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Abstract: Abstract Let R be the space of real numbers with the ordinary topology. Define \(x\star_{1} y = xy \,\,(x, y \in \mathbf{R})\) and \(x\star_{2} y = {\rm max}\{1, x\}+{\rm max}\{1, y\}\,\,(x, y \in \mathbf{R})\) . We show that there is no cancellative continuous semigroup operation which is distributed by \(\star_{i}\,\,(i=1,2)\) . Conversely we show that there is no cancellative continuous semigroup operation which is distributive over \(\star_{i}\,\,(i=1,2)\) . Moreover we discuss the above arguments for a null semigroup operation on R. PubDate: 2022-12-21

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Abstract: Abstract Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for \(\varepsilon=3.26\cdot 10^{-7}\) , large \(k\) and \(k^{2-\varepsilon} \le N\le k^{2+\varepsilon}\) , we construct \(n \times N\) RIP matrices of order \(k\) with \(k = \Omega( n^{1/2+\varepsilon/4)}\) . PubDate: 2022-12-19

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Abstract: Abstract Let \(\mathcal{G}\) be an abelian group, \( \mathcal{A} \) a \( C^* \) -algebra and \( \mathcal{M} \) a pre-Hilbert \( \mathcal{A} \) -module with an \( \mathcal{A} \) -valued inner product \( \langle.,.\rangle \) . We show if a function \( f \colon \mathcal{G}\rightarrow \mathcal{M} \) satisfies the inequality $$ \langle f(x)+f(y),f(x)+f(y)\rangle \leq \langle f(x+y),f(x+y)\rangle,\quad x,y\in \mathcal{G}, $$ then \( f \) is additive. We also prove that for functions \( f \colon \mathcal{G}\rightarrow \mathcal{M} \) , the inequality $$\begin{aligned} & \langle 2f(x)+2f(y)-f(x-y),2f(x)+2f(y)-f(x-y)\rangle \\ &\quad\quad\quad\quad \leq \langle f(x+y),f(x+y)\rangle,\quad x,y\in \mathcal{G}, \end{aligned}$$ implies \( f \) is quadratic. These results enable us to prove the equivalence of a functional inequality and the Drygas functional equation. In addition, we investigate the stability problem associated with these functional inequalities. Finally, we give some examples of quadratic and Drygas functions. PubDate: 2022-12-19

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Abstract: In this paper we begin by presenting an equilibrium result for abstract economies for majorized type maps defined on Hausdorff topological vector spaces. The ideas here motivate new results for maximal and coincidence points for collectively multivalued maps. PubDate: 2022-12-19

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Abstract: Abstract We first provide new sufficient conditions for the existence of minima of functions defined on 0-complete partial metric spaces. We then apply the obtained results to derive some fixed point results for single-valued and set-valued mappings that improve and generalize several well-known results in the literature. Examples are given to illustrate our findings. PubDate: 2022-11-24

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Abstract: Abstract We construct a convex set A with cardinality 2n and with the property that an element of the difference set \(A-A\) can be represented in n different ways. We also show that this construction is optimal by proving that for any convex set A, the maximum possible number of representations an element of \(A-A\) can have is \(\lfloor A /2 \rfloor\) . PubDate: 2022-11-23

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Abstract: Abstract Recently, Altun et al. [1] introduced the notion of p-proximal contractions and p-proximal contractive mappings and discussed about best proximity point results for this two classes of mappings. After that, Gabeleh and Markin [3] showed that the best proximity point theorem for p-proximal contractions proved in [1] follows from the same conclusion fixed point theory. In this short note, we show that the best proximity point theorem for p-proximal contractive mappings follows from the corresponding fixed point result for metric spaces. PubDate: 2022-11-23

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Abstract: Abstract The purpose of this paper is to present a new atomic decomposition for a dense class of weighted Hardy spaces \(H_{w}^{p}(\mathbb R^n)\) via the discrete Calderón-type reproducing formula and the weighted Littlewood–Paley–Stein theory, where \(w\in A_\infty\) is a Muckenhoupt's weight and \(0<p<\infty\) . Our results can recover and improve the known ones in the literature by avoiding using the maximal function characterization and the Calderón–Zygmund decomposition. Moreover, we give a new proof of the weighted Hardy spaces estimates for generalized Calderón–Zygmund operators in terms of the atomic decomposition and the vector-valued singular integral operator theory. Although the theory of \(H_{w}^{p}(\mathbb R^n)\) is well known, we give new and simpler proofs, which in turn is amenable to utilization in general and nonclassical settings. PubDate: 2022-11-23

