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In this paper, we study the concomitants of m-generalized order statistics (m-GOSs) and m-dual generalized order statistics (m-DGOSs) from bivariate Cambanis family with nonzero parameter values as an extension of several recent papers. Moreover, we derive some information measures, namely the Shannon entropy, Kullback–Leibler (KL) divergence and Fisher information number (FIN) for the concomitants of m-GOSs, when $$m>-1,$$ and record values, for $$m=-1.$$ Also, the joint distribution of concomitants of m-GOSs and record values for this family are studied. Besides, some useful recurrence relations between moments of concomitants are obtained. Finally, the ordinary order statistics (OOSs), record values and sequential order statistics (SOSs) as subclasses of m-GOSs, as well as the progressive type II censored order statistics (POSs) as a more general subclass of GOSs, are separately discussed.

In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

In this paper, we mainly study the geometric model of the derived category of gentle one-cycle algebras provided by Opper, Plamondon and Schroll. We provide a realization of AAG-invariant on the surface, which is slightly different from the realization in their paper, and deduce a standard form of marked surfaces of gentle one-cycle algebras under derived equivalences. As an application, we classify those derived-unique gentle one-cycle algebras.

This paper aims to discuss the mathematical details in Lewis’ model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Levy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis’ model and validate his results for particular Levy processes. We also simply give the analyticity conditions for the characteristic function of the Carr–Geman–Madan–Yor model and a simple derivation of the characteristic function of Kou’s double exponential model.

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

In this paper, by the affine analogue of the fundamental theorem for Euclidean planar curves, we classify the affine curves with constant affine curvatures. Note that we use the fully affine group and not the equi-affine subgroup consisting of area-preserving affine transformations. (Caution: much of the literature omits the “equi-” in their treatment.) According to the equivariant method of moving frames, explicit formulas for the generating affine differential invariants and invariant differential operators are constructed. At the same time, by using the fact that the affine transformation group GA$$(2,\mathbb {R})$$ can factor as a product of two subgroup $$B\cdot \mathrm{SE}(2,\mathbb {R})$$ and the moving frame of the subgroup SE$$(2,\mathbb {R})$$, we build the moving frame of GA$$(2,\mathbb {R})$$ and obtain the relations among invariants of group GA$$(2,\mathbb {R})$$ and its subgroup SE$$(2,\mathbb {R})$$. Applying the affine curvature to recognize affine equivalent objects is considered in the last part of this paper.

In this paper, we study the concomitants of m-dual generalized order statistics (and consequently m-generalized order statistics) from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution as an extension of several recent papers. Some well-known information measures, the Shannon entropy, the Kullback–Leibler distance and the Fisher information number, are derived. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. Finally, an application of these results is given for bivariate generalized exponential distribution.

In this paper, a general five-step discrete-time Zhang neural network is proposed for time-varying nonlinear optimization (TVNO). First, based on the bilinear transform and Routh–Hurwitz stability criterion, we propose a general five-step third-order Zhang dynamic (ZeaD) formula, which is shown to be convergent with the truncation error $$O(\tau ^3)$$ with $$\tau >0$$ the sampling gap. Second, based on the designed ZeaD formula and the continuous-time Zhang dynamic design formula, a general five-step discrete-time Zhang neural network (DTZNN) model with quartic steady-state error pattern is presented for TVNO, which includes many multi-step DTZNN models as special cases. Third, based on the Jury criterion, we derive the effective domain of step size in the DTZNN model, which determines its convergence and convergence speed. Fourth, a nonlinear programming is established to determine the optimal step size. Finally, simulation results and discussions are given to validate the theoretical results obtained in this paper.

In this paper, we consider the following logarithmic Choquard equation $$\begin{aligned} - \,\Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~\ \text{ in }~\ {\mathbb {R}}^n, \end{aligned}$$where $$n\ge 3, \lambda >0, 2 2$$, this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for $$n\ge 3$$ by using the variational method.

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl-Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

International audience; In this paper, we first introduce the concept and properties of ω-periodic limit process. Then we apply specific criteria obtained to investigate asymptotically ω-periodic mild solutions of a Stochastic Differential Equation driven by a Brownian motion. Finally, we give an example to show usefulness of the theoritical results that we obtain in the paper. MSC : 34C25, 34C27, 60H30, 34 F05.

