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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.

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 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 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.

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 $$\lceil (2k)/3\rceil +1$$ .

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.

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.

The aim of this paper is to study the Wold-type decomposition in the class ofm-isometries. One of our main results establishes an equivalent condition for an analyticm-isometry to admit the Wold-type decomposition for$$m\ge 2$$m≥2. In particular, we introduce thek-kernel condition which we use to characterize analyticm-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for$$m\ge 2$$m≥2. As a result, we also show thatm-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to unilateral weighted shifts. We also provide a characterization ofm-isometric unilateral operator valued weighted shifts with positive and commuting weights.

Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let $$ Z_n(\cdot ) $$ be the counting measure which counts the number of particles of generation n in a given set. Revesz (1994) studied the convergence rates in the central and local limit theorems for $$ Z_n(\cdot ) $$ in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for $$ Z_n(\cdot ) $$ in the case the underlying motion is governed by a general non-lattice random walk on $${\mathbb {R}}^d$$ with the characteristic function of the motion law satisfying the weak Cramer condition.

In this paper, we obtain the local regularity estimates in Besov spaces of weak solutions for the following parabolic p-Laplacian equations: $$\begin{aligned} u_{t}-\text {div} ~\! a \left( Du, x,t \right) =\text {div}~ {\mathbf {F}} \end{aligned}$$ under some proper assumptions on the functions a and $${\mathbf {F}}$$ . Moreover, we would like to point out that our results improve the known results for such equations.

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.

In this paper, we give a description of Lie (Jordan) triple derivations and generalized Lie (Jordan) triple derivations of an arbitrary triangular algebra $${\mathfrak {A}}$$ through a triangular algebra $${\mathfrak {A}}^{0},$$ where $${\mathfrak {A}}^{0}$$ is a triangular algebra constructed from the given triangular algebra $${\mathfrak {A}}$$ using the notion of maximal left (right) ring of quotients such that $${\mathfrak {A}}$$ is the subalgebra of $${\mathfrak {A}}^{0}$$ having the same unity.

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, Soltes 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.

The connective eccentricity index is a degree-distance-based graph invariant, while the modified second Zagreb index is a degree-based graph invariant. The Parikh word representable graph is a new class of graphs G(w), which corresponds to words w that are finite sequence of symbols. In this paper, we first present explicit formulas for the connective eccentricity index and modified second Zagreb index of the Parikh word representable graphs corresponding to binary core words of the form aub over a binary alphabet $$\{a,\,b\}$$ , respectively. Then, we investigate the relationship between the connective eccentricity index and modified second Zagreb index and obtain some comparative results about these two indices.

This paper is concerned with the large time decay rates of the two-dimensional (2D) micropolar equations with zero angular viscosity. Based on the generalized Fourier splitting methods and low frequency effect analysis, we firstly obtain the solutions decay as $$\Vert u\Vert _{L^2}+ \Vert \omega \Vert _{L^2}\le C(1+t)^{-\frac{1}{2}}$$ . Moreover, by exploring the new structure of the system, we obtain a new improved decay rates $$\Vert \omega \Vert _{L^2}+ \Vert \nabla u\Vert _{L^2}\le C(1+t)^{-1}$$ . Our methods here are also available to the time decay issue of the complex fluid flows with partial dissipation.

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.

In this paper, we give some vanishing theorems for harmonic p-forms on complete noncompact Riemannian manifolds satisfying a weighted p-Poincare inequality with nonnegative scalar curvature and under pointwise curvature pinching conditions which are bounded from above by the weight function.

In this paper, we prove that any projection in the convex hull of three surjective linear isometries on $$\mathrm {AC}(X)$$ is a generalized bi-circular projection, where $$\mathrm {AC}(X)$$ denotes the Banach space of all absolutely continuous functions on a compact subset of $$\mathbb {R}$$ with at least two points. We also show that the trivial projections are the only projections on $$\mathrm {AC}(X)$$ which can be represented as the average of three surjective linear isometries.

In the present paper, we devote our aspiration to some initial and final value problems for a class of space-fractional diffusion equation with time-dependent diffusivity factor. For the initial value problem (IVP), we investigate the stability of the solution concerning the data and the fractional order. For the final value problem, we prove the ill-posedness and suggest a filter method to regularize the problem. Explicit convergence rate of Holder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are performed to demonstrate the effectiveness of the proposed method.

