
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Abstract. 
DOI: 10.14232/actasm019028x
AMS Subject Classification
(1991): 01A70
Keyword(s):
András Krámli
Received March 18, 2019. (Registered under 28/2019.)
Marianna Bolla,
Fatma Abdelkhalek,
Máté Baranyi

957

Abstract. This versatile topic goes back to the inventions of Gauss, Markov, and Gibbs, whose ideas are incorporated in graphical models and regression graphs. Later, the geneticist S. Wright (19231934) and the philosopher and computer scientist J. Pearl (19861987) developed the tools, but their notation is too complicated to formulate the mathematical background. Here we mainly follow the uptodate discussion of statisticians S. Lauritzen and N. Wermuth, and try to juxtapose the directedundirected and discretecontinuous cases.
DOI: 10.14232/actasm0183314
AMS Subject Classification
(1991): 62H99, 68T30
Keyword(s):
graphical models,
loglinear models,
Markov random fields,
covariance selection,
recursive linear regressions
Received September 10, 2018. (Registered under 81/2018.)
Vinayak Joshi,
Shubhangi Kavishwar

5968

Abstract. In this paper, we define $z$ideals in bounded lattices. A separation theorem for the existence of prime $z$ideals is proved in distributive lattices. As a consequence, we prove that every $z$ideal is the intersection of some prime $z$ideals. Lastly, we prove a characterization of dually semicomplemented lattices.
DOI: 10.14232/actasm0160122
AMS Subject Classification
(1991): 06B10, 06D75
Keyword(s):
$z$ideals,
Baer ideal,
$0$ideal,
closed ideal,
minimal prime ideal,
maximal ideal,
dense ideal,
dually semicomplemented lattice
Received February 22, 2016 and in final form September 3, 2018. (Registered under 12/2016.)
George Grätzer,
Harry Lakser

6996

Abstract. Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the joinirreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representation of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.
DOI: 10.14232/actasm0170609
AMS Subject Classification
(1991): 06B10, 06A06
Keyword(s):
finite lattice,
principal congruence,
ordered set
Received October 10, 2017, and in revised form November 26, 2017. (Registered under 60/2017.)
Abstract. The goal of the paper is to transfer some order properties of starordered Rickart *rings to Baer semigroups. A focal Baer semigroup $S$ is a semigroup with 0 expanded by two unary idempotentvalued operations, $\lt $ and $\rt $, such that the left (right) ideal generated by $x\lt $ (resp., $x\rt $) is the left (resp., right) annihilator of $x$. $S$ is said to be symmetric if the ranges of the two operations coincide and $p\lt = p\rt $ for every $p$ from the common range $P$. Such a semigroup is shown to be $P$semiabundant. If it is also Lawson reduced, then $P$ is an orthomodular lattice under the standard order of idempotents, and a restricted version of Drazin star partial order can be defined on $S$. The lattice structure of $S$ under this order is shown to be similar, in several respects, to that of starordered Rickart *rings.
DOI: 10.14232/actasm0173195
AMS Subject Classification
(1991): 20M25, 20M10, 06F99
Keyword(s):
Baer semigroup,
closed idempotent,
orthomodular lattice,
Rickart ring,
Rickart *ring,
star order
Received November 6, 2017 and in final form April 14, 2018. (Registered under 69/2017.)
Taras Banakh,
Volodymyr Gavrylkiv