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Abstract: Abstract Continuing our previous work in [23], we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong [6], we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and 0-concordance of 2-knots. PubDate: 2022-11-23

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Abstract: Abstract In ZF (i.e. the Zermelo–Fraenkel set theory without the Axiom of Choice (AC)), we investigate the set-theoretic strength of a generalized version of Hindman's theorem and of certain weaker forms of this theorem, which were introduced by Fernández-Bretón [8], with respect to their interrelation with several weak choice principles. In this direction, we determine the status of (this general version of) Hindman's theorem (and of weaker forms) in certain permutation models of \(\mathbf{ZFA} + \neg\mathbf{AC}\) and transfer the results to ZF, strengthen some results of [8] and settle a related open problem from Howard and Rubin [10]; thus filling the gap in information in both [8] and [10]. PubDate: 2022-11-23

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Abstract: Abstract The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in \( z <1\) , then $$\max_{ z =1} P'(z) \leq \frac{n}{2}\max_{ z =1} P(z) .$$ The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in \( z <K, K\leq 1\) is still not settled. Motivated by the above open problem, in this paper we prove some inequalities for a class of polynomials having no zeros in \( z <K, K\leq 1\) , and all these zeros lie either on a ray L emanating from the origin or zeros are symmetrically placed along this line L. Several important consequences are also discussed. Our results are the best possible and examples for the equality cases have been presented. PubDate: 2022-11-21

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Abstract: Abstract We discuss two hybrid mean value problems by using the properties of Dirichlet L-functions. The first one is involving a new sum analogue of Dedekind sum and Hurwitz zeta-function, while the second is about Hardy sum S5(h, p) and general Kloosterman sum \(K( n,r,\lambda;p) \) , attached a Dirichlet character \(\lambda\) modulo p. Consequently, we give several exact computational formulas, which generalize some conclusions of this context. PubDate: 2022-11-21

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Abstract: Abstract Let \(\{a_{n}, n \geq 1\}\) be a sequence of positive numbers and \(\{X, X_{n}, n \geq 1\}\) be a sequence of identically distributed random variables. The strong law of large numbers and complete convergence for the partial sums of the random sequence \(\{X, X_{n}, n \geq 1\}\) are established under the mild moment condition \(\sum\nolimits^{\infty}_{n=1} \mathbb{P}( X > a_{n})<\infty\) and under general dependence conditions. These results generalize and extend some known works. PubDate: 2022-11-21

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Abstract: Abstract Let \(\Lambda_{\pi}(n)\) denote the \(n\) th coefficient in the Dirichlet series expansion of the logarithmic derivative of \(L(s,\pi)\) associated with an automorphic irreducible cuspidal representation of \(\mathrm{GL}_{m}\) over \(\mathbb{Q}\) . In this paper, for all \(\alpha\) of irrational type 1 lying in the interval \([0,1]\) , we investigate the best possible estimate for the sum \(\sum _{n\le x} \Lambda_{\pi}(n)e(n\alpha)\) under a certain assumption. And we consider the metric result on the exponential sum involving automorphic \(L\) -functions without any assumptions. Let \(\Lambda(n)\) be the von Mangoldt function. Then as an application, for \(\varepsilon>0\) and all \(0<\alpha<1\) in a set of full Lebesgue measure (depending on \(\pi\) ), we obtain \(\sum _{n\le x} \Lambda(n)\lambda_{\pi}(n)e(n\alpha)=O(x^{\frac{5}{6}+\varepsilon})\) . PubDate: 2022-11-21

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Abstract: Abstract The aim of this paper is to obtain some characterizations (or representations) of set-valued solutions defined on an Abelian monoid with values in a Hausdorff topological vector space of a new class of functional equations. PubDate: 2022-11-21

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Abstract: Abstract Let \((X, \mathcal{B}, \mu, T)\) be a dynamical system where X is a compact metric space with Borel \(\sigma\) -algebra \(\mathcal{B}\) , and \(\mu\) is a probability measure that is ergodic with respect to the homeomorphism \(T \colon X \to X\) . We study the following differentiation problem: Given \(f \in C(X)\) and \(F_k \in \mathcal{B}\) , where \(\mu(F_k) > 0\) and \(\mu(F_k) \to 0\) , when can we say that $$\lim_{k \to \infty} \frac{\int_{F_k} ( \frac{1}{k} \sum_{i = 0}^{k - 1} T^i f ) \, \, \mathrm{d} \mu}{\mu(F_k)} = \int f \, \mathrm{d} \mu \ '$$ We establish some sufficient conditions for these sequences to converge. PubDate: 2022-11-21

