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In this paper, we study the nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. Global bifurcation of nontrivial solutions of this problem is investigated. We prove the existence of two families of unbounded continua of the set of solutions to this problem bifurcating from points and intervals of the line of trivial solutions. Moreover, it is shown that these continua are contained in classes of functions possessing oscillating properties of the eigenfunctions of the corresponding linear problem and their derivatives.

In this paper, we prove that $$cmo^{p}(\mathbb {R}^{n})$$ and $$\Lambda _{n(\frac{1}{p}-1)}$$ , the dual spaces of local Hardy space $$h^{p}(\mathbb {R}^{n})$$ , are coincide with equivalent norms for $$\frac{n}{n+1}

In this paper, we study the following nonlinear Schrodinger–Bopp–Podolsky system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+q\phi u=f(u)&{}\\ -\Delta \phi +a^2\Delta ^2\phi =4\pi u^2 \end{array} \right. \ \,\mathrm{in }\,{\mathbb {R}}^3, \end{aligned}$$ where $$a>0$$ , $$q>0$$ and $$V\in {\mathcal {C}}({\mathbb {R}}^3,{\mathbb {R}})$$ . By means of the variational methods, we prove the existence of infinitely many nontrivial solutions, the existence of a ground state solution for $$f(u)=|u|^{p-2}u+h(u)$$ with $$p\in [4,6)$$ and the existence of at least one positive solution for $$f(u)=P(x)u^5+\mu |u|^{p-2}u$$ with $$p\in (2,6)$$ under some certain assumptions.

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $\psi$-divisible. Since the existence of $\psi$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the concept of $\psi$-normal divisible group, i.e. a group for which the $\psi$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $\psi$-normal divisible groups which are neither simple nor nilpotent., Comment: 10 pages

For a connected graph G, the Mostar index Mo(G) and the irregularity irr(G) are defined as $$Mo(G)=\sum _{uv\in E(G)}|n_u-n_v|$$ and $$ irr (G)=\sum _{uv\in E(G)}|d_u-d_v|$$ , respectively, where $$d_u$$ is the degree of the vertex u of G and $$n_u$$ denotes the number of vertices of G which are closer to u than to v for an edge uv. In this paper, we focus on the difference $$\Delta M(G)=Mo(G)- irr (G)$$ of graphs G. For trees T of order n, we characterize the minimum and second minimum $$\Delta M(T)$$ of T and the minimum $$\Delta M(Tr(T))$$ of the triangulation graphs Tr(T). The parity of $$\Delta M$$ of cactus graphs is also reported. The effect on $$\Delta M$$ is studied for two local operations of subdivision and contraction of an edge in a tree. A formula for $$\Delta M(S(T))$$ of the subdivision trees S(T) and the upper and lower bounds on $$\Delta M(S(T))- \Delta M(T)$$ are determined with the corresponding extremal trees T. Moreover, three related open problems are proposed to $$\Delta M$$ of graphs.

In the present paper, we show that $$\vartheta $$ -convergent double series spaces $${\mathcal {CS}}_\vartheta $$ and the series space $$\mathcal {BV}$$ of double sequences of bounded variation are BDK-spaces and investigate their $$AK(\vartheta )$$ -space property, where $$\vartheta \in \{p,bp,r\}$$ .

Examining the asymptotic growth behavior of holomorphic and meromorphic functions has significant importance in complex analysis. Estimations of upper bounds for the rate of growth of entire functions are mostly guaranteed. However, the analog estimations for lower bounds of the rate of growth are not always attainable. In this paper, we give the lower rate of growth of the generalized Hadamard product of two entire axially monogenic functions. It has also been shown that the product entire axially monogenic function is of regular growth or perfectly regular growth when its constituent entire axially monogenic functions possess these properties. The investigation of both upper and lower bounds by means of linear transmutation is also provided. Furthermore, some applications related to approximation theory are outlined.

The main purpose of this paper is twofold. First, a class of polyharmonic mappings with positive Jacobian and being univalent in the unit disk $${\mathbb {D}}$$ are constructed. Second, we state and prove several theorems about the radius of univalence of a class of polyharmonic mappings, which gives a partial generalized answer to the radius problem posed by Abu Muhanna (Complex Var Elliptic Equ 53:745–751, 2008).

