ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. As the main result of the paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, the paper contains the following three observations. First, every coatomless three-generated lattice has at least one atom. Second, we give some sufficient conditions implying that a three-generated lattice has at most three atoms. Third, we present a three-generated meet-distributive lattice with four atoms.

DOI: 10.14232/actasm-020-769-4

AMS Subject Classification (1991): 06B99

Keyword(s): three-generated lattice, number of atoms, coatom, atomless lattice, herringbone lattice, $n$-distributive lattice, 2-distributive lattice, non-finitely presented lattice, convex geometry, meet-distributive lattice, semidistributive lattice, semimodular lattice

received 9.1.2020, revised 8.10.2020, accepted 4.12.2020. (Registered under 19/2020.)

Abstract. Let $f(x)\in \mathbb Z [x]$ be monic and irreducible over $\mathbb Q $, with $\deg (f)=n$. Let $K=\mathbb Q (\theta )$, where $f(\theta )=0$, and let $\mathbb Z _K$ denote the ring of integers of $K$. We say $f(x)$ is \emph{non-monogenic} if $ \{1,\theta ,\theta^2,\ldots , \theta ^{n-1} \}$ is not a basis for $\mathbb Z_K$. By extending ideas of Ratliff, Rush and Shah, we construct infinite families of non-monogenic trinomials.

DOI: 10.14232/actasm-021-463-3

AMS Subject Classification (1991): 11R04; 11R09, 12F05

Keyword(s): monogenic, trinomial, irreducible

received 13.2.2021, accepted 2.3.2021. (Registered under 213/2021.)

Abstract. A set $A$ of nonnegative integers is called a Sidon set if there is no Sidon 4-tuple, i.e., $(a,b,c,d)$ in $A$ with $a+b=c+d$ and $\{a, b\}\cap \{c, d\}=\emptyset $. Cameron and Erdős proposed the problem of determining the number of Sidon sets in $[n]$. Results of Kohayakawa, Lee, Rödl and Samotij, and Saxton and Thomason have established that the number of Sidon sets is between $2^{(1.16+o(1))\sqrt {n}}$ and $2^{(6.442+o(1))\sqrt {n}}$. An $\alpha $-generalized Sidon set in $[n]$ is a set with at most $\alpha $ Sidon 4-tuples. One way to extend the problem of Cameron and Erdős is to estimate the number of $\alpha $-generalized Sidon sets in $[n]$. We show that the number of $(n/\log ^4 n)$-generalized Sidon sets in $[n]$ with additional restrictions is $2^{\Theta (\sqrt {n})}$. In particular, the number of $(n/\log ^5 n)$-generalized Sidon sets in $[n]$ is $2^{\Theta (\sqrt {n})}$. Our approach is based on some variants of the graph container method.

DOI: 10.14232/actasm-018-777-z

AMS Subject Classification (1991): 05A16, 05D05

Keyword(s): the graph container method, generalized Sidon set

received 1.3.2018, revised 9.8.2020, accepted 29.10.2020. (Registered under 27/2018.)

Abstract. We study the ai-semiring variety defined by the identity $x^n\approx x$. We show that some subvarieties of this variety are determined by certain properties of some binary relations and provide equational basis for them. Also, we provide models of the free objects in some subvarieties of this variety.

DOI: 10.14232/actasm-021-164-8

AMS Subject Classification (1991): 08B15, 08B20, 16Y60, 20M07

Keyword(s): Burnside ai-semiring, congruence, free object, lattice, variety

received 15.1.2021, accepted 23.3.2021. (Registered under 164/2021.)

