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Mathematics and statistics journals publish papers on the theory and application of mathematics, statistics, and probability. Most mathematics journals have a broad scope that encompasses most mathematical fields. These commonly include logic and foundations, algebra and number theory, analysis (including differential equations, functional analysis and operator theory), geometry, topology, combinatorics, probability and statistics, numerical analysis and computation theory, mathematical physics, etc.

# Mathematics and Statistics

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## Extended Cauchy–Schwarz Inequality and its Application for the two-Class Fisher Discriminant Analysis

Studia Scientiarum Mathematicarum Hungarica
Authors:
Maciej Sablik
and
Katarzyna Stapor

We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

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## Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Studia Scientiarum Mathematicarum Hungarica
Author:
Hua Wang

Let be a Schrödinger operator on the Heisenberg group $ℍ n$ , where $Δ ℍ n$ is the sublaplacian on $ℍ n$ and the nonnegative potential V belongs to the reverse Hölder class $B q$ with $q ∈ [ Q / 2 , + ∞ )$ . Here is the homogeneous dimension of $ℍ n$ . Assume that ${ e − s L } s > 0$ is the heat semigroup generated by $L$ . The Lusin area integral $S L ; α$ and the Littlewood–Paley–Stein function $g λ , L *$ associated with the Schrödinger operator $L$ are deﬁned, respectively, by

$S L ; α ( f ) ( u ) : = ∬ Γ α ( u ) s d d s e − s L f ( υ ) 2 d υ d s s Q / 2 + 1 1 / 2 ,$

where

$Γ α ( u ) : = { ( υ , s ) ∈ ℍ n × ( 0 , + ∞ ) : u − 1 υ < α s } ,$

and

Where $λ ∈ (0,+ ∞ )$ is a parameter. In this article, the author shows that there is a relationship between $S L ; α$ and the operator $g λ , L *$ and for any $1 ≤ p < ∞$ , the following inequality holds true:

$S L ; 2 j ( f ) L p ℍ n ≤ C 2 j Q / 2 + 2 j Q / p s L ( f ) L p ( ℍ n ) .$

Based on this inequality and known results for the Lusin area integral $S L ; 1$ , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function $g λ , L *$ on $L p ( ℍ n )$ . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator $L$ on $ℍ n$ . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator $g λ , L *$ acting on the Morrey spaces for an appropriate choice of $λ > 0$ . It can be shown that the same conclusions hold for the operator $g λ , L *$ on generalized Morrey spaces as well.

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## Some Classes of Polynomials Satisfying Sendov’s Conjecture

Studia Scientiarum Mathematicarum Hungarica
Authors:
,
, and
Robert Bentley Gardner

In this paper, a relationship between the zeros and critical points of a polynomial p(z) is established. The relationship is used to prove Sendov’s conjecture in some special cases.

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## Stationary Solution of a Fluid Queue Driven by a Queue with Chain Sequence Rates and Controlled Input

Studia Scientiarum Mathematicarum Hungarica
Authors:
Susairaj Sophia
and
Babu Muthu Deepika

A ﬂuid queueing system in which the ﬂuid ﬂow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a ﬂuid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

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

Studia Scientiarum Mathematicarum Hungarica
Authors:
,
Tayebe Lal Shateri
, and

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 $α ∩ ( β ∘ γ ∘ β ) ⊆ _ α β ∘ γ ∘ α β ∘ γ$ … (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 CA n denote the class of cyindric algebras of dimension n. Roughly CA n 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 RCA n of representable CA n s reflects algebraically the semantics of Ln . Members of RCA n 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 CA n has a finite equational axiomatization, RCA n is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CA n 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 RCA n 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 Nr n CA ω ∩ CRCA n $⊆ ¯$ K $⊆ ¯$ S c Nr n CA n +3, where CRCA n is the class of completely representable CA n s, and S c denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that S d RaCA ω $⊆ ¯$ K $⊆ ¯$ S c RaCA5 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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