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being uncolored. At each discrete time interval, a colored vertex with exactly one uncolored neighbor forces this uncolored neighbor to be colored. The initial set S is a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if G is a connected, cubic, claw-free graph of order $$n \ge 6$$, then $$Z(G) \le \alpha (G) + 1$$ where $$\alpha (G)$$ denotes the independence number of G. Further we prove that if $$n \ge 10$$, then $$Z(G) \le \frac{1}{3}n + 1$$. Both bounds are shown to be best possible.

The aim of this paper is to consider the relation between $$\mathsf {Lie}$$-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional Lie algebras, these concepts are not identical, even if the pairs of Leibniz algebras are $$\mathsf {Lie}$$-stem. Moreover, throughout the paper, we provide some conditions under which $$\mathsf {Lie}$$-isoclinism and isomorphism of $$\mathsf {Lie}$$-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well.

The purpose of this paper is by using a generalized forward–backward splitting method to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi-variational inclusions with accretive mappings and the set of fixed points for a $$\lambda $$ -strictly pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. As applications, we utilize our results to study the approximation problem of solutions to a system of variational inequalities, accretive variational inequality problem and convex minimization problem in Banach spaces.

In a recent paper, Das introduced the graph $$\mathcal {I}n(\mathbb {V})$$ , called subspace inclusion graph on a finite dimensional vector space $$\mathbb {V}$$ , where the vertex set is the collection of nontrivial proper subspaces of $$\mathbb {V}$$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of $$\mathcal {I}n(\mathbb {V})$$ when the base field is arbitrary, and he also studied some other properties of $$\mathcal {I}n(\mathbb {V})$$ when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of $$\mathcal {I}n(\mathbb {V})$$ . In this paper, we give the answer to the open problem.

In this paper, several new estimates of the minimum H-eigenvalue for weakly irreducible nonsingular $$\mathcal {M}$$ -tensors, including the new Brauer-type estimates and the new S-type estimates, are derived. It is proved that the new estimates are tighter than some existing ones and numerical examples are given to verify this fact. The other main result of this paper is to provide a sharper Ky Fan-type theorem which is better than the original Ky Fan theorem for the nonsingular $$\mathcal {M}$$ -tensors.

In this paper, we give new existence criteria for the inverse along an element by means of Drazin inverses, one-sided annihilator ideals, idempotents and one-sided invertibility of certain elements in semigroups and rings, respectively. In addition, we take advantage of idempotents and one-sided principal ideals to characterize the inner inverse along an element in a ring. Furthermore, their expressions are shown. Next, we obtain Cline’s formula for the inverse along an element. Finally, we study the inverse of the product along an element in a semigroup. In particular, some results in this paper recover the consequences of the classical generalized inverses such as group inverse, Drazin inverse, and Moore-Penrose inverse.

In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature.

It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Holder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Holder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Holder continuous.

In this paper, we are interested in the brush number of a graph—a concept introduced by McKeil, Messinger, Nowakowski and Pralat. Our main aim in this paper is to study the brush number of the Cartesian product of a tree with a path, and a tree with a cycle. Based on an optimal cleaning process of a tree, we describe cleaning processes that give upper bounds to the brush number of $$T\times P_n$$ and $$T\times C_n$$ . We also show that these bounds are tight if T is a star, a double star and some generalisation of a star.

This paper studies the steady boundary layer flow over an impermeable moving vertical flat plate with convective boundary condition at the left side of the flat plate. The governing partial differential equations are transformed into a system of ordinary (similarity) differential equations by using corresponding similarity variables. These equations were then solved numerically using the function bvp4c from Matlab for different values of the Rayleigh number Ra, the convective heat transfer parameter $$\gamma $$ , and the Prandtl number Pr. This paper demonstrates that a similarity solution is possible if the convective boundary condition heat transfer is associated with the hot or cooled fluid on the left side of the flat plate proportional to $$x^{-1/4}$$ . For the sake of comparison of the numerical results, the case of the static flat plate $$(\sigma =0)$$ has been also studied. For the case of a moving flat plate $$(\sigma =1)$$ , it is shown that the solutions have two branches in a certain range of the positive (assisting flow) and negative (opposing flow) values of the Rayleigh number Ra. In order to test the physically available solutions, a stability analysis has been also performed. The effects of the governing parameters on the skin friction, heat transfer, wall temperature, velocity and temperature profiles, as well as on the streamlines and isotherms are investigated. Comparison with results from the open literature shows a very good agreement.

