ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In this paper, Gröbner rings are studied. In particular, we prove constructively that Gröbner rings are stably coherent. Moreover, we prove that a valuation ring is Gröbner if and only if it is both coherent and archimedean, answering (in the multivariate case) an open question.

DOI: 10.14232/actasm-013-514-3

AMS Subject Classification (1991): 13C10, 19A13, 14Q20, 03F65

Keyword(s): Gröbner bases, coherent rings, monomial orders, Syzygy modules

Received February 18, 2013, and in revised form June 13, 2013. (Registered under 14/2013.)

Abstract. We introduce a new type of `poly-Cauchy polynomials' defined by a certain generating function. These polynomials are generalizations of the classical Cauchy polynomials and poly-Cauchy numbers. We give their explicit expression and prove basic properties; the addition formula, iterated integral expression, differential relations and recurrence formula. We also give new type zeta functions associated with the poly-Cauchy polynomials.

DOI: 10.14232/actasm-013-761-9

AMS Subject Classification (1991): 05A15, 11B68, 11B75, 11M41

Keyword(s): generalized Bernoulli polynomials, poly-Bernoulli numbers, zeta functions, poly-Cauchy numbers, poly-Cauchy polynomials

Received February 13, 2013, and in revised form January 7, 2014. (Registered under 11/2013.)

Abstract. It is well known that for every congruence $\Theta\in\rm Con(\mathcal A)$ and each surjective homomorphism $h\colon\mathcal A\to\mathcal B$, the image $h(\Theta )=\{(h(a),h(b)) \mid(a,b)\in\Theta \}$ is a tolerance on $\mathcal B$. We study algebras and classes of algebras whose every tolerance is a homomorphic image of a congruence. In particular, we prove that every homomorphic image of a congruence on $\mathcal A$ is a congruence on $\mathcal B$ if and only if $\mathcal A$ is $3$-permutable. Let $\mathcal K$ be a class of algebras such that every tolerance on $\mathcal B\in\mathcal K$ is a homomorphic image of a congruence of an algebra that belongs to $\mathcal K$. Then $\mathcal K$ is tolerance factorable if and only if each $\mathcal B\in\mathcal K$ is factorable by the tolerance $\Theta\circ \Phi\circ \Theta $ for all $\Theta,\Phi\in \Con(\mathcal B)$. This result is extended for a strongly tolerance factorable variety.

DOI: 10.14232/actasm-012-861-x

AMS Subject Classification (1991): 08A30, 08B99

Keyword(s): tolerance, congruence, tolerance factorable algebra, strongly tolerance factorable variety, TImC-property, free algebra, $3$-permutability

Received December 19, 2012, and in revised form June 10, 2013. (Registered under 111/2012.)

Abstract. The quasivariety consisting of all the algebras of a given type that can be embedded as a subalgebra into some algebra which has a transitive automorphism group contains the variety of all idempotent algebras of the given type: every idempotent algebra \(\mathbf B\) can be embedded as a retract into an algebra which has a transitive automorphism group and which is simultaneously a subdirect power of \(\mathbf B\) and a direct limit of powers of \(\mathbf B\). This result applies in particular to lattices, bands, Steiner quasigroups and so on.

DOI: 10.14232/actasm-013-271-3

AMS Subject Classification (1991): 08A35, 08C15, 20B27; 06Bxx, 20Mxx, 20N05

Keyword(s): idempotent algebra, transitive automorphism group, symmetry, quasivariety

Received March 25, 2013, and in revised form November 14, 2013. (Registered under 21/2013.)

Abstract. The intrinsic structure of binary fields poses a challenging complexity problem from both hardware and software point of view. Motivated by applications to modern cryptography, we describe some simple techniques aimed at performing computations over binary fields using systems with limited resources. This is particularly important when such computations must be carried out by means of very small and simple machines. The algorithms described in the present paper provide an increased efficiency in computations, when compared to the previously known algorithms for the arithmetic over prime fields.