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or a "lollipop", or an interval. As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real., Comment: 14 pages, 1 figure

In this paper, we generalize the notion of the ergodic shadowing property to the iterated function systems and prove some related theorems on this notion. In addition, we give an example to show that there is an iterated function system which has the ergodic shadowing property but not weakly mixing. Moreover, we show that ergodic shadowing property implies the average shadowing property for iterated function systems.

A Roman dominating function on a graph G is a function $$f:V(G)\rightarrow \{0,1,2\}$$ satisfying the condition that every vertex u for which $$f(u) = 0$$ is adjacent to at least one vertex v for which $$f(v) =2$$ . A Roman dominating function f is called a restrained Roman dominating function if the induced subgraph of G by the vertices with label 0 has no isolated vertex. The weight of a restrained Roman dominating function is the value $$\omega (f)=\sum _{u\in V(G)} f(u)$$ . The minimum weight of a restrained Roman dominating function of G is called the restrained Roman domination number of G and denoted by $$\gamma _{rR}(G)$$ . In this paper, we show that for any graph G of order $$n\ge 5$$ , $$6\le \gamma _{rR} (G)+\gamma _{rR} ({\overline{G}})\le n+5$$ and characterize all the extremal graphs. In addition, we classify all graphs with large restrained Roman domination number.

In this paper, we study the following Chern–Simons–Schrodinger equation $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\omega u+\lambda \Big (\frac{h^{2}(|x|)}{|x|^{2}}+ \int _{|x|}^{+\infty }\frac{h(s)}{s}u^{2}(s)\hbox {d}s\Big )u=g(u) \quad \text{ in }\ {\mathbb {R}}^{2},\\ \displaystyle u\in H_r^1({\mathbb {R}}^{2}), \end{array}\right. } \end{aligned}$$ where $$\omega ,\lambda >0$$ and $$h(s)=\frac{1}{2}\int _{0}^{s}ru^{2}(r)\hbox {d}r$$ . Since the nonlinearity g is asymptotically 5-linear at infinity, there would be a competition between g and the nonlocal term. By constrained minimization arguments and the quantitative deformation lemma, we prove the existence of least energy sign-changing radial solution, which changes sign exactly once. Further, we study the concentration of the least energy sign-changing radial solutions as $$\lambda \rightarrow 0$$ .

In this paper, we introduce semi-doubly stochastic ( $${\mathcal {SDS}}$$ ) operators on $$L^1(X,\mu )$$ . The Ryff’s theorem extended to sigma-finite measure space using semi-doubly stochastic operators on $$L^1(X,\mu )$$ .

Let $$K_n-C_{k}$$ denote the graph obtained from the complete graph $$K_n$$ by deleting k edges of a cycle $$C_k$$ , that is, a nearly complete graph. In this paper, we completely determine the structure of the sandpile group of $$K_{n}-C_{k}$$ for $$3\le k\le n-2$$ .

In this paper, we introduce a space of $$\theta $$ -admissible distributions denoted by $${\mathcal {A}}_\theta ^*$$ as well as the notion of $$\theta $$ -admissible operators. We study the regularity properties of the classical conditional expectation acting on $${\mathcal {A}}_\theta ^*$$ and acting on $${\mathcal {L}}({\mathcal {A}}_\theta ,{\mathcal {A}}_\theta ^*)$$ which is the space of linear continuous operators from $${\mathcal {A}}_\theta $$ into $${\mathcal {A}}_\theta ^*$$ . An integral representation with respect to the coordinate system of the quantum white noise (QWN) derivatives and their adjoints $$\{D_t^{\pm }, D_t^{\pm *},\,t\in {\mathbb {R}}\}$$ of such conditional expectation is given. Then, we give a quantum white noise counterpart of the Clark formula. Finally, we introduce the QWN Hitsuda–Skorokhod integrals. Such integrals are shown to be QWN martingales using a new notion of QWN conditional expectation.

