
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

361361
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Ivan Chajda,
Helmut Länger

361374

Abstract. The relationship between MValgebras and semirings was described by A. Di Nola and B. Gerla in [DG] and [Ge]. An analogous result was proved by the authors for commutative basic algebras and socalled near semirings in [CL]. In order to generalize this approach to noncommutative basic algebras, we introduce the concept of right near semirings and point out some of their basic properties. We obtain a representation of basic algebras by socalled coupled right near semirings in a way similar to that obtained in [CL].
DOI: 10.14232/actasm014307z
AMS Subject Classification
(1991): 03G25, 06D35, 16Y30, 16Y60
Keyword(s):
basic algebra,
right near semiring,
coupled right near semiring,
monotonous basic algebra,
weakly monotonous basic algebra
Received July 18, 2014, and in final form January 5, 2015. (Registered under 57/2014.)
Eszter K. Horváth,
Branimir Šešelja,
Andreja Tepavčevič

375380

Abstract. For an integer $n\geq1$, an $n$ary latticevalued Boolean function is a map from the $n$th direct power of the 2element Boolean lattice to a bounded lattice. In terms of closure systems and cuts, we characterize latticevalued Boolean functions that can be given by linear combinations of elements of the codomain lattice.
DOI: 10.14232/actasm0143311
AMS Subject Classification
(1991): 06E30, 06B23, 06B99, 06D99, 06A15
Keyword(s):
isotone,
Boolean function,
lattice operations,
closure systems
Received December 11, 2014, and in revised form December 23, 2014. (Registered under 81/2014.)
Abstract. In an earlier paper, to describe how a congruence spreads from a prime interval to another in a finite lattice, I introduced the concept of primeperspectivity and its transitive extension, primeprojectivity and proved the Primeprojectivity Lemma. In this paper, I specialize the Primeprojectivity Lemma to slim, planar, semimodular lattices to obtain the Swing Lemma, a very powerful description of the congruence generated by a prime interval in this special class of lattices.
DOI: 10.14232/actasm0157571
AMS Subject Classification
(1991): 06C10, 06B10
Keyword(s):
primeperspective,
congruence,
congruenceperspective,
perspective,
prime interval
Received February 4, 2015, and in revised form April 21, 2015. (Registered under 7/2015.)
Miklós Dormán,
Géza Makay,
Miklós Maróti,
Róbert Vajda

399424

Abstract. The aim of this paper is to give an overview about monoidal intervals on three and fourelement sets. Furthermore, two uncountable monoidal intervals on threeelement sets are presented in the paper, and we describe some infinite families of collapsing monoids.
DOI: 10.14232/actasm0140283
AMS Subject Classification
(1991): 08A40
Keyword(s):
collapsing monoid,
monoidal interval
Received April 6, 2014, and in revised form June 16, 2015. (Registered under 28/2014.)
Abstract. We make a conjecture about integer powers which states that for any integer $n\geq2$, the $n^{th}$ power of any arbitrary integer, including zero, can be expressed `primitively' and `nontrivially', in infinitely many different ways as the sum or difference of $(n + 1)$ number of other nonzero, but not necessarily distinct integral $n^{th}$ powers. The conjecture is established for squares, cubes (partly) and biquadrates, and is open for the remaining cases. Finally, a few more questions are raised for further investigation.
DOI: 10.14232/actasm0133192
AMS Subject Classification
(1991): 11D41, 11P05
Keyword(s):
diophantine equation,
conjecture on integer powers (coip),
Waringtype problems
Received October 10, 2013, and in revised form November 3, 2014. (Registered under 69/2013.)
I. Kátai,
B. M. Phong

431436

Abstract. Let $f$ and $g$ be completely additive functions, $\delta(n)=g([\sqrt{2}n])f(n)C, C\in\rr $. If $\lim_{x\to\infty }{1\over x}\sharp\{n\le x  \ \delta(n)\ > \epsilon\}=0$ for every $\epsilon >0$, then $f(n)=g(n)=A\log n$, where $A={2C/{\log2}}$.
DOI: 10.14232/actasm014327y
AMS Subject Classification
(1991): 11K65, 11N37, 11N64
Keyword(s):
completely additive functions,
multiplicative group
Received November 24, 2014, and in revised form March 29, 2015. (Registered under 77/2014.)
Fatma AlKharousi,
Alan J. Cain,
Victor Maltcev,
Abdullahi Umar

