ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. By H. Strietz, 1975, and G. Czédli, 1996, the complete lattice $\Equ(A)$ of all equivalences is four-generated, provided the size $|A|$ is an accessible cardinal. Results of I. Chajda and G. Czédli, 1996, G. Takách, 1996, T. Dolgos, 2015, and J. Kulin, 2016, show that both the lattice $\Quo(A)$ of all quasiorders on $A$ and, for $|A|\leq\aleph _0$, the lattice $\Tran(A)$ of all transitive relations on $A$ have small generating sets. Based on complicated earlier constructions, we derive some new results in a concise but not self-contained way.

DOI: 10.14232/actasm-016-056-2

AMS Subject Classification (1991): 06B99

Keyword(s): quasiorder lattice, lattice of preorders, minimum-sized generating set, four-generated lattice, lattice of transitive relations

Received October 4, 2016, and in revised form October 22, 2016. (Registered under 56/2016.)

Abstract. The Swing Lemma, due to G. Grätzer for slim semimodular lattices and extended by G. Czédli, G. Grätzer, and H. Lakser for all planar semimodular lattices, describes the congruence generated by a prime interval in an efficient way. Here we present a new, direct proof of this lemma, which is shorter than the earlier ones. Also, motivated by the Swing Lemma and mechanical pinball games with flippers, we construct an online game called Swing Lattice Game.

DOI: 10.14232/actasm-016-036-3

AMS Subject Classification (1991): 06C10

Keyword(s): Swing Lemma, Swing Lattice Game, semimodular lattice, planar lattice, lattice congruence

Received July 15, 2016, and in revised form March 14, 2017. (Registered under 36/2016.)

Abstract. We present two identities in two variables under which every lattice admitting a unary operation becomes a uniquely complemented distributive lattice. We show that the distributive law can be easily syntactically derived from these two identities.

DOI: 10.14232/actasm-016-514-2

AMS Subject Classification (1991): 06C15, 06D05

Keyword(s): lattice with complementation, uniquely complemented lattice, distributive lattice, free lattice

Received March 9, 2016, and in final form June 19, 2016. (Registered under 14/2016.)

Abstract. It is known that varieties of semilattice-ordered semigroups are in one-to-one correspondence with the ordered pairs $(\rho,[ ])$ where $\rho $ is a fully invariant congruence on the free semigroup on a countably infinite set and $[ ]$ is a $\rho $-admissible closure operator. We find all admissible closure operators for varieties of left normal bands. Using the obtained results we describe all varieties of semilattice-ordered left normal bands by admissible closure operators. We solve the identity problem for all varieties of semilattice-ordered normal bands.

DOI: 10.14232/actasm-016-777-4

AMS Subject Classification (1991): 06F05, 08B15, 20M07

Keyword(s): variety, semilattice-ordered semigroup, normal band, admissible closure operator

Received April 29, 2016, and in revised form October 20, 2016. (Registered under 27/2016.)

Abstract. We investigate the existence of simultaneous representations of real numbers $x$ in bases $1< q_1<\cdots < q_r$, $r\geq2$, with a finite digit set $A\subset{\msbm R}R $. We prove that if $A$ contains both positive and negative digits, then each real number has infinitely many common expansions. In general the bases depend on $x$. If $A$ contains the digits $-1,0,1$, then there exist two non-empty open intervals $I,J$ such that for any fixed $q_1\in I$ each $x\in J$ has common expansions for some bases $q_1<\cdots < q_r$.

DOI: 10.14232/actasm-015-080-0

AMS Subject Classification (1991): 11A63, 11B83

Keyword(s): simultaneous Rényi expansions, interval filling sequences

Received November 11, 2015, and in revised form February 1, 2016. (Registered under 80/2015.)

Abstract. A \emph{cyclic polygon} is a convex $n$-gon inscribed in a circle. If, in addition, one of its sides is a diameter of the circle, then the polygon will be called \emph{Thalesian}. Up to permutation, a Thalesian $n$-gon is determined by the \emph{lengths} of its non-diametric sides. It is also determined by the \emph{distances} of its non-diametric sides from the center of its circumscribed circle. We prove that the Thalesian $n$-gon in general can be constructed with straightedge and compass neither from these lengths if $n\geq4$, nor from these distances if $n\geq5$. An analogous statement for the constructibility of cyclic $n$-gons from the side lengths was found by P. Schreiber in 1993; his statement was first proved by the present author and Á. Kunos in 2015. The 2015 paper could only prove the non-constructibility of cyclic $n$-gons from the distances for $n$ even; here we extend this result for all $n\geq5$.