Let $$p\ge 1, \ell \in \mathbb {N}, \alpha ,\beta >-1$$ and $$\varpi =(\omega _0,\omega _1, \ldots , \omega _{\ell -1})\in \mathbb {R}^{\ell }$$ . Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: $$\begin{aligned} \Vert f \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}:= \left( \sum _{k=0}^{\ell -1} \left| f^{(k)}(\omega _{k})\right| ^{p} + \int _{-1}^{1} \left| f^{(\ell )}(x)\right| ^{p} \mathrm{d}\mu ^{\alpha ,\beta }(x)\right) ^{\frac{1}{p}}, \end{aligned}$$ where $$ \mathrm{d}\mu ^{\alpha ,\beta }(x)=(1-x)^{\alpha } (1+x)^{\beta }\mathrm{d}x$$ . Obviously, $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle 2}}= \sqrt{\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}}$$ , where $$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$ is the inner product $$\begin{aligned} \langle f,g \rangle _{{\scriptscriptstyle \mathsf {s}}}:= \sum _{k=0}^{\ell -1} f^{(k)}(\omega _{k}) \, g^{(k)}(\omega _{k}) + \int _{-1}^{1} f^{(\ell )}(x) \,g^{(\ell )}(x) \mathrm{d}\mu ^{\alpha ,\beta }(x). \end{aligned}$$ In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type $$L^p$$ , in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$ and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in $$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$ norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to $$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$ .

A subset D of the vertex set of a graph $$\Gamma $$ is a dominating set for $$\Gamma $$ if each vertex of $$\Gamma $$ is either in D or has a neighbor in D. The size of a minimum cardinality dominating set of a graph is its domination number. In this paper, we initiate the study of domination in a well-known family of rather symmetric tetravalent graphs known as the Rose window graphs. We compare their domination number to the domination number of their spanning generalized Petersen subgraphs, which have been studied quite extensively in the literature.

Solving linear system of differential equations by Jordan canonical form needs the change in the real field to complex and then retrieve the complex solutions to the real ones. B. Malesevic, D. Todoric, I. Jovovic, and S. Telebakovic suggest that it is more convenient to apply the rational canonical form than the Jordan canonical form. They reduce the linear system $$DY=AY$$ to higher order differential equations $$f_j(D)z=0$$ , where the polynomials $$f_1,f_2,\ldots ,f_k$$ appear in the rational canonical form of A. ( $$D=d/dx$$ .) The present paper proves the converse of their theorems and discovers certain dimensionality relations between the solution spaces of the two types of differential equation problems. The approach is pure algebraic and leaves the door open for extensions to formal differential equations.

A subset D of the vertex set V(G) of a graph G is called a [1, k]-dominating set if every vertex from $$V-D$$ is adjacent to at least one vertex and at most k vertices of D. A [1, k]-dominating set with minimum number of vertices is called a $$\gamma _{[1,k]}(G)$$ -set, and the number of its vertices is called the [1, k]-domination number of G and is denoted by $$\gamma _{[1,k]}(G)$$ . In this paper, we express the computation of the [1, k]-domination number of lexicographic products $$G\circ H$$ as an optimization problem over certain partitions of V(G). Nonetheless, in special cases explicit formulas are possible.

The three-dimensional magnetohydrodynamics equation with damping is considered in this paper. Global attractor of the 3D magnetohydrodynamics equations with damping is proved for $$4\le \beta 0$$ .

In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form $$\begin{aligned} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{aligned}$$ for all $$x,y \in X$$ , where m is a fixed scalar such that $$m \ne 0,1$$ , and f is a map from a quasi-normed space X to a quasi-Banach space Y over the same field with X by applying the alternative fixed point theorem.

The Jensen inequality is one of the most important inequalities in theory of inequalities, and numerous results are devoted to this inequality. This inequality has many applications in several fields. This article is devoted to present a new interesting refinement of the well-known Jensen inequality and to give applications for means. The paper also aims to achieve numerous applications in information theory ; in particular, a comprehensive detail has been given for Zipf’s law and obtained bounds for Zipf–Mandelbrot entropy. At the end of the article, a more general refinement of Jensen inequality is presented associated with m finite sequences.

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is $$O((M + 1)n)$$ O ( ( M + 1 ) n ) , where $$M+1$$ M + 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require $$O((M+1)^3 n^3)$$ O ( ( M + 1 ) 3 n 3 ) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

Weaving Hilbert space frames have been introduced recently by Bemrose et al. to deal with some problems in distributed signal processing. In this paper, we survey this topic from the viewpoint of the duality principle. In this regard, not only we obtain new properties in weaving frame theory related to dual frames but also we bring up new approaches for manufacturing pairs of woven frames. Specifically, we give some sufficient conditions under which a frame with its canonical dual, alternate duals or approximate duals constitute some concrete pairs of woven frames. Moreover, we provide some approaches for constructing weaving frames by using small perturbations and present a condition where different operators preserve the weaving property. As a consequence, the canonical duals of two woven frames are woven.

