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Abstract: The Fibonacci cube of dimension n, denoted as \(\Gamma _n\) , is the subgraph of the hypercube induced by vertices with no consecutive 1s. The Lucas cube \(\Lambda _n\) is the cyclic version of \(\Gamma _n\) . The irregularity of a graph G is the sum of \( d(x)-d(y) \) over all edges \(\{x,y\}\) of G. In two recent papers based on the recursive structure of \(\Gamma _n\) , it is proved that the irregularity of \(\Gamma _n\) and \(\Lambda _n\) is two times the number of edges of \(\Gamma _{n-1}\) and 2n times the number of vertices of \(\Gamma _{n-4}\) , respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind, we give a bijective proof of both results. For these two graphs, we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach, we determine the number of edges and the sequence of degrees of the cube complement of \(\Gamma _n\) . PubDate: 2021-11-01

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Abstract: In order to solve the general multidimensional perturbed oscillatory system \(y'' + \varOmega y = f(y, y')\) with \(K\in {\mathbb {R}}^{d\times d}\) , the order conditions for the ERKN (extended Runge–Kutta–Nyström) methods and some effective ERKN methods were presented in the literature. These methods integrate exactly the multidimensional unperturbed oscillator \(y'' + \varOmega y = 0\) . In this paper, we analyze the stability of ERKN methods for general oscillatory second-order initial value problems whose right-hand-side functions depend on both y and \(y'\) . Based on the linear test model \(y''(t)+\omega ^2y(t)+\mu y'(t)=0\) with \(\mu <2\omega \) , further discussion and analysis on the linear stability of ERKN methods for general oscillatory problems are presented. A new conception of \(\alpha \) -stability region is proposed to investigate how well the numerical methods respect the damping rate of the general oscillatory systems. It gains more insight to the numerical methods when applied to the systems involving \(y'\) . Numerical experiments are carried out to show the significance of the theory. PubDate: 2021-11-01

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Abstract: We study the existence and asymptotic behavior of least energy sign-changing solutions for a N-Laplacian equation of Kirchhoff type with critical exponential growth in \(\mathbb {R}^N\) $$\begin{aligned} \left\{ \begin{aligned}&- \bigg (a+b\int _{\mathbb {R}^N} \nabla u ^N\mathrm{d}x\bigg )\Delta _N u+V( x ) u ^{N-2}u= f( x ,u), \\&u\in W^{1,N}(\mathbb {R}^N), \\ \end{aligned} \right. \end{aligned}$$ where \(a,b>0\) are constants, \(\Delta _Nu=\text {div}( \nabla u ^{N-2}\nabla u)\) , and V(x) is a smooth function. Under some suitable assumptions on \(f\in C(\mathbb {R}^N\times \mathbb {R})\) , we apply the constraint minimization argument to establish a least energy sign-changing solution \(u_b\) with precisely two nodal domains. Moreover, we show that the energy of \(u_b\) is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of \(u_b\) as \(b\searrow 0^+\) . Our results generalize the existing ones to the N-Kirchhoff equation with critical growth. PubDate: 2021-11-01

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Abstract: Given a finite group G, the permutability graph of non-normal subgroups \(\Gamma _{N}(G)\) is given by all proper non-normal subgroups of G as vertex set and two vertices H and K are joined if \(HK = KH\) . Rajkumar and Devi generalized \(\Gamma _{N}(G)\) to the permutability graph of subgroups \(\Gamma (G)\) , extending the vertex set to all proper subgroups of G (removing the assumption of being non-normal) and keeping the same criterion to join two vertices. We consider a natural counterpart for \(\Gamma (G)\) and \(\Gamma _{N}(G)\) , that is, the subgroups lattice \(\mathrm {L}(G)\) of G, and introduce the non-permutability graph of subgroups \(\Gamma _{\mathrm {L}(G)}\) ; its vertices are now given by the set \(\mathrm {L}(G)-{\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))\) , where \({\mathfrak {C}}_{\mathrm {L}(G)}(\mathrm {L}(G))\) denotes the smallest sublattice of \(\mathrm {L}(G)\) containing all permutable subgroups of G, and we join two vertices H, K of \(\Gamma _{\mathrm {L}(G)}\) if and only if \(HK\ne KH\) . We study classical invariants of \(\Gamma _{\mathrm {L}(G)}\) , showing generalizations of previous results in the literature. PubDate: 2021-11-01

