
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Gábor Czédli,
Júlia Kulin

312

Abstract. By H. Strietz, 1975, and G. Czédli, 1996, the complete lattice $\Equ(A)$ of all equivalences is fourgenerated, 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 selfcontained way.
DOI: 10.14232/actasm0160562
AMS Subject Classification
(1991): 06B99
Keyword(s):
quasiorder lattice,
lattice of preorders,
minimumsized generating set,
fourgenerated lattice,
lattice of transitive relations
Received October 4, 2016, and in revised form October 22, 2016. (Registered under 56/2016.)
Gábor Czédli,
Géza Makay

1329

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/actasm0160363
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.)
Ivan Chajda,
R. Padmanabhan

3134

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/actasm0165142
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 semilatticeordered semigroups are in onetoone 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 semilatticeordered left normal bands by admissible closure operators. We solve the identity problem for all varieties of semilatticeordered normal bands.
DOI: 10.14232/actasm0167774
AMS Subject Classification
(1991): 06F05, 08B15, 20M07
Keyword(s):
variety,
semilatticeordered semigroup,
normal band,
admissible closure operator
Received April 29, 2016, and in revised form October 20, 2016. (Registered under 27/2016.)
Vilmos Komornik,
Marco Pedicini,
Attila Pethő

5160

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 nonempty 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/actasm0150800
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 nondiametric sides. It is also determined by the \emph{distances} of its nondiametric 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 nonconstructibility of cyclic $n$gons from the distances for $n$ even; here we extend this result for all $n\geq5$.
DOI: 10.14232/actasm0150728
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.)
Mihály Bessenyei,
Ágnes Konkoly,
Gréta Szabó

7181

Abstract. In this paper, we study singlevariable 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/actasm0165269
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.)
Clément de Seguins Pazzis

83111

Abstract. Let $V$ be an infinitedimensional vector space over a field. In a previous article [dSPSum4], we have shown that every endomorphism of $V$ splits into the sum of four squarezero 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 finiterank 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 squarezero ones, and we prove that every endomorphism of $V$ is a linear combination of three idempotents.
DOI: 10.14232/actasm0163191
AMS Subject Classification
(1991): 15A24, 16B50
Keyword(s):
infinitedimensional vector space,
endomorphism,
decomposition,
squarezero endomorphism,
idempotent
Received November 18, 2016. (Registered under 69/2016.)
V. M. Petechuk,
J. V. Petechuk

113123

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 socalled $(\ast )$ 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/actasm016004x
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.)
Feng Qi,
Mansour Mahmoud

125141

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/actasm016813x
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.)
D. Borgohain,
S. Naik

143154

Abstract. In this paper we study the twoparameter 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/actasm0165426
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.)
Yunus E. Zeytuncu

155164

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 PrékopaLeindler inequality.
DOI: 10.14232/actasm0155820
AMS Subject Classification
(1991): 32A25, 32A36
Keyword(s):
weighted Bergman projection,
PrékopaLeindler inequality,
Reinhardt domain
Received November 22, 2015, and in revised form January 5, 2016. (Registered under 82/2015.)
John E. McCarthy,
James E. Pascoe

165175

Abstract. The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The JuliaCarathéodory 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/actasm016311x
AMS Subject Classification
(1991): 32A40
Keyword(s):
JuliaCaratheodory 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 FibonacciDyck 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/actasm0153230
AMS Subject Classification
(1991): 46L80, 37B40, 46L55, 37B10
Keyword(s):
$C^*$algebra,
Ktheory,
CuntzKrieger algebra,
$\lambda $graph system,
Dyck shift,
FibonacciDyck 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/actasm0150643
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.)
Yue Wu,
Yixin Yang

215221

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/actasm016550y
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 UHFsubalgebra $C$ 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/actasm0160249
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/actasm0160520
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/actasm0150884
AMS Subject Classification
(1991): 47A13
Keyword(s):
complex Grothendieck constant,
von Neumann inequality,
VaropoulosKaijser polynomial
Received December 18, 2015, and in final form February 17, 2016. (Registered under 88/2015.)
Eva A. GallardoGutiérrez,
Jonathan R. Partington,
Daniel J. Rodríguez

271290

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/actasm0158377
AMS Subject Classification
(1991): 47A15
Keyword(s):
righttranslation invariant subspaces
Received December 17, 2015. (Registered under 87/2015.)
M. R. Jabbarzadeh,
M. Jafari Bakhshkandi

291299

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/actasm0158250
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 ErdősSzekeres 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/actasm017502z
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.)
János Marcell Benke,
Gyula Pap

313344

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/actasm016506x
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.)

345353
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