Abstract. Let $\mathbb F $ be a field and let $k,n_1,\ldots ,n_k$ be positive integers with $n_1+\cdots +n_k=n\geqslant 2$. We denote by ${\cal T}_{n_1,\ldots ,n_k}$ a block triangular matrix algebra over $\mathbb F $ with unity $I_n$ and center $Z({\cal T}_{n_1,\ldots ,n_k})$. Fixing an integer $1<r\leq n$ with $r\neq n$ when $\left |\mathbb F \right |=2$, we prove that an additive map $\psi \colon {\cal T}_{n_1,\ldots ,n_k}\rightarrow {\cal T}_{n_1,\ldots ,n_k}$ satisfies $ \psi (A)A-A\psi (A)\in Z({\cal T}_{n_1,\ldots ,n_k})$ for every rank $r$ matrices $A\in {\cal T}_{n_1,\ldots ,n_k}$ if and only if there exist an additive map $\mu \colon {\cal T}_{n_1,\ldots ,n_k}\rightarrow \mathbb F $ and scalars $\lambda ,\alpha \in \mathbb F $, in which $\alpha \neq 0$ only if $r=n$, $n_1=n_k=1$ and $\left |\mathbb F \right |=3$, such that $ \psi (A)=\lambda A+\mu (A)I_n+\alpha (a_{11}+a_{nn})E_{1n} $ for all $A=(a_{ij})\in {\cal T}_{n_1,\ldots ,n_k}$, where $E_{ij}\in {\cal T}_{n_1,\ldots ,n_k}$ is the matrix unit whose $(i,j)$th entry is one and zero elsewhere. Using this result, a complete structural characterization of commuting additive maps on rank $s>1$ upper triangular matrices over an arbitrary field is addressed.

DOI: 10.14232/actasm-020-586-y

AMS Subject Classification (1991): 15A03, 15A04, 16R60

Keyword(s): centralizing map, commuting map, block triangular matrix, rank, functional identity

received 6.8.2020, revised 10.8.2020, accepted 13.8.2020. (Registered under 86/2020.)

Abstract. We consider some properties of integrals considered by Hardy and Koshliakov that have connections to the digamma function. We establish a new general integral formula that provides a connection to the polygamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.

DOI: 10.14232/actasm-020-664-3

AMS Subject Classification (1991): 11M06, 33C15

Keyword(s): Fourier integrals, Riemann xi function, digamma function

received 21.6.2020, revised 29.10.2020, accepted 2.11.2020. (Registered under 664/2020.)

Abstract. We derive a new bound for the spectral variations of matrices explicitly expressed via the entries of the considered matrices. In the appropriate situations our results are considerably sharper than the well-known bounds.

DOI: 10.14232/actasm-020-566-z

AMS Subject Classification (1991): 15A18, 15A42

Keyword(s): matrices, perturbations

received 6.6.2020, revised 19.12.2020, accepted 14.1.2021. (Registered under 66/2020.)

Abstract. We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\in \mathbb {C}$ and $n$-dimensional spatial variable $x\in \mathbb {C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$. We prove in particular that the inhomogeneity of the equation and the formal solution are together $s$-Gevrey for any $s\geq 1$. In the opposite case $s<1$, we show that the solution is generically $1$-Gevrey while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s'$-Gevrey for no $s'<1$.

DOI: 10.14232/actasm-020-571-9

AMS Subject Classification (1991): 35C10, 35K05, 35K55, 40A30, 40B05

Keyword(s): Gevrey order, heat equation, inhomogeneous partial differential equation, nonlinear partial differential equation, formal power series, divergent power series

received 21.3.2020, revised 30.9.2020, accepted 29.10.2020. (Registered under 321/2020.)

Abstract. We study a class of quasilinear elliptic systems in Sobolev spaces. We prove the existence of a weak solution via Young measures.

DOI: 10.14232/actasm-020-910-z

AMS Subject Classification (1991): 35J57, 35D30, 46E30

Keyword(s): quasilinear elliptic systems, weak solutions, Young measures

received 10.4.2020, revised 29.3.2021, accepted 7.4.2021. (Registered under 410/2020.)

Abstract. In this article, we introduce and study Catalan almost convergent sequence spaces by using the Catalan matrix. We obtain topological properties and the $\beta $-dual of the resulting sequence spaces. Finally, we define the Catalan core of complex-valued sequences and give some inclusion theorems related to the Catalan core.