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Abstract: This paper is devoted to an inverse space-dependent source problem for space-fractional diffusion equation. Furthermore, we show that this problem is ill-posed in the sense of Hadamard, i.e., the solution (if it exists) does not depend continuously on the data. In addition, we propose a simplified generalized Tikhonov regularization method and prove the corresponding convergence estimates by using a priori regularization parameter choice rule and a posteriori parameter choice rule, respectively. Finally, numerical examples are carried to support the theoretical results and illustrate the effectiveness of the proposed method. PubDate: 2021-11-01

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Abstract: A graph is NIC-planar if it can be drawn in the plane so that there is at most one crossing per edge and two pairs of crossing edges share at most one common end vertex. A (p, 1)-total k-labelling of a graph G is a function f from \(V(G)\cup E(G)\) to the color set \(\{0,1,\ldots ,k\}\) such that \( f(x)-f(y) \ge 1\) if \(xy\in E(G)\) , \( f(e_1)-f(e_2) \ge 1\) if \(e_1\) and \(e_2\) are two adjacent edges in G, and \( f(x)-f(e) \ge p\) if a vertex x is incident with the edge e. The minimum k such that G has a (p, 1)-total k-labelling is the (p, 1)-total labelling number of G. In this paper, we prove that the (p, 1)-total labelling number ( \(p\ge 2\) ) of every NIC-planar graph G is at most \(\Delta (G)+2p-2\) provided that \(\Delta (G)\ge 6p+4.\) PubDate: 2021-11-01

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Abstract: Given a finite graph G, the maximum length of a sequence \((v_1,\ldots ,v_k)\) of vertices in G such that each \(v_i\) dominates a vertex that is not dominated by any vertex in \(\{v_1,\ldots ,v_{i-1}\}\) is called the Grundy domination number, \(\gamma _\mathrm{gr}(G)\) , of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that \(\gamma _\mathrm{gr}(G) \ge \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}\) holds for every connected k-regular graph of order n different from \(K_{k+1}\) and \(\overline{2C_4}\) . The bound in the case \(k=3\) reduces to \(\gamma _\mathrm{gr}(G)\ge \frac{n}{2}\) , and we characterize the connected cubic graphs with \(\gamma _\mathrm{gr}(G)=\frac{n}{2}\) . If G is different from \(K_4\) and \(K_{3,3}\) , then \(\frac{n}{2}\) is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound. PubDate: 2021-11-01

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Abstract: In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy–Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent “radial” definitions of these local operators. Moreover, using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces. PubDate: 2021-11-01

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Abstract: In this paper, we give a new product formula : $$\begin{aligned} {\mathsf {H}}^t(E) {\mathsf {H}}^s(F)\le c_1 {\mathsf {H}}^{t+s}(E\times F)\le & {} c_2 {\mathsf {H}}^s(E) {\mathsf {P}}^t(F)\\\le & {} c_3 {\mathsf {P}}^{t+s}(E\times F) \le c_4 {\mathsf {P}}^s(E) {\mathsf {P}}^t(F). \end{aligned}$$ where \(E\subseteq \mathbb {R}^d\) , \(F\subseteq \mathbb {R}^l\) , \(t,s\ge 0\) and \( {\mathsf {H}}^t\) and \({\mathsf {P}}^s\) denote, respectively, the lower and upper Hewitt–Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt–Stromberg dimensions \({\mathsf {b}}(E\times F)\) and \({\mathsf {B}}(E\times F)\) in terms of the Hewitt–Strombeg dimensions of E and F. PubDate: 2021-11-01