Fourth-order tensors play a fundamental role in signal processing, wireless communication systems, image processing, data analysis and higher-order statistics. In this paper, we introduce a Z-identity tensor and establish two Z-eigenvalue inclusion sets for fourth-order tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness of fourth-order symmetric tensors. Further, we propose upper bounds on the Z-spectral radius of fourth-order nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms under suitable conditions.

In this paper, we show that the Caratheodory function $$\varphi _{\mathrm{Ne}}(z)=1+z-z^3/3$$ maps the open unit disk $$\mathbb {D}$$ onto the interior of the nephroid, a 2-cusped kidney-shaped curve, $$\begin{aligned} \left( (u-1)^2+v^2-\frac{4}{9}\right) ^3-\frac{4 v^2}{3}=0, \end{aligned}$$ and introduce new Ma–Minda-type function classes $$\mathcal {S}^*_{\mathrm{Ne}}$$ and $$\mathcal {C}_{\mathrm{Ne}}$$ associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete–Szego problems are discussed for the classes $$\mathcal {S}^*_{\mathrm{Ne}}$$ and $$\mathcal {C}_{\mathrm{Ne}}$$ . Moreover, for $$\beta \in \mathbb {R}$$ and some analytic function p(z) satisfying $$p(0)=1$$ , we prove certain subordination implications of the first-order differential subordination $$1+\beta {zp'(z)}/{p^j(z)}\prec \varphi _{\mathrm{Ne}}(z),j=0,1,2$$ , and obtain sufficient conditions for some geometrically defined function classes available in the literature.

The irregularity of a graph G is the sum of $$|\mathrm{deg}(u) - \mathrm{deg}(v)|$$ over all edges uv of G. In this paper, this invariant is considered on $$\pi $$ -permutation graphs, Fibonacci cubes, and trees. An upper bound on the irregularity of $$\pi $$ -permutations graphs is given, and $$\pi $$ -permutation graphs that attain the equality are characterized. The concept of the irregularity is extended to arbitrary edge subsets and applied to permutation edges of $$\pi $$ -permutation graphs. An exact formula for the irregularity of Fibonacci cubes is proved. An upper bound on the irregularity of trees in terms of the diameter is given, and trees that attain the equality are characterized.

Let $$({{\mathcal {X}}},d,\mu )$$ be a non-homogeneous metric measure space satisfying the so-called upper doubling and the geometrically doubling conditions in the sense of Hytonen. Under the assumption that the dominating function $$\lambda $$ satisfies weak reverse doubling conditions, in this paper, the authors prove that the generalized homogeneous Littlewood–Paley g-function $${\dot{g}}_{r} (r\in [2,\infty ))$$ is bounded from the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ into the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ for $$\beta \in (0,1)$$ , and the commutator $${\dot{g}}_{r,b}$$ generated by the $$b\in {\mathrm{Lip}}_{\beta }(\mu )$$ and the $${\dot{g}}_{r}$$ is bounded on the Lebesgue space $$L^{p}(\mu )$$ with $$p\in (1,+\infty )$$ . Furthermore, the boundedness of the $${\dot{g}}_{r}$$ and the commutator $${\dot{g}}_{r,b}$$ on generalized Morrey space $$L^{p,\phi }(\mu )$$ is also obtained, respectively.

In this paper, based on Alikhanov’s L2- $$1_\sigma $$ high-order approximation and anisotropic finite element methods, a fully discrete scheme for time fractional variable coefficient diffusion equations on anisotropic meshes is presented. Firstly, we prove that the discrete scheme is unconditionally stable in $$H^1$$ -norm, then the results of convergence in $$L_2$$ -norm and superclose in $$H^1$$ -norm are derived by combining interpolation with projection, and then, the superconvergence in $$H^1$$ -norm is obtained by using interpolation post-processing technique. In addition, it is worth mentioning the key technology of combining interpolation and projection. If interpolation or projection is used alone, the results of this article cannot be obtained. Finally, numerical examples are provided to verify the correctness of theoretical analysis.

