4,892 research outputs found
Congruences modulo prime powers of Hecke eigenvalues in level
We continue the study of strong, weak, and -weak eigenforms introduced by
Chen, Kiming, and Wiese. We completely determine all systems of Hecke
eigenvalues of level modulo , showing there are finitely many. This
extends results of Hatada and can be considered as evidence for the more
general conjecture formulated by the author together with Kiming and Wiese on
finiteness of systems of Hecke eigenvalues modulo prime powers at any fixed
level. We also discuss the finiteness of systems of Hecke eigenvalues of level
modulo , reducing the question to the finiteness of a single eigenvalue.
Furthermore, we answer the question of comparing weak and -weak eigenforms
and provide the first known examples of non-weak -weak eigenforms.Comment: 28 pages; Minor revisio
Characterizing Behavioural Congruences for Petri Nets
We exploit a notion of interface for Petri nets in order to design a set of net combinators. For such a calculus of nets, we focus on the behavioural congruences arising from four simple notions of behaviour, viz., traces, maximal traces, step, and maximal step traces, and from the corresponding four notions of bisimulation, viz., weak and weak step bisimulation and their maximal versions. We characterize such congruences via universal contexts and via games, providing in such a way an understanding of their discerning powers
A theory of theta functions to the quintic base
Properties of four quintic theta functions are developed in parallel with
those of the classical Jacobi null theta functions. The quintic theta functions
are shown to satisfy analogues of Jacobi's quartic theta function identity and
counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of
Sign Formulas. The resulting library of quintic transformation formulas is used
to describe series multisections for modular forms in terms of simple matrix
operations. These efforts culminate in a formal technique for deducing
congruences modulo powers of five for a variety of combinatorial generating
functions, including the partition function. Further analysis of the quintic
theta functions is undertaken by exploring their modular properties and their
connection to Eisenstein series. The resulting relations lead to a coupled
system of differential equations for the quintic theta functions
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