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Abstract: Abstract Partially balanced incomplete block (PBIB)-designs are well known to be the generalization of combinatorial 2-designs. In this paper, we first construct PBIB-designs from diametral paths of distance-regular graphs, which generalizes the result for strongly regular graphs. Furthermore, for Q-polynomial distance-regular graphs associated with regular semilattices, we obtain the construction of PBIB-designs through descendents with fixed dual width. PubDate: 2024-02-20

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Abstract: Abstract A set of vertices X of a graph G is a strong edge geodetic set if, to any pair of vertices from X, we can assign one (or zero) shortest path between them, such that every edge of G is contained in at least one on these paths. The cardinality of a smallest strong edge geodetic set of G is the strong edge geodetic number \(\mathrm{sg_e}(G)\) of G. In this paper, the strong edge geodetic number of complete multipartite graphs is determined. Graphs G with \(\mathrm{sg_e}(G) = n(G)\) are characterized and \(\mathrm{sg_e}\) is determined for Cartesian products \(P_n\,\square \, K_m\) . The latter result in particular corrects an error from the literature. PubDate: 2024-01-31

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Abstract: Abstract Let \(X=X_1\cup \cdots \cup X_s\subset \mathbb {P}^n\) , \(n\ge 4\) , be a general union of smooth non-special curves with \(X_i\) of degree \(d_i\) and genus \(g_i\) and \(d_i\ge \max \{2g_i-1,g_i+n\}\) if \(g_i>0\) . We prove that X has maximal rank, i.e., for any \(t\in \mathbb {N}\) either \(h^0(\mathcal {I}_X(t))=0\) or \(h^1(\mathcal {I}_X(t))=0\) if it is so in a few explicit cases in \(\mathbb {P}^4\) . We also prove an unconditional weaker result, maximal rank up to a positive integer \(\delta _n\) . PubDate: 2024-01-31

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Abstract: Abstract Suppose that x is a sufficiently large number and \(j\ge 2\) is any integer. Let \(L(s, \textrm{sym}^j f)\) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL \(_{2}(\mathbb {Z})\) . Also, let \(\lambda _{\textrm{sym}^j f}(n)\) be the n-th normalized Dirichlet coefficient of \(L(s, \textrm{sym}^j f)\) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients \(\lambda _{\textrm{sym}^j f}(n)\) over two sparse sequences of positive integers, which improves previous results. PubDate: 2024-01-31

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Abstract: Abstract In this paper, we consider the biharmonic Choquard equation with the nonlocal term on the weighted lattice graph \({\mathbb {Z}}^N\) , namely for any \(p>1\) and \(\alpha \in (0,\,N)\) $$\begin{aligned} \Delta ^2u-\Delta u+V(x)u=\left( \sum _{y\in {\mathbb {Z}}^N,\,y\not =x}\frac{ u(y) ^p}{d(x,\,y)^{N-\alpha }}\right) u ^{p-2}u, \end{aligned}$$ where \(\Delta ^2\) is the biharmonic operator, \(\Delta \) is the \(\mu \) -Laplacian, \(V:{\mathbb {Z}}^N\rightarrow {\mathbb {R}}\) is a function, and \(d(x,\,y)\) is the distance between x and y. If the potential V satisfies certain assumptions, using the method of Nehari manifold, we prove that for any \(p>(N+\alpha )/N\) , there exists a ground state solution of the above-mentioned equation. Compared with the previous results, we adopt a new method to finding the ground state solution from mountain-pass solutions. PubDate: 2024-01-31

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Abstract: Abstract This paper deals with a pseudo-parabolic equation with singular potential and variable exponents. First, we determine the existence and uniqueness of weak solutions in Sobolev spaces with variable exponents. Second, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely using the initial energy. Third, we obtain lower and upper bounds of blow-up time for all possible initial energy. The results in this paper are compatible with the corresponding problems with constant exponents. Part results of the paper extend the recent ones in Lian et al. (J Differ Equ 269:4914–4959, 2020), Xu and Su (J Funct Anal 264:2732–2763, 2013), and Liu and Yu (J Funct Anal 274:1276–1283, 2018). PubDate: 2024-01-29

