ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $\symm{\ell }$ and for any $n \geq\ell $, the set of $n$-permutations involving only members of $G$ as $\ell $-patterns is a subgroup of $\symm{n}$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.

DOI: 10.14232/actasm-017-510-4

AMS Subject Classification (1991): 08A40, 05A05

Keyword(s): permutation patterns, Galois connections

Received February 1, 2017. (Registered under 10/2017.)

Abstract. Formulas for the numbers of distinct nets of prisms, antiprisms, and similar polyhedra are presented.

DOI: 10.14232/actasm-016-789-y

AMS Subject Classification (1991): 05A05, 54A25

Keyword(s): nets, polyhedra, symmetry

Received July 25, 2016, and in revised form March 24, 2017. (Registered under 39/2016.)

Abstract. \emph{Finite convex geometries} are combinatorial structures. It follows from a recent result of M. Richter and L. G. Rogers that there is an infinite set $\Trr $ of planar convex polygons such that $\Trr $ with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of $\Trr $ to a finite subset in a natural way. For a (small) nonnegative $\epsilon < 1$, a differentiable convex simple closed planar curve $S$ will be called an \emph{almost-circle of accuracy} $1-\epsilon $ if it lies in an annulus of radii $0< r_1\leq r_2$ such that $r_1/r_2 \geq1-\epsilon $. Motivated by Richter and Rogers' result, we construct a set $\Tczk $ such that (1) $\Tczk $ contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) $\Tczk $ with respect to the geometric convex hull operator is a locally convex geometry; (3) $\Tczk $ is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive $\epsilon\in \real $ and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets $E$ of $\Tczk $ such that each $E$ consists of almost-circles of accuracy $1-\epsilon $ and the convex geometry in question is represented by restricting the convex hull operator to $E$. The affine-disjointness of $E_1$ and $E_2$ means that, in addition to $E_1\cap E_2=\emptyset $, even $\psi(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $\psi $.

DOI: 10.14232/actasm-016-044-8

AMS Subject Classification (1991): 05B25; 06C10, 52A01

Keyword(s): abstract convex geometry, anti-exchange system, differentiable curve, almost-circle

Received August 23, 2016, and in revised form June 27, 2017. (Registered under 44/2016.)

Abstract. For a finite distributive lattice $D$, let us call $Q \ci D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$ correspond to $Q$ under this isomorphism. We find a necessary condition for representability by principal congruences and prove that for finite distributive lattices with a join-irreducible unit element this condition is also sufficient.

DOI: 10.14232/actasm-017-036-7

AMS Subject Classification (1991): 06B10

Keyword(s): congruence lattice, principal congruence, join-irreducible congruence, finite distributive lattice, principal congruence representable set

Received May 22, 2017, and in revised form August 1, 2017. (Registered under 36/2017.)

Abstract. Let $f\colon{\msbm R} \to\C $ be a Lebesgue integrable function on the real line ${\msbm R} :=(-\infty, \infty )$ and consider its trigonometric integral defined by $I(x):= \int_{{\msbm R} } f(t) e^{itx} dt$, $x\in{\msbm R} $. We give sufficient conditions in terms of certain integral means of $f$ to ensure that $I(x)$ belong to one of the Zygmund classes $\Zyg(\alpha )$ and zyg$(\alpha )$ for some $0< \alpha\le 2$. In the particular case $\alpha =1$, our theorems are the nonperiodic versions of those of A. Zygmund on the smoothness of the sum of trigonometric series (see in [bib2] and also [bib3, on pp. 320--321]). Our method of proof is essentially different from that used by A. Zygmund. We establish interesting interrelations between the order of magnitude of certain initial integral means and those of certain tail integral means of the function $f$.

DOI: 10.14232/actasm-017-514-6

AMS Subject Classification (1991): 42A16, 42A38, 26A16

Keyword(s): trigonometric series and integrals, Lipschitz classes Lip(a) and lip(a), Zygmund classes Zyg(a) and zyg(a), initial integral means, tail integral means

Received February 14, 2017, and in revised form April 2, 2017. (Registered under 14/2017.)