DOI: 10.14232/actasm-020-793-6

AMS Subject Classification (1991): 40A05, 40A30

Keyword(s): Catalan numbers, almost convergence, sequence spaces, core theorems

received 12.10.2020, revised 21.1.2021, accepted 30.1.2021. (Registered under 43/2020.)

Abstract. We present the injectivity and support results of the Radon transform for the double fibrations of semisimple symmetric spaces in the setting of the inclusion incidence relation which generalizes the setting of our previous result in [Ish2]. We also generalize the projection slice theorem which relates the Radon transform with the Fourier transforms on semisimple symmetric spaces.

DOI: 10.14232/actasm-020-164-4

AMS Subject Classification (1991): 44A12, 43A85

Keyword(s): Radon transform, symmetric space, Fourier transform

received 13.1.2020, revised 19.3.2021, accepted 20.3.2021. (Registered under 164/2020.)

Abstract. For every $m\geq 2$, let $\mathbb {R}^m_{\|\cdot \|}$ be $\mathbb {R}^m$ with a norm $\|\cdot \|$ such that $|{ext} B_{\mathbb {R}^m_{\|\cdot \|}}|=2m$. For every $n\geq 2,$ we devote ourselves to the description of the sets of extreme and exposed points of the closed unit balls of ${\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})$ and ${\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$, where ${\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})$ is the space of $n$-linear forms on $\mathbb {R}^m_{\|\cdot \|}$, and ${\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$ is the subspace of ${\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})$ consisting of symmetric $n$-linear forms. Let ${\mathcal F}={\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})$ or ${\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|}).$ First we classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We obtain $\big |{ext}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\big |=2^{(m^n)}$ and $\big | {ext}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}\big |=2^{o{dim}({\mathcal L}_s(^n\mathbb {R}^m_{\||\cdot \||}))}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that ${ext}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}={ext}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\cap {\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$ and $ o{exp}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}= o{exp}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\cap {\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$.

DOI: 10.14232/actasm-020-824-2

AMS Subject Classification (1991): 46A22

Keyword(s): multilinear forms, symmetric multilinear forms, extreme points, exposed points

received 24.8.2020, revised 8.12.2020, accepted 14.12.2020. (Registered under 824/2020.)

Abstract. In this paper, we introduce and study the class of almost (L) limited operators, these are operators from a Banach space into a Banach lattice whose adjoint carries disjoint (L) and weak* null sequences to norm null ones. We establish some characterizations of this class of operators, and present some connections between this class of operators and almost limited operators. After that, we prove that almost (L) limited operators from a Banach lattice into a $\sigma $-Dedekind complete one which are lattice homomorphism are exactly operators whose adjoint carries almost (L) sets into L weakly compact ones. Finally, we derive some characterizations of a $\sigma $-Dedekind complete Banach lattice whose dual has order continuous norm.

DOI: 10.14232/actasm-020-564-y

AMS Subject Classification (1991): 46A40, 46B40, 46B42

Keyword(s): (L) set, almost (L) set, almost (L) limited operator, order continuous norm

received 17.5.2020, revised 19.10.2020, accepted 2.11.2020. (Registered under 564/2020.)

Abstract. A bounded linear operator $T\colon H_1\rightarrow H_2$, where $H_1,H_2$ are Hilbert spaces, is said to be norm attaining if there exists a unit vector $x\in H_1$ such that $\|Tx\|=\|T\|$ and absolutely norm attaining (or $\mathcal {AN}$-operator) if $T|M\colon M\rightarrow H_2$ is norm attaining for every closed subspace $M$ of $H_1$. \par We prove a structure theorem for positive operators in $\beta (H):=\{T\in \mathcal B(H): T|_{M}\colon M\rightarrow M$ is norm attaining for all $M\in \mathcal R_{T}\}$, where $\mathcal R_T$ is the set of all reducing subspaces of~$T$. We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class.