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Abstract: In this paper, we consider a nonlinear Schrödinger equation involving the fractional Laplacian with Dirichlet condition: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{\frac{\alpha }{2}}u+A(x)u=f(x,u,\nabla u) \ \text{ in } \ \Omega ,\\ u>0, \ \text{ in }\ \Omega ; \ u\equiv 0, \ \text{ in }\ \mathbb R^n\backslash \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a domain (bounded or unbounded) in \(\mathbb R^n\) which is convex in \(x_1\) -direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in \(x_1\) -direction in the left half domain of \(\Omega \) . Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space \(\mathbb R^n_+\) . PubDate: 2021-11-01

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Abstract: Let \(w=(w_0,w_1, \dots ,w_l)\) be a vector of nonnegative integers such that \( w_0\ge 1\) . Let G be a graph and N(v) the open neighbourhood of \(v\in V(G)\) . We say that a function \(f: V(G)\longrightarrow \{0,1,\dots ,l\}\) is a w-dominating function if \(f(N(v))=\sum _{u\in N(v)}f(u)\ge w_i\) for every vertex v with \(f(v)=i\) . The weight of f is defined to be \(\omega (f)=\sum _{v\in V(G)} f(v)\) . Given a w-dominating function f and any pair of adjacent vertices \(v, u\in V(G)\) with \(f(v)=0\) and \(f(u)>0\) , the function \(f_{u\rightarrow v}\) is defined by \(f_{u\rightarrow v}(v)=1\) , \(f_{u\rightarrow v}(u)=f(u)-1\) and \(f_{u\rightarrow v}(x)=f(x)\) for every \(x\in V(G){\setminus }\{u,v\}\) . We say that a w-dominating function f is a secure w-dominating function if for every v with \(f(v)=0\) , there exists \(u\in N(v)\) such that \(f(u)>0\) and \(f_{u\rightarrow v}\) is a w-dominating function as well. The (secure) w-domination number of G, denoted by ( \(\gamma _{w}^s(G)\) ) \(\gamma _{w}(G)\) , is defined as the minimum weight among all (secure) w-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs \(G\circ H\) are related to \(\gamma _w^s(G)\) or \(\gamma _w(G)\) . For the case of the secure domination number and the weak Roman domination number, the decision on whether w takes specific components will depend on the value of \(\gamma _{(1,0)}^s(H)\) , while in the case of the total version of these parameters, the decision will depend on the value of \(\gamma _{(1,1)}^s(H)\) . PubDate: 2021-11-01

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Abstract: In this paper, we show the existence of the Lelong–Demailly numbers of positive S-plurisubharmonic currents. Moreover, a valid definition of \(\mathrm{{dd}}^{c}g \wedge T\) is obtained for plurisubharmonic currents T and unbounded plurisubharmonic functions g. The importance of this definition comes from the sharpness of our condition on the Hausdorff dimension of the locus points of g. PubDate: 2021-11-01

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Abstract: Let \(G=(V(G),E(G))\) be a simple graph. A set \(S \subseteq V(G)\) is a strong edge geodetic set if there exists an assignment of exactly one shortest path between each pair of vertices from S, such that these shortest paths cover all the edges E(G). The cardinality of a smallest strong edge geodetic set is the strong edge geodetic number \(\mathrm{sg_e}(G)\) of G. In this paper, the strong edge geodetic problem is studied on the Cartesian product of two paths. The exact value of the strong edge geodetic number is computed for \(P_n \,\square \,P_2\) , \(P_n \,\square \,P_3\) and \(P_n \,\square \,P_4\) . Some general upper bounds for \(\mathrm{sg_e}(P_n \,\square \,P_m)\) are also proved. PubDate: 2021-11-01