DOI: 10.14232/actasm-015-072-8

AMS Subject Classification (1991): 51M04, 12D05

Keyword(s): inscribed polygon, cyclic polygon, circumscribed polygon, compass and ruler, straightedge and compass, Thalesian polygon

Received September 17, 2015. (Registered under 72/2015.)

Abstract. In this paper, we study single-variable linear functional equations that involve one unknown function and a finite set of known functions forming a group under composition. The main results present the complete description of the solution set, and provide an alternative and effective approach for some known facts. In the proofs, the standard methods of Linear Algebra, of Group Theory, and the Axiom of Choice play a key role.

DOI: 10.14232/actasm-016-526-9

AMS Subject Classification (1991): 39B22, 15A03, 15A06, 20F38

Keyword(s): functional equations of a single variable, rank of matrices, permutations of matrices, finite groups, cosets of finite groups

Received April 19, 2016, and in revised form July 21, 2016. (Registered under 26/2016.)

Abstract. Let $V$ be an infinite-dimensional vector space over a field. In a previous article [dSPSum4], we have shown that every endomorphism of $V$ splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study decompositions into sums of three endomorphisms with prescribed split annihilating polynomials with degree $2$. Except for endomorphisms that are the sum of a scalar multiple of the identity and of a finite-rank endomorphism, we achieve a simple characterization of such sums. In particular, we give a simple characterization of the endomorphisms that split into the sum of three square-zero ones, and we prove that every endomorphism of $V$ is a linear combination of three idempotents.

DOI: 10.14232/actasm-016-319-1

AMS Subject Classification (1991): 15A24, 16B50

Keyword(s): infinite-dimensional vector space, endomorphism, decomposition, square-zero endomorphism, idempotent

Received November 18, 2016. (Registered under 69/2016.)

Abstract. We give a description of the isomorphism classes of matrix groups over commutative rings with 1 and that have dimension more than 3 and containing the group of elementary transvections. We characterize those homomorphisms of matrix groups, which satisfy the so-called $(\ast )$ condition. Such homomorphisms can be constructed with the help of the standard homomorphism. We apply the characterization obtained to the description of the above class of matrix groups.

DOI: 10.14232/actasm-016-004-x

AMS Subject Classification (1991): 20H25, 20K30, 20G20, 15A30

Keyword(s): linear group over ring, homomorphism of matrix group, elementary subgroup over ring

Received January 26, 2016. (Registered under 4/2016.)

Abstract. In the paper, the authors bound the gamma function, the digamma function, and the harmonic numbers in terms of the exponent of fractional functions and several trigonometric functions such as the tangent, hyperbolic tangent, secant, and cosecant functions.

DOI: 10.14232/actasm-016-813-x

AMS Subject Classification (1991): 33B15, 26A48, 26D05, 33B10

Keyword(s): bound, gamma function, digamma function, trigonometric function, fractional function, exponential function, inequality, harmonic number, monotonicity

Received November 1, 2016. (Registered under 63/2016.)

Abstract. In this paper we study the two-parameter family of generalized Ces?ro operators $\mathcal{P}^{b,c}$ on the space of Cauchy transforms $K$. In [cs], Siskasis and Cima obtained the boundedness of the Ces?ro operator and $\alpha $-Ces?ro operator on the space of Cauchy transforms. Motivated by this we obtain the boundedness of the operators $\mathcal{P}^{b,c}$ on $K$ for $b+1>c>0$ as well as an upper bound of its norm. Also an alternate method for boundedness of $\mathcal{P}^{b,c}$ has been obtained by finding the adjoint of $\mathcal{P}^{b,c}$.

DOI: 10.14232/actasm-016-542-6

AMS Subject Classification (1991): 33C05, 30E20, 30H99, 46E15, 47B38

Keyword(s): Gaussian hypergeometric function, boundedness, spaces of Cauchy transforms

Received August 12, 2016, and in final form October 11, 2016. (Registered under 42/2016.)

Abstract. We prove a general version of [Boas84, Theorem 4.1] to obtain Sobolev estimates for weighted Bergman projections on convex Reinhardt domains by using the Prékopa--Leindler inequality.

