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

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

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

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 $p ( z ) ≡ z n p ( 1 / z )$

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

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

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

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

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

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

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

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

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

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

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

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

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

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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# Domain representable Lindelöf spaces are cofinally Polish

## Abstract

We prove that, for any cofinally Polish space X, every locally finite family of non-empty open subsets of X is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of σ-compact spaces. It turns out that, for a topological group G whose space has the Lindelöf Σ-property, the space G is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

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# Korselt rational bases of prime powers

Author: Nejib Ghanmi

## Abstract

Let N be a positive integer, $A$ be a subset of ℚ and $α=α1α2∈A\{0, N}$. N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α 2 pα 1 divides α 2 Nα 1 for every prime divisor p of N. By the Korselt set of N over $A$, we mean the set $A−KS(N)$ of all $α∈A\{0, N}$ such that N is an α-Korselt number.

In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set $A-KS(ql)$ and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of $A=ℚ∩[−1, 1[$ is studied where we prove that $(ℚ∩[−1, 1[)-KS(ql)$ is empty if and only if l = 2.

Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

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# Lemniscate convexity of generalized Bessel functions

Authors: Vibha Madaan, Ajay Kumar and V. Ravichandran

## Abstract

Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and “normalized” Bessel function up(z) = up,b,c(z) satisfies the inequalities ∣(1 + (zup(z)/up(z)))2 − 1∣ &lt; 1 or ∣((zu p(z))′/up(z))2 − 1∣ &lt; 1. We also determine the condition on these parameters so that $−(4(p+(b+1)/2)/c)up'(x)≺1+z$. Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination $1+(zhμ,p''(z)/hμ,q'(z))≺1+z$. Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed.

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# On exponential decay of the variance of BLUE for the mean of a stationary sequence

Authors: Nikolay Babayan and Mamikon S. Ginovyan

## Abstract

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.

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# On the multi-dimensional modal logic of substitutions

Authors: Tarek Sayed Ahmed and Mohammad Assem Mahmoud

## Abstract

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

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# Perfect Pell and Pell–Lucas numbers

Authors: Jhon J. Bravo and Florian Luca

## Abstract

The Pell sequence $(Pn)n=0∞$ is given by the recurrence Pn = 2Pn −1 + Pn −2 with initial condition P 0 = 0, P 1 = 1 and its associated Pell-Lucas sequence $(Qn)n=0∞$ is given by the same recurrence relation but with initial condition Q 0 = 2, Q 1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

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# A remark on large gaps between primes in arithmetic progressions

Author: Deniz Ali Kaptan

## Abstract

We record an implication between a recent result due to Li, Pratt and Shakan and large gaps between arithmetic progressions.

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# Ruin probability for discrete risk processes

Author: Ivana Geček Tuden

## Abstract

We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

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# Stronger forms of sensitivity on product dynamical system via Furstenberg families

Authors: Rahul Thakur and Ruchi Das

## Abstract

In this paper, it has been investigated that how various stronger notions of sensitivity like 𝓕-sensitive, multi-𝓕-sensitive, (𝓕1, 𝓕2)-sensitive, etc., where 𝓕, 𝓕1, 𝓕2 are Furstenberg families, are carried over to countably infinite product of dynamical systems having these properties and vice versa. Similar results are also proved for induced hyperspaces.

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# A variant of the Fejér–Jackson inequality

Authors: Horst Alzer and Man Kam Kwong

## Abstract

We prove: For all natural numbers n and real numbers x ∈ [0, π] we have $−548130−585≤∑k=1n(−1)k+1(sin((2k−1)x)2k−1+sin(2kx)2k)$.

The sign of equality holds if and only if n = 2 and x = 4π/5.

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# Cohomology of the vector fields lie algebras on ℝℙ1 acting on bilinear differential operators

Author: Abdaoui Meher

## Abstract

Let Vect (ℝℙ1) be the Lie algebra of smooth vector fields on ℝℙ1. In this paper, we classify $aff(1)$-invariant linear differential operators from Vect (ℝℙ1) to $Dλ,μ;ν$ vanishing on $aff(1)$, where $Dλ,μ;ν :=Homdiff(ℱλ⊗ℱμ;ℱν)$ is the space of bilinear differential operators acting on weighted densities. This result allows us to compute the first differential $aff(1)$-relative cohomology of Vect (ℝℙ1) with coefficients in $Dλ,μ;ν$.

