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Abstract: Abstract A new formula expressing explicitly the integrals, antidifference, of discrete orthogonal polynomials \(\{P_{n}(x):\) Hahn, Meixner, Kravchuk, and Charlier \(\}\) of any degree in terms of \(P_{n}(x)\) themselves are proved. Other formulae for the expansion coefficients of general-order difference integrations \(\nabla ^{-s}f(x),\,\Delta ^{-s}f(x),\) \(\nabla ^{-s}[x^{\ell }\nabla ^{q}f(x)]\,\) and \(\Delta ^{-s}[x^{\ell }\Delta ^{q}f(x)],\) of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. PubDate: 2022-06-01

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Abstract: Abstract A k-hypergraph with vertex set V and edge set E is called t-regular if every t-element subset of V lies in the same number of elements of E. In this note, we give the necessary and sufficient conditions for the existence of some new families of 3-regular self-complementary k-hypergraphs for \(k=4, 5, 6, 7\) . PubDate: 2022-06-01

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Abstract: Abstract A set D of vertices in a graph G is a locating-dominating set if for every two vertices u, v of \(V-D\) the sets \(N_{G}(u)\cap D\) and \(N_{G}(v)\cap D\) are non-empty and different. The locating-domination number \(\gamma _{L}(G)\) is the minimum cardinality of a locating-dominating set of G. A graph G is \(\gamma _{L}\) -dot-critical if contracting any edge of G decreases its locating-domination number. Let \(k\ge 3\) be an integer. A graph G is called k- \(\gamma _{L}\) -dot-critical if G is \(\gamma _{L}\) -dot-critical with \(\gamma _{L}\left( G\right) =k.\) In this paper, we characterize all connected 3- \(\gamma _{L}\) -dot-critical graphs. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we introduce the unbalanced crown graphs \({\mathcal {C}}_{m,n}\) , and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained. PubDate: 2022-06-01

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Abstract: Abstract For a graph \(\Gamma \) , let \(\gamma (\Gamma ),\) \(\gamma _{t}(\Gamma )\) , and \(\gamma _{tR2}(\Gamma )\) denote the domination number, the total domination number, and the total Roman \(\{2\}\) -domination number, respectively. It was shown in Abdollahzadeh Ahangar et al. (Discuss Math Graph Theory, in press) that for each nontrivial connected graph \(\Gamma ,\) \(\gamma _{t}(\Gamma )\le \gamma _{tR2}(\Gamma )\le 3\gamma (\Gamma ).\) The problem that arises naturally is to characterize the graphs attaining each bound. For the left inequality, we establish a necessary and sufficient condition for nontrivial connected graphs \(\Gamma \) with \(\gamma _{tR2} (\Gamma )=\gamma _{t}(\Gamma ),\) and we characterize those graphs that are \(\{C_{3},C_{6}\}\) -free or block. For the right inequality, we present a necessary condition for nontrivial connected graphs \(\Gamma \) with \(\gamma _{tR2}(\Gamma )=3\gamma (\Gamma ),\) and we characterize those graphs that are diameter-2 or trees. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we propose a lower bound sequence for \(\tau \left( A \circ {A^{ - 1}}\right) \) , the minimum eigenvalue of Hadamard product of an M-matrix and its inverse by constructing a vector x and a constant k such that \(A \circ {A^{ - 1}}x\ge kx\) . We prove the convergence of the lower bound sequence, which is an improvement on some of the existing results. By introducing a parameter and modifying constantly x and k, a more precise lower bound sequence is obtained. An example is given to show that the truth value of the minimum eigenvalue can be obtained by applying the new theorem to some kind of cyclic matrix. And several numerical experiments are given to demonstrate that the new bounds are sharper than some existing ones in most cases. PubDate: 2022-06-01

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Abstract: Abstract This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we determine the complex-valued solutions of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y) \end{aligned}$$ for all \(x,y \in M\) , where M is a monoid, \(\sigma \) : \(M\longrightarrow M\) is an involutive automorphism and \(\tau \) : \(M\longrightarrow M\) is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2-dimensional irreducible representations of M. PubDate: 2022-06-01