DOI: 10.14232/actasm-015-582-0

AMS Subject Classification (1991): 32A25, 32A36

Keyword(s): weighted Bergman projection, Prékopa--Leindler inequality, Reinhardt domain

Received November 22, 2015, and in revised form January 5, 2016. (Registered under 82/2015.)

Abstract. The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

DOI: 10.14232/actasm-016-311-x

AMS Subject Classification (1991): 32A40

Keyword(s): Julia--Caratheodory Theorem, boundary behavior of holomorphic functions, function theory on the bidisk

Received October 25, 2016, and in revised form February 22, 2017. (Registered under 61/2016.)

Abstract. The Fibonacci--Dyck shift $D_F$ is a subsystem of the Dyck shift $D_2$ constrained by the Fibonacci matrix $ F = \bigl[\begin{smallmatrix} 1 & 1 1 & 0

DOI: 10.14232/actasm-015-323-0

AMS Subject Classification (1991): 46L80, 37B40, 46L55, 37B10

Keyword(s): $C^*$-algebra, K-theory, Cuntz--Krieger algebra, $\lambda $-graph system, Dyck shift, Fibonacci--Dyck shift, subshift

Received September 25, 2015, and in revised form April 23, 2016. (Registered under 73/2015.)

Abstract. We consider the multidimensional analogues of the Lorentz spaces and weighted Lebesgue spaces. For functions monotone in each variable in these spaces we study integrability properties of their Fourier transforms.

DOI: 10.14232/actasm-015-064-3

AMS Subject Classification (1991): 42B10

Keyword(s): Fourier transform, monotone functions, anisotropic Lorentz spaces, weighted Lebesgue spaces

Received August 25, 2015, and in revised form December 13, 2015. (Registered under 64/2015.)

Abstract. In order to explore the submodules in the Hardy space over the bidisk, a new equivalence relation which is based on the core operator, namely congruence, was introduced in [Yan2005]. As we know, the structure of the submodules in $H^2(\mathbb{D}^2)$ is very complicated. So it is beneficial to study some good examples of submodules. This paper studies the congruence of the Hardy submodule over the bidisk. It will be shown that the congruence of two inner sequences based submodules can be totally described by the ratio of their inner sequences.

DOI: 10.14232/actasm-016-550-y

AMS Subject Classification (1991): 47A13, 46E20

Keyword(s): core operator, congruence, inner sequence based submodule

Received September 16, 2016, and in revised form November 16, 2016. (Registered under 50/2016.)

Abstract. Let $\CalH $ be a finite dimensional Hilbert space and $V$ a multiplicative unitary operator on $\CalHt $. Baaj and Skandalis showed that $V$ induces a finite dimensional $C^*$-Hopf algebra $H$ and its dual $C^*$-Hopf algebra $H^0$. Applying their results, Cuntz constructed a coaction $\lambda $ of $H$ on the Cuntz algebra $\CalO(\CalH )$, which is generated by $\CalH $. Let $\lambda |_C$ be its restriction to a canonical UHF-subalgebra $C$ of $\CalO(\CalH )$, which is a coaction of $H$ on $C$. In this paper, we shall show that $\lambda |_C$ is an approximately representable coaction of $H$ on $C$ with the Rohlin property.

DOI: 10.14232/actasm-016-024-9

AMS Subject Classification (1991): 46L05, 16T05

Keyword(s): $C^*$-algebras, finite dimensional $C^*$-Hopf algebras, approximately representable, multiplicative unitary, the Rohlin property

Received April 14, 2016, and in revised form September 1, 2016. (Registered under 24/2016.)

Abstract. Let $\th\in H^\i $ be an inner function with two spectral points on the unit circle $\T $, and let us consider the numerical range $W(S(\th ))$ of the truncated shift $S(\th )$. The question, whether the shape of $W(S(\th ))$ determines $\th $, is studied. Several conditions are given, when this is true, reducing the problem to a particular case.

DOI: 10.14232/actasm-016-052-0

AMS Subject Classification (1991): 30H15, 30J05, 47A12, 47A20

Keyword(s): numerical range, truncated shift, inner function, Herglotz class

Received September 22, 2016, and in revised form March 4, 2017. (Registered under 52/2016.)