An Italian dominating function on a graph G with vertex set V(G) is a function $$f :V(G) \rightarrow \{0,1,2\}$$ having the property that for every vertex v with $$f(v) = 0$$ , at least two neighbors of v are assigned 1 under f or at least one neighbor of v is assigned 2 under f. The weight of an Italian dominating function f is the sum of the values assigned to all the vertices under f. The Italian domination number of G, denoted by $$\gamma _{I}(G)$$ , is the minimum weight of an Italian dominating of G. It is known that if G is a connected graph of order $$n \ge 3$$ , then $$\gamma _{I}(G) \le \frac{3}{4}n$$ . Further, if G has minimum degree at least 2, then $$\gamma _{I}(G) \le \frac{2}{3}n$$ . In this paper, we characterize the connected graphs achieving equality in these bounds. In addition, we prove Nordhaus–Gaddum inequalities for the Italian domination number.

In this paper, we study a Cucker–Smale-type flocking model with distributed time delays, in which individuals interact with each other through general communication weights, and delays are information processing and reactions of individuals. By constructing two dissipative differential inequalities on position and velocity together with a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to two classes of different communication weights. All results are novel and can be illustrated by numerical simulations using some concrete influence functions.

In this paper, we deal with systems of wave and pseudo-hyperbolic equations. Some semilinear equations for hypoelliptic operators on $${\mathbb {R}}^{n}$$ , Heisenberg groups, stratified Lie groups and graded Lie groups are studied. In particular, we obtain nonexistence results for nonlinear hyperbolic and pseudo-hyperbolic equations and systems on graded Lie groups. Also, we show Kato-type exponents for systems of pseudo-hyperbolic equations for Rockland operators on graded Lie groups.

Computing the crossing number of a graph is NP-hard. In the paper, for a disconnected 6-vertex graph $$Q =C_{5}\cup K_{1}$$ , we obtain the crossing number $$Z(6,n)+\lfloor \frac{n}{2}\rfloor $$ of the graph $$Q_{n}$$ which is the join product of Q with the discrete graph by introducing the “rotation” method. Moreover, we give crossing numbers for join products of Q with the path and the cycle. Besides, we also get directly crossing numbers for join products of some connected 6-vertex graphs with the path and the cycle, some of which were studied by M. Klesc, D. Kravecova, and J. Petrillova.

In this paper, we consider the existence of extreme points of analytic $$\alpha $$ -normal functions. First, in terms of $$\alpha $$ -normal unit-valued set, we obtain a necessary condition for a little $$\alpha $$ -normal function to be an extreme point of $${\mathcal {N}}_{0, 1}^{\alpha }$$ , where $${\mathcal {N}}_{0, 1}^{\alpha }$$ is the unit ball of the class of little $$\alpha $$ -normal functions in the unit disk; second, we prove that the $$\alpha $$ -normal unit-valued set of a little $$\alpha $$ -normal function in $${\mathcal {N}}_{0, 1}^{\alpha }$$ consists of several simple closed pairwise disjoint analytic curves and several isolated points.

In this paper, we mainly discuss the constructions of some new K-g-frames which differ from the existing methods. Meanwhile, we use the relation between a positive operator and the frame operator of a K-g-frame to yield a new K-g-frame. We also obtain a necessary and sufficient condition to generate a new K-g-frame. In addition, we correct some recent results which were obtained by Huang and Leng. In the end, we give an equivalent characterization to construct some new tight K-g-frames by two given g-Bessel sequences. Our results generalize and improve some remarkable results.

In this paper, we investigate a nonlinear differential inclusion with Dirichlet boundary conditions containing a weak Laplace operator of fractional orders (defined via the spectral decomposition of the Laplace operator $$-{\varDelta }$$ - Δ under Dirichlet boundary conditions). Using variational methods, we characterize solutions of such a problem. Our approach is based on tools from convex analysis (properties of a Legendre–Fenchel transform).

In this paper, we develop and study the theory of weighted fundamental groups of weighted simplicial complexes. When all weights are 1, the weighted fundamental group reduces to the usual fundamental group as a special case. We also study weighted versions of classical theorems like van Kampen's theorem. In addition, we also investigate the abelianization, lower central series and applications of weighted fundamental groups., Comment: 20 pages

In this paper, we consider the strong solutions to the incompressible viscous, resistive Hall-MHD system. Firstly, we show that the solutions to the Hall-MHD system depend continuously on the initial data in $$H^s({\mathbb {R}}^3)$$ , $$s>\frac{3}{2}$$ . In addition, we obtain that the solutions to the Hall-MHD system converge to the solutions of the MHD system as the Hall effect coefficient tends to zero.

