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Abstract: Abstract The net Laplacian matrix of a signed graph \(\Gamma \) is defined as \(N_{\Gamma }=D_{{\Gamma }}^{\pm }-A_{{\Gamma }}\) , where \(D_{{\Gamma }}^{\pm }\) and \(A_{{\Gamma }}\) are the diagonal matrix of net degrees and the adjacency matrix of \(\Gamma \) , respectively. In this paper, we introduce several corona-like products of two signed graphs, and compute the characteristic polynomial of the corresponding net Laplacian matrix either in the general case or in a case when one of constituents (or factor graphs) is regular. The corresponding net Laplacian eigenvalues are explicitly computed under an additional assumption requiring particular signature for one constituent. For a real vector \(\textbf{b}\) , the pair \((N_\Gamma , \textbf{b})\) is controllable if \(N_\Gamma \) has no eigenvector orthogonal to \(\textbf{b}\) . As an application of the obtained results, we give necessary and sufficient conditions for the controllability of each product, again either in the general case or a particular case imposing regularity and/or particular signature. PubDate: 2024-08-10
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Abstract: Abstract For a metric continuum X and a point \(p\in X\) , the hyperspace of arcs in X containing p, Arcs(p, X), is defined as the set containing \(\{p\}\) and all arcs in X that contain the point p, we endow this hyperspace with the Hausdorff metric. In this paper we introduce the property of Kelley by arcs; a continuum X has the property of Kelley by arcs provided that for each \(p\in X\) , for each \(A\in Arcs(p, X)\) , and for each sequence of points \(\{p_{n}\}_{n\in \mathbb {N}}\) that converges to p, there exists a sequence \(\{A_{n}\}_{n\in \mathbb {N}}\) that converges to A such that \(A_{n}\in Arcs(p_{n}, X)\) , for each \(n\in \mathbb {N}\) . We show that in the class of finite graphs the property of Kelley by arcs characterizes the fact of being either an arc or a simple closed curve. We prove that in arc-continua the property of Kelley by arcs is equivalent to the property of Kelley. Also, we prove that homogeneous continua have the property of Kelley by arcs. Moreover, we prove that in the class of dendroids the property of Kelley by arcs implies the property of Kelley, and we show that the arc is the only dendroid with the property of Kelley by arcs. Finally, we study the property of Kelley by arcs in the class of compactifications of the ray \([0, \infty )\) with remainder a finite graph or a dendroid. PubDate: 2024-08-08
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Abstract: Abstract In this paper, we study inverse limits with a single upper semi-continuous function F such that it is the union of mappings defined from a compact metric space X into itself. We prove that if Dom(F) is a totally disconnected space, then \(\displaystyle \lim _{\longleftarrow }(X,F)\) is homeomorphic to the Cantor set. This gives a partial answer to a problem posed by Ingram (Topol Appl 299:1–11, 2021), and it answers a question asked by Capulín et al. (Topol Proc 60:71–80, 2022). PubDate: 2024-08-07
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Abstract: Abstract The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of Gauss hypergeometric functions and extend the formalism to certain generalized forms of these functions. It is shown that suggested approach is particularly efficient for evaluating integrals involving hypergeometric functions and their combination with other special functions. PubDate: 2024-08-03
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Abstract: Abstract The 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by \(\nu _2(n)\) . In this paper, we prove that Euler’s partition function p(n) can be expressed in terms of \(\nu _2(n)\) . Our approach allows us to express the sum of positive divisors of n in terms of \(\nu _2(n)\) . We introduce the notion of 2-adic color partition and provide a new combinatorial interpretation for Euler partition function p(n). Connections between partitions and the game of m-Modular Nim with two heaps are presented in this context. PubDate: 2024-07-29
