We examine the connection of two graph parameters, the size of a minimum feedback arcs set and the acyclic disconnection. A feedback arc set of a directed graph is a subset of arcs such that after deletion the graph becomes acyclic. The acyclic disconnecti ...
We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation x2 + y2 + z2 = xyz modulo prime numbers greater than 7. The proof uses Euler characteristic and an enumeration of the short cycles in these gr ...
We investigate generalizations along the lines of the Mordell-Lang conjecture of the author's p-adic formal Manin-Mumford results for n-dimensional p-divisible formal groups F. In particular, given a finitely generated subgroup (sic) of F(Q(p)) and a close ...
In this thesis we consider the problem of estimating the correlation of Hecke eigenvalues of GL2 automorphic forms with a class of functions of algebraic origin defined over finite fields called trace functions. The class of trace functions is vast and inc ...
Information retrieval (IR) systems such as search engines are important for people to find what they need among the tremendous amount of data available in their organization or on the Internet. These IR systems enable users to search in a large data collec ...
Recent years have witnessed a rise in real-world data captured with rich structural information that can be better depicted by multi-relational or heterogeneous graphs.
However, research on relational representation learning has so far mostly focused on th ...
Recent years have witnessed a rise in real-world data captured with rich structural information that can be conveniently depicted by multi-relational graphs. While inference of continuous node features across a simple graph is rather under-studied by the c ...
Let G be a drawing of a graph with n vertices and e > 4n edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvatal, Newborn, Szemeredi and Leighton, the nu ...
An abstract topological graph (briefly an AT-graph) is a pair A = (G, X) where G = (V, E) is a graph and X. E2 is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from X crosses exac ...
In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that ...
An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turan problem, first studied by Pach and Tardos, asks for the maximum number ex(
We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Polya-Vinogradov range). We then derive applications to the second moment of cus ...
A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one ...
This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A \emph{simple topological graph} is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any t ...
It is shown that fora constant t is an element of N, every simple topological graph on n vertices has 0(n) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of th ...
We construct "generalized Heegner cycles" on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-functio ...
Let K be a global field of characteristic not 2. The embedding problem for maximal tori in a classical group G can be described in terms of algebras with involution. The aim of this paper is to give an explicit description of the obstruction group to the H ...
This contribution draws practical implications of a recently published estimation of the tensile ductility in laminated composites made of two ductile materials, typically metals or alloys, which harden as both the strain and the strain-rate increase. To t ...
We study sums over primes of trace functions of l-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothe ...
We consider the Node-weighted Steiner Forest problem on planar graphs. Demaine et al. showed that a generic primal-dual algorithm gives a 6-approximation. We present two different proofs of an approximation factor of~3. Then, we draw a connection to Goem ...