DOI: 10.14232/actasm-012-813-7

AMS Subject Classification (1991): 14G50, 11T55

Keyword(s): binary field, cryptography, limited system

Received August 27, 2012, and in revised form January 10, 2013. (Registered under 63/2012.)

Abstract. Let $P,P^\prime $ be preprojective and $I,I^\prime $ preinjective Kronecker modules. Working with the extension monoid product, we give conditions for the existence of short exact sequences of the form $0\to P\to I\to I^\prime \to0$ (and dually for $0\to P^\prime \to P\to I\to0$). We show that the existence of these short exact sequences is equivalent with the existence of certain short exact sequences of preinjective (respectively preprojective) Kronecker modules, hence they obey the combinatorial rule described in [SzSz2].

DOI: 10.14232/actasm-012-315-9

AMS Subject Classification (1991): 16G20

Keyword(s): Kronecker algebra, Kronecker module, extension monoid product

Received August 30, 2012, and in revised form April 23, 2013. (Registered under 65/2012.)

Abstract. We investigate the group of normalized units of the group algebra $\mathbb Z_p^eG$ of a finite abelian $p$-group $G$ over the ring $\mathbb Z_p^e$ of residues modulo $p^e$ with $e\geq1$.

DOI: 10.14232/actasm-013-510-1

AMS Subject Classification (1991): 16S34, 16U60, 20C05

Keyword(s): group algebra, unitary unit, symmetric unit

Received January 31, 2013, and in revised form March 5, 2013. (Registered under 10/2013.)

Abstract. We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements $\gamma_1,\gamma_2$ in this group is Diophantine if there is a number $A$ such that a product of length $l$ of elements of the set $\{\gamma_1,\gamma_2,\gamma_1^-1,\gamma_2^-1\}$ is either the unit element or of distance at least $A^-l$ from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension~$0$.

DOI: 10.14232/actasm-013-757-6

AMS Subject Classification (1991): 22E25, 30C15, 11C08

Keyword(s): solvable Lie groups, Diophantine property, roots of polynomials, zeros of polynomials

Received January 24, 2013, and in revised form March 3, 2013. (Registered under 7/2013.)

Abstract. The Volterra operator $V\colon f(x)\to\int _0^xf(t) dt$ on $C[0,1]$ or $L^p(0,1)$ ($1\leq p< \infty $) is characterized as the unique bounded linear operator on the space satisfying the algebraic condition $(*)$ $[S,V]=V^2$, $Ve=Se$, where $S$ is the multiplication operator $f(x)\to xf(x)$, and $e$ is the function with constant value $1$. Similarly, if $S_\alpha $ is the multiplication operator $f\to\alpha f$ on $C[0,1]$, where $\alpha $ is a given injective real-valued $C[0,1]$-function of bounded variation vanishing at $0$, then the Stieltjes--Volterra operator $f(x)\to\int _0^xf(t) d\alpha(t)$ on $C[0,1]$ is characterized as the unique bounded linear operator on the space satisfying the above condition with $S=S_\alpha $. For $1< p< \infty $, the Riemann--Liouville semigroup is characterized as the unique regular semigroup $V(\cdot )$ on $\msbm C^+$ acting in $L^p(0,1)$, whose boundary group's type is less than $\pi $, and for which $V:=V(1)$ satisfies Relation~$(*)$.

DOI: 10.14232/actasm-012-570-7

AMS Subject Classification (1991): 47D03, 26A42, 47G10, 97I50

Keyword(s): Volterra operator, Volterra Communication Relation, Stieltjes--Volterra operator, $C_0$-semigroup, regular semigroup, boundary group, type (of $C_0$-semigroup), Riemann--Liouville semigroup, Uniqueness theorem

Received September 7, 2012, and in revised form February 14, 2013. (Registered under 70/2012.)

Abstract. We investigate Fubini-type properties of the space of functions of bounded Hardy--Vitali-type $p$-variation. This leads us to consider mixed norm spaces of bivariate functions whose linear sections have bounded $p$-variation in the sense of Wiener.

