4,892 research outputs found

    Congruences modulo prime powers of Hecke eigenvalues in level 11

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    We continue the study of strong, weak, and dcdc-weak eigenforms introduced by Chen, Kiming, and Wiese. We completely determine all systems of Hecke eigenvalues of level 11 modulo 128128, 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 11 modulo 99, reducing the question to the finiteness of a single eigenvalue. Furthermore, we answer the question of comparing weak and dcdc-weak eigenforms and provide the first known examples of non-weak dcdc-weak eigenforms.Comment: 28 pages; Minor revisio

    Characterizing Behavioural Congruences for Petri Nets

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    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

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    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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