437445

Abstract. We prove that the monoids $\begin{align*} \mathrm{Mon}\langle a,b,c,d :\;& a^nb=0, ac=1, db=1, dc=1,\\ & dab=1, da^2b=1,\ldots, da^{n1}b=1\rangle \end{align*}$ are congruencefree for all $n\geq 1$. This provides a new countable family of finitely presented congruencefree monoids, bringing us one step closer to understanding the monoid version of the BooneHigman Conjecture. We also provide examples showing that finitely presented congruencefree monoids may have quadratic Dehn function.
DOI: 10.14232/actasm013028z
AMS Subject Classification
(1991): 20M05, 20M10
Keyword(s):
BooneHigman Conjecture,
congruencefree,
finitely presented,
rewriting systems
Received May 1, 2013, and in revised form May 21, 2015. (Registered under 28/2013.)
Janusz Matkowski,
Zsolt Páles

447456

Abstract. In this paper we characterize generalized quasiarithmetic means, that is means of the form $M(x_1,\dots,x_n):=(f_1+\cdots +f_n)^{1}(f_1(x_1)+\cdots +f_n(x_n))$, where $f_1,\dots,f_n\colon I\to{\msbm R} $ are strictly increasing and continuous functions. Our characterization involves the Gauss composition of the cyclic meantype mapping induced by $M$ and a generalized bisymmetry equation.
DOI: 10.14232/actasm0150287
AMS Subject Classification
(1991): 39B40
Keyword(s):
generalized quasiarithmetic mean,
bisymmetry,
characterization
Received April 9, 2015, and in revised form June 2, 2015. (Registered under 28/2015.)
Rafael Dahmen,
Helge Glöckner

457468

Abstract. Consider a smooth vector field $f\colon{\msbm R} ^n\to{\msbm R} ^n$ and a maximal solution $\gamma\colon ]a,b[ \to{\msbm R} ^n$ to the ordinary differential equation $x'=f(x)$. It is a wellknown fact that, if $\gamma $ is bounded, then $\gamma $ is a global solution, i.e., $ ]a,b[ ={\msbm R} $. We show by example that this conclusion becomes invalid if ${\msbm R} ^n$ is replaced with an infinitedimensional Banach space.
DOI: 10.14232/actasm0142717
AMS Subject Classification
(1991): 34C11; 26E20, 34A12, 34G20, 37C10, 3401
Keyword(s):
ordinary differential equation,
smooth dynamical system,
autonomous system,
Banach space,
finite life time,
maximal solution,
bounded solution,
relatively compact set,
tubular neighborhood,
nearest point
Received March 25, 2014. (Registered under 21/2014.)
Margit Pap,
Ferenc Schipp

469482

Abstract. In this paper we give an overview of the discretization results connected to MalmquistTakenaka systems for the unit disc and upper halfplane. We prove that the discretization nodes on the real line have similar properties like the discretization nodes on the unit circle: for example they satisfy some equilibrium conditions and they are stationary points of some logarithmic potential. The problems whether they are the minimum of a logarithmic potential is formulated and solved in a special case.
DOI: 10.14232/actasm0157656
AMS Subject Classification
(1991): 42C05, 33C50, 33A65, 41A20, 30H10, 42B30, 65T99
Keyword(s):
Hardy spaces,
MalmquistTakenaka systems,
discrete orthogonality,
equilibrium conditions
Received February 17, 2015, and in revised form March 18, 2015. (Registered under 15/2015.)
László Csizmadia,
László Hatvani

483502

Abstract. The equation $x''+a^{2}(t)x=0$ with $ a(t) :=\begin{cases} \sqrt{\frac{g}{l\varepsilon }} &\text{ if $2kT\leq t<(2k+1)T$,}\\ \sqrt{\frac{g}{l+\varepsilon }} &\text{ if $(2k+1)T\leq t<(2k+2)T$, $(k=0,1,\dots )$,}\end{cases} $ is considered, where $g$ and $l$ denote the constant of gravity and the length of the pendulum, respectively; $\varepsilon >0$ is a parameter measuring the intensity of swinging. Concepts of solutions going away from the origin and approaching to the origin are introduced. Necessary and sufficient conditions are given in terms of $T$ and $\varepsilon $ for the existence of solutions of these types, which yield conditions for the existence of $2T$periodic and $4T$periodic solutions as special cases. The domain of instability, i.e., the Arnold tongues of parametric resonance are deduced from these results.
DOI: 10.14232/actasm0155109
AMS Subject Classification
(1991): 34D20, 70J40; 34A26, 34A37, 34C25
Keyword(s):
second order linear differential equations,
step function coefficients,
periodic coefficients,
impulsive effects,
periodic solutions,
parametric resonance,
swinging
Received February 16, 2015, and in revised form October 14, 2015. (Registered under 10/2015.)
András Szijártó,
Jenő Hegedűs

