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# Limiting distributions associated with moments of exponential Brownian functionals

Studia Scientiarum Mathematicarum Hungarica
Authors: Y. Hariya and M. Yor

During the last decade, a number of explicit results about the distributions of exponential functionals of Brownian motion with drift: have been obtained, often originating with the works of D. Dufresne.

In the present paper, we rely extensively on these results to show the existence of limiting measures as$T→∞$, when the law of ${Bt+μt,0≤_t≤_T}$ is perturbed by the Radon-Nikodym density consisting of either of the normalized functionals exp $(−αAT(μ))$ or $1/(AT(μ))m$. The results exhibit different regimes according to whether in the first case, and to a partition of the $(μ,m)$-plane in the second case.

Although a large number of similar studies have been made for, say, one-dimensional diffusions, the present study, which focuses upon Brownian exponential functionals, appears to be new.

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# Automorphisms of Finite Order of Soluble Groups of Finite Rank

Studia Scientiarum Mathematicarum Hungarica
Author: Bertram A. F. Wehrfritz

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

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# Borel Directions and the Uniqueness of Algebroid Functions Dealing with Multiple Values

Studia Scientiarum Mathematicarum Hungarica
Authors: Yang Tan and Qingcai Zhang

In this paper, we investigate the uniqueness of algebroid functions in angular domain by the method of conformal mapping. We discuss the relations between the Borel directions and uniquenss with the multiple values of algebroid functions and obtain some results which extend some uniqueness results of meromorphic functions to that of algebroid functions.

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# Estimates of Functions with Transformed Double Fourier Series

Studia Scientiarum Mathematicarum Hungarica
Author: Boris V. Simonov

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

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# Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

Studia Scientiarum Mathematicarum Hungarica
Authors: Jinjiang Li, Min Zhang, and Haonan Zhao

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+ s) exceptions, all even positive integers up to N can be represented in the form $p12+p22+p33+p43+p56+p66,$

where p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are prime numbers.

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# Existence Results for a Class of p(x)-Kirchhoff Problems

Studia Scientiarum Mathematicarum Hungarica
Author: Mostafa Allaoui

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:

$−M∫Ω1p(x)∇up(x)dx+∫∂Ωβ(x)p(x)∇up(x)dσdiv(∇up(x)-2∇u)=f(x,u)inΩ,$

Where $β∈L∞(∂Ω)$ and $f:Ω×ℝ→ℝ$ satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

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# A Note on Weakly 𐒎 -Subgroups

Studia Scientiarum Mathematicarum Hungarica
Author: Changwen Li

The major aim of the note is to give new brief proofs of the results in the paper “The influence of weakly H -subgroups on the structure of finite groups” [Studia Scientiarum Mathematicarum Hungarica, 51 (1), 27–40 (2014)].

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# On the Maximal Operators of T Means with Respect to Walsh–Kaczmarz System

Studia Scientiarum Mathematicarum Hungarica
Authors: Nata Gogolashvili and George Tephnadze

In this paper we prove and discuss some new (Hp, Lp,∞) type inequalities of the maximal operators of T means with monotone coefficients with respect to Walsh–Kaczmarz system. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out. In particular, we apply these results to prove a.e. convergence of such T means.

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# Sticky Polymatroids on At Most Five Elements

Studia Scientiarum Mathematicarum Hungarica
Author: László Csirmaz

The sticky polymatroid conjecture states that any two extensions of the polymatroid have an amalgam if and only if the polymatroid has no non-modular pairs of flats. We show that the conjecture holds for polymatroids on five or less elements.

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Mathematica Pannonica
Author: Péter Berkics

A linear operator on a Hilbert space $ℍ$, in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if $k+l:k,l∈GS∩GS*=ℍ$.

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

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# Conceptions of Topological Transitivity on Symmetric Products

Mathematica Pannonica
Authors: Franco Barragán, Sergio Macías, and Anahí Rojas

Let X be a topological space. For any positive integer n, we consider the n-fold symmetric product of X, ℱn(X), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : XX, we consider the induced functions ℱn(ƒ): ℱn(X) → ℱn(X). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ+-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++, semi-open and irreducible. In this paper we study the relationship between the following statements: ƒM and ℱn(ƒ) ∈ M.

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# Dual Pairs of Matroids with Coefficients in Finitary Fuzzy Rings of Arbitrary Rank

Mathematica Pannonica
Author: Walter Wenzel

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

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# Lower Estimate of Clique Size via Edge Coloring

Mathematica Pannonica
Authors: Balázs Király and Sándor Szabó

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

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# Pascal’s Hexagram and Desargues Configurations

Mathematica Pannonica
Author: Jaydeep Chipalkatti

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to six conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

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# The Principal Fibre Bundle on Lorentzian Almost R-para Contact Structure

Mathematica Pannonica
Authors: Lovejoy S. Das and Mohammad Nazrul Islam Khan

The purpose of this paper is to study the principal fibre bundle (P, M, G, π p ) with Lie group G, where M admits Lorentzian almost paracontact structure (Ø, ξ p, ηp, g) satisfying certain condtions on (1, 1) tensor field J, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map π * is the isomorphism.

