We prove a necessary and sufficient condition for a function being a polynomial function over a finite, commutative, unital ring. Further, we give an algorithm running in quasilinear time that determines whether or no...
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We prove a necessary and sufficient condition for a function being a polynomial function over a finite, commutative, unital ring. Further, we give an algorithm running in quasilinear time that determines whether or not a function given by its function table can be represented by a polynomial, and if the answer is yes then it provides one such polynomial. (C) 2017 Elsevier B.V. All rights reserved.
There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In...
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There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.
We describe the unary polynomial functions on the non-solvable groups G with SL(n, q) less than or equal to G less than or equal to GL(n, q) and on their quotients G/Y with Y less than or equal to Z(G), and we compute...
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We describe the unary polynomial functions on the non-solvable groups G with SL(n, q) less than or equal to G less than or equal to GL(n, q) and on their quotients G/Y with Y less than or equal to Z(G), and we compute the size of the inner automorphism near-ring I(G/Y). We compare this near-ring to the endomorphism near-ring E(G/Y), and we obtain a full characterization of those G and Y for which I(G/Y) = E(G/Y) holds. For the case Y= polynomial functions, this characterization yields that we have E(G) = I(G) if and only if G=SL(n, q). We investigate the automorphism near-ring A(G), and we show that for all non-solvable groups G with SL(n, q) less than or equal to Gless than or equal toGL(n, q), we have I(G)=A(G). Our results are based on a description of the polynomial functions on those non-abelian finite groups G that satisfy the following conditions: G' = G", G/Z(G) is centerless, and there is no normal subgroup N of G with G' boolean AND Z(G) < N < G'.
A congruence preserving function on a subdirect product of two finite Mal'cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projec...
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A congruence preserving function on a subdirect product of two finite Mal'cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A x B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A x B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A x B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.
polynomial functions on abelian groups are studied from the point of view of spectral synthesis. It is proved that the torsion free rank of an abelian group is finite if and only if each complex generalized polynomial...
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A map from a group G to itself is called polynomial if it can be written as a product consisting of constant functions, the identity, and the function g -> g(-1). We discuss several concepts related to polynomialit...
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ISBN:
(纸本)9789806560604
A map from a group G to itself is called polynomial if it can be written as a product consisting of constant functions, the identity, and the function g -> g(-1). We discuss several concepts related to polynomiality on linear groups. Furthermore we give a necessary topological condition for a map to be polynomial. As a consequence, we prove that transposition is in general not polynomial.
Linear modulation methods are highly sensitive to high power amplifier nonlinearities. An adaptive predistortion lineariser using polynomial functions is described. An important problem faced in optimisation of nonlin...
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Linear modulation methods are highly sensitive to high power amplifier nonlinearities. An adaptive predistortion lineariser using polynomial functions is described. An important problem faced in optimisation of nonliner circuits such as predistortion linearisers is that of convergence into a local minimum. This results from the nonquadratic shape of the objective functions. A solution to this problem is presented using a postdistorter with similar polynomial functions as the predistorter. Sample of the signals at the power amplifier output and predistorter input are demodulated. These demodulated signals are used for estimation of the postdistorter polynomial coefficients. The objective functions used in the estimation are quadratic functions of the coefficients being estimated resulting in a rapid convergence to the global minimum. The coefficients of the predistorter polynomials are set from those of the postdistorter. Computer simulation results for the proposed lineariser are presented. These results show 50 dB spectrum spreading improvement. Further, unlike previously reported linearisers, the proposed lineariser is insensitive to the demodulator gain and phase imperfections. The performance of the proposed lineariser structured with fifth order polynomial functions is also compared with that for earlier polynomial type linearisers.
Relevancy transformation operators (RET operators) have been widely used in fuzzy systems modelling and the construction of weighted aggregation functions. Several construction methods ofREToperators based on differen...
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Relevancy transformation operators (RET operators) have been widely used in fuzzy systems modelling and the construction of weighted aggregation functions. Several construction methods ofREToperators based on different aggregation functions such as t-norm, t-conorm and copula, have been proposed. In this paper, the attention is paid to the expression of RET operators, which is an important feature from an application the point of view. polynomial RET operators are introduced as those RET operators in the form of polynomial functions of two variables. A complete characterisation of polynomial RET operators of degree less than 4 are presented.
Let R be a finite commutative ring. The set F(R) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units F(R)(x) is just the set of all unit-valued polynomial functions....
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Let R be a finite commutative ring. The set F(R) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units F(R)(x) is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on R[x]/(x(2)) = R[alpha], the ring of dual numbers over R, and show that the group P-R(R[alpha]), consisting of those polynomial permutations of R[alpha] represented by polynomials in R[x], is embedded in a semidirect product of F(R)(x) by the group P(R) of polynomial permutations on R. In particular, when R = F-q, we prove that P-Fq(F-q[alpha]) congruent to P(F-q) (sic)(theta) F(F-q)(x). Furthermore, we count unit-valued polynomial functions on the ring of integers modulo p(n) and obtain canonical representations for these functions.
The classical result of L. Szekelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Szekelyhidi's result may be generalized to eq...
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The classical result of L. Szekelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Szekelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F(x + y) - F(x) - F(y) = yf(x) + xf(y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1-2):143-154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.
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