DOI: 10.14232/actasm-012-363-z

AMS Subject Classification (1991): 26A45, 26B35, 46E35

Keyword(s): Hardy--Vitali variation, $p$-variation, mixed norm spaces, embeddings

Received December 28, 2012. (Registered under 113/2012.)

Abstract. In the spaces $S^p$ of functions of several variables, $2\pi $-periodic in each variable, we study the approximative properties of operators $A^\triangle _\varrho,r$ and $P^\triangle _\varrho,s$, which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes $S^pH_\omega $.

DOI: 10.14232/actasm-012-837-8

AMS Subject Classification (1991): 42B05, 26B30, 26B35

Keyword(s): space $S^p$, classes $H_\omega $, linear methods

Received October 30, 2012. (Registered under 87/2012.)

Abstract. A one-dimensional version of the Poisson summation formula for functions of bounded variation due to R.M. Trigub is extended to the multivariate case under minimal assumptions on functions.

DOI: 10.14232/actasm-013-518-5

AMS Subject Classification (1991): 42B10, 42B05, 42B35, 26B30

Keyword(s): Fourier integral, trigonometric series, bounded variation

Received March 5, 2013, and in revised form October 5, 2013. (Registered under 18/2013.)

Abstract. We determine when the average of two isometries on a Banach space of analytic vector valued functions is a projection. We also characterize the generalized bi-circular projections on a Banach space of absolutely continuous vector valued functions.

DOI: 10.14232/actasm-013-781-8

AMS Subject Classification (1991): 30D05

Keyword(s): bi-circular projections, Hardy spaces, isometry, convex combination of isometries

Received May 21, 2013, and in revised form February 14, 2014. (Registered under 31/2013.)

Abstract. A result of K. Stroethoff and D. Zheng provides on the polydisk a necessary and sufficient condition for the Toeplitz product $T_fT_{\overline g}$ to be bounded on the unweighted Bergman space. We consider the same question on the vector weighted Bergman space of the unit polydisc.

DOI: 10.14232/actasm-012-777-1

AMS Subject Classification (1991): 32A36, 47B35

Keyword(s): Bergman spaces, reproducing kernel, Toeplitz operator

Received May 2, 2012, and in final form August 20, 2014. (Registered under 27/2012.)

Abstract. In this paper we give a characterization for the uniform exponential stability of evolution families $\{\Phi(t,t_0)\}_{t\geq t_0}$ on $\mathbb R_+$ that are not a priori required to be bounded, using the hypothesis that the pairs of function spaces $(L^1(X),L^\infty (X))$ and $(\mathcal C_{00}(\mathbb R_+,X),\mathcal C(\mathbb R_+,X))$ are admissible to the evolution families.

DOI: 10.14232/actasm-012-562-2

AMS Subject Classification (1991): 34D05, 34D09

Keyword(s): evolution family, admissibility, uniform exponential stability, asymptotic property

Received August 21, 2012. (Registered under 62/2012.)

Abstract. In the present paper we give some characterizations for the stability and instability of the evolution families by using the Lyapunov function, in Banach spaces. These results are a generalization of those obtained by N.U. Ahmed in [1] for the case of $C_0$-semigroups.

DOI: 10.14232/actasm-012-566-4

AMS Subject Classification (1991): 34D05, 34D09, 47D06, 93D20

Keyword(s): evolution family, Lyapunov operator inequality, exponential stability, exponential instability

Received August 31, 2012, and in revised form April 25, 2014. (Registered under 66/2012.)

Abstract. We analyse a four-dimensional compartmental system that describes the spread of ectoparasites and a disease carried by them in a population. We identify three threshold parameters that determine which of the four potential equilibria exist. These parameters completely characterize the stability properties of the equilibria and also the global behaviour of solutions. We provide a detailed description of the global attractor in each possible scenario. The key mathematical tools of the proofs are Lyapunov--LaSalle theory, persistence theory, Poincaré--Dulac criteria and unstable manifolds. In the most complicated case, the global attractor consists of four equilibria and various heteroclinic orbits connecting those equilibria, forming a two-dimensional manifold in the phase space.