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# Counting surface branched covers

Authors: Carlo Petronio and Filippo Sarti

## Abstract

To a branched cover f between orientable surfaces one can associate a certain branch datum $D(f)$, that encodes the combinatorics of the cover. This $D(f)$ satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum $D$ there exists a branched cover f such that $D(f)=D$. One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.

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# Good congruences on abundant semigroups with quasi-ideal adequate transversals

Author: Aifa Wang

## Abstract

The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal S° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts I, S° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

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# Monomorphisms in categories of firm acts

Authors: Valdis Laan and Ülo Reimaa

## Abstract

We prove that in the category of firm acts over a firm semigroup monomorphisms co-incide with regular monomorphisms and we give an example of a non-injective monomorphism in this category. We also study conditions under which monomorphisms are injective and we prove that the lattice of subobjects of a firm act over a firm semigroup is isomorphic to the lattice unitary subacts of that act.

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# On the problem of Pillai with Padovan numbers and powers of 3

## Abstract

Let {P n}n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1, P 2 = 1, and Pn +3 = Pn +1 + Pn for all n ≥ 0. In this paper, we find all integers c admitting at least two representations as a difference between a Padovan number and a power of 3.

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# Pre-Schwarzian norm for linear operators of uniformly convex functions of order α and type β

Authors: Jacek Dziok and Hanaa M. Zayed

## Abstract

By making use of the pre-Schwarzian norm given by

$‖f‖ = supz∈U (1−|z|2) |f′′(z)f′(z)|,$
we obtain such norm estimates for Hohlov operator of functions belonging to the class of uniformly convex functions of order α and type β. We also employ an entirely new method to generalize and extend the results of Theorems 1, 2 and 3 in . Finally, some inequalities concerning the norm of the pre-Schwarzian derivative for Dziok-Srivastava operator are also considered.

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# Various notions of represetability for cylindric and polyadic algebras

Author: Tarek Sayed Ahmed

## Abstract

For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if $A ∈PEAα$ is atomic, then for any n < ω, the n-neat reduct of $A$, in symbols $ℜrnA→B$, is a completely representable PEAn (regardless of the representability of $A$). That is to say, for all non-zero $a∈ℜrnA$, there is a $Ba∈SetPEAn$ and a homomorphism $fa:ℜrnA→B$ such that fa(a) ≠ 0 and $fa(∑X) =∪x∈Xfa(x)$ for any $X ⊂= A$ for which $∑X$ exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElS c Nr n CA ω, where El denotes ‘elementary closure’ and S c is the operation of forming complete subalgebras.

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# Almost sure central limit theorems for weighted dependent sequences

Author: Khurelbaatar Gonchigdanzan

## Abstract

Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ kn, n ≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

$limn→∞1log n∑k=1n1kI(∑i=1kwkiXi≦x)=12π∫−∞xe−12t2dt a.s..$

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# Associative rings in which nilpotents form an ideal

Author: Orest D. Artemovych

## Abstract

It is shown that if N(R) is a Lie ideal of R (respectively Jordan ideal and R is 2-torsion-free), then N(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

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# Commutators of sublinear operators in grand morrey spaces

Authors: Vakhtang Kokilashvili, Alexander Meskhi and Humberto Rafeiro

## Abstract

In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

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# Eigenstructure for binomial operators

Authors: Shifen Wang and Chungou Zhang

## Abstract

In this article, the eigenvalues and eigenvectors of positive binomial operators are presented. The results generalize the previously obtained ones related to Bernstein operators. Illustrative examples are supplied.

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# Groups with restrictions on proper uncountable subgroups

Authors: Francesco De Giovanni and Marco Trombetti

## Abstract

A group G is called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to investigate the structure of uncountable groups of cardinality ℵ in which all proper subgroups of cardinality ℵ are metahamiltonian. It is proved that such a group is metahamiltonian, provided that it has no simple homomorphic images of cardinality ℵ. Furthermore, the behaviour of elements of finite order in uncountable groups is studied in the second part of the paper.