119137

Abstract. A subset $B$ of a group $G$ is called a {\em difference basis} of $G$ if each element $g\in G$ can be written as the difference $g=ab^{1}$ of some elements $a,b\in B$. The smallest cardinality $B$ of a difference basis $B\subset G$ is called the {\em difference size} of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\tfrac{\Delta[G]}{\sqrt{G}}$ is called the {\em difference characteristic} of $G$. Using properties of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge11$, any finite Abelian $p$group $G$ has difference characteristic $\eth[G]<\frac{\sqrt{p}1}{\sqrt{p}3}\cdot\sup _{k\in\IN }\eth[C_{p^k}]< \sqrt{2}\cdot\frac {\sqrt{p}1}{\sqrt{p}3}$. Also we calculate the difference sizes of all Abelian groups of cardinality less than $96$.
DOI: 10.14232/actasm017586x
AMS Subject Classification
(1991): 05B10, 05E15, 16L99, 16Z99, 20D60, 20K01
Keyword(s):
finite group,
Abelian group,
difference basis,
difference characteristic
Received December 28, 2017 and in final form May 20, 2018. (Registered under 86/2017.)
Imre Kátai,
Bui Minh Phong

139145

Abstract. 
DOI: 10.14232/actasm0182791
AMS Subject Classification
(1991): 11K65, 11N37, 11N64
Keyword(s):
multiplicative functions,
completely multiplicative functions
Received March 17, 2018 and in final form June 18, 2018. (Registered under 29/2018.)
Gerd Jensen,
Christian Pommerenke

147170

Abstract. In this paper a concise but systematic account of the relations between Shabat polynomials and plane trees is given, including the famous equivalence theorem for plane trees attributed to RiemannBelyiGrothendieckShabat. Furthermore, some analytical features of Shabat polynomials and metric properties of Shabat trees are discussed.
DOI: 10.14232/actasm0178216
AMS Subject Classification
(1991): 05C05, 05C10, 30C10, 30C85, 30F99
Keyword(s):
Shabat polynomials,
plane trees,
conformal mapping
Received November 11, 2017 and in final form February 7, 2018. (Registered under 71/2017.)
Gopal Dutta,
P. Veeramani

171180

Abstract. In 2013, JiménezMelado and LlorensFuster proved that the renorming of $\ell ^2$, $x=\max\{\x\_2,p(x)\}$, where $p$ is a seminorm on $\ell ^2$ satisfying certain conditions, has the weak fixed point property. In this paper, we generalize this result for a Banach space having normal structure and Schauder basis. From this, we derive that every Banach space having normal structure and Schauder basis has an equivalent renorming that lacks asymptotic normal structure but has the weak fixed point property.
DOI: 10.14232/actasm0173394
AMS Subject Classification
(1991): 47H09, 47H10, 46B20
Keyword(s):
nonexpansive mappings,
weak fixed point property
Received December 30, 2017 and in final form April 28, 2018. (Registered under 89/2017.)
Horst Alzer,
Man Kam Kwong

181187

Abstract. Let $F_{\alpha,\beta }(x)=\beta E_{\beta } (x^{\beta })\alpha E_{\alpha }(x^{\alpha })$, where $E_{\alpha }$ denotes the MittagLeffler function. We prove that if $\alpha, \beta\in (0,1]$, then $F_{\alpha,\beta }$ is completely monotonic on $(0,\infty )$ if and only if $\alpha\leq \beta $. This extends a result of T. Simon, who proved in 2015 that $F_{\alpha,1}$ is completely monotonic on $(0,\infty )$ if $\alpha\in (0,1]$. Moreover, we apply our monotonicity theorem to obtain some functional inequalities involving $F_{\alpha,\beta }$.
DOI: 10.14232/actasm0182635
AMS Subject Classification
(1991): 26A48, 33E12
Keyword(s):
MittagLeffler function,
completely monotonic,
functional inequalities
Received January 17, 2018. (Registered under 13/2018.)
Francisco Javier GarcíaPacheco,
Soledad MorenoPulido