Abstract. Let $[z-w]$ be the smallest invariant subspace of $H^2$ over the bidisk containing $z-w$. Let $M$ be an invariant subspace satisfying $[z-w]\subsetneqq M \subset H^2$. We denote by $F^M_z$ the compression operator of the multiplication operator by $z$ on $M\ominus w M$ which is called the fringe operator of $M$. It is proved that $F^M_z$ is Fredholm and ${\rm ind} F^M_z=-1$. Its generalizations are also given.

DOI: 10.14232/actasm-017-012-6

AMS Subject Classification (1991): 47A15, 32A35, 47B35

Keyword(s): Hardy space over the bidisk, invariant subspace, fringe operator, Fredholm operator, Fredholm index, Bergman space

Received February 4, 2017, and in revised form April 7, 2017. (Registered under 12/2017.)

Abstract. The generalized fractional integral operator and the Caputo type generalized fractional derivative operator are defined which contain the Riemann--Liouville integral operator, the Hadamard fractional integral operator, the Caputo fractional derivative operator and the Caputo type Hadamard fractional derivative operator as special cases. General solutions (the explicit solutions) of the impulsive Caputo type generalized fractional differential equations are given. Applying our results, existence results of solutions of boundary value problems for an impulsive fractional differential equations involved with the Caputo type generalized fractional derivatives are established. Examples and some remarks on recent published papers are presented to illustrate the main theorems.

DOI: 10.14232/actasm-016-793-1

AMS Subject Classification (1991): 34A37, 34K15

Keyword(s): higher order Caputo type generalized fractional differential equation, general solution, impulse effect, Caputo derivative, boundary value problem

Received August 23, 2016. (Registered under 43/2016.)

Abstract. The paper studies the robustness of a general notion of trichotomy, referred to as ``nonuniform $(h,k,\mu,\nu )$-trichotomy''. This means that any sufficiently small linear perturbation of a nonuniform $(h,k,\mu,\nu )$-trichotomy has the same asymptotic behavior as the original trichotomy. This result will be obtained as a consequence of two other important results: the robustness of dichotomy and the equivalence between a trichotomy and two dichotomies.

DOI: 10.14232/actasm-016-785-9

AMS Subject Classification (1991): 34D05, 34D09, 34D10, 93B35, 93C73

Keyword(s): dichotomy, trichotomy, robustness, evolution operator

Received July 13, 2016, and in revised form February 21, 2017. (Registered under 35/2016.)

Abstract. We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in{\msbm Z}^N$, $N\ge3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset\{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so-called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\subset M$ the Weyl multiplier for convergence of these series has the form $W(\nu )=\prod_{j=1}^{N-k} \log(|\nu_{{\alpha }_j}|+2)$, where $\alpha_j\in M\setminus J_k $, $\nu =(\nu_1,\dots,\nu_N)\in{{\msbm Z}}^N$. So, the ``one-dimensional'' Weyl multiplier $\log(|\cdot |+2)$ presents in $W(\nu )$ only on the places of ``free'' (nonlacunary) components of the vector $\nu $. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M. Kojima in the classes $L_p$, $p>1$, and by D. K. Sanadze, Sh. V. Kheladze in Orlicz class. Note, that presence of two or more ``free'' components in the index $n$ (as follows from the results by Ch. Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\geq3$ even in the class of continuous functions.

DOI: 10.14232/actasm-017-275-8

AMS Subject Classification (1991): 42B05

Keyword(s): multiple trigonometric Fourier series, convergence almost everywhere, lacunary sequence, Weyl multipliers

Received April 21, 2017. (Registered under 25/2017.)

Abstract. Let $\{P_n\}_{n=0}^\infty $ be an orthogonal polynomial sequence on the real line with respect to a probability measure $\mu $ with compact and infinite support and $D_N=\sum_{n=0}^N P_nh_n$ the $N$th element of the Dirichlet kernel, where $h_n=(\int P_n^2d\mu )^{-1}$. We are investigating the $r$th integer power $D_N^r$ and prove for special orthogonal polynomials that in the case $r\in\mathbb {N}\setminus\{1\}$ the sequence $\{D_N^r\}_{N=0}^\infty $ gives rise to an approximate identity. This applies for example for Jacobi polynomials.

DOI: 10.14232/actasm-016-072-z

AMS Subject Classification (1991): 42C05, 42C10

Keyword(s): orthogonal polynomials, approximate identities, Dirichlet kernel

Received December 21, 2016, and in revised form May 24, 2017. (Registered under 72/2016.)