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# The odd Nadarajah-Haghighi family of distributions: properties and applications

Authors: Abraão D. C. Nascimento, Kássio F. Silva, Gauss M. Cordeiro, Morad Alizadeh, Haitham M. Yousof and G. G. Hamedani

## Abstract

We study some mathematical properties of a new generator of continuous distributions called the Odd Nadarajah-Haghighi (ONH) family. In particular, three special models in this family are investigated, namely the ONH gamma, beta and Weibull distributions. The family density function is given as a linear combination of exponentiated densities. Further, we propose a bivariate extension and various characterization results of the new family. We determine the maximum likelihood estimates of ONH parameters for complete and censored data. We provide a simulation study to verify the precision of these estimates. We illustrate the performance of the new family by means of a real data set.

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# On quasi-radical of near-ring of polynomials

Authors: Ebrahim Hashemi, Fatemeh Shokuhifar and Abdollah Alhevaz

## Abstract

The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R 0[x] equals to the set of all nilpotent elements of R 0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R 0[x] is a subset of the intersection of all maximal left ideals of R 0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R 0[x] coincides with the intersection of all maximal left ideals of R 0[x]. Moreover, we prove that the quasi-radical of R 0[x] is the greatest quasi-regular (right) ideal of it.

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# On weakly ℌ-embedded subgroups and p-nilpotence of finite groups

## Abstract

Let G be a finite group and H a subgroup of G. We say that H is an -subgroup of G if NG (H) ∩ HgH for all gG; H is called weakly -embedded in G if G has a normal subgroup K such that HG = HK and HK is an -subgroup of G, where HG is the normal clousre of H in G, i. e., HG = 〈Hg|gG〉. In this paper, we study the p-nilpotence of a group G under the assumption that every subgroup of order d of a Sylow p-subgroup P of G with 1 < d < |P| is weakly -embedded in G. Many known results related to p-nilpotence of a group G are generalized.

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# Polynomial index in discrete valuation rings

Authors: Mohamed E. Charkani and Abdulaziz Deajim

## Abstract

Let R be a discrete valuation ring, $p$ its nonzero prime ideal, PR[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the $p$-adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo $p$. By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

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# Rings in which every regular locally principal ideal is projective

Authors: Rachida El Khalfaoui and Najib Mahdou

## Abstract

In this article, we study the class of rings in which every regular locally principal ideal is projective called LPP-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.

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# Approximation of functions by some exponential operators of max-product type

## Abstract

In this paper we study the uniform approximation of functions by a generalization of the Picard and Gauss-Weierstrass operators of max-product type in exponential weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. We extend and improve previous results.

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# Existence and uniqueness of solutions of an A-harmonic elliptic equation

Author: Mouna Kratou

## Abstract

This paper deals with the existence and uniqueness of weak solution of a problem which involves a class of A-harmonic elliptic equations of nonhomogeneous type. Under appropriate assumptions on the function f, our main results are obtained by using Browder Theorem.

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# The odd log-logistic gompertz lifetime distribution: Properties and applications

## Abstract

In this paper, we introduce a new three-parameter generalized version of the Gompertz model called the odd log-logistic Gompertz (OLLGo) distribution. It includes some well-known lifetime distributions such as Gompertz (Go) and odd log-logistic exponential (OLLE) as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the kth order moment, moment generating function and the quantile measure are provided. We discuss maximum likelihood estimation of the OLLGo parameters as well as three other estimation methods from one observed sample. The flexibility and usefulness of the new distribution is illustrated by means of application to a real data set.

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# On copure projective and cotorsion modules

Authors: Wei Ren and Duocai Zhang

## Abstract

Let R be an IF ring, or be a ring such that each right R-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of QF rings, perfect rings and noetherian rings are given.

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# On the convergence of Cesàro means of negative order of Vilenkin-Fourier series

Author: Gvantsa Shavardenidze

## Abstract

In 1971 Onnewer and Waterman establish a sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In this paper we consider different classes of functions of generalized bounded oscillation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesàro means of negative order.