In the present paper, we study linear operators \(\Delta \) from the algebra of \(2\times 2\) matrices \({\mathbb {M}}_2({\mathbb {C}})\) into its tensor square. Each such kind of mapping defines a quadratic operator on the state space of \({\mathbb {M}}_2({\mathbb {C}})\). We know that q-purity of quasi quantum quadratic operators (q.q.o.) is equivalent to the invariance of the unite sphere under the corresponding quadratic operator. Therefore, in the paper, we consider quadratic operators, which preserve the unit circle, and show that the corresponding quasi q.q.o. cannot be not positive. Note that this is a much weaker condition than the q-purity of quasi q.q.o. Moreover, we will classify q-pure circle preserving quadratic operators into three disjoint classes (non isomorphic). Moreover, we are able to show that quasi q.q.o. corresponding to the first class is block positive. Note that the block positivity is weaker than positivity. This kind of operator, i.e., not positive but block-positive operator allows us to detect that the given state on \({\mathbb {M}}_2({\mathbb {C}})\otimes {\mathbb {M}}_2({\mathbb {C}})\) is either entangled or not. The obtained results will allow us to verify whether a given mapping is positive or not. This finding suggests us to produce a class of non-positive mappings. Moreover, it will shed some light in finding entanglement states.

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

In this paper, we investigate sparsity regularization for the diffusion coefficient identification problem. Here, the regularization method is incorporated with the energy functional approach. The advantages of our approach are to deal with convex minimization problems. Therefore, the well-posedness of the problem is obtained without requiring regularity property of the parameter. The convexity of regularized problems also allows use the fast algorithms developed recently. Furthermore, the convergence rates of the method are obtained under a simple source condition. The main results of this paper are the well-posedness and convergence rates of sparsity regularization. We also obtain some new results of the continuity and the differentiability of related operators.

In this paper, we consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where $$\gamma $$ is a quotient of odd positive integers, $$\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n$$ with $$r,\ p,\ e$$ , and $$q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}$$ are nondecreasing continuous functions on $$\mathbb {R}$$ and $$\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.$$ Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

This paper presents a reliable approach for solving linear systems of ordinary and fractional differential equations. First, the FDEs or ODEs of a system with initial conditions to be solved are transformed to Volterra integral equations. Then Taylor expansion for the unknown function and integration method are employed to reduce the resulting integral equations to a new system of linear equations for the unknown and its derivatives. The fractional derivatives are considered in the Riemann–Liouville sense. Some numerical illustrations are given to demonstrate the effectiveness of the proposed method in this paper.

The aim of this paper is to study the demographic impact of malaria on the host population and investigate the possibility to eliminate it with a strategy based on prevention (e.g., the use of bed nets). In this paper, a deterministic model for the transmission of malaria (with variable human and mosquito population sizes) has been studied. The model equilibria have been found. Their local and global stability have been analyzed. An exact formula for the critical case fatality required to stop the growth of the host has been found. The conditions to eliminate malaria with a strategy based on the use of bed nets have been found.

Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v to be the number of different colors that are used on edges incident to v, and denote it by $$d^c(v)$$ . In a previous paper, we showed that if $$d^c(v)\ge k$$ (color degree condition) for every vertex v of G, then G has a rainbow path of length at least $$\lceil (k+1)/2\rceil $$ . Later, in another paper we first showed that if $$k\le 7$$ , G has a rainbow path of length at least $$k-1$$ , and then, based on this we used induction on k and showed that if $$k\ge 8$$ , then G has a rainbow path of length at least $$\lceil (3k)/5\rceil +1$$ . In 2010, Gyarfas and Mhalla showed that in any proper edge-colored complete graph $$K_n$$ , there is a rainbow path with no less than $$(2n+1)/3$$ vertices. In the present paper, by using a simpler approach we further improve the result by showing that if $$k\ge 8$$ , G has a rainbow path of length at least $$\lceil (2k)/3\rceil +1$$ .

