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13
Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
"... We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integrodifferential algebras. The algebraic treatment of boundary problems brings up ..."
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Cited by 9 (8 self)
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We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integrodifferential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integrodifferential operators, is used for both stating and solving linear boundary problems. The other structure, called integrodifferential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integrodifferential polynomials for generating an automated proof establishing a canonical simplifier for integrodifferential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.
Weighted lattice polynomials
 Discrete Mathematics
"... We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete ..."
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Cited by 8 (8 self)
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We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. Key words: weighted lattice polynomial, lattice polynomial, bounded distributive lattice, discrete Sugeno integral. 1
On the number of polynomial functions on nilpotent groups of class 2
, 1998
"... In the case of a nilpotent group of class 2 a certain invariant of the group, the length defined by S. D. Scott can be used to determine the number of polynomial functions on the group. Sharp upper and lower bounds for this invariant are determined. It is shown how the length of a group can be deter ..."
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Cited by 5 (1 self)
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In the case of a nilpotent group of class 2 a certain invariant of the group, the length defined by S. D. Scott can be used to determine the number of polynomial functions on the group. Sharp upper and lower bounds for this invariant are determined. It is shown how the length of a group can be determined from a set of generating elements and the length of all pgroups up to order p 4 is determined as an application. 1 Introduction All the groups we are treating here are finite. The centre of a group G will be denoted by Z(G). For the definitions of nilpotency and commutators read the corresponding chapters in [Hup67]. We deal with polynomial functions on groups of nilpotency class 2, which will be described in the next theorem. The general definitions of a polynomial and a polynomial function can be found in [LN73]. Polynomial functions on a group G with pointwise addition and composition form a nearring, the polynomial nearring on G, denoted by P (G). # This study was kindly sup...
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dicks ..."
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Cited by 2 (0 self)
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
When Are Weak Permutation Polynomials Strong?
"... . For a commutative finite ring with identity R, the two definitions of permutation polynomial in several indeterminates over R coincide if and only if R is a direct sum of finite fields. Amer. Math. Soc. 1991 Mathematics Subject Classification: 11T06, 13M10; Secondary: 13B25. All rings considered a ..."
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Cited by 2 (2 self)
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. For a commutative finite ring with identity R, the two definitions of permutation polynomial in several indeterminates over R coincide if and only if R is a direct sum of finite fields. Amer. Math. Soc. 1991 Mathematics Subject Classification: 11T06, 13M10; Secondary: 13B25. All rings considered are commutative and finite, and ring always means ring with identity. A polynomial f 2 R[x] is said to be a permutation polynomial (abbreviated PP) if the function it defines on R through substitution, r 7! f(r), is a permutation. This notion has been generalized to polynomials in several indeterminates in two different ways. We will characterize the rings for which the two coincide. (For quotient rings of the integers this has been done by Kaiser and Nobauer [1].) We write the cartesian product of n copies of a set S as S !n? , to avoid confusion with the power S n , if S happens to be an ideal. An mtuple of polynomials (f 0 ; f 1 ; : : : ; f m\Gamma1 ) in n indeterminates over R indu...
An automated confluence proof for an infinite rewrite system parametrized over an integrodifferential algebra
 2010. Proceedings of ICMS 2010, LNCS
"... In this paper we present an automated proof for the confluence of a rewrite system for integrodifferential operators (given in Table 1). We also outline a generic prototype implementation of the integrodifferential polynomials—the key tool for this proof—realized using the Theorema system. With it ..."
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Cited by 2 (1 self)
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In this paper we present an automated proof for the confluence of a rewrite system for integrodifferential operators (given in Table 1). We also outline a generic prototype implementation of the integrodifferential polynomials—the key tool for this proof—realized using the Theorema system. With its generic functor mechanism—detailed in Section 2—we are able to provide a formalization of the theory of integrodifferential
Description of the project: Clones on groups 1
, 2006
"... Abstract: For an arbitrary algebraic structure (universal algebra), polynomial functions are those that can be obtained from the constant functions and the projection operations using the operations of this algebraic structure. This concept is a true generalization of the wellknown concept of polyn ..."
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Abstract: For an arbitrary algebraic structure (universal algebra), polynomial functions are those that can be obtained from the constant functions and the projection operations using the operations of this algebraic structure. This concept is a true generalization of the wellknown concept of polynomial functions on commutative rings (e.g., polynomial functions on the real numbers) and has been studied since the 50s of the last century. “Rings are particular algebraic structures whose operations include the operations of a group. Such algebras have a group reduct or, using a dual point of view, they are expanded groups. We will mainly focus on them. We aim to solve open problems in universal algebra that are connected to the following: Is there a finite group that has more than countably many expansions such that their sets of polynomial functions are pairwise distinct? Our starting point is a conjecture by P. Idziak: On a fixed finite set whose size is squarefree there exist only finitely many clones containing a group operation and all constant functions. While E. Aichinger and the author already proved this conjecture to
A topological property of polynomial functions on GL(2, R)
, 2006
"... A function from the group GL(2, R) to itself is called polynomial if it can be written as some product of constant functions, the identity function, and the function that maps every element to its inverse. We give a necessary topological condition for a function to be polynomial. As a consequence we ..."
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A function from the group GL(2, R) to itself is called polynomial if it can be written as some product of constant functions, the identity function, and the function that maps every element to its inverse. We give a necessary topological condition for a function to be polynomial. As a consequence we prove that transposition is not polynomial.