DOI: 10.14232/actasm-013-004-y

AMS Subject Classification (1991): 37B25, 37C70, 92D30

Keyword(s): compartmental system, ectoparasites, global dynamics, Lyapunov function, persistence, global attractor

Received January 9, 2013. (Registered under 4/2013.)

Abstract. The aim of the paper is to generalize two fundamental theorems of K.~Tandori. The coefficient sequences in his theorems are classical monotone decreasing. We moderate the classical monotonicity to locally almost monotonicity assumption.

DOI: 10.14232/actasm-013-777-5

AMS Subject Classification (1991): 42A20, 42A16

Keyword(s): orthogonal series, coefficient conditions, locally almost monotonicity

Received April 25, 2013, and in revised form June 3, 2013. (Registered under 27/2013.)

Abstract. For $1 \leq p < \infty $, we denote by $\AC ^p [0, 1]$ the space of all absolutely continuous functions on the interval $[0, 1]$ whose derivatives belong to $L^p [0, 1]$. Under the assumption that ${\rm AC}^p [0, 1]$ is equipped with the norm $\| f \|_\sigma = |f(0)| + \| f' \|_L^p$ or $\| f \|_m = \max\{ |f(0)|, \| f' \|_L^p \}$, we characterize the surjective linear isometries on ${\rm AC}^p [0, 1]$.

DOI: 10.14232/actasm-012-327-3

AMS Subject Classification (1991): 46B04; 46E15

Keyword(s): linear isometry, absolutely continuous function

Received September 24, 2012, and in revised form December 3, 2012. (Registered under 77/2012.)

Abstract. We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Omega,X)$, with $\Omega$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.

DOI: 10.14232/actasm-012-060-2

AMS Subject Classification (1991): 47B38; 47B15, 46B99, 47A65

Keyword(s): generalized bi-circular projections, projections as combination of finite order operators, reflections, isometric reflections, isometries

Received August 10, 2012, and in revised form October 10, 2012. (Registered under 60/2012.)

Abstract. If $T$ is a Hilbert space contraction, then $T^*nT^n\mathop\to ^sA$, where $A$ is a nonnegative contraction. The strong limit $A$ is a projection if and only $T=G\oplus V$, where $G$ is a strongly stable contraction and $V$ is an isometry. This article is an expository paper on the class of contractions $T$ for which $A$ is a projection. After surveying such a class, it is shown that it is quite a large class. Indeed, it includes (i) all contractions whose adjoint has property PF, and also (ii) all contractions whose intersection of the continuous spectrum of its completely nonunitary direct summand with the unit circle has Lebesgue measure zero. Some new questions are investigated as well. For instance, is $A$ a projection for every biquasitriangular contraction $T$? If so, then every contraction not in class $\mathcal C_{00}$ has a nontrivial invariant subspace.

DOI: 10.14232/actasm-013-255-6

AMS Subject Classification (1991): 47A45; 47A15

Keyword(s): partially isometric contractions, biquasitriangular operators, invariant subspaces

Received January 17, 2013, and in final version March 11, 2013. (Registered under 5/2013.)

Abstract. Let $ R$ be a power bounded operator, and let $\cal M$ be its invariant subspace such that $ R|_\cal M$ is similar to a unilateral shift of finite multiplicity and $P_\cal M^\perp R|_\cal M^\perp $ is similar to a $C_0$-contraction. Then $R$ is similar to a contraction.

DOI: 10.14232/actasm-014-267-4

AMS Subject Classification (1991): 47A45, 47A65, 47B99

Keyword(s): power bounded operator, similarity, contraction, unilateral shift, $C_0$-contraction

Received February 21, 2014, and in final form September 1, 2014. (Registered under 17/2014.)