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Abstract: The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate’s trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators. PubDate: 2021-11-01

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Abstract: This paper investigates a single server queueing system with an infinite waiting space in which customers are arrived according to renewal process and are served in batches of random size under continuous-time batch Markovian service process. We first determine the vector probability generating function of the system-length distribution at pre-arrival epoch. The system-length distribution at pre-arrival epoch is extracted in terms of zeros of the related characteristic polynomial of the vector probability generating function. By the Markov renewal theory argument, we determine the system-length distribution at random epoch. We also derive the system-length distribution at post-departure epoch using the ‘rate in = rate out’ argument. Finally, some numerical results are exhibited for different inter-arrival time distribution to demonstrate the system performance measures and correctness of analytical results. PubDate: 2021-11-01

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Abstract: The Wiener index is one of the most widely studied parameters in chemical graph theory. It is defined as the sum of the lengths of the shortest paths between all unordered pairs of vertices in a given graph. In 1991, Šoltés posed the following problem regarding the Wiener index: Find all graphs such that its Wiener index is preserved upon removal of any vertex. The problem is far from being solved, and to this day, only one graph with such property is known: the cycle graph on 11 vertices. In this paper, we solve a relaxed version of the problem, proposed by Knor et al. in 2018. For a given k, the problem is to find (infinitely many) graphs having exactly k vertices such that the Wiener index remains the same after removing any of them. We call these vertices good vertices, and we show that there are infinitely many cactus graphs with exactly k cycles of length at least 7 that contain exactly 2k good vertices and infinitely many cactus graphs with exactly k cycles of length \(c \in \{5,6\}\) that contain exactly k good vertices. On the other hand, we prove that G has no good vertex if the length of the longest cycle in G is at most 4. PubDate: 2021-11-01

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Abstract: We study the inclusion relation of the triangular ratio metric balls and the Cassinian metric balls in subdomains of \(\mathbb {R}^n\) . Moreover, we study distortion properties of Möbius transformations with respect to the triangular ratio metric in the punctured unit ball. PubDate: 2021-11-01

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Abstract: The \(\tau \) -Keiper/Li coefficients attached to a function F are closely related to its zero-free regions. However, the absence of a closed formula for calculating these coefficients makes them challenging to use. Motivated by Voros approach, we introduce the discretized \(\tau \) -Keiper/Li coefficients. A finite sum representation derived for these coefficients is useful for numerical calculations. Representation in terms of zeros of the corresponding function is basis for analytic considerations. We prove that the violation of \(\tau /2\) -generalized Riemann hypothesis implies oscillations of corresponding discretized \(\tau \) -Li coefficients with power-growing amplitudes. Results are obtained for the class \({\mathcal {S}}^{\sharp \flat }(\sigma _0, \sigma _1)\) , which contains the Selberg class, the class of all automorphic L-functions, the Rankin–Selberg L-functions, as well as products of suitable shifts of those functions. PubDate: 2021-11-01

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Abstract: In this paper, a new extragradient algorithm is presented to solve the pseudomonotone equilibrium problem with a Bregman–Lipschitz-type condition. The superiority of this algorithm is that it can be performed without any precedent information about the Bregman–Lipschitz coefficients. The weak convergence of the algorithm is determinate under mild assumption, and the strong convergence will be established as the bifunction equilibrium is satisfied in different additional assumptions. In conclusion, we can use the algorithm to find a solution of the variational inequality problem. At the end, several numerical examples are exhibited that demonstrate the efficiency of our method compared to the related methods in the studies. PubDate: 2021-11-01

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Abstract: In this paper, we consider a complex multi-objective fractional programming problem (CMFP). We establish the necessary optimality conditions of problem (CMFP) in the sense of Pareto optimality and derive its sufficient optimality conditions using generalized convexity. Finally, we construct the parametric dual problem to the primal problem (CMFP) and their duality theorems. PubDate: 2021-11-01