This paper deals with certain algebraic systems called polygroups. A polygroup is a completely regular, reversible in itself hypergroup. The concept of topological polygroups is a generalization of the concept of topological groups. In this paper, we present the concept of topological hypergroups and prove some properties. Then, we define the notion of topological polygroups. By considering the relative topology on subpolygroups we prove some properties of them. Finally, the topological isomorphism theorems of topological polygroups are proved.

For \(\alpha , \gamma \ge 0\) and \(\beta 0, \, \, \, z\in E \end{aligned}$$ for some \(\phi \in {\mathbb {R}}\). For \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\), we consider the integral transform $$\begin{aligned} V_{\lambda }(f)(z):=\int _{0}^{1}\lambda (t)\frac{f(tz)}{t}\mathrm{d}t, \end{aligned}$$ where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). In a very recent paper, Ali et al. (J Math Anal Appl 385:808–822, 2012) discussed the starlikeness of the integral transform \(V_{\lambda }(f)\) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). The aim of present paper is to find conditions on \(\lambda (t)\) such that \(V_{\lambda }(f)\) is starlike of order \(\delta \) (\(0\le \delta \le 1/2\)) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). As applications, we study various choices of \(\lambda (t)\), related to classical integral transforms.

A connected graph \(G\) is optimal-\(\kappa \) if the connectivity \(\kappa (G)=\delta (G)\), where \(\delta (G)\) is the minimum degree of \(G\). It is super-\(\kappa \) if every minimum vertex cut isolates a vertex. An optimal-\(\kappa \) graph \(G\) is \(m\)-optimal-\(\kappa \) if for any vertex set \(S\subseteq V(G)\) with \(|S|\le m\), \(G-S\) is still optimal-\(\kappa \). The maximum integer of such \(m\), denoted by \(O_{\kappa }(G)\), is the vertex fault tolerance of \(G\) with respect to the property of optimal-\(\kappa \). The concept of vertex fault tolerance with respect to the property of super-\(\kappa \), denoted by \(S_{\kappa }(G)\), is defined in a similar way. In a previous paper, we have proved that \(\min \{\kappa _1(G)-\delta (G),\delta (G)-1\}\le O_\kappa (G)\le \delta (G)-1\) and \(\min \{\kappa _1(G)-\delta (G)-1,\delta (G)-1\}\le S_\kappa (G)\le \delta (G)-1\). We also have \(S_\kappa (G)\le O_\kappa (G)\le \delta (G)-1\). In this paper, we study the realizability problems concerning the above three bounds. By construction, we proved that for any non-negative integers \(a,b,c\) with \(a\le b\le c\), (i) there exists a graph \(G\) such that \(\kappa _1(G)-\delta (G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\); (ii) there exists a graph \(G\) with \(\kappa _1(G)-\delta (G)-1=a\), \(S_{\kappa }(G)=b\), and \(\delta (G)-1=c\); and (iii) there exists a graph \(G\) such that \(S_{\kappa }(G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\).

The power graph $${\fancyscript{P}}(G)$$ of a group $$G$$ is the graph which has a vertex set of the group elements and two elements are adjacent if one is a power of the other. Chakrabarty, Ghosh, and Sen proved the main properties of the undirected power graph of a finite group. The aim of this paper is to generalize some results of their work and presenting some counterexamples for one of the problems raised by these authors. It is also proved that the power graph of a $$p$$ -group is $$2$$ -connected if and only if the group is a cyclic or generalized quaternion group and if $$G$$ is a nilpotent group which is not of prime power order then the power graph $${\fancyscript{P}}(G)$$ is $$2$$ -connected. We also prove that the number of edges of the power graph of the simple groups is less than or equal to the number of edges in the power graph of the cyclic group of the same order. This partially answers to a question in an earlier paper. Finally, we give a complete classification of groups in which the power graph is a union of complete graphs sharing a common vertex.