Hamming graph is known to be an important class of graphs, and it is a challenge to obtain algorithms that recognize whether a given graph G is a Hamming graph. Let G be a group and $$S\subseteq G$$ be a nonempty subset of G. The Cayley graph with respect to S is a graph whose vertex set is G and arcset is the set of pairs (u, v) such that $$v=su$$ for some element $$s\in S$$ . This graph is denoted by $$\mathrm{Cay}(G,S)$$ .Let $$R=\oplus _{i}R_{i}$$ be a graded ring, S be the set of homogeneous elements of R, $$S'$$ a subset of S, and $$S''=\oplus _{i\ge k}R_{i}$$ . In this paper, with a different view, we study $$\mathrm{Cay}(R, S')$$ as a generalization of $$\mathrm{Cay}(R, S)$$ to obtain a new point of view to study Cartesian products of complete graphs (Hamming graph). In particular, we show that any Hamming graph over sets of prime power sizes is isomorphic to $$\mathrm{Cay}(R,S')$$ for some graded ring R and a subset $$S'\subseteq S$$ . Also we study $$\mathrm{Cay}(R, S'')$$ as another Cayley graph over graded rings and obtain relations between this graph and total, cototal and counit graphs.

Linear complementary dual (LCD) codes are linear codes whose intersections with their duals are trivial. In this paper, characterizations of LCD codes with respect to the symplectic inner product, i.e. symplectic LCD codes, over finite fields are given. Some methods for constructing symplectic LCD codes and symplectic LCD MDS codes are presented. As an application, a class of symplectic LCD MDS codes is constructed by employing Vandermonde matrices, and the corresponding MDS maximal entanglement entanglement-assisted quantum error-correcting codes (EAQECCs) are constructed.

This paper investigates the nonlinear backward fractional diffusion equations (BFDEs) having a variable order and variable diffusion coefficient. It is well-known that the BFDEs are ill-posed in the sense of Hadamard. Moreover, we investigate the problem taking into account the disturbance both variable diffusion coefficient and variable order. This may lead to some common regularization strategy to failure. Therefore, we propose a regularization method for this kind of problem. Under some appropriate regularity assumptions of the exact solution, a convergence estimate of Holder-type is proved.

The commuting graph $$\varDelta (G)$$ of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if $$xy = yx$$ . In this paper, first we discuss some properties of $$\varDelta (G)$$ . We determine the edge connectivity and the minimum degree of $$\varDelta (G)$$ and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of $$\varDelta (G)$$ are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of $$\varDelta (G)$$ are equal. Further, we investigate the commuting graph of the semidihedral group $$SD_{8n}$$ . In this connection, we discuss various graph invariants of $$\varDelta (SD_{8n})$$ including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $$\varDelta (SD_{8n})$$ .

The independence polynomial of a graph G is $$I(G;x)=\sum _{k=0}^{\alpha (G)} s_{k}\cdot x^{k}$$ , where $$s_{k}$$ and $$\alpha (G)$$ denote the number of independent sets of cardinality k and the independence number of G, respectively. We say that a cycle is a $$\tilde{3}$$ -cycle if its length is divisible by 3, otherwise a non- $$\tilde{3}$$ -cycle. Define $$\phi (G)$$ to be the decycling number of G. Engstrom proved that $$|I(G;-1)| \le 2^{\phi (G)}$$ for any graph G. In this paper, we first prove that $$|I(G;-1)|\le 2^{\beta (G)}-\beta (G)$$ for graphs with non- $$\tilde{3}$$ -cycles, where $$\beta (G)$$ is the cyclomatic number of G. Infinitely many examples show that there do exist graphs satisfying $$\beta (G)=\phi (G)$$ and containing non- $$\tilde{3}$$ -cycles. In such a case, it improves the Engstrom’s result. Furthermore, $$|I(G;-1)|\le 2^{k}$$ provided that all cycles of G are pairwise disjoint, where k is the number of $$\tilde{3}$$ -cycles of G. This provides a new perspective on estimating the independence polynomial at -1 of many special graphs. In the case G contains no vertices of degree 1, $$|I(G;-1)|\le 2^{\beta (G)}-1$$ if $$\beta (G)\ge 2$$ , and $$|I(G;-1)|\le 2^{\beta (G)-1}$$ if G contains a non- $$\tilde{3}$$ -cycle.

The class of $$\{P_1,P_2\}$$ -Nekrasov matrices, defined in terms of permutation matrices $$P_1$$ and $$P_2$$ , is a generalization of the well-known class of Nekrasov matrices. In this paper, some computable error bounds for linear complementarity problems (LCPs) of $$\{P_1,P_2\}$$ -Nekrasov matrices are given, which depend only on the entries of the involved matrices and can be used to obtain the perturbation bounds of $$\{P_1,P_2\}$$ -Nekrasov matrices LCPs. Besides, some sufficient conditions ensuring that the subdirect sum of $$\{P_1,P_2\}$$ -Nekrasov matrices lies in the same class are also provided.

