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Abstract: Abstract In this study, we investigate a damped p-Laplacian wave equation with logarithmic nonlinearity, given by $$\begin{aligned} u_{tt}+\Delta ^2 u -\Delta _{p} u+(g*\Delta u)(t)-\Delta u_{t}+\eta (t)u_{t}= u ^{\gamma -2}u\ln u \ \ \textrm{in}\ \ \Omega \times {\mathbb {R}}^{+}, \end{aligned}$$ where \(\gamma>p>2\) and \(\Omega \subset {\mathbb {R}}^{n}\) . By making appropriate assumptions on the relaxation function g and the initial data, we establish the occurrence of finite time blow-up for solutions at varying initial energy levels. For sub-critical initial energy, we obtain blow-up solutions within the framework of potential wells in combination with concavity arguments. We also demonstrate that under suitable conditions, solutions with arbitrarily high positive initial energy will blow up. Furthermore, we discuss lifespan estimates for blowing up solutions. In addition, we provide a general stability analysis of the solution energy. Our results in this work complement and extend the previous work of Pereira et al. (Math Methods Appl Sci 46:8831–8854, 2023) in which the blow-up and decay results were obtained for the case \(\gamma =p\) . PubDate: 2023-11-11
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Abstract: Abstract We construct transmutations that intertwine the operator of fractional differentiation of distributed order and the first-order derivative. Fractional differentiation is considered in two forms, in the sense of Riemann-Liouville and in the sense of Gerasimov-Caputo. For each of them, a corresponding transmutation operator is constructed. Distributed differentiation is given using a Lebesgue-Stieltjes measure. This covers both continuously and discretely distributed order fractional operators, as well as their combinations. In the case when the measure is concentrated at a point, the found transmutation operators coincide with the Stankovich transforms. Therefore, the constructed operators, as well as a special function arising at that, are generalizations of the Stankovich transforms and the Wright function to the case of distributed parameters. The found transmutations make it possible to find solutions for evolutionary equations of distributed order in terms of the corresponding first-order equations. PubDate: 2023-11-11
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Abstract: Abstract The present article is devoted to one generalization of the classical Salem function such that defined in terms of the \(P_3\) -representations of real numbers from the closed interval [0, 1]. The main attention is given to such properties of the presented generalization as the continuity, the monotonicity, and map properties, as well as to self-affine, self-similar, and integral properties. In addition, modeling this generalization by functional equations, properties of its graph, and differential properties are considered. PubDate: 2023-11-11
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Abstract: Abstract This work concerns the description of all possible global dynamics on the Poincaré disc associated to a family of planar semi-homogeneous polynomial vector fields of the form \(X= ((x+y)y, x^3+Bx^2y+Cxy^2+Dy^3)\) with three real parameters B, C, D where \(B-C+D\ne 1\) . PubDate: 2023-11-11
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Abstract: Abstract In the current work, we obtain a family of two-step with-memory methods for solving nonlinear equations. The proposed methods have three function evaluations and six-order convergence. Also, these methods are based on the weight function and have an efficiency index of 1.81712. We solve nonlinear equations with simple roots using with-memory methods. In the following, we have specified the weight function by the highest absorption region. We show several numerical tests and compare our technique with a selection of known methods, using the basins of attraction and numerical computations to confirm the efficiency. PubDate: 2023-11-11
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Abstract: Abstract Let A, B be disjoint sets such that \(A\cup B= [1,2n]\subset {\mathbb {Z}}\) and \(\vert A\vert = \vert B\vert = n\) . Let us call \(m (A, B)=\max _{t \in {\mathbb {Z}}}\vert (t+B)\cap A \vert \) and consider \(M(n):=\min \limits _{(A,B)} m(A,B)\) (over all partitions with \(A \cup B=\left[ 1,2n\right] \) ). There are well-known upper and lower bounds of M(n). In this paper we studied a variation of this problem, i.e. we considered a finite abelian group G with \(\vert G \vert =k\) , we define M(G) which is analogous to M(n) and we obtained upper and lower bounds for M(G). PubDate: 2023-11-11
