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# A Type of Orthonormal bases on 2-*-inner Product Spaces

Studia Scientiarum Mathematicarum Hungarica

In this paper, we deﬁne an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2-*-inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an A-2-inner product space.

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# Another characterization of congruence distributive varieties

Studia Scientiarum Mathematicarum Hungarica
Author: Paolo Lipparini

## Abstract

We provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity $α∩(β∘γ∘β)⊆_ αβ∘γ∘αβ∘γ$ … (k factors) holds in 𝓥, for some natural number k.

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# Atom Canonicity and First Order Definability in Classes of Algebras of Relations

Studia Scientiarum Mathematicarum Hungarica
Author: Tarek Sayed Ahmed

## Abstract

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of Ln. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever $A$ ∈ V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of Lω,ω, fails dramatically for Ln even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that NrnCAω ∩ CRCAn $⊆¯$ K $⊆¯$ S cNrnCAn +3, where CRCAn is the class of completely representable CAns, and S c denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that S dRaCAω $⊆¯$ K $⊆¯$ S cRaCA5 is not elementary, where S d denotes the operation of forming dense subalgebra.

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# Banach algebras satisfying certain chain conditions on closed ideals

Studia Scientiarum Mathematicarum Hungarica
Authors: Abdullah Alahmari, Falih A. Aldosray, and Mohamed Mabrouk

## Abstract

Let 𝔄 be a unital Banach algebra and its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ is finite dimensional. We also prove that a C *-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

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# Characterizations of the Cauchy distribution associated with integral transforms

Studia Scientiarum Mathematicarum Hungarica
Author: Kazuki Okamura

## Abstract

We give two new simple characterizations of the Cauchy distribution by using the Möbius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.

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# Integral inequalities in a generalized context

Studia Scientiarum Mathematicarum Hungarica
Authors: Péter Kórus, Luciano M. Lugo, and Juan E. Nápoles Valdés

## Abstract

In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions.

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# Lemniscate and exponential starlikeness of regular Coulomb wave functions

Studia Scientiarum Mathematicarum Hungarica
Author: İbrahim Aktaş

## Abstract

In this study, a normalized form of regular Coulomb wave function is considered. By using the differential subordinations method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponential starlike in the open unit disk, respectively. In additon, by using the relationship between the regular Coulomb wave function and the Bessel function of the first kind we give some conditions for which the classical Bessel function of the first kind is lemniscate and exponential starlike in the unit disk 𝔻.

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# Ninety years of k-tridiagonal matrices

Studia Scientiarum Mathematicarum Hungarica
Authors: Carlos M. da Fonseca, Victor Kowalenko, and László Losonczi

## Abstract

This survey revisits Jenő Egerváry and Otto Szász’s article of 1928 on trigonometric polynomials and simple structured matrices focussing mainly on the latter topic. In particular, we concentrate on the spectral theory for the first type of the matrices introduced in the article, which are today referred to as k-tridiagonal matrices, and then discuss the explosion of interest in them over the last two decades, most of which could have benefitted from the seminal article, had it not been overlooked.

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# On power integral bases for certain pure number fields defined by x 24 – m

Studia Scientiarum Mathematicarum Hungarica

## Abstract

Let K = ℚ(α) be a number field generated by a complex root α of a monic irreducible polynomial f(x) = x 24m, with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.

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# Quadrilateral cell graphs of the normalizer with signature (2,4,∞)

Studia Scientiarum Mathematicarum Hungarica
Authors: Nazli Yazici Gözütok and Bahadir Özgür Güler

## Abstract

In this study, we investigate suborbital graphs G u,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on $ℚ∧$ using the group $ГΒ+$ (N) and then obtain some properties of the graphs arising from this action.

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