503526

Abstract. We study the transversal vibrations $u=u(x,t)\in C^2(\mathbb{R}^2)$ of the infinite $x\in(\infty,\infty )$ string under the external force $f(x,t)$ for all $t \in(\infty,\infty )$, when the classical D'Alembert's formula with Duhamel's principle describes the whole vibration process using the initial data $u_{t=0}=\varphi, u_t_{t=0}=\psi $. In our case the vibration process can be completely described, provided we know both the position and the speed of the string at some $t_0 \in\mathbb {R}$. We will show that certain choices of $t_0$ are suitable for solving some control and observability problems (including also some mixed problems for the semiinfinite $x\in[0,\infty )$ string). In the second part of the paper a minimal restriction on the righthand side function $f(x,t)$ is presented that guarantees the property $v(x,t)\in C^2(\mathbb{R}^2)$ for the solution $v$ of the problem $$ v_{tt}a^2 v_{xx}=f(x,t), \quad v_{t=t_0}=v_t_{t=t_0}=0, $$ namely $f\in C(\mathbb{R}^2)$ and $f_t \in C(\mathbb{R}^2)$.
DOI: 10.14232/actasm0140525
AMS Subject Classification
(1991): 35L05, 35A09, 35Q93
Keyword(s):
string vibrations,
classical solutions,
observation problems,
smoothness of the solutions
Received July 3, 2014, and in revised form January 12, 2015. (Registered under 52/2014.)
Zoltán Daróczy,
Vilmos Totik

527534

Abstract. A functional equation involving pairs of means is considered. It is shown that there are only constant solutions if continuous differentiability is assumed, and there may be nonconstant everywhere differentiable solutions. Various other situations are considered, where less smoothness is assumed on the unknown function.
DOI: 10.14232/actasm0158051
AMS Subject Classification
(1991): 39B22
Keyword(s):
functional equation,
means
Received June 4, 2014, and in revised form September 21, 2015. (Registered under 55/2015.)
Abstract. Using summability theory, we obtain restricted convergence of the inverse continuous wavelet transform at Lebesgue points for functions from the $L_p$ and Wiener amalgam spaces.
DOI: 10.14232/actasm0155308
AMS Subject Classification
(1991): 42C40; 42C15, 42B08, 42A38, 46B15
Keyword(s):
continuous wavelet transform,
Wiener amalgam spaces,
$\theta $summability,
inversion formula,
restricted convergence
Received April 22, 2015. (Registered under 30/2015.)
Abstract. It is well known that the Walsh functions form an orthonormal system on the unit interval. The weighted Walsh functions are also orthonormal with respect to some weight function. The almost everywhere convergence of the $(C,1)$ means of WalshFourier series of integrable functions is an old result of Fine [fine]. The main aim of this paper is to generalize this a.e. convergence relation with respect to weighted Walsh systems.
DOI: 10.14232/actasm0150245
AMS Subject Classification
(1991): 42C10
Keyword(s):
weighted Walsh system,
Fejér means,
almost everywhere convergence
Received March 24, 2015. (Registered under 24/2015.)
Abstract. In this paper we consider integrability conditions for dyadic maximal Walsh series. Namely, we give a condition on the coefficients of a Walsh series which is sufficient for the series being the WalshFourier series of a function belonging to the dyadic Hardy space. In the classical trigonometric case the analogous question involves the real periodic Hardy space. Then the problem leads to integrability conditions on both the trigonometric series and its conjugate, which in fact can be reduced to integrability conditions for cosine and sine series.
DOI: 10.14232/actasm015032x
AMS Subject Classification
(1991): 42C10; 30H10
Keyword(s):
WalshPaley system,
Hardy spaces,
integrability conditions,
dyadic maximal function
Received May 4, 2015, and in revised form August 14, 2015. (Registered under 32/2015.)
Robert F. Allen,
Katherine C. Heller,
Matthew A. Pons

