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Abstract  

This paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups: a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szép product of an arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary, not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some new results concerning the strong actions of P 2 on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these repesentations and show that the structure of the representation can be explained by studying the binary representations of the numbers

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}$$ \end{document}
. We also raise some questions about strong representations of the polycyclic monoids on free abelian groups.

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representations of finite groups , (Springer-Verlag, 1977). Serre J.-P. Linear representations of finite groups 1977

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Dudek, W. A. and Trokhimenko, V. S. , Representations of Menger (2, n )-semi-groups by multiplace functions, Commun. Algebra 34 (2006), 259–274. MR 2194765 Trokhimenko V. S

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. , Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl. , 294 (2004), 399–411. Srivastava H M Some families of hypergeometric

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We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.

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Acta Mathematica Hungarica
Authors: Shigeki Akiyama, Tibor Borbély, Horst Brunotte, Attila Pethő and Jörg M. Thuswaldner

Summary We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

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representations in algebraic logic , Journal of Symbolic Logic, , 62 ( 3 ) ( 1997 ) p. 816 – 847 . [16] Hirsch , R. and Hodkinson , I. , Completions and complete representations

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Abstract  

We present several series and product representations for γ, π, and other mathematical constants. One of our results states that, for all real numbers µ s>0, we have

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma = \sum\limits_{k = 0}^\infty {\frac{1} {{(1 + \mu )^{k + 1} }}\sum\limits_{m = 0}^k {\left( {_m^k } \right)} \left( { - 1} \right)^m \mu ^{k - m} S(m),}$$ \end{document}
where S(m) = ∑k=1 1/2k+m.

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Abstract  

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C*-algebras and to the branching function systems of Bratteli and Jorgensen.

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Abstract  

We shall investigate several properties of the integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt}$$ \end{document}
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

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