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# Revisiting the universal linear algebraic model for the characteristic two case

Author: Máté L. Juhász

## Abstract

In , a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes (). This formalism was independent from the underlying field, providing an extension and general approach to other fields, such as finite fields. Some steps were taken even for the characteristic 2 case.

In this article, we undertake the study of the characteristic 2 case in more detail. In particular, the concept of virtual quadratic spaces is used (), and a similar result is achieved for finite fields of characteristic 2 as for other fields. Some differences from the non-characteristic 2 case are also pointed out.

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# Roots in extensions of domains or monoids

Author: Gerhard Angermüller

## Abstract

In this note connections between root extensions of monoids and some finiteness conditions on monoids are studied, giving new proofs and generalizing results of Etingof, Malcolmson and Okoh for domains. In the same spirit, results of Jedrzejewicz and Zielinski on root-closed extensions of domains are generalized and sharpened to monoids. Using the same methods, a criterion for being a completely integrally closed domain is generalized to monoids.

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# Some new hermite-hadamard type inequalities and their applications

Authors: Artion Kashuri and Rozana Liko

## Abstract

In this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex function f and for (s, m)-convex function f in the second sense in conformable fractional integral forms. Second, by using five new integral identities, we present some new Riemann-Liouville fractional trapezoid and midpoint type inequalities. Third, using these results, we present applications to f-divergence measures. At the end, some new bounds for special means of different positive real numbers and new error estimates for the trapezoidal and midpoint formula are provided as well. These results give us the generalizations of the earlier results.

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# Ball characterizations in spaces of constant curvature

Authors: Jesus Jerónimo-Castro and Endre Makai, Jr.

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C + 2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds).

Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds).

We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points S d for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

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# Extended exponentiated Nadarajah-Haghighi model: Mathematical properties, characterizations and applications

Authors: Morad Alizadeh, Mahdi Rasekhi, Haitham M. Yousof, Thiago G. Ramires and G. G. Hamedani

In this article, a new four-parameter model is introduced which can be used in mod- eling survival data and fatigue life studies. Its failure rate function can be increasing, decreasing, upside down and bathtub-shaped depending on its parameters. We derive explicit expressions for some of its statistical and mathematical quantities. Some useful characterizations are presented. Maximum likelihood method is used to estimate the model parameters. The censored maximum likelihood estimation is presented in the general case of the multi-censored data. We demonstrate empirically the importance and exibility of the new model in modeling a real data set.

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# On ergodic properties of operator nets on the predual of von neumann algebras

Author: Nazife Erkurşun Özcan

In this paper, we proved theorems which give the conditions that special operator nets on a predual of von Neumann algebras are strongly convergent under the Markov case. Moreover, we investigate asymptotic stability and existence of a lower-bound function for such nets.

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# Reconstruction of martingales and applications to multiple Haar series

Authors: Mamikon Ginovyan and Karen Keryan

Reconstruction theorems for martingales with respect to regular filtration are proved provided that the majorant of the martingale satisfies some specified condition. The ob-tained results are applied to obtain formulas for restoration of coeffcients for multiple Haar series.

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# A unified lattice path approach to distributions related to Markov dependent trials

Authors: Babita Goyal and Kanwar Sen

For fixed integers n(= 0) and μ, the number of ways in which a moving particle taking a horizontal step with probability p and a vertical step with probability q, touches the line Y = n+μX for the first time, have been counted. The concept has been applied to obtain various probability distributions in independent and Markov dependent trials.

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# Weak compactness of direct sums in locally convex cones

We discuss the weakly compact subsets of direct sum cones for the upper, lower and symmetric topologies and investigate the X-topologies of the weak upper, lower and sym-metric compact subsets of direct sum cones on product cones.

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# Efficient online packing of 4-dimensional cubes into the unit cube

Authors: Janusz Januszewski and Łukasz Zielonka

Any sequence of 4-dimensional cubes of total volume not greater than 1/8 can be online packed into the unit cube.

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# Existence of the first eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere

Authors: Mariusz Bodzioch, Mikhail Borsuk and Sebastian Jankowski

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

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