575587

Abstract. In this paper, we study the multiplication operators on \(S^2\), the space of analytic functions on the open unit disk \(\D\) whose first derivative is in \(H^2\). Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. Finally, we prove that the isometric multiplication operators are precisely those induced by a constant function of modulus one.
DOI: 10.14232/actasm0142759
AMS Subject Classification
(1991): 47B38, 46E20, 47B32, 30H10
Keyword(s):
multiplication operator,
$S^2$,
Hardy space
Received March 29, 2014, and in revised form July 19, 2014. (Registered under 25/2014.)
Abstract. Spectral mapping theorems are proved for residual sets and quasianalytic spectral sets of polynomially bounded operators.
DOI: 10.14232/actasm015295z
AMS Subject Classification
(1991): 47A45, 47A60
Keyword(s):
spectral mapping theorem,
polynomially bounded operator,
quasianalytic operator
Received June 24, 2015, and in revised form August 24, 2015. (Registered under 45/2015.)
Abstract. In this paper we provide a new proof of the operator version of Julia's lemma which was proven by Ky Fan in [fan6]. The advantage of our method is that we can actually improve the inequality [fan6, p. 242 (a)].
DOI: 10.14232/actasm0140766
AMS Subject Classification
(1991): 47A63
Keyword(s):
positive operators,
strictly positive operators
Received November 20, 2014, and in final form February 24, 2015. (Registered under 76/2014.)
Abstract. We prove that if $T$ is an $m$isometry on a Hilbert space and $b(z)$ is an inner function, then $b(T)$ is also an $m$isometry. This work is motivated by Bermúdez, Mendoza and Martinón [BMM] where it was proved that if $T$ is an $(m,p)$isometry on a Banach space, then $T^{r}$ is also an $(m,p)$isometry for any positive integer $r.$ We also prove several functional calculus formulas for a single operator or the product of two commuting operators on Hilbert spaces and Banach spaces. Results for classes of operators on Hilbert spaces such as hypercontractions in Agler [A2], hyperexpansions in Athavale [At2] and alternating hyperexpansion in Sholapurkar and Athavale [ShT] are obtained by using these formulas. Finally those classes of operators are introduced on Banach spaces.
DOI: 10.14232/actasm0145503
AMS Subject Classification
(1991): 47A60, 47A80, 47B99
Keyword(s):
isometry,
$(m,
p)$isometry,
functional calculus,
hypercontraction,
hyperexpansion,
Banach space
Received June 20, 2014, and in revised form April 1, 2015. (Registered under 50/2014.)
Gábor Czédli,
Ádám Kunos

643683

Abstract. We study \emph{convex cyclic polygons}, that is, inscribed $n$gons. Starting from P. Schreiber's idea, published in 1993, we prove that these polygons are not constructible from their \emph{side lengths} with straightedge and compass, provided $n$ is at least five. They are nonconstructible even in the particular case where they only have \emph{two} different \emph{integer} side lengths, provided that $n\neq6$. To achieve this goal, we develop two tools of separate interest. First, we prove a \emph{limit theorem} stating that, under reasonable conditions, geometric constructibility is preserved under taking limits. To do so, we tailor a particular case of Puiseux's classical theorem on some generalized power series, called \emph{Puiseux series}, over algebraically closed fields to an analogous theorem on these series over real square root closed fields. Second, based on \emph{Hilbert's irreducibility theorem}, we give a \emph{rational parameter theorem} that, under reasonable conditions again, turns a nonconstructibility result with a transcendental parameter into a nonconstructibility result with a rational parameter. For $n$ even and at least six, we give an elementary proof for the nonconstructibility of the cyclic $n$gon from its side lengths and, also, from the \emph{distances} of its sides from the center of the circumscribed circle. The fact that the cyclic $n$gon is constructible from these distances for $n=4$ but nonconstructible for $n=3$ exemplifies that some conditions of the limit theorem cannot be omitted.
DOI: 10.14232/actasm0152593
AMS Subject Classification
(1991): 51M04, 12D05
Keyword(s):
inscribed polygon,
cyclic polygon,
circumscribed polygon,
compass and ruler,
straightedge and compass,
geometric constructibility,
Puiseux series,
power series,
holomorphic function,
field extension,
Hilbert's irreducibility theorem
Received February 13, 2015. (Registered under 9/2015.)
Abstract. A Minkowski geometry is Euclidean if and only if the altitudes of any trigon are concurrent. A Minkowski geometry is Euclidean if and only if the perpendicular bisectors of any trigon are concurrent.
DOI: 10.14232/actasm0155180
AMS Subject Classification
(1991): 53A35; 51M09, 52A20
Keyword(s):
Minkowski geometry,
circumcenter,
classification
Received February 24, 2015, and in revised form March 26, 2015. (Registered under 18/2015.)
Árpád Kurusa,
Tibor Ódor

699714

Abstract. Several affirmative answers are given in any dimension for Ulam's question about bodies floating stable in every direction if the body floats like a ball and its floating body is spherical.
DOI: 10.14232/actasm0148018
AMS Subject Classification
(1991): 53C65
Keyword(s):
floating body,
sections,
caps,
weight,
ball,
sphere,
isoperimetric inequality
Received June 23, 2014, and in final form March 7, 2015. (Registered under 51/2014.)

715728
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