189201

Abstract. A modulus for the BishopPhelpsBollobás property for operators (BPBpo) is formerly introduced in the literature that characterizes whether a pair of Banach spaces enjoys the BPBpo and that also provides the best possible value of the BPBpo for a given pair of Banach spaces that enjoys it. We use it also to show that the BPBpo is hereditary to a class of complemented subspaces that strictly includes the $M$summands. We also provide an equivalent reformulation of this modulus. Finally, the continuity properties of this modulus are also discussed.
DOI: 10.14232/actasm0187655
AMS Subject Classification
(1991): 47A05, 46B20
Keyword(s):
BishopPhelpsBollobás modulus,
continuous linear operator,
complemented subspace
Received January 19, 2018 and in final form April 24, 2018. (Registered under 15/2018.)
M. Mackey,
P. Mellon

203214

Abstract. We study iterates, $f^n$, of a fixedpoint free compact holomorphic map $f\colon B\rightarrow B$ where $B$ is the open unit ball of any $JB^*$triple of finite rank. These spaces include $L(H,K)$, $H,K$ Hilbert, dim$(H)$ arbitrary, dim$(K)< \infty $, or any classical Cartan factor or $C^*$algebra of finite rank. Apart from the Hilbert ball, the sequence of iterates $(f^n)_n$ does not generally converge (locally uniformly on $B$) and little is known of accumulation points. We present a short proof of a Wolff theorem for $B$ and establish key properties of the resulting $f$invariant subdomains. We define a concept of closed convex holomorphic hull, $\mathop{\rm Ch}(x)$, for $x \in\partial B$ and prove the following. There is a unique tripotent $u$ in $\partial B$ such that all constant subsequential limits of $(f^n)_n$ lie in $\mathop{\rm Ch}(u)$. As a consequence we also get a short proof of the classical Hilbert ball results.
DOI: 10.14232/actasm018518z
AMS Subject Classification
(1991): 47H10, 32M15; 32H50, 58C10
Keyword(s):
iteration,
bounded symmetric domain,
DenjoyWolff theorem
Received February 8, 2018 and in final form March 8, 2018. (Registered under 18/2018.)
Emmanuel Fricain,
Andreas Hartmann,
William T. Ross

215230

Abstract. In this paper we discuss the multipliers between range spaces of coanalytic Toeplitz operators.
DOI: 10.14232/actasm0187697
AMS Subject Classification
(1991): 30J05, 30H10, 46E22
Keyword(s):
Hardy spaces,
inner functions,
model spaces,
multipliers,
de BrangesRovnyak spaces
Received February 8, 2018 and in final form March 21, 2018. (Registered under 19/2018.)
H. Ezzahraoui,
M. Mbekhta,
E. H. Zerouali

231248

Abstract. We extend in this paper the notion of Cauchy dual to operators with closed range. We then give several useful properties of Cauchy duals extending the case of leftinvertible operators. As a consequence, we show that a weak concavity concept of an operator induces a corresponding weak hyponormality of its Cauchy dual.
DOI: 10.14232/actasm0180202
AMS Subject Classification
(1991): 47B20
Keyword(s):
closed range,
hyperexpansive operators,
Cauchy dual,
MoorePenrose inverse
Received February 10, 2018 and in final form June 28, 2018. (Registered under 20/2018.)
Abstract. We characterize meromorphic function fields closed by partial derivatives in $n$ variables.
DOI: 10.14232/actasm0185307
AMS Subject Classification
(1991): 32A20; 32W50
Keyword(s):
algebraic addition theorem,
degenerate Abelian functions,
BriotBouquet type partial differential equations
Received March 22, 2018 and in final form March 6, 2019. (Registered under 30/2018.)
Abstract. In this paper an algorithm based on polyphase matrices for constructing a pair of orthogonal wavelet frames is suggested, and a general form for all orthogonal tight wavelet frames on local fields of positive characteristic is described. Moreover, an investigation regarding their properties by means of the Fourier transform is carried out.
DOI: 10.14232/actasm0187854
AMS Subject Classification
(1991): 42C40; 42C15. 43A70, 11S85
Keyword(s):
wavelet frame,
orthogonality,
framelet symbol,
polyphase matrix,
extension principle,
Fourier transform,
local field
Received April 9, 2018. (Registered under 35/2018.)
Fritz Gesztesy,
Konrad Schmüdgen