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Abstract: Abstract The conjecture posed by Marco Buratti, Peter Horak and Alex Rosa states that a (multiset) list L of \(v-1\) positive integers not exceeding \(\lfloor v/2\rfloor \) is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set \(\{0, 1,\ldots ,v-1\}\) if and only if for every divisor d of v, the number of multiples of d appearing in L is at most \(v-d\) . A list L is called realizable if there exists such Hamiltonian path \(P=(x_0,x_1\ldots ,x_{n-1})\) of the complete graph with n vertices whose edge lengths is the given list L. In particular, if \(x_0=0\) the realization is called standard; also, if \(x_{n-1}=1\) the realization is called unitary; and finally, it there exists \(i\in \{0,\ldots ,n-2\}\) such that the vertices \(\{x_{i},x_{i+1}\}=\{n-2,n-1\}\) , then the realization is called strong. In this paper, we give a complete solution when the underlying set is \(\{1,2,4\}\) , using unitary/strong linear realizations. In addition, we construct some interesting linear realizations of size 3, 4 and 5 obtained from some (standard) unitary/strong linear realizations whose underlying sets are \(\{1,2\}\) , \(\{1,3\}\) , \(\{1,x+1\}\) (where \(x\ge 3\) is an odd positive integer), \(\{1,2,4\}\) and \(\{1,2,4,8\}\) . PubDate: 2023-11-11
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Abstract: Abstract Partitions wherein the even parts appear in two different colors are known as cubic partitions. Recently, Merca introduced and studied the function A(n), which is defined as the difference between the number of cubic partitions of n into an even number of parts and the number of cubic partitions of n into an odd number of parts. In particular, using Smoot’s RaduRK Mathematica package, Merca proved the following congruences by finding the exact generating functions of the respective functions. For all \(n\ge 0\) , $$\begin{aligned} A(9n+5)&\equiv 0 \pmod 3,\\ A(27n+26)&\equiv 0 \pmod 3. \end{aligned}$$ Using generating function manipulations and dissections, da Silva and Sellers proved these congruences and two infinite families of congruences modulo 3 arising from these congruences. In this paper, by employing Ramanujan’s theta function identities, we present simplified formulas of the generating functions from which proofs of the congruences of Merca as well as those of da Silva and Sellers follow quite naturally. We also study analogous partition functions wherein multiples of k appear in two different colors, where \(k\in \{3,5,7,23\}\) . PubDate: 2023-11-11
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Abstract: Abstract The article develops operational calculus based on a differential operator with piecewise constant matrix coefficients. The core of a matrix Laplace transformation is the exhibitor of a matrix argument of \(e^{ - Apt }. \) Difference from a Laplace transformation consists that the matrix Laplace transformation affects a vector function; the image is a vector function also. The definition of the Laplace matrix transformation with a division point is given, and its properties and applications to systems of differential equations with piecewise constant coefficients are studied. The Mellin inversion formula for the matrix integral Laplace transform is proved. Significant differences from the scalar case are established; for example, the theorem on the shift of the image argument is valid under the assumption of permutation of matrices. The technique of applying the matrix Laplace transform for solving systems of differential equations and higher order differential equations with piecewise constant coefficients is developed. A method for solving differential equations and systems with piecewise constant coefficients using matrix Laplace transform has been developed. The solution of the vector analog of the problem for the thermal field distribution of a semi-infinite rod is found. PubDate: 2023-10-28
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Abstract: Abstract The Laplacian perturbed by surface in \({\mathbb {R}}^3\) is suggested as a background of solvable model for scattering of ultrasound waves by a cell with inserted nanoparticles. The cell is considered as a domain with a boundary formed by a potential supported by a surface. Nanoparticles are presented by point-like potentials. The asymptotics of the Green function with the singularity at the surface is obtained. Resonances induced by point-like potentials are studied. An application of the result to explanation of selective cancer cell membrane destruction in ultrasonic field is discussed. PubDate: 2023-10-28
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Abstract: Abstract In this paper, a new obstruction to an extension for GKM-graphs (in sense of Guillemin and Zara) is given. For a j-independent GKM-graph, we give a comparison result between the subposets of rank \(<j\) in the face posets for the orbit space of the GKM-action and for the GKM-graph. An example of a k-independent (n, k)-type GKM-graph is constructed for any \(n\ge k\ge 2\) by taking a quotient of a certain periodic GKM-graph. As an application of the above results, we prove that such a GKM-graph has no nontrivial extensions (for any \(n\ge k\ge 2\) ) and cannot be realized by a GKM-manifold (for any \(n=k=3\) or \(n\ge k\ge 4\) ). PubDate: 2023-10-24