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Abstract: Abstract A weaker variant of the selection principle \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\Omega ),\) namely \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega }),\) is investigated in this article. We present situations where \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\Omega )\) behaves differently from \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega }).\) Few characterization results are obtained by considering mappings into the Baire space. Several results are presented concerning critical cardinalities. In particular, we perform investigations assuming near coherence of filters (NCF) and semifilter trichotomy. Besides, \({{\,\mathrm{U_{fin}}\,}}({\mathcal {O}},\overline{\Omega })\) is characterized using weakly groupable and related covers. We also exhibit certain game theoretic observations. PubDate: 2024-01-27

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Abstract: Abstract Linearization is a standard approach in the computation of eigenvalues, eigenvectors and invariant subspaces of matrix polynomials and rational matrix valued functions. An important source of linearizations are the so called Fiedler linearizations, which are generalizations of the classical companion forms. In this paper the concept of Fiedler linearization is extended from square regular to rectangular rational matrix valued functions. The approach is applied to Rosenbrock functions arising in mathematical system theory. PubDate: 2024-01-12

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Abstract: Abstract Let \(\Phi \) be a field of prime characteristic p and let G be a finite group. We develop an equivalence relation between the set of isomorphism types of indecomposable (simple) KG-modules, where K is any finite subfield of \(\Phi \) , and relate the equivalence classes to the set of isomorphism types of indecomposable (resp. simple) \(\Phi G\) -modules. When \(\Phi \) is the algebraic closure of a field F of order p, we study indecomposable (resp. simple) \(\Phi G-\) modules and obtain a classification of the isomorphism types of simple \(\Phi G\) -modules and a new formula for the number of such types in each equivalence class. PubDate: 2024-01-05

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Abstract: Abstract A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space \(\mathbb {R}^{p,q}\) . We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value $$\begin{aligned} \varvec{x}-\varvec{y} &=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \\&\quad -(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{aligned}$$ instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions. PubDate: 2024-01-04

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Abstract: In this paper, we analyze various classes of multi-dimensional Stepanov almost automorphic type functions in general metric. We clarify the main structural properties for the introduced classes of metrically Stepanov almost automorphic type functions, providing also some applications to the abstract Volterra integro-differential equations. PubDate: 2024-01-04

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Abstract: Abstract In this work we prove, under symmetry and convexity assumptions on the domain \(\Omega \) , the non- degeneracy at zero of the Hessian matrix of the Robin function for the spectral fractional Laplacian. This work extends to the fractional setting the results of M. Grossi concerning the classical Laplace operator. PubDate: 2024-01-04

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Abstract: Abstract In this article, we present a numerical analysis of the Korteweg-de Vries (KdV) and Regularized Long Wave (RLW) equations using a finite difference space-time method. The KdV and RLW equations are partial differential equations that describe the behavior of long shallow water waves. We show that the finite difference space-time method is an effective way to solve these equations numerically, and we compare the results with those obtained using explicit method and generalized finite difference (GFD) formulae. Our results indicate that the finite difference space-time method provides accurate and stable solutions for both the KdV and RLW equations. PubDate: 2024-01-04

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Abstract: Abstract Let \(\text {S}\) be a finite set, each element of which receives a color. A rainbow t-set of \(\text {S}\) is a t-subset of \(\text {S}\) in which different elements receive different colors. Let \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \) denote the set of all rainbow t-sets of \(\text {S}\) , let \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \) represent the union of \(\left( {\begin{array}{c}\text {S}\\ i\end{array}}\right) \) for \(i=0,\ldots , t\) , and let \(2^\text {S}\) stand for the set of all rainbow subsets of \(\text {S}\) . The rainbow inclusion matrix \(\mathcal {W}^{\text {S}}\) is the \(2^\text {S}\times 2^{\text {S}}\) (0, 1) matrix whose (T, K)-entry is one if and only if \(T\subseteq K\) . We write \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\) for the \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix and the \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix of \(\mathcal {W}^{\text {S}}\) , respectively, and so on. We determine the diagonal forms and the ranks of \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\) . We further calculate the singular values of \(\mathcal {W}_{t,k}^{\text {S}}\) and construct accordingly a complete system of \((0,\pm 1)\) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let \(\mathcal {D}^{\text {S}}_{t,k}\) denote the integral lattice orthogonal to the rows of \(\mathcal {W}_{\le t,k}^{\text {S}}\) and let \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\) denote the orthogonal lattice of \(\mathcal {D}^{\text {S}}_{t,k}\) . We make use of Frankl rank to present a \((0,\pm 1)\) basis of PubDate: 2023-12-13