291293

Abstract. We reprove a recent result of Z. Sebestyén and Zs. Tarcsay [9]: If $T^*T$ and $TT^*$ are selfadjoint, then $T$ is closed.
DOI: 10.14232/actasm018295y
AMS Subject Classification
(1991): 47B25; 47B65
Keyword(s):
von Neumann's theorem
Received May 10, 2018. (Registered under 45/2018.)
Geetanjali M. Phatak,
V. M. Sholapurkar

295311

Abstract. M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a nonnegative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2({\msbm R}_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space ${\cal H}$ of vectorvalued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with ${\cal H}$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertible, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t$.
DOI: 10.14232/actasm0185463
AMS Subject Classification
(1991): 47B20, 47B37; 47A10, 46E22
Keyword(s):
weighted translation semigroup,
completely alternating,
completely hyperexpansive,
analytic,
operator valued weighted shift
Received May 10, 2018 and in final form January 31, 2019. (Registered under 46/2018.)
Abstract. The question if every polynomially bounded operator is similar to a contraction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In [gam16], cyclic polynomially bounded operators which are not similar to contractions were constructed. The construction was based on a perturbation of a sequence of finitedimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, studied earlier by Pisier. In this paper, a cyclic polynomially bounded operator $T_0$ is constructed so that $T_0$ is not similar to a contraction and $\omega_a(T_0)={\msbm O}$. Here $\omega_a(z)=\exp(a\frac{z+1}{z1})$, $z\in{\msbm D}$, $a>0$, and ${\msbm D}$ is the open unit disk. To obtain such a $T_0$, a slight modification of the construction from [gam16] is needed.
DOI: 10.14232/actasm018797y
AMS Subject Classification
(1991): 47A60; 47A65, 47A16, 47A20
Keyword(s):
polynomially bounded operator,
similarity,
contraction,
unilateral shift,
isometry,
$C_0$contraction,
$C_0$operator
Received May 15, 2018 and in final form February 6, 2019. (Registered under 47/2018.)
Abstract. In this paper we study a conjugation on a Banach space $\x $ and show properties of operators concerning conjugation $C$ and show spectral properties of such operators. Next we show spectral properties of an $(m,C)$symmetry (isometry) operator $T$ on a complex Banach space $\x $. We prove that, for a $C$doubly commuting pair $(T,S)$, if $T$ is an $(m,C)$symmetry (isometry) and $S$ is an $(n,C)$symmetry (isometry), then $T + S$ and $TS$ are $(m + n  1,C)$symmetries (isometries).
DOI: 10.14232/actasm018801y
AMS Subject Classification
(1991): 47A05; 47B25, 47B99
Keyword(s):
Banach space,
linear operator,
conjugation,
spectrum
Received June 4, 2018 and in final form August 31, 2018. (Registered under 51/2018.)
Abstract. Let $K_0$ be a compact convex subset of the plane $\preal $, and assume that whenever $K_1\subseteq\preal $ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. FejesTóth. A theorem of L. FejesTóth from 1967 states that the assumption above holds for $K_0$ if and only if $K_0$ is a disk. In a paper that appeared in 2017, the present author introduced a new concept of crossing, and proved that L. FejesTóth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive than the old one. Therefore, L. FejesTóth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a minisurvey shows that this purely geometric paper has precursors in combinatorics and, mainly, in lattice theory.
DOI: 10.14232/actasm0185222
AMS Subject Classification
(1991): 52C99; 52A01, 06C10
Keyword(s):
compact convex set,
circle,
characterization of circles,
disk,
crossing,
abstract convex geometry,
AdarichevaBolat property,
boundary of a compact convex set,
supporting line,
slideturning,
lattice
Received February 18, 2018. (Registered under 22/2018.)

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