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Abstract: Abstract In recent years, the study of piecewise linear differential systems has gained importance due to their applications. Many diverse natural phenomena are already modeled by this kind of system, like in physics, biology, economics, etc. It is common knowledge that limit cycles are the primary research focus in the qualitative theory of piecewise differential systems. Planar systems have been extensively taken into account in the majority of publications examining the existence and maximum number of limit cycles of piecewise differential systems. Still, a few investigations are looking into this issue in \(\mathbb {R}^3\) . This study aims to answer the problem of the existence and the maximum number of limit cycles for a class of discontinuous piecewise differential systems in \(\mathbb {R}^3\) formed by two Karabut systems and having one of the cylinders \(C_i =\{(x,y,z)\in \mathbb {R}^3:f_i(x,y,z)=0\}\) with \(i=1,2\) as the switching manifold where \(f_1(x,y,z)=z-x^2\) and \(f_2(x,y,z)=x^2+y^2-1\) . The exact maximum number of limit cycles in this class of discontinuous piecewise differential systems is difficult to determine. Nevertheless, we find that there cannot be more than two limit cycles when the separation surface is the cylinder \(C_1\) and at most four limit cycles when we separate with the cylinder \(C_2\) . Additionally, we give instances for any separation surface that guarantees the achievement of this maximum. PubDate: 2024-07-26
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Abstract: Abstract The projective span of a smooth manifold is defined to be the maximal number of linearly independent tangent line fields. We initiate a study of projective span, highlighting its relationship with the span, a more classical invariant. We calculate the projective span for all Wall manifolds, which are certain mapping tori of Dold manifolds. PubDate: 2024-07-26
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Abstract: Abstract We introduce new generalized q-deformed coherent states (q-CS) by replacing the q-factorial of \([n]_q!\) in the series expansion of the classical q-CS by the generalized factorial \(x_n^{q,\alpha }!\) where \(x_n^{q,\alpha }=(1+\alpha q^{n-1})[n]_q\) . We use the shifted operators method based on the sequence \(x_n^{q,\alpha }\) to obtain a realization in terms of Al-Salam–Chihara polynomials for the basis vectors of the Fock space carrying the constructed q-CS. These new states interpolate between the q-CS of Arik–Coon type ( \(\alpha =0\) , \(0<q<1\) ) and a set of coherent states of Barut–Girardello type for the Meixner–Pollaczek oscillator ( \(\alpha \ne 0\) , \(q\rightarrow 1\) ). We also discuss their associated Bargmann-type transforms. As application, we introduce a generalization of the Euler probability distribution and we derive its main statistical parameters. PubDate: 2024-07-24
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Abstract: Abstract Let G be a graph and H a graph possibly with loops. We will say that a graph G is an H-colored graph if and only if there exists a function \(c:E(G)\longrightarrow V(H)\) . A cycle \((v_1,\ldots ,v_k,v_1)\) is an H-cycle if and only if \((c(v_1 v_2),\ldots ,c(v_{k-1}v_k),\) \(c(v_kv_1), c(v_1 v_2))\) is a walk in H. Whenever H is a complete graph without loops, an H-cycle is a properly colored cycle. In this paper, we work with an H-colored complete graph, namely G, with local restrictions given by an auxiliary graph, and we show sufficient conditions implying that every vertex in V(G) is contained in an H-cycle of length 3 (respectively 4). As a consequence, we obtain some well-known results in the theory of properly colored walks. PubDate: 2024-07-16
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Abstract: Abstract Let \((P_n)_{n\ge 0}\) and \((R_n )_{n\ge 0}\) be the Padovan and Perrin sequences, respectively. Let \(b\ge 2\) be an integer. In this paper, we study the Diophantine equations \(P_{n}=b^{d}R_{m}+R_{k}\) and \(R_{n}=b^{d}P_{m}+P_{k}\) in non-negative integers (n, m, k), where d denotes the number of digits of \(R_k\) and \(P_k\) in base b, respectively. Furthermore, we will see that in the range \(2\le b\le 100\) the number 170,625 is the largest Padovan number which can be represented as a concatenation of two Perrin numbers, on the other hand the number 101,639 is the highest Perrin number which can be a concatenation of two Padovan numbers. PubDate: 2024-07-08
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Abstract: Abstract This paper is a continuation of recent studies of the class of Hadamard–Bergman operators presented in the works of Karapetyants and Samko (Complex Anal Oper Theory 14:77, 2020). We consider a specific weighted case in which we can provide more detailed conditions for the boundedness in weighted Lebesgue spaces. As the main result, we prove boundedness theorems (sufficient conditions and also necessary conditions for positive kernels) for weighted (variable) Hadamard–Bergman operators using the technique of operators with homogeneous kernels, developed earlier in real analysis. PubDate: 2024-07-05
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Abstract: Abstract In this paper, we consider one-dimensional linear swelling porous thermoelastic soils system mixture with second sound, we first prove the well-posedness of the system by using the Lumer–Phillips Theorem. Also, we establish the exponential stability of the solution by introducing an appropriate Lyapunov functional. PubDate: 2024-07-05