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# Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

Mathematica Pannonica
Authors: Ákos Beke, Sándor Szabó, and Bogdán Zavalnij

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

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# Dispersion’s Uncertainty Principles Associated with the Directional Short-Time Fourier Transform

Studia Scientiarum Mathematicarum Hungarica
Authors: Siwar Hkimi, Hatem Mejjaoli, and Slim Omri

We introduce the directional short-time Fourier transform for which we prove a new Plancherel’s formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, logarithmic uncertainty principle, Faris–Price uncertainty principles and Donoho–Stark’s uncertainty principles.

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# The Extended Beta Generator of Distributions: Properties and Applications

Studia Scientiarum Mathematicarum Hungarica
Authors: Gauss M. Cordeiro, Thiago G. Ramires, Edwin M. M. Ortega, and Rodrigo R. Pescim

We deﬁne the extended beta family of distributions to generalize the beta generator pioneered by Eugene et al. [10]. This paper is cited in at least 970 scientiﬁc articles and extends more than ﬁfty well-known distributions. Any continuous distribution can be generalized by means of this family. The proposed family can present greater ﬂexibility to model skewed data. Some of its mathematical properties are investigated and maximum likelihood is adopted to estimate its parameters. Further, for different parameter settings and sample sizes, some simulations are conducted. The superiority of the proposed family is illustrated by means of two real data sets.

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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 $ℍn$, where $Δℍn$ is the sublaplacian on $ℍn$ and the nonnegative potential V belongs to the reverse Hölder class $Bq$ with $q∈[Q/2,+∞)$. Here is the homogeneous dimension of $ℍn$. Assume that ${e−sL}s>0$ is the heat semigroup generated by$L$. The Lusin area integral $SL;α$ and the Littlewood–Paley–Stein function $gλ,L*$ associated with the Schrödinger operator $L$ are deﬁned, respectively, by

$SL;α(f)(u):=∬Γα(u)sddse−sLf(υ)2dυdssQ/2+11/2,$

where

$Γα(u):={(υ,s)∈ℍn×(0,+∞):u−1υ<αs},$

and

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

$SL;2j(f)Lpℍn≤C2jQ/2+2jQ/psL(f)Lp(ℍn).$

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

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# Some Classes of Polynomials Satisfying Sendov’s Conjecture

Studia Scientiarum Mathematicarum Hungarica

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

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 L n. 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 L ω,ω, fails dramatically for L n even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that $NrnCAω∩CRCAn⊆¯K⊆¯ScNrnCAn+3$ , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that $SdRaCAω⊆¯K⊆¯ScRaCA5$ is not elementary, where Sd 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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# The unit balls of $ℒ ( n l ∞ m )$ and $ℒ s ( n l ∞ m )$

Studia Scientiarum Mathematicarum Hungarica
Author: Sung Guen Kim

## Abstract

For n,m≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$, where $ℒ(ln∞m)$ is the space of n-linear forms on $ℝm$ with the supremum norm, and $ℒs(ln∞m)$ is the subspace of $ℒ(ln∞m)$ consisting of symmetric n-linear forms. First we classify the extreme points of the unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$, respectively. We show that ext $Bℒ(ln∞m)$ ⊂ ext $Bℒ(ln∞m+1)$, which answers the question in []. We show that every extreme point of the unit balls of $ℒ(ln∞m)$ and $ℒs(ln∞m)$ is exposed, correspondingly. We also show that
$expBℒS(ln∞2)=expBℒ(ln∞2)∩ℒs(ln∞2)$

and $expBℒs(l2∞m+1)≠expBℒ(l2∞m+1)∩ℒs(l2∞m+1),$

which answers the questions in [].

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# Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs

Studia Scientiarum Mathematicarum Hungarica
Authors: Hassen Cheriha, Yousra Gati, and Vladimir Petrov Kostov

## Abstract

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r + ,r ), where r + is the number of positive roots and r the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r + ,r ) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for $(s;r0+,r0−;r1+,r1−;…;rd−1+,rd−1−)$, where$ri+$ (resp.$ri−$) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

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# A note on the ternary purely exponential diophantine equation A x + B y = C z with A + B = C 2

Studia Scientiarum Mathematicarum Hungarica
Authors: Elif kizildere, Maohua le, and Gökhan Soydan

## Abstract

Let l,m,r be fixed positive integers such that 2$∤$| l, 3$∤$ lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm 2 − 1,(lr)lm 2 + 1} > 30, then the equation (rlm 2 − 1)x + ((lr)lm 2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

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# Numerical solution of linear differential equations by Walsh polynomials approach

Studia Scientiarum Mathematicarum Hungarica
Authors: György Gát and Rodolfo Toledo

## Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

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# On sharpening of inequalities for a class of polynomials satisfying $p ( z ) ≡ z n p ( 1 / z )$

Studia Scientiarum Mathematicarum Hungarica
Authors: Ritu Dhankhar, Narendra Kumar Govil, and Prasanna Kumar

## Abstract

Let $p(z)=∑j=0najzj$ be a polynomial of degree n. Further, let$M(p, R)=max|z|=R≥1|p(z)| ,$ and $‖p‖=M(p, 1)$. Then according to the well-known Bernstein inequalities, we have $‖p′‖≤n‖p‖$ and $M(p,R)≤Rn‖p‖$. It is an open problem to obtain inequalities analogous to these inequalities for the class of polynomials satisfying p(z) ≡ z n p(1/z). In this paper we obtain some inequalites in this direction for polynomials that belong to this class and have all their coefficients in any sector of opening γ, where 0 $≤_$ γ < π. Our results generalize and sharpen several of the known results in this direction, including those of Govil and Vetterlein [3], and Rahman and Tariq [12]. We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds.

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# On star Lindelöf spaces

Studia Scientiarum Mathematicarum Hungarica
Authors: Wei-Feng Xuan and Yan-Kui Song

## Abstract

In this paper, we prove that if X is a space with a regular G δ-diagonal and X 2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that $∩${g 2(n, x): nω} = {x} for each xX then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying (X) = κ then e(X) $≦$ 22κ; and if X is Hausdorff and we(X) = (X) = κsubset of a space then e(X) $≦$ 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω 1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.

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# Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

Studia Scientiarum Mathematicarum Hungarica
Authors: Gert Vegter and Mathijs Wintraecken

## Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

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# Some properties of harmonic numbers

Studia Scientiarum Mathematicarum Hungarica
Authors: Bing-Ling Wu and Xiao-Hui Yan

## Abstract

Let H n be the n-th harmonic number and let v n be its denominator. It is known that v n is even for every integer $n>=2$. In this paper, we study the properties of H n and prove that for any integer n, v n = e n(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.

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# Some remarks on the midrange crossing constant

Studia Scientiarum Mathematicarum Hungarica
Authors: Éva Czabarka, Inne Singgih, Laszlό Székely, and Zhiyu Wang

## Abstract

We verify an upper bound of Pach and Tóth from 1997 on the midrange crossing constant. Details of their$89π2$ upper bound have not been available. Our verification is different from their method and hinges on a result of Moon from 1965. As Moon’s result is optimal, we raise the question whether the midrange crossing constant is $89π2$.

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# Zeros of the Riemann zeta-function in the discrete universality of the Hurwitz zeta-function

Studia Scientiarum Mathematicarum Hungarica
Author: Antanas Laurinčikas

## Abstract

Let 0 < γ 1 < γ 2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + k h, α), h > 0, with parameter α such that the set {log(m + α): m$ℕ0$} is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γ k} is applied.

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# Characterization of automorphisms of twisted tensor biproducts

Studia Scientiarum Mathematicarum Hungarica
Authors: Wang Xing, Chen Quanguo, and Wang Dingguo

## Abstract

We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

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# The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

Studia Scientiarum Mathematicarum Hungarica
Authors: Tahir S. Gadjiev, Vagif S. Guliyev, and Konul G. Suleymanova

## Abstract

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

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# The first cohomology group of some operator algebras on Hilbert C*-modules

Studia Scientiarum Mathematicarum Hungarica
Authors: Hoger Ghahramani and Saman Sattari

## Abstract

Let X be a Hilbert C*-module over a C*-algebra B. In this paper we introduce two classes of operator algebras on the Hilbert C*-module X called operator algebras with property $k$ and operator algebras with property ℤ, and we study the first (continuous) cohomology group of them with coefficients in various Banach bimodules under several conditions on B and X. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

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# Integral bases of pure fields with square-free parameter

Studia Scientiarum Mathematicarum Hungarica
Author: László Remete

## Abstract

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type $ℚ(nm)$ is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field $ℚ(nm)$, where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of $ℚ(nm)$ is repeating periodically in m with period length depending on n.

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# Jordan's totient function and trigonometric sums

Studia Scientiarum Mathematicarum Hungarica
Author: Wenchang Chu

## Abstract

Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function.

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# An L-function free proof of Hua's Theorem on sums of five prime squares

Studia Scientiarum Mathematicarum Hungarica
Author: Claus Bauer

## Abstract

We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

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# Character varieties of even classical pretzel knots

Studia Scientiarum Mathematicarum Hungarica
Author: Haimiao Chen

## Abstract

For each even classical pretzel knot P(2k 1 + 1, 2k 2 + 1, 2k 3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

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# A Construction of Cohen-Macaulay Graphs

Studia Scientiarum Mathematicarum Hungarica
Authors: Safyan Ahmad and Shamsa Kanwal

## Abstract

We present a technique to construct Cohen–Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen–Macaulay paths.

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