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Abstract: Abstract A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step. PubDate: 2023-10-21
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Abstract: Abstract In this paper, we give some applications of the Hankel-type Segal–Bargmann transform \(\mathscr {B}_\alpha \) to the field of uncertainty inequalities and to the field of partial differential equations. The resolution of the time-dependent Schrödinger equations and the time-dependent Dirac equations is based on the techniques of the transmutation operators on the Hankel-type Fock space \(\mathscr {F}_{\alpha ,*}(\mathbb {C}^d)\) . PubDate: 2023-10-21
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Abstract: Abstract Let \(N=K(\root 3 \of {D})\) be a cubic Kummer extension of the cyclotomic field \(K={\mathbb {Q}}(\zeta _3),\) containing a primitive cube root of unity \(\zeta _3,\) with cube free integer radicand \(D>1.\) Denote by f the conductor of the abelian extension N/K, and by \(N^{*}\) the relative genus field of N/K. The aim of the present work is to find out all positive integers D and conductors f such that the genus group \({\text {Gal}}\left( N^{*}/N\right) \cong {\mathbb {Z}}/3{\mathbb {Z}}\times {\mathbb {Z}}/3{\mathbb {Z}}\) is elementary bicyclic. PubDate: 2023-10-21
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Abstract: Abstract In this present investigation, with the help of the Sălăgean differential operator and the Hadamard product, we define new subclasses of analytic functions. We investigate sharp upper bounds for these subclasses. The results presented in this paper have been shown to generalize and improve some recent work of Frasin and Darus (Internat J Math Math Sci 24(9): 577–581, 2000). PubDate: 2023-10-11
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Abstract: Abstract We study in the current paper an optimization problem subject to a controlled system with maximal monotone operators and integral perturbation, in Hilbert spaces. First, we establish a new existence and uniqueness theorem for a coupled system by two first-order differential inclusions governed by maximal monotone operators and single-valued perturbations. Then, we minimize an integral functional over the controls acting in the state of the operators into the coupled system under consideration. PubDate: 2023-10-07
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Abstract: Abstract The interaction of a moving circular cylinder of radius R with the arbitrary circulation \(\gamma \) and two parallel vortex filaments of identical intensity are considered. The stability of a system of two vortex filaments and a cylinder located in the middle between them is studied. The filaments rotate around the cylinder at a constant angular velocity. The problem depends on three parameters \((q,a,\gamma )\) : \(q=R^2/R_0^2,\) where \(2R_0\) is the distance between filaments, \(0<q<1,\) the added mass of the cylinder \(a>0\) and the circulation around cylinder \(\gamma \ne 0.\) The case \(\gamma =2\) was studied earlier. The purpose of this paper is to study the influence of circulation \(\gamma \) in the problem under consideration. The eigenvalues of the linearization matrix are studied for all parameter values. For fixed values of \(\gamma ,\) the parameter plane (q, a) is divided into areas of two types: the area of instability, when the linearization matrix has at least one eigenvalue in the right half-plane, and the linear stability area—all eigenvalues lie on the imaginary axis. The results of the study are consistent with the limiting case of a fixed cylinder (for \(a\rightarrow \infty \) ). PubDate: 2023-10-07
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Abstract: Abstract This study aims for the global existence and asymptotic stability of solutions for a class of nonlinear viscoelastic higher-order p(x)-Laplacian equation. First, we prove the global existence of solutions in the appropriate range of the variable exponents and next, by using Martinez’s Lemma, we prove the asymptotic stability of solutions. Our results extend and improve the earlier results in the literature. PubDate: 2023-10-06
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Abstract: Abstract When the classical Hamburger moment problem has solutions, it has either exactly one solution or infinitely many solutions. Correspondingly, the moment problem is said to be either determinate or indeterminate. In terms of Jacobi operators, this dichotomy translates into the operator being either selfadjoint or symmetric nonselfadjoint. In this work, we present a new criterion for the determinate–indeterminate classification which hinges on bases of representation (in Akhiezer–Glazman terminology) for Jacobi operators so that the corresponding matrices have a certain structure. PubDate: 2023-10-04
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