Abstract. The inequality $C_2(n)\leq2 K^\C_G$, where $K_G^\C $ is the complex Grothendieck constant and $C_2(n)=\sup\big \{\|p(\boldsymbol T)\|:\|p\|_{\D ^n,\infty }\leq1, \|\boldsymbol T\|_{\infty } \leq1 \big\},$ for each $n\in\N,$ is due to Varopoulos. Here the supremum is taken over all commuting $n$-tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions and all complex polynomials $p$ in $n$ variables of degree at most $2$ and of supremum norm at most $1$ over the polydisc. We show that $C_2(n)\leq\frac {3\sqrt{3}}{4} K^\C_G$ for each $n\in\N.$

DOI: 10.14232/actasm-015-088-4

AMS Subject Classification (1991): 47A13

Keyword(s): complex Grothendieck constant, von Neumann inequality, Varopoulos-Kaijser polynomial

Received December 18, 2015, and in final form February 17, 2016. (Registered under 88/2015.)

Abstract. A remarkable theorem of Domar asserts that the lattice of the invariant subspaces of the right shift semigroup $\{S_{\tau }\}_{\tau\geq 0}$ in $L^2({\msbm R}R _+, w(t)dt)$ consists of just the \textit{``standard invariant subspaces''} whenever $w$ is a positive continuous function in ${\msbm R}R _+$ such that (1) $\log w$ is concave in $[c,\infty )$ for some $c\geq0$, (2) $\lim_ {t\to\infty }\frac{-\log w(t)}{t}=\infty,$ and $\lim_ {t\to\infty }\frac{\log |\log w(t)|-\log t}{\sqrt{\log t}}=\infty.$ We prove an extension of Domar's Theorem to a strictly wider class of weights $w$, answering a question posed by Domar in [Do3].

DOI: 10.14232/actasm-015-837-7

AMS Subject Classification (1991): 47A15

Keyword(s): right-translation invariant subspaces

Received December 17, 2015. (Registered under 87/2015.)

Abstract. In this note we study reducing subspaces for weighted composition operators defined on $L^2(\Sigma )$. Some necessary and sufficient conditions are given for such operators to have two types of reducing subspaces of the forms $L^2(\Sigma_A)$ and $L^2(\mathcal{A})$. This is basically discussed by using conditional expectation properties.

DOI: 10.14232/actasm-015-825-0

AMS Subject Classification (1991): 47B37, 47B38

Keyword(s): reducing subspace, weighted composition operators, conditional expectation

Received October 16, 2015. (Registered under 75/2015.)

Abstract. Very recently Richter and Rogers proved that any convex geometry can be represented by a family of convex polygons in the plane. We shall generalize their construction and obtain a wide variety of convex shapes for representing convex geometries. We present an Erdős--Szekeres type obstruction, which answers a question of Czédli negatively, that is general convex geometries cannot be represented with ellipses in the plane. Moreover, we shall prove that one cannot even bound the number of common supporting lines of the pairs of the representing convex sets. In higher dimensions we prove that all convex geometries can be represented with ellipsoids.

DOI: 10.14232/actasm-017-502-z

AMS Subject Classification (1991): 52A01, 52C45

Keyword(s): finite convex geometries, convex set

Received January 10, 2017, and in revised form April 10, 2017. (Registered under 2/2017.)

Abstract. We study local asymptotic properties of likelihood ratios of certain Heston models. We distinguish three cases: subcritical, critical and supercritical models. For the drift parameters, local asymptotic normality is proved in the subcritical case, only local asymptotic quadraticity is shown in the critical case, while in the supercritical case not even local asymptotic quadraticity holds. For certain submodels, local asymptotic normality is proved in the critical case, and local asymptotic mixed normality is shown in the supercritical case. As a consequence, asymptotically optimal (randomized) tests are constructed in cases of local asymptotic normality. Moreover, local asymptotic minimax bound, and hence, asymptotic efficiency in the convolution theorem sense are concluded for the maximum likelihood estimators in cases of local asymptotic mixed normality.

DOI: 10.14232/actasm-016-506-x

AMS Subject Classification (1991): 60H10, 91G70, 60F05, 62F12

Keyword(s): Heston model, local asymptotic quadricity, local asymptotic mixed normality, local asymptotic normality, asymptotically optimal tests, local asymptotic minimax bound for estimators, asymptotic efficiency in the convolution theorem sense

Received February 1, 2016, and in final form August 17, 2016. (Registered under 6/2016.)