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Abstract: Abstract This note deals with three problems posed in the 1930s by two prominent members of the Polish school of mathematics. The first problem is known as the isometric Banach conjecture and deals with the simplest characterization of Hilbert spaces among Banach spaces. It was stated in 1932 by Stephan Banach in his book Théorie des opérations linéaires, 1987. The second is Problem 68 in the Scottish Book, 2015, posed by Stanislaw Ulam in 1936, and the third is the well known floating body problem stated by Stanislaw Ulam in 1935 and which is also known as Problem 19 of the Scottish Book. PubDate: 2024-07-04
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Abstract: Abstract The Mitsch order is already known as a natural partial order for semigroups and rings. The purpose of this paper is to further study of the Mitsch order on modules by investigating basic properties via endomorphism rings. And so, this study also contributes to the results related to the orders on rings. As a module theoretic analog of the Mitsch order, we show that this order is a partial order on arbitrary modules. Among others, lattice properties of the Mitsch order and the relations between the Mitsch order and the other well-known orders, such as the minus order, the Jones order, the direct sum order, and the space pre-order on modules, are studied. In particular, we prove that the minus order is the Mitsch order and we supply an example to show that the converse does not hold in general. PubDate: 2024-07-03
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Abstract: Abstract Let S be a semigroup with an involution \(\tau \) (meaning that it is an involutive anti-automorphism of S)and let \(\mathbb {H}\) be the skew field of quaternions. Our main goal is to solve d’Alembert’s quaternion functional equation $$\begin{aligned} L(xy)+L(x\tau (y))=2L(x)L(y),\,\,\,x,y\in S, \end{aligned}$$ where \(L:S\rightarrow \mathbb {H}\) is the unknown function. PubDate: 2024-07-02
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Abstract: Abstract In this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category \(\mathcal {C}.\) We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n- \(\mathcal {X}\) -tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n- \(\mathcal {X}\) tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as \(\mathcal {X}\) -complete and \(\mathcal {X}\) -hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of n- \(\mathcal {X}\) -tilting theory in Argudin Monroy and Mendoza Hernández (2021). PubDate: 2024-07-01
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Abstract: Abstract In this paper, we prove the monotonicity property of ratios for the Nuttall Q-function. As a direct consequence, the monotonicity properties of ratios for the generalized Marcum Q-function and the confluent hypergeometric function are derived. Next, new bounds for the Nuttall Q-function are established. In particular, new bounds for the generalized Marcum Q-function are presented. Furthermore, numerical computations and graphical representations are discussed. PubDate: 2024-07-01
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Abstract: Abstract We prove the null controllability of a one-dimensional degenerate parabolic equation with drift and a singular potential. Here, we consider a weighted Neumann boundary control at the left endpoint, where the potential arises. We use a spectral decomposition of a suitable operator, defined in a weighted Sobolev space, and the moment method by Fattorini and Russell to obtain an upper estimate of the cost of controllability. We also obtain a lower estimate of the cost of controllability using a representation theorem for analytic functions of exponential type. PubDate: 2024-06-28
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Abstract: Abstract In this paper, we propose a topological social choice model in an equivariant setting. We generalize the Chichilnisky–Heal impossibility theorem to the realm of proper actions of almost-connected locally compact groups. We also improve one result stated in Juárez-Anguiano (Topol Appl 279:107246, 2020) about the preservation of equivariant ANR’s by symmetric powers. PubDate: 2024-06-25
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Abstract: Abstract This paper aims to investigate the existence result for a new class of \(\Phi \) -Caputo-type fractional differential Langevin equation involving the p-Laplacian operator. We develop these result with the help of the theory of p-Laplacian operator, and by making use of some basic proprieties of fractional calculus. By applying Schaefer’s fixed point theorem, we established the existence result. As application, we give an example to demonstrate our theoretical result. PubDate: 2024-06-25
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