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Abstract: Abstract The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of \( {\textrm{d}}\left[ {x}_{a}(s) + {\mathfrak {g}}(s, {x}_{a}(s - \omega (s)))\right] =\left[ {\mathfrak {I}}{x}_a(s) + {\mathfrak {f}}(s, {x}_a(s -\varrho (s)))\right] {\textrm{d}}s + \varsigma (s){\textrm{d}}\varpi ^{{\mathbb {H}}}(s),\) \(0\le s\le {\mathcal {T}}\) , \({x}_a(s) = \zeta (s),\ -\rho \le s\le 0. \) The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. PubDate: 2023-12-10

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Abstract: Abstract Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the \(j^*\) -metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres. PubDate: 2023-11-30 DOI: 10.1007/s41980-023-00837-w

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Abstract: Abstract In this paper, we show that for all \(v\equiv 0,1\) (mod 5) and \(v\ge 15\) , there exists a super-simple (v, 5, 2) directed design. Moreover, for these parameters, there exists a super-simple (v, 5, 2) directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes. PubDate: 2023-11-26 DOI: 10.1007/s41980-023-00835-y

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Abstract: Abstract In this paper, we consider a relativistic Abelian Chern–Simons equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u=\lambda \left( a(b-a)e^{u}-b(b-a)e^{v}+a^{2}e^{2u}-abe^{2v}+b(b-a)e^{u+v}\right) +4\pi \sum \limits _{j=1}^{N_{1}} \delta _{p_{j}},\\ \Delta v=\lambda \left( -b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^{2}e^{2v}+b(b-a)e^{u+v}\right) +4\pi \sum \limits _{j=1}^{N_{2}} \delta _{q_{j}}, \end{array} \right. \end{aligned}$$ on a connected finite graph \(G=(V, E)\) , where \(\lambda >0\) is a constant; \(a>b>0\) ; \(N_{1}\) and \(N_{2}\) are positive integers; \(p_{1}, p_{2}, \ldots , p_{N_{1}}\) and \(q_{1}, q_{2}, \ldots , q_{N_{2}}\) denote distinct vertices of V. Additionally, \(\delta _{p_{j}}\) and \(\delta _{q_{j}}\) represent the Dirac delta masses located at vertices \(p_{j}\) and \(q_{j}\) . By employing the method of constrained minimization, we prove that there exists a critical value \(\lambda _{0}\) , such that the above equation admits a solution when \(\lambda \ge \lambda _{0}\) . Furthermore, we employ the mountain pass theorem developed by Ambrosetti–Rabinowitz to establish that the equation has at least two solutions when \(\lambda >\lambda _{0}\) . PubDate: 2023-11-24 DOI: 10.1007/s41980-023-00830-3

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Abstract: Abstract In this paper, we present the complete list of \(U_{q}(sl_{2})\) -symmetries on quantum polynomial algebra \(k_q[x^{\pm 1},y]\) in the case that the action of the generator K of \(U_{q}(sl_{2})\) is a non-toric automorphism. The conditions for the isomorphism of such structures are explored as well. PubDate: 2023-11-23 DOI: 10.1007/s41980-023-00832-1

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Abstract: Abstract Let \(\mathcal {C}\) be an abelian monoidal category. It is proved that the nilpotent category \({\text {Nil}}(\mathcal {C})\) of \(\mathcal {C}\) admits almost monoidal structure except the unit axiom. As an application, it is proved that Hom and Tensor functors exist over \({\text {Nil}}(\mathcal {C})\) and tensor–hom adjunction remains true over the nilpotent category of the category of finite-dimensional vector spaces, which develops some recent results on this topic. PubDate: 2023-11-20 DOI: 10.1007/s41980-023-00831-2