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Abstract: Abstract The double Roman domination can be described as a strengthened defense strategy. In an empire, each city can be protected by at most three troops. Every city having no troops must be adjacent to at least two cities with two troops or one city with three troops. Every city having one troop must be adjacent to at least one city with more than one troop. Such an assignment is called a double Roman dominating function (DRDF) of an empire/a graph. The minimum number of troops under such an assignment is the double Roman domination number, denoted as \(\gamma _{dR}\) . Shao et al. (2018) determine the exact value of \(\gamma _{dR}(P(n,1))\) . Jiang et al. (2018) determine \(\gamma _{dR}(P(n,2))\) . In this article, we investigate the double Roman domination number of P(n, k) for \(k\ge 3\) . We determine the exact value of \(\gamma _{dR}(P(n,k))\) for \(n\equiv 0(\bmod 4)\) and \(k\equiv 1(\bmod 2)\) , and present an improved upper bound of \(\gamma _{dR}(P(n,k))\) for \(n\not \equiv 0(\bmod 4)\) or \(k\not \equiv 1(\bmod 2)\) . Our results imply P(n, 3) for \(n\equiv 0(\bmod 4)\) is double Roman which can partially answer the open question present by Beeler et al. (2016). PubDate: 2022-06-01

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Abstract: Abstract A group G is called to be EMN-group (rep. ESMN-group) if all maximal subgroup (rep. second maximal subgroup) of G of even order are nilpotent. In this paper, we mainly investigate the structure of EMN-groups and ESMN-groups. PubDate: 2022-06-01

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Abstract: Abstract Today, real-world problems modeling is the first step in controlling, analyzing, and optimizing them. One of the applied techniques for modeling some of these problems is the queueing theory. Usually, the conditions such as lack of space, feedback, admission limits, etc. are the inseparable parts of these problems. This paper deals with modeling and analyzing the steady-state behavior of an \(M^{X} /G/ 1\) retrial queueing system with two phases of heterogeneous services and general retrial time. The arriving batches join the system with dependent admission due to the server state. If the customers find the server busy, they join the orbit to repeat their request. Although, the first phase of service is essential for all customers, any customer has three options after the completion of the \(i\) -th phase \(\left( {i = 1,2} \right)\) . They may take the \(\left( {i + 1} \right)\) -th phase of service with probability \({\uptheta }_{{\text{i}}}\) , otherwise, return the orbit with probability \(p_{i} \left( {1 - \theta_{i} } \right)\) or leave the system with probability \({ }\left( {1 - p_{i} } \right)\left( {1 - \theta_{i} } \right)\) . In this paper, after finding the steady-state distributions and the probability generating functions of the system and orbit size, some important performance measures are found. Then, the sensitivity analysis of performance measures and cost rate is done concerning arrival/retrial rates, feedback probabilities, and state-dependent admission in a telecommunication system. PubDate: 2022-06-01

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Abstract: Abstract Let (a, b, c) be a primitive Pythagorean triple. Set \(a = m^2-n^2 \) , \(b=2mn\) , and \(c=m^2+n^2\) with m and n positive coprime integers, \( m>n \) and \(m \not \equiv n \pmod 2 \) . A famous conjecture of Jeśmanowicz asserts that the only positive integer solution to the Diophantine equation \(a^x+b^y=c^z\) is \((x,y,z)=(2,2,2).\) A solution \((x,y,z) \ne (2,2,2)\) of this equation is called an exceptional solution. In this note, we will prove that for any \( n>0 \) there exists an explicit constant c(n) such that if \( m> c(n) \) , the above equation has no exceptional solution when all x,y and z are even. Our result improves that of Fu and Yang (Period Math Hung 81(2):275–283, 2020). As an application, we will show that if \(4 \mid \!\mid m\) and \(m > c(n) \) , then Jeśmanowicz’ conjecture holds. PubDate: 2022-06-01