Abstract. We present sufficient conditions in order (the space of) a Riesz operator $T$ in a Hilbert space $H$ have a Jordan--Schur basis with respect to a scalar product equivalent to the original one. This is related to Schur's lemma for a compact operator, which is an extension of Schur's classical theorem on unitary triangularization in a finite dimensional space. The finite dimensional case is also studied.

DOI: 10.14232/actasm-012-092-8

AMS Subject Classification (1991): 47B06, 47B40; 15A21

Keyword(s): Riesz operator, Schur's lemma, unitary triangularization, equivalent scalarproduct, Jordan--Schurbasis, spectraloperator, resolutionof the identity

Received October 10, 2012. (Registered under 92/2012.)

Abstract. In 1979, T. Ando posed the following question: suppose $E$ and $F$ are two projection-valued measures defined on an algebra $\Sigma $ of subsets of $\Omega $, which satisfy $$ \|E(\Delta )-F(\Delta )\|\le1-\delta, \Delta\in \Sigma, $$ for some $\delta >0$. Does there exist a unitary operator $u$ such that $u^*E(\Delta )u=F(\Delta )$ for all $\Delta\in \Sigma $? He knew that the answer was affirmative if both measures were strongly $\sigma $-additive and maximal (i.e. $E$ and $F$ have cyclic vectors). In this note, we show that the answer is also affirmative if both measures take values in a common finite von Neumann algebra.

DOI: 10.14232/actasm-012-080-1

AMS Subject Classification (1991): 47B15, 47C15

Keyword(s): spectral measure, unitary equivalence, finite algebra

Received October 10, 2012. (Registered under 80/2012.)

Abstract. A densely defined operator $T$ acting between Hilbert spaces is shown to be closed if and only if $T^*T$ and $TT^*$ are both selfadjoint operators on the corresponding Hilbert spaces. This is an extension of the classical von Neumann theorem [vonNeumann1930] on the selfadjointness of $T^*T$ whenever $T$ is closed.

DOI: 10.14232/actasm-013-283-x

AMS Subject Classification (1991): 47B25, 47B65

Keyword(s): Hilbert space, closed operators, selfadjoint operators, von Neumann theorem

Received May 8, 2013, and in revised form May 16, 2013. (Registered under 33/2013.)

Abstract. We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions with general weights. As applications, we also characterize bounded and compact weighted composition operators between weighted Banach space of analytic functions and Bloch-type spaces.

DOI: 10.14232/actasm-013-502-9

AMS Subject Classification (1991): 47B38; 47B33

Keyword(s): differentiation operators, weighted composition operators, weighted Banach space of analytic functions, Bloch-type spaces

Received January 3, 2013, and in final form January 29, 2013. (Registered under 2/2013.)

Abstract. Some recent papers formulated sufficient conditions for the decomposition of matrix variances [LZ-PD, PD-TG]. A statement was that if we have one or two observables, then the decomposition is possible. In this paper we consider an arbitrary finite set of observables and we present a necessary and sufficient condition for the decomposition of the matrix variances.

DOI: 10.14232/actasm-013-789-z

AMS Subject Classification (1991): 81Q10

Keyword(s): matrix

Received July 1, 2013, and in final form December 12, 2013. (Registered under 39/2013.)

Abstract. This paper deals with the structure of the boundary of a closed starshaped set in Euclidean space. Based on related results, we derive continuity properties of radial functions of starshaped sets and also new results on dispensable points of such sets. This yields new basic insights in the geometry of starshaped sets, since the notion of dispensable points of starshaped sets is a direct generalization of that of extreme points of convex sets.

DOI: 10.14232/actasm-013-275-5

AMS Subject Classification (1991): 52A07, 52A20, 52A30

Keyword(s): dispensable point, extreme point, infinity cone, opposite cone, radial function, starshaped set, upper semicontinuity

Received April 16, 2013, and in revised form September 12, 2013. (Registered under 25/2013.)