DOI: 10.14232/actasm-020-426-9

AMS Subject Classification (1991): 47A15, 47A46; 47A10, 47A58

Keyword(s): compact operator, norm attaining operator, $\mathcal {AN}$-operator, reducing subspace

received 26.9.2020, revised 2.2.2021, accepted 5.2.2021. (Registered under 926/2020.)

Abstract. In this article we study bounded operators $T$ on a Banach space $X$ which satisfy the discrete Gomilko--Shi-Feng condition $\int _{0}^{2\pi }|\langle R(re^{it},T)^{2}x,x^*\rangle |dt \leq \frac {C}{(r^2-1)}\norme {x}\norme {x^*},\quad r>1, x\in X, x^* \in X^*$.

DOI: 10.14232/actasm-020-040-y

AMS Subject Classification (1991): 47A60, 46B28, 42B35

Keyword(s): $\gamma $-boundedness, power bounded operators, functional calculus, Besov spaces

received 9.10.2020, revised 26.11.2020, accepted 6.12.2020. (Registered under 40/2020.)

Abstract. We extend some inequalities for normal matrices and positive linear maps related to the Russo-Dye theorem. The results cover the case of some positive linear maps $\Phi $ on a von Neumann algebra ${\mathcal {M}}$ such that $\Phi (X)$ is unbounded for all nonzero $X\in {\mathcal {M}}$.

DOI: 10.14232/actasm-020-671-1

AMS Subject Classification (1991): 47A63, 46L52

Keyword(s): positive linear maps, operator inequalities, $\tau $-measurable operators

received 21.4.2020, revised 6.1.2021, accepted 13.1.2021. (Registered under 421/2020.)

Abstract. We show that the abstract operator-theoretic (general) Trotter--Kato formulae yield the convergence of numerical methods used for solving differential equations. These methods combine operator splitting procedures with certain time discretisation schemes which should be consistent, strongly A-stable, positive rational approximations of the exponential function. We also show that it is possible to apply more numerical steps in one splitting time step and the convergence results remain true.

DOI: 10.14232/actasm-020-140-3

AMS Subject Classification (1991): 47D06, 65M12

Keyword(s): operator semigroups, operator splittings, time discretisation schemes, Trotter--Kato product formula, numerical solution of differential equations

received 9.11.2020, revised 13.1.2021, accepted 2.2.2021. (Registered under 140/2020.)

Abstract. In this work, we give a simple approach to a priori estimates for the singular parabolic p(x,t)-Laplace Dirichlet problem in domain $\Omega $ with the boundary $\partial \Omega \in C^{1+\beta }$ with $\beta \in (0,1)$.

DOI: 10.14232/actasm-019-762-1

AMS Subject Classification (1991): 65M15, 65M60, 35K65

Keyword(s): finite elements, variable exponent, $p(x, t)$-Laplacian, regularized problem

received 31.12.2019, revised 1.2.2021, accepted 1.2.2021. (Registered under 262/2019.)

Abstract. Let $X_{1,n}\le \cdots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail with tail index $\gamma $. Given known constants $d_{i,n}$, $1\le i\le n$, consider the weighted power sums $\sum ^{k_n}_{i=1}d_{n+1-i,n}\log ^pX_{n+1-i,n}$, where $p>0$ and the $k_n$ are positive integers such that $k_n\to \infty $ and $k_n/n\to 0$ as $n\to \infty $. Under some constraints on the weights $d_{i,n}$, we prove asymptotic normality for the power sums over the whole heavy-tail model. We apply the obtained result to construct a new class of estimators for the parameter $\gamma $.

DOI: 10.14232/actasm-020-323-9

AMS Subject Classification (1991): 60F05, 62G32

Keyword(s): tail index, regular variation, weighted power sum, maximum domain of attraction

received 3.7.2020, revised 28.1.2021, accepted 29.1.2021. (Registered under 73/2020.)