ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The relationship between MV-algebras 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 so-called near semirings in [CL]. In order to generalize this approach to non-commutative 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 so-called coupled right near semirings in a way similar to that obtained in [CL].

DOI: 10.14232/actasm-014-307-z

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.)

Abstract. For an integer $n\geq1$, an $n$-ary lattice-valued Boolean function is a map from the $n$-th direct power of the 2-element Boolean lattice to a bounded lattice. In terms of closure systems and cuts, we characterize lattice-valued Boolean functions that can be given by linear combinations of elements of the co-domain lattice.

DOI: 10.14232/actasm-014-331-1

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 prime-perspectivity and its transitive extension, prime-projectivity and proved the Prime-projectivity Lemma. In this paper, I specialize the Prime-projectivity 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/actasm-015-757-1

AMS Subject Classification (1991): 06C10, 06B10

Keyword(s): prime-perspective, congruence, congruence-perspective, perspective, prime interval

Received February 4, 2015, and in revised form April 21, 2015. (Registered under 7/2015.)

Abstract. The aim of this paper is to give an overview about monoidal intervals on three- and four-element sets. Furthermore, two uncountable monoidal intervals on three-element sets are presented in the paper, and we describe some infinite families of collapsing monoids.

DOI: 10.14232/actasm-014-028-3

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 `non-trivially', in infinitely many different ways as the sum or difference of $(n + 1)$ number of other non-zero, 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/actasm-013-319-2

AMS Subject Classification (1991): 11D41, 11P05

Keyword(s): diophantine equation, conjecture on integer powers (coip), Waring-type problems

Received October 10, 2013, and in revised form November 3, 2014. (Registered under 69/2013.)

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/actasm-014-327-y

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.)

Abstract. We prove that the monoids $\begin{align*} \mathrm{Mon}\langle a,b,c,d :\;& a^nb=0, ac=1, db=1, dc=1,\\ & dab=1, da^2b=1,\ldots, da^{n-1}b=1\rangle \end{align*}$ are congruence-free for all $n\geq 1$. This provides a new countable family of finitely presented congruence-free monoids, bringing us one step closer to understanding the monoid version of the Boone--Higman Conjecture. We also provide examples showing that finitely presented congruence-free monoids may have quadratic Dehn function.

DOI: 10.14232/actasm-013-028-z

AMS Subject Classification (1991): 20M05, 20M10

Keyword(s): Boone-Higman Conjecture, congruence-free, finitely presented, rewriting systems

Received May 1, 2013, and in revised form May 21, 2015. (Registered under 28/2013.)

Abstract. In this paper we characterize generalized quasi-arithmetic 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 mean-type mapping induced by $M$ and a generalized bisymmetry equation.

DOI: 10.14232/actasm-015-028-7

AMS Subject Classification (1991): 39B40

Keyword(s): generalized quasi-arithmetic mean, bisymmetry, characterization

Received April 9, 2015, and in revised form June 2, 2015. (Registered under 28/2015.)

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 well-known 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 infinite-dimensional Banach space.

DOI: 10.14232/actasm-014-271-7

AMS Subject Classification (1991): 34C11; 26E20, 34A12, 34G20, 37C10, 34--01

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.)

Abstract. In this paper we give an overview of the discretization results connected to Malmquist--Takenaka systems for the unit disc and upper half-plane. 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/actasm-015-765-6

AMS Subject Classification (1991): 42C05, 33C50, 33A65, 41A20, 30H10, 42B30, 65T99

Keyword(s): Hardy spaces, Malmquist--Takenaka systems, discrete orthogonality, equilibrium conditions

Received February 17, 2015, and in revised form March 18, 2015. (Registered under 15/2015.)

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/actasm-015-510-9

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.)

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 semi-infinite $x\in[0,\infty )$ string). In the second part of the paper a minimal restriction on the right-hand 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/actasm-014-052-5

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.)

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 non-constant everywhere differentiable solutions. Various other situations are considered, where less smoothness is assumed on the unknown function.

DOI: 10.14232/actasm-015-805-1

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/actasm-015-530-8

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 Walsh--Fourier 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/actasm-015-024-5

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 Walsh--Fourier 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/actasm-015-032-x

AMS Subject Classification (1991): 42C10; 30H10

Keyword(s): Walsh--Paley system, Hardy spaces, integrability conditions, dyadic maximal function

Received May 4, 2015, and in revised form August 14, 2015. (Registered under 32/2015.)

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/actasm-014-275-9

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/actasm-015-295-z

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/actasm-014-076-6

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/actasm-014-550-3

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.)

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 non-constructible 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 non-constructibility result with a transcendental parameter into a non-constructibility result with a rational parameter. For $n$ even and at least six, we give an elementary proof for the non-constructibility 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 non-constructible for $n=3$ exemplifies that some conditions of the limit theorem cannot be omitted.

DOI: 10.14232/actasm-015-259-3

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/actasm-015-518-0

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.)

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/actasm-014-801-8

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.)