Topic: Hubs and Authorities
Abstract: We introduce the idea of Hub and Authority rankings inside large scale networks with appropriate historical context, and introduce a new form for calculating Hubs and Authorities by turning a directed network into a bipartite network, along with efficient computational tools to evaluate these rankings in large scale networks.
Title: Zonotopes and Zonal Diagrams
Abstract: A zonotope is the Minkowski sum of a finite number of closed line segments in some Euclidean space. Peter McMullen's 1971 paper "On Zonotopes" describes the technique of zonal diagrams as a tool for studying combinatorial properties of zonotopes. Zonal diagrams, which are analogous to Gale diagrams for general polytopes and to central diagrams for centrally symmetric polytopes, have proved more effective than these techniques for the particular study of zonotopes.
In particular, zonal diagrams yield relationships between zonotopes with n zones of dimensions d and n-d, and lead to the enumeration of all combinatorial types of d-zonotopes with at most d+2 zones. In this master's talk, we will discuss zonotopes and zonal diagrams, describe the combinatorial results mentioned above, and briefly consider the relationship between zonotopes and hyperplane arrangements.
December 4, Thursday
Title: Material tensors and pseydotensors of weakly-textured polycrystals with orientation measure defined on the orthogonal group
Abstract: Material properties of polycrystalline aggregates should manifest the influence of crystallographic texture as defined by the orientation distribution function (ODF). A representation theorem on material tensors of weakly-textured polycrystals was established by Man and Huang (2012), by which a given material tensor can be expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a number of undetermined material parameters. Man and Huang's theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with (single) crystal symmetry defined by a proper point group. In the present study we consider ODFs defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover pseudotensors and polycrystals with crystal symmetry defined by any improper point group. This extension is important because many materials, including common metals such as aluminum, copper, iron, have their group of crystal symmetry being an improper point group. We present the restrictions on texture coefficients imposed by crystal symmetry for all the 21 improper point groups and we illustrate the extended representation theorem by its application to elasticity.
Title: On Artin's Conjecture for diagonal forms
Abstract: Emil Artin conjectured that any form of degree d in more than d^2 variables with coefficients in a p-adic field K has a nontrivial zero in K. Terjanian provided a counterexample to the conjecture, and many more have been found afterwards. However, in the case of diagonal forms, the result is known to hold for K=\mathbb{Q}_p. The conjecture for diagonal forms over arbitrary p-adic fields remains unproved. We investigate partial results.
Title: Using the method of layer potentials to solve a mixed boundary value problem
Abstract: Following the exposition by William McLean in his book Strongly Elliptic Systems and Boundary Integral Equations, we use the method of layer potentials to show that on a bounded Lipschitz domain, the mixed problem for Laplace’s equation is equivalent to a 2 × 2 system of boundary integral equations.
For Mathematics Doctoral Students who pre-registered through the DGS's Office.
For Mathematics Doctoral Students who pre-registered through the DGS's Office.
For Mathematics Doctoral Students who pre-registered through the DGS's Office.
For Mathematics Doctoral Students who pre-registered through the DGS's Office.
For Mathematics Doctoral students who pre-registered through the DGS's Office.
