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The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a nice representation of the modular form associated to each elliptic curve. Here we provide explicit representations of the modular forms associated to certain Legendre form elliptic curves as linear combinations of quotients of Dedekind’s eta-function. We also give congruences for some of the modular forms’ coefficients in terms of Gaussian hypergeometric functions.
We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically.
We prove a transformation equation satisfied by a set of holomorphic functions with rational Fourier coefficients of cardinality arising from modular forms. This generalizes the classical transformation property satisfied by modular forms with rational coefficients, which only applies to a set of cardinality for a given weight.
The Johnson graph is defined as the graph whose vertices are the -subsets of the set , where two vertices are adjacent if they share exactly elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS for short) do not seem to have been studied before. We give some necessary conditions and some sufficient conditions for a graph to be JIS, including: in a JIS graph, any two maximal cliques share at most two vertices; all trees, cycles, and complete graphs are JIS; disjoint unions and Cartesian products of JIS graphs are JIS; every JIS graph of order is an induced subgraph of for some . This last result gives an algorithm for deciding if a graph is JIS. We also show that all JIS graphs are edge move distance graphs, but not vice versa.
We consider a weighted least squares finite element approach to solving convection-dominated elliptic partial differential equations, which are difficult to approximate numerically due to the formation of boundary layers. The new approach uses adaptive mesh refinement in conjunction with an iterative process that adaptively adjusts the least squares functional norm. Numerical results show improved convergence of our strategy over a standard nonweighted approach. We also apply our strategy to the steady Navier–Stokes equations.
We explore the properties of , the graph of equivalence classes of zero-divisors of a commutative Noetherian ring . We determine the possible combinations of diameter and girth for the zero-divisor graph and the equivalence class graph , and examine properties of cut-vertices of .
Holte introduced a matrix as a transition matrix related to the carries obtained when summing numbers base . Since then Diaconis and Fulman have further studied this matrix proving it to also be a transition matrix related to the process of -riffle shuffling cards. They also conjectured that the matrix is totally nonnegative. In this paper, the matrix is written as a product of a totally nonnegative matrix and an upper triangular matrix. The positivity of the leading principal minors for general and is proven as well as the nonnegativity of minors composed from initial columns and arbitrary rows.
For graphs and with totally ordered vertex sets, a function mapping the vertex set of to the vertex set of is an order-preserving homomorphism from to if it is nondecreasing on the vertex set of and maps edges of to edges of . In this paper, we study order-preserving homomorphisms whose target graph is the complete graph on vertices. By studying a family of graphs called nonnesting arc diagrams, we are able to count the number of order-preserving homomorphisms (and more generally the number of order-preserving multihomomorphisms) mapping any fixed graph to the complete graph .
This paper describes a permutation notation for the Weyl groups of type and . The image in the permutation group is presented as well as an analysis of the structure of the group. This description enables faster computations in these Weyl groups which will prove useful for a variety of applications.
Let be the number of quintic number fields whose Galois closure has Galois group and whose discriminant is bounded by . By a conjecture of Malle, we expect that for some constant . The best upper bound currently known is , and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is . Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for quartic fields in terms of a similar norm equation.
The Legendre family of elliptic curves has the remarkable property that both its periods and its supersingular locus have descriptions in terms of the hypergeometric function . In this work we study elliptic curves and elliptic integrals with respect to the hypergeometric functions and , and prove that the supersingular -invariant locus of certain families of elliptic curves are given by these functions.