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Abstract: Abstract Let \(S= \{ p_1, \ldots, p_s\}\) be a finite, non-empty set of distinct prime numbers and \((U_{n})_{n \geq 0}\) be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and \(w = (w_k, \ldots, w_1)\in\mathbb{Z}^k, w_1, \ldots, w_k\neq 0\) we define \((U_j^{(k, w)})_{j\geq 1}\) an increasing sequence composed of integers of the form \( w_kU_{n_k} +\cdots + w_1U_{n_1} \) , \( n_k>\cdots >n_1\) . Under certain assumptions, we prove that for any \(\varepsilon >0\) , there exists an integer \(n_{0}\) such that \([U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},\) \({\rm for}\, j > n_0\) , where \([m]_S\) denotes the S-part of the positive integer m. On further assumptions on \((U_{n})_{n \geq 0}\) , we also compute an effective bound for \([U_j^{(k, w)}]_S\) of the form \((U_j^{(k,w)})^{1-c}\) , where c is a positive constant depending only on r, \(a_1\) , . . ., \(a_r\) , \(U_0\) , . . ., \(U_{r-1}\) , \(w_1\) , . . ., \(w_k\) and S. PubDate: 2022-11-21

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Abstract: Abstract A word \(w\in F_r\) is said to satisfy a probability gap if there exists a constant \(\delta_w < 1\) such that, for any finite group \(G\) , if the probability that \(w(g_1, g_2, \ldots, g_r)= 1\) in \(G\) is at least \(\delta_w\) , then \(w\) is an identity in \(G\) . Moreover we saythat a group \(G\) has the \(w_t\) -property if however \(r\) subsets \(X_1, . . . ,X_r\) of \(G\) are chosen with \( X_1 = \cdots = X_r =t\) , there exists \((g_1, \ldots,g_r)\in X_1 \times \cdots \times X_r\) such that \(w(g_1, \ldots,g_r)=1\) . We prove that if \(w\) satisfies a probability gap, then for every positive integer \(t\) there exists a constant \(c_t\) such that if a finite group \(G\) satisfies the \(w_t\) -property, then \( G \leq c_t\) or \(w\) is an identity in \(G\) . PubDate: 2022-10-01

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Abstract: Abstract A family \(\mathcal F\) of k-subsets of { \(1,2,\ldots,n\) } is called t-intersecting if \( F\cap F' \geq t\) for all \(F,F'\in \mathcal F\) . A set E is called an r-sunflower shadow of \(\mathcal F\) if one can choose r members \(F_1, F_2, \dots, F_r\) of \(\mathcal F\) containing E and \(F_1\setminus E,\, F_2\setminus E,\dots, {F_r\setminus E}\) are pairwise disjoint. Let $$\mathcal D(n,k,t,\ell,r)= \Big\{D\in \bigl({{[n]}\atop {k}} \bigr) : D\cap [t+(2r-2)\ell] \geq t+(r-1)\ell\Big\}. $$ Motivated by our recent work [6] on intersecting families without unique shadow, we show that for \(\ell\leq t,\, k\geq t+(r-1)\ell\) and \(n\geq n_0(k),\, \mathcal D(n,k,t,\ell,r)\) is the only family attaining the maximum size among all t-intersecting families with all their \(\ell\) th shadows being r-sunflower. PubDate: 2022-10-01

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Abstract: Abstract The property that we have termed generalized Bassian is a natural concept for many areas of algebra, namely the existence of an injective homomorphism \(A\to A/I\) for an object (group, ring, module, algebra, etc.) \(A\) with a normal sub-object \(I\) (normal subgroup, ideal, submodule, etc.) forces that \(I\) is a direct summand of \(A\) . It is a common generalization of the question is it possible to embed an object (group, ring, module, algebra, etc.) in a proper homomorphic image of itself, originally raised by Bass [3]. Here we study the generalized concept for Abelian groups and achieve a certain deep although not complete characterization of all Abelian groups satisfying this generalized property. PubDate: 2022-10-01