The k-rainbow index $$rx_k(G)$$ of a connected graph G was introduced in 2010. As a natural counterpart of the k-rainbow index, the concept of k-vertex-rainbow index $$rvx_k(G)$$ of a connected graph G was introduced in 2016. In this paper, we mainly investigate the k-vertex-rainbow index of a connected graph G and obtain some sharp bounds for $$rvx_k(G)$$ . In particular, by using Lovasz Local Lemma and dominating sets, we prove that a connected graph G with n vertices and minimum degree $$\delta $$ has $$rvx_3(G) < \frac{11n}{\delta }+16$$ . In general, the investigation of k-vertex-rainbow index of a graph may be harder than that of k-rainbow index of a graph. For example, it is noticed that $$rx_2(G)\le rx_3(G)\le \cdots \le rx_n(G)$$ for every connected graph G of order n, but we show that this monotonicity of vertex-rainbow index does not hold in some graph classes by illustrating a counterexample which is a cycle of order n. As a byproduct, we get the exact values of k-vertex-rainbow index for a cycle of order n when $$k=3,n-4,n-3,n-2,n-1$$ and n, respectively.

In this paper, we study homological classification of semigroups with local units depending on properties around projectivity and flatness of acts over them, the results form a new progress in this research area. First, we show that Hom functors and tensor functors in the category of acts over semigroups are left and right exact, respectively. Then we characterize semigroups with local units over which all acts have some property with respect to (principal) weak flatness and torsion freeness of acts. Finally, we establish relationships between several different properties and Rees short exact sequence of acts.

We deal with the Cauchy problem of nonhomogeneous magnetic Benard system with zero resistivity in $${\mathbb {R}}^2$$ . There are few results until now about well-posedness to such model with non-negative density. In the present paper, by delicate weighted energy method, we derive the local existence of a unique strong solution. In particular, the initial density is allowed to vanish and even has compact support.

Let $$K_i$$ be plane convex bodies with the perimeters $$L_{K_i}$$ and areas $$A_{K_i}$$ for $$i=1,2$$ , respectively. In this paper, the lower bounds of the mixed isoperimetric deficit $$\Delta _{K_1,K_2}=L_{K_1}^2L_{K_2}^2-16\pi ^2 A_{K_1}A_{K_2}$$ are obtained; these bounds involve the areas enclosed by the evolute and the Wigner caustic, Hausdorff and $$L_2$$ distances between two convex bodies.

Let X, Y be independent continuous univariate random variables with unknown distributions. Suppose we observe two independent random samples $$X'_1, \ldots , X'_n$$ and $$Y'_1, \ldots , Y'_m$$ from the distributions of $$X' = X+\zeta $$ and $$Y'=Y+\eta $$ , respectively. Here $$\zeta $$ , $$\eta $$ are random noises and have known distributions. This paper is devoted to an estimation for unknown cumulative distribution function (cdf) $$F_{X+Y}$$ of the sum $$X+Y$$ on the basis of the samples. We suggest a nonparametric estimator of $$F_{X+Y}$$ and demonstrate its consistency with respect to the root mean squared error. Some upper and minimax lower bounds on convergence rate are derived when the cdf’s of X, Y belong to Sobolev classes and when the noises are Fourier-oscillating, supersmooth and ordinary smooth, respectively. Particularly, if the cdf’s of X, Y have the same smoothness degrees and $$n=m$$ , our estimator is minimax optimal in order when the noises are Fourier-oscillating as well as supersmooth. A numerical example is also given to illustrate our method.

In this paper, an SEIRS epidemic model is proposed, incorporating appropriate compartments as isolated exposed class and diagnosed class, relevant to interventions. In the scenario of constant isolated proportion, the qualitative analyses are carried out in terms of the basic reproduction number $$\mathscr {R}_0$$ . The sensitivity of $$\mathscr {R}_0$$ with respect to model parameters is discussed. The Pontryagin’s Maximum Principle is applied to characterize the optimal control problem analytically, aiming at finding the optimal value of the control to minimize the cost of interventions. A general explicit expression for the optimal control is obtained. Numerical simulations are performed to illustrate analytical results.

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and $$f(x_1,\ldots ,x_n)$$ a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and $$f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}$$ . If $$f(x_1,\ldots ,x_n)$$ is not central valued on R and $$\begin{aligned} d(F^2(u)u-G(u^2))=0 \end{aligned}$$ for all $$u\in f(I)$$ , then all possible forms of the maps F, G and d are described in the present paper.

The aim of this paper is to show that fixed point theorem in cones can be applied to singular equations. Using the positivity of Green’s function and the external force e(t), we prove the existence of a positive periodic solution for a damped singular equation with sub-linearity, semi-linearity and super-linearity conditions, and these results are applicable to weak and strong singularities. As applications, we consider the existence of a positive periodic solution for nonlinear elasticity model and Ermakov–Pinney equation