Abstract. Let $M$ be a compact Riemannian manifold and let $C^{1}(M,\mathbb{R}^{d})$ be the space of all $C^1$-maps of $M$ to the $d$-dimensional Euclidean space $\mathbb{R}^d$ with the $C^1$-topology. For $p \in[1,\infty ]$ and a compact submanifold $K$ of $M$, we define a norm $\| \cdot\| _{\langle M,K;p\rangle }$ on $C^{1}(M,\mathbb{R}^{d})$ by $ \| f \| _{\langle M,K;p\rangle } = (\| f|K \| _{\infty }^{p} + \| Df \| _{\infty }^{p})^{1/p} $ for $f \in C^{1}(M,\mathbb{R}^{d})$, where $Df$ denotes the derivative of $f$ (the norm $\| Df \| _{\infty }$ will be defined in Section 1). For two pairs $(M,K)$, $(N,L)$ of Riemannian manifolds and their submanifolds, we characterize a surjective linear $\| \cdot\| _{\langle M,K;p\rangle }-\| \cdot\| _{\langle N,L;p\rangle }$-isometry $T\colon C^{1}(M,\mathbb{R}^{d}) \to C^{1}(N,\mathbb{R}^{d})$, satisfying some regularity conditions, as a modified weighted composition operator whose symbol is a Riemannian homothety $\varphi\colon N \to M$. If $K$ and $L$ are not singletons, then $\varphi $ is a Riemannian isometry and satisfies $\varphi(L) = K$. The result indicates that the isometries of $C^1$-function spaces with respect to the above norms determine not only the isometry type of the ambient manifold but also the {\it embedding type} of the submanifolds up to isometry. Applying this result we study deformations of isometry groups associated with some perturbations of norms on $C^{1}(M,\mathbb{R}^{d})$. Aspects of these deformations naturally depend on isometry groups of the underlying manifolds.

DOI: 10.14232/actasm-016-323-4

AMS Subject Classification (1991): 46E15, 55R25, 53C99

Keyword(s): isometry, weighted composition operator, Banach--Stone theorem, Banach bundle, continuously differentiable functions, Riemannian manifold, embedding

Received December 25, 2016, and in revised form April 24, 2017. (Registered under 73/2016.)

Abstract. Let $M$ be a compact Riemannian manifold and let $C^{1}(M,\mathbb{R}^{d})$ be the space of all $C^1$-maps of $M$ to $\mathbb{R}^d$. For $p \in[1,\infty ]$, we introduce a norm $\| \cdot\| _{D,p}$ on $C^{1}(M,\mathbb{R}^{d})$ of the form $ \| f \| _{D, p} = (\| f \| _{\infty }^{p} + \| Df \| _{\infty }^{p})^{1/p}, f \in C^{1}(M,\mathbb{R}^{d}) $, where $Df$ denotes the derivative of $f$ (see Section 1 for the precise definition of the norm). We prove that every surjective linear isometry $T\colon C^{1}(M,\mathbb{R}^{d})\to C^{1}(M,\mathbb{R}^{d})$ which maps the constant functions to the constant functions is a generalized weighted composition operator in that $ Tf(x) = U(f(\varphi(x))), f \in C^{1}(M,\mathbb{R}^{d}), x \in M $, for a Riemannian isometry $\varphi\colon M \to M$ of $M$ and a linear isometry $U\colon\mathbb {R}^{d} \to\mathbb {R}^{d}$. This is an analogue of the classical Banach--Stone theorem for $C^1$-function spaces and partly extends previous results [BotelhoJamison], [K3] to function spaces over higher dimensional spaces.

DOI: 10.14232/actasm-016-283-9

AMS Subject Classification (1991): 46E15, 55R25

Keyword(s): isometry, weighted composition operator, Banach--Stone theorem, Banach bundle, continuously differentiable functions

Received June 18, 2016, and in revised form March 19, 2017. (Registered under 33/2016.)

Abstract. The Burkholder--Davis--Gundy inequalities and the sharp maximal function inequalities for martingales are established for rearrangement-invariant quasi-Banach function spaces.