We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder's, Cauchy-Schwarz's and Minkowski's inequalities., This is a preprint of a paper whose final and definite form will appear in the Bulletin of the Malaysian Mathematical Sciences Society. Paper submitted 12-Feb-2013; revised 07-May-2013; accepted for publication 05-Jun-2013

The Hermite matrix polynomials have been generalized in a number of ways, and many of these generalizations have been shown to be important tools in applications. In this paper, we introduce a new generalization of the Hermite matrix polynomials and present the recurrence relations and the expansion of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions \(\exp (xB)\), \(\sin (xB)\), \(\cos (xB)\), \(\cosh (xB)\) and \(\sinh (xB)\) in terms of these generalized Hermite matrix polynomials and thus prove that many of the seemingly different generalizations of the Hermite matrix polynomials may be viewed as particular cases of the two-variable polynomials introduced here. The generalized Chebyshev and Legendre matrix polynomials have also been introduced in this paper in terms of these generalized Hermite matrix polynomials.

The purpose of this paper is to present a new iteration method based on the hybrid method in mathematical programming, extragradient method, and Mann’s method for finding a common element of the solution set of equilibrium problems, the solution set of variational inequality problems for a monotone, Lipschitz continuous mapping and the set of fixed points for a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by this process. The results in this paper generalize and extend some well-known strong convergence theorems in the literature.

In this paper, we give some properties of the generalized derivation algebra \(\mathrm{GDer}(T)\) of a Hom–Lie triple systems T. In particular, we prove that \(\mathrm{GDer}(T) = \mathrm{QDer}(T) + \mathrm{QC}(T)\), the sum of the quasiderivation algebra and the quasicentroid. We also prove that \(\mathrm{QDer}(T)\) can be embedded as derivations in a larger Hom–Lie triple system. General results on centroids of Hom–Lie triple systems are also developed in this paper.

In this paper, we classify all isolated, completely isolated, and (left/right) convex subsemigroups of $${\mathcal {C}}_{n}$$ , the semigroup of all order-preserving and decreasing transformations on $$X_{n}=\{1,\ldots ,n\}$$ under its natural order. Moreover, we find the rank of each convex subsemigroup of $${\mathcal {C}}_{n}$$ .

In this paper, we study ad-nilpotent elements of semiprime rings R with involution $$*$$ whose indices of ad-nilpotence differ on $${{\,\mathrm{Skew}\,}}(R,*)$$ and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients $$Q_m^s(R)$$ of R, a remains ad-nilpotent of the original indices in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ and $$Q_m^s(R)$$ . There exists an idempotent $$e\in Q_m^s(R)$$ that orthogonally decomposes $$a=ea+(1-e)a$$ and either ea and $$(1-e)a$$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 0 modulo 4), or ea and $$(1-e)a$$ have different indices of ad-nilpotence (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 3 modulo 4). Furthermore, we show that $$Q_m^s(R)$$ has a finite $${\mathbb {Z}}$$ -grading induced by a $$*$$ -complete family of orthogonal idempotents and that $$eQ_m^s(R)e$$ , which contains ea, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence n.

If a graph G is chordal, Ha and Van Tuyl proved that $${{\,\mathrm{reg}\,}}(G) = \nu _0(G)$$ , where $$\nu _0(G)$$ is the induced matching number of G. In this paper, we produce a lower bound for $${{\,\mathrm{reg}\,}}(G)$$ which coincides with the result of Ha and Van Tuyl in the case of chordal graphs. We also consider the case G is vertex-decomposable.

Given a graph G and an edge $$e=xy$$ in G, let $$n_x(e)$$ and $$n_y(e)$$ be the number of vertices that have the distance to x less than that to y, and the number of vertices that have the distance to y less than that to x, respectively. The contribution of e is defined as $$|n_x(e)-n_y(e)|$$ . The Mostar index of G is the sum of all edge contributions in G. A segment in a given tree T is a path, each of whose inner vertices has degree exactly 2, and none of whose two ends has the degree 2. The segment sequence of T is the length sequence of all segments in T. In this paper, we focus on the tree set with a fixed segment sequence, and the tree set with a fixed size together with a fixed segment number. We completely determine the graphs with the greatest Mostar index among the two sets, respectively. The graphs with the least Mostar index among the second set are also identified.

