds and the mapping class group of surfaces and related moduli spaces. Concepts studied will include aspects of infinite group theory such as the lower central series, word and conjugacy problems, and the Fox differential calculus. Other (related) topics include braid group representations and classical knot and link invariants (e.g., Alexander polynomials). Of particular interest will be the lower central and solvable series of a group and computations for semi-direct products.
The applications chosen for this incarnation of the course will have little overlap (primarily braid theory) with the Fall 2003 course taught by Prof. Cohen although there will be a significant overlap in tools -- particularly the free differential calculus and the lower central series.
This course is a continuation of the fall 2004 Algebraic Number Theory course. If you have had a basic Algebraic Number Theory course at another time you are welcome to join the class. The course will start with an introduction to complete fields (the p-adics) and move on to other advanced topics including some class field theory. We will again do some computations using the MAGMA computer program.
Classical electrodynamics will be studied in two different mathematical contexts. First, we introduce the Maxwell equations with an understanding of how they arise from physical experiment. We will concentrate mostly on the linear equations and their behavior in periodic media (photonic crystals). The mathematical context for this is the spectral theory of unbounded differential operators and compact integral operators. Second, we will go into the fundamental physical principles axiomatically and understand the Maxwell-Lorentz space-time relation. The mathematical context for this is differential geometry.
This course provides a basic introduction to finite dimensional, continuous time, deterministic control systems at the graduate level. The course is intended for PhD students in applied mathematics and engineering graduate students with a solid background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory course emphasizing controllability and stabilization.
In the last 25 years a theory has emerged that deals with the non spatial localization of the Fourier transform when the frequency is well localized. Wavelet theory provides a framework to construct an orthonormal basis that has localization properties both in space and frequency. It uses translation and dilation to zoom into a given part of the spatial variable. Wavelet theory lies in the intersection among harmonic analysis, signal processing, image processing, and scientific calculation.
This course will present an overview of wavelet theory and its applications. It will deal with the following topics: introduction to Fourier series and integrals; Gabor analysis, or the windowed Fourier transform; Hilbert spaces, orthonormal basis and Riesz basis; the continuous wavelet transform in one and several dimensions; multi-resolution approximations; Haar wavelets and Daubechies wavelets; construction of wavelets in one dimension; and wavelet sets.
There are three parts of the course. The first gives an introduction into standard semigroup theory with view to the solution of linear time-independent initial-value problems, in particular parabolic and hyperbolic linear partial differential equations (PDE) with time-independent coefficients. The second gives an introduction into abstract linear evolution equations of the `hyperbolic' type, which also covers the case of time-dependent coefficients. The final part introduces into the solution of non-linear (quasi-linear) evolution equations of the hyperbolic type, which in particular allow the treatment of quasi-linear PDE, thereby covering the majority of PDE relevant in physics and engineering. Lecture notes for the whole course will be online available during the class. For the first part also any standard text on semigroup can be used, like Pazy, A. 1983, Semigroups of linear operators and applications to partial differential equations, Springer, New York.
ICHD3 labels cross-references between headache disorders through its numerical diagnostic codes. We exploited this consistency computationally to generate the above graphical representation without the use of natural language-processing algorithms. To make sure no parsing errors exist, the data were manually reviewed and verified by the author.
graph theory: Edge coloring of a complete graph is one of my favorite pictorial proofs. Eulerian tour. Euler characteristic (of a planar graph) is related. What else? Shortest path would require discussion of algorithm. Hall's marriage theorem is surprising and neat but I don't think I can prove it from scratch in an hour.
combinatorics: Start with $N \choose m$ and then stars-and-bars for sure. What are your favorite (elementary) examples? Can Burnside's lemma be proved from scratch (at highschool level, without group theory) in an hour?
The $2$-dimensional case of Sperner's lemma is a good introduction to graph theory. The proof is self-contained aside from using the Handshaking lemma, which is also easy to show. Take a triangle and subdivide it into several smaller triangles. Call a triangle minimal if it contains no other triangles. If whenever two minimal triangles touch, they share a side, then we call this subdivision a triangulation. (This is to avoid the possibility that the side of one minimal triangle is part of a side of another triangle.)
There are five minimal triangles that have a vertex of each color. Sperner's lemma states that there must always be a minimal triangle with a node of each color, and in fact, there must be an odd number of such triangles. To see this, we will construct a graph from this triangulation. We do so by associating a vertex to each face, including the outside face, and draw an edge between two vertices if the side shared by their corresponding triangles has two different colors. See the picture below.
The internal vertices can have degree either $0$, $2$, or $3$. The degree is $0$ if the nodes of the corresponding triangle all have the same color, $2$ if the nodes use two different colors, and $3$ if the nodes all use different colors. We want to show that there are an odd number of internal vertices of odd degree. The Handshaking Lemma tells us that a graph must have an even number of vertices with odd degree, so if we can show that the outside vertex has odd degree, then the lemma follows. For example, the outside vertex in our picture has degree $7$.
To see this, look at any side of the original graph, say the red-green side. Moving from the red node to the green node, that side contributes $1$ whenever the colors switch from red to green or green to red. Since we start with a red node and end with a green node, there must be an odd number of switches. Hence, each side of the original triangle makes an odd contribution to the degree of the outside vertex so the outside vertex has odd degree. By the Handshaking Lemma, Sperner's Lemma follows.
You did probability before. Even without cryptography, I think it's fascinating that one can pretend to have probability using a pseudorandom number generating algorithm. The simplest algorithms are very simple, but still depend on basic results in modulo arithmetic, finite fields, etc. You could teach some basic statistical tests on PRNG quality if you want.
I've always been a fan of the coin problem. It's basically a generalization of the question, "what is the largest score in American football that cannot be achieved (say, ignoring safeties)?" It's got a built-in hook with the story of the "McNugget numbers". It very naturally motivates $\gcd(a_1, \ldots, a_n)$ and you can discuss existence of the Frobenius number. As I recall there's a nice graphical proof of the formula for two coins. The topic typically uses some basic number theory like Bezout's lemma, but if anything that's a feature and not a bug. 2ff7e9595c
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