On aq-Analogue of thep-Adic Log Gamma Functions and Related Integrals

Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a combinatorial criterion for representations of C to admit a theory of Grobner bases. From this, we obtain a criterion for noetherianity of representations. The second gives a combinatorial criterion for a general "rationality" result for Hilbert series of representations of C. This criterion connects to the theory of formal languages, and makes essential use of results on the generating functions of languages, such as the transfer-matrix method and the Chomsky-Schutzenberger theorem. 
Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example: we give a new, more robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb), and a family of natural generalizations, are noetherian; we give an easy proof of a generalization of the Lannes-Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of $\Delta$-modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

Gr\"obner methods for representations of combinatorial categories

Given a category C of a combinatorial nature, we study the following fundamental question: how does the combinatorial behavior of C affect the algebraic behavior of representations of C? We prove two general results. The first gives a combinatorial criterion for representations of C to admit a theory of Grobner bases. From this, we obtain a criterion for noetherianity of representations. The second gives a combinatorial criterion for a general “rationality” result for Hilbert series of representations of C. This criterion connects to the theory of formal languages, and makes essential use of results on the generating functions of languages, such as the transfer-matrix method and the Chomsky–Schutzenberger theorem. Our work is motivated by recent work in the literature on representations of various specific categories. Our general criteria recover many of the results on these categories that had been proved by ad hoc means, and often yield cleaner proofs and stronger statements. For example: we give a new, more robust, proof that FI-modules (originally introduced by Church–Ellenberg–Farb), and a family of natural generalizations, are noetherian; we give an easy proof of a generalization of the Lannes–Schwartz artinian conjecture from the study of generic representation theory of finite fields; we significantly improve the theory of ∆modules, introduced by Snowden in connection to syzygies of Segre embeddings; and we establish fundamental properties of twisted commutative algebras in positive characteristic.

/pdf/grobner-methods-for-representations-of-combinatorial-2zht5p7koh.pdf

Gröbner methods for representations of combinatorial categories

An extension of the formalism of quantum mechanics to the case where the canonical variables and functions are valued in a field of p‐adic numbers is considered. A new p‐adic integral calculus is used for the realization of the Gauss representation in p‐adic quantum mechanics.

p‐adic quantum mechanics with p‐adic valued functions

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, in. Eluding in optical code-division multiple-access communication systems and in constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback. In this paper we construct optimal constant weight cyclically permutable codes with length n, weight w, and a minimum Hamming distance 2w-2. Some of these codes coincide with the well-known design called a difference family. Some of the constructions use combinatorial structures with other applications in coding. >

Constructions for optimal constant weight cyclically permutable codes and difference families

We discuss the properties of $p$-adic analogues of the logistic and Smale horseshoe maps, and adapt them to form possible practical pseudo-random number generators. The properties of these practical modifications are studied in detail in the case $p=2$.

/pdf/p-adic-chaos-and-random-number-generation-55jkwjvgrx.pdf

{$p$}-adic chaos and random number generation

We give a characterization of algebraic functions over a field of positive characteristic and we then deduce that the Hadamard product of two algebraic series in several commutative variables over a field of positive characteristic is again algebraic.

Chris F. Woodcock

Papers

{$p$}-adic chaos and random number generation

Algebraic Functions Over a Field of Positive Characteristic and Hadamard Products

Convolutions on the ring of p-adic integers

The Noether numbers for cyclic groups of prime order

On the transcendence of certain series