In this paper, we study the following nonlinear fractional Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b\int _{\mathbb {R}^{3}}|(-\Delta )^{\frac{s}{2}}u|^{2}\mathrm{d}x\right) (-\Delta )^{s}u+\lambda V(x)u=|u|^{p-2}u, &{} {\mathrm{in }}\mathbb {R}^3, \\ u\in H^{s}(\mathbb {R}^3), \end{array} \right. \end{aligned}$$ where $$s\in (\frac{3}{4}, 1)$$ , $$a, b, \lambda $$ are positive parameters, $$2

In this paper, we first prove that the connected cubic edge-transitive bi-Cayley graphs over a generalized dihedral group have girth 6. Using this, a complete classification is given of these graphs.

In this paper, we consider the following critical p-biharmonic equation involving Hardy potential $$\begin{aligned} \begin{aligned} \Delta _p^2u-\Delta _qu-\mu \frac{|u|^{p-2}u}{|x|^{2p}}=|u|^{p_{*}-2}u,\quad {{x\in \mathbb {R}^{N}}}, \end{aligned} \end{aligned}$$ where $$2\leqslant p

In this paper, we show the advantages of using Formal Concept Analysis in representing FS-domains. Based on the notion of contractive operators on formal contexts, we propose a new notion of FS-contexts. We prove that FS-formal concepts of FS-contexts are concrete representations of FS-domains, just as formal concepts of classical formal contexts are concrete representations of complete lattices. Moreover, we prove that the category of FS-contexts with G-formal connections is equivalent to that of FS-domains with Scott-continuous functions.

Given a graph G, the burning number of G is the smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ is a burning sequence of G. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A t-unicyclic graph is a unicycle graph in which the unique vertex in the graph with degree greater than two has degree $$ t + 2 $$ . In this paper, we first present the bounds for the burning number of t-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for $$t\le 2$$ .

As the generalization of the digraph, the directed hypergraph (dihypergraph), denoted by $${H}$$ , is considered in this paper. First, we introduce the notions of the zeta function of $${H}$$ and dihypergraph coverings over $${H}$$ . Next, several expressions for the zeta function of $${H}$$ are given, and all dihypergraph coverings over $${H}$$ are generated by permutation voltage assignments on the incidence bipartite digraph or the edge-colored digraph of $${H}$$ . Final, we derive two explicit decomposition formulae for the zeta function of any dihypergraph covering $$\overline{{H}}$$ over $${H}$$ , which turn out that the zeta function of $${H}$$ divides the zeta function of $$\overline{{H}}$$ .

The objective of this paper is to investigate the third and fourth Hankel determinants for the class of functions with bounded turning associated with Bernoulli’s lemniscate. The fourth Hankel determinants for 2-fold symmetric and 3-fold symmetric functions are also studied.

Glrac semigroups are common generalizations of left GC-lpp semigroups and left inverse semigroups. And, such a semigroup is a left restriction semigroup if and only if the projection set is a semilattice. So, glrac semigroup is also a generalization of left restriction semigroup. Permissible subsets of a glrac semigroup are introduced in this paper. In terms of permissible subsets, we define (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups. Many characterizations of (uniquely) factorizable glrac semigroups and (uniquely) almost factorizable glrac semigroups are obtained. As their applications, we establish the structures of uniquely factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups) and uniquely almost factorizable left GC-lpp semigroups (left inverse semigroups, inverse semigroups, ample semigroups, left restriction semigroup, restriction semigroups). Our results enrich and extend the related results of almost factorizable restriction semigroups.

This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form $$\begin{aligned} \begin{aligned} \triangle {\mathtt {z}_{\mathtt {j}}}+\frac{(\mathtt {N}-2)^2r_0^{2\mathtt {N}-2}}{\vert x\vert ^{2\mathtt {N}-2}}\mathtt {z}_\mathtt {j}+\varphi (\vert x\vert )\mathtt {g}_{\mathtt {j}}(\mathtt {z}_{\mathtt {j}+1})=0,~\mathtt {R}_12,$$ $$0