DOI: 10.14232/actasm-012-817-9

AMS Subject Classification (1991): 60G42, 46E35, 60G46

Keyword(s): martingale inequalities, Burkholder--Davis--Gundy's inequalities, sharp maximal function, rearrangement-invariant quasi-Banach function spaces

Received September 10, 2012, and in revised form April 20, 2017. (Registered under 67/2012.)

Abstract. The aim of this note is to generalize the notion of Fredholm operator to an arbitrary $C^*$-algebra. Namely, we define ``finite type'' elements in an axiomatic way, and also we define a Fredholm type element $a$ as such an element of a given $C^*$-algebra for which there are finite type elements $p$ and $q$ such that $(1-q)a(1-p)$ is ``invertible''. We derive an index theorem for such operators. In Applications we show that many well-known operators are special cases of our theory. Those include: classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert $C^*$-modules over a unital $C^*$-algebra in the sense of Mishchenko and Fomenko.

DOI: 10.14232/actasm-015-526-5

AMS Subject Classification (1991): 47A53, 46L08, 46L80

Keyword(s): $C^*$-algebra, Fredholm operators, $K$ group, index

Received April 8, 2015, and in final form July 26, 2017. (Registered under 26/2015.)

Abstract. A new proof of a representation result of weakly continuous local semigroups of symmetric operators obtained by A. Klein and L. J. Landau is given. As an application, a representation result as Laplace transform of operator valued functions of positive type, which generalizes a result given by D. V. Widder, is obtained.

DOI: 10.14232/actasm-017-040-x

AMS Subject Classification (1991): 47D03, 47B25, 44A10

Keyword(s): local semigroup of operators, symmetric, self-adjoint, infinitesimal generator

Received June 12, 2017, and in revised form September 3, 2017. (Registered under 40/2017.)

Abstract. For a power $q$ of a prime $p$, the Artin--Schreier--Mumford curve $ASM(q)$ of genus $\gg =(q-1)^2$ is the nonsingular model $\cX $ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c, c\neq0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin--Schreier--Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|< 1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{\gg }(\sqrt{\gg }+1)^2$, far away from the Hurwitz bound $84(\gg -1)$ valid in zero characteristic; see [Co-Ka2003-1,Co-Ka-Ko2001,Co-Ka2004]. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $\Automorph(\cX )$ of all automorphisms of $\cX $ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan [mv1982]; see also [AK]. Furthermore, we show that $ASM(q)$ has a nonsingular model $\cY $ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.

DOI: 10.14232/actasm-017-757-9

AMS Subject Classification (1991): 14H37, 14H05

Keyword(s): algebraic curves, algebraic function fields, positive characteristic, automorphism groups

Received January 30, 2017, and in revised form September 25, 2017. (Registered under 7/2017.)

Abstract. Let $K_0$ be a compact convex subset of the plane $\preal $, and assume that $K_1\subseteq\preal $ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\preal\to \preal $. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are contained in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in\set {0,1,2}$ and a $k\in\set {0,1}$ such that $K_{1-k}$ is contained in the convex hull of $K_k\cup(\set{A_0,A_1, A_2}\setminus\set {A_j})$. Here we prove that this property characterizes disks among compact convex subsets of the plane. In fact, we prove even more since we replace ``similar'' by ``isometric'' (also called ``congruent''). Circles are the boundaries of disks, so our result also gives a characterization of circles.

DOI: 10.14232/actasm-016-570-x

AMS Subject Classification (1991): 52C99, 52A01

Keyword(s): convex hull, circle, abstract convex geometry, anti-exchange system, Carathéodory's theorem, carousel rule, boundary of a compact convex set, lattice

Received December 13, 2016, and in revised form May 10, 2017. (Registered under 70/2016.)

Abstract. Kira Adaricheva and Madina Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in\set {0,1,2}$ and $k\in\set {0,1}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\set{A_0,A_1, A_2}\setminus\set {A_j})$. We give a short new proof for this result, and we point out that a straightforward generalization for spheres fails.

DOI: 10.14232/actasm-016-307-7

AMS Subject Classification (1991): 52C99, 52A01

Keyword(s): convex hull, circle, sphere, abstract convex geometry, anti-exchange system, Carathéodory's theorem, carousel rule

Received October 8, 2016, and in revised form May 17, 2017. (Registered under 57/2016.)