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Abstract: Abstract In this article, a weighted problem under boundary Dirichlet condition in the unit ball of \({\mathbb {R}}^{2}\) is considered. The non-linearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. It is proved that there is a nontrivial positive weak solution to this equation. In the critical case, the compactness condition is not satisfied but a suitable asymptotic condition is used to avoid the non-compactness levels to the energy functional. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we give a full classification of pseudo-Riemannian Lie groups of dimension four with non-trivial recurrent curvature tensor (i.e., non-locally symmetric). Then we investigate Ricci solitons on recurrent curvature Lie groups. Locally conformally flat examples of our classification have also been presented. PubDate: 2022-06-01

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Abstract: Abstract In this note we give a simple version (with a simple proof) of Noether’s Normalization Lemma which implies Zariski’s Lemma and Hilbert’s Nullstellensatz (weak form) more naturally. PubDate: 2022-06-01

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Abstract: Abstract A ring R is said to be a pz-ring (resp., psz-ring) whenever every prime ideal of R is a z-ideal (resp., sz-ideal). In this article, we introduce and investigate these concepts. We show that X is a P-space if and only if C(X) is a pz-ring; if and only if \(\frac{C(X)}{I}\) is a pz-ring for every nonzero ideal I of C(X). Also, we show that if X is a compact space and I is an ideal of C(X), then the quotient ring \(\frac{C(X)}{I}\) is a pz-ring if and only if I is a finite intersection of maximal ideals. We prove that R[[x]] is never a pz-ring. Also, we introduce a new class of ideals in \(\prod _{\lambda \in \Lambda }R_\lambda \) denoted by \({\mathcal {I}}({\mathcal {F}},\{I_\lambda \})\) , where \({\mathcal {F}}\) is a filter on \(\Lambda \) and then we show that \(R_\lambda \) is a pz-ring for every \(\lambda \in \Lambda \) if and only if every semiprime ideal of the form \({\mathcal {I}}({\mathcal {F}},\{I_\lambda \})\) is a z-ideal, where \(\mathcal F\) is an ultrafilter. Moreover, in addition to the main statements of the article, it is proved that an ideal I of R is an sz-ideal if and only if (I, x) is an sz-ideal in R[x]. Also, we show that I is a semiprime ideal in R if and only if I[x] is an sz-ideal in R[x]; if and only I[x] is a z-ideal in R[x]. Using this fact, we present a class of simple examples that shows the sum of two z-ideals is not a z-ideal, in general. PubDate: 2022-06-01

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Abstract: Abstract The class of close-to-convex functions are univalent and so its subclasses. For normalised analytic functions defined on the unit disk, four subclasses of close-to-convex functions are considered and Hermitian-Toeplitz determinants for these classes are investigated. All the results presented in this article are sharp. PubDate: 2022-06-01

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Abstract: Abstract In this article, we introduce generalized reverse derivations in semirings and present conditions that lead to the commutativity of additively inverse semirings. PubDate: 2022-06-01

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Abstract: Abstract Given a nilpotent Lie algebra L of dimension \(\le 6\) on an arbitrary field of characteristic \(\ne 2\) , we show a direct method to detect whether L is capable or not via computations on the size of its nonabelian exterior square \(L \wedge L\) . For dimensions higher than 6, we show a result of general nature, based on the evidences of the low dimensional case, but also on the evidences of large families of nilpotent Lie algebras, namely the generalized Heisenberg algebras. Indeed, we detect the capability of \(L \wedge L\) via the size of the Schur multiplier \(M(L/Z^\wedge (L))\) of \(L/Z^\wedge (L)\) , where \(Z^\wedge (L)\) denotes the exterior center of L. PubDate: 2022-06-01

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Abstract: Abstract In this paper, we study the cotorsion dimensions and cotorsion triples in the recollements of abelian categories. The main results are that recollements induce new (resp. complete hereditary) cotorsion triples from the middle category and that the cotorsion dimensions are bounded under certain conditions. As an application, the cotorsion triples in the recollements of module categories with respect to triangular matrix algebras are recovered. PubDate: 2022-06-01