For $$\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}$$ , a real quaternion matrix A is said to be $$\eta $$ -Hermitian if $$A=A^{\eta *},$$ where $$A^{\eta *}=-\eta A^{*}\eta $$ , and $$A^{*}$$ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an $$\eta $$ -Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general $$\eta $$ -Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results.

The main aim of this paper is to consider various notions of (dense) $$m_{n}$$ -distributional chaos of type s and (dense) reiterative $$m_{n}$$ -distributional chaos of type s for general sequences of linear not necessarily continuous operators in Frechet spaces. Here, $$(m_{n})$$ is an increasing sequence in $$[1,\infty )$$ satisfying $$\liminf _{n\rightarrow \infty }\frac{m_{n}}{n}>0$$ and s could be $$0,1,2,2+,2\frac{1}{2},3,1+,2-,2_{Bd},2_{Bd}+$$ . We investigate $$m_{n}$$ -distributionally chaotic properties and reiteratively $$m_{n}$$ -distributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Frechet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.

In this paper, we consider the following class of fractional Hamiltonian systems $$\begin{aligned} \begin{aligned}&_xD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{x}u(x))+L(x)u(x)=\nabla W(x,u(x)) + f(x),\quad x\in \mathbb {R}\\&\quad u\in H^{\alpha }(\mathbb {R},\mathbb {R}^N), \end{aligned} \end{aligned}$$ where $$\alpha \in (1/2,1)$$ , $$L\in C(\mathbb {R},\mathbb {R}^{N^2})$$ is a symmetric positive definite matrix, $$W\in C^1(\mathbb {R}\times \mathbb {R}^N,\mathbb {R})$$ is superquadratic and even in u. By using Bolle’s perturbation method in critical point theory, we prove the existence of infinitely many solutions in spite of the lack of the symmetry of this problem. Moreover, we study the spectral properties of the operator $${_{x}}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{x})+L(x)$$ .

For $$n\in \mathbb {N}$$ , let $$\mathcal {C}_{n}$$ be the semigroup of all order-preserving and decreasing transformations on $$X_{n}=\{1,\ldots ,n\}$$ , under its natural order, and let $$N(\mathcal {C}_{n})$$ be the set of all nilpotent elements of $$\mathcal {C}_n$$ and let $$\mathrm {Fix\, }(\alpha ) = \{ x\in X_n: x\alpha =x\}$$ for any transformation $$\alpha $$ . An element a of a finite semigroup is called m-potent (m-nilpotent) element if $$a^{m+1}=a^{m}$$ ( $$a^{m}=0$$ ) and $$a,a^2,\ldots , a^m$$ are distinct. In this paper, we obtain a formulae for the number of m-nilpotent elements and so the number of m-potent elements in $$N(\mathcal {C}_{n})$$ for $$1\le m \le n-1$$ . Moreover, for any subset Y of $$X_n$$ , we obtain a formulae for the number of m-potent elements of $$\mathcal {C}_{n,Y}= \{ \alpha \in \mathcal {C}_n: \mathrm {Fix\, }(\alpha )=Y\}$$ .

In this paper, we present some new identities for (alternating) multiple zeta values and (alternating) Euler sums by using the method of iterated-integral representations of series. In particular, we prove five new evaluations of (alternating) mixed Euler sums via (alternating) multiple zeta values. Some interesting consequences and illustrative examples are considered.

In this paper, we study the oscillation of the second-order Emden–Fowler neutral differential equations with advanced and delay arguments $$\begin{aligned} (r(t)(x(t)+p(t)x(\tau (t)))')'+q(t)x^\gamma (\sigma (t))=0,\ \quad t\geqslant t_0, \end{aligned}$$ where $$\tau (t)\leqslant t$$ , $$\sigma (t)\geqslant t$$ . Sufficient conditions for oscillation of the equation are obtained by using the inequality principle, comparison principle and Riccati transform. Two examples are given to illustrate the results.

In this paper, we consider a stochastic delayed two-species competition chemostat model with the Monod growth response function. First, we verify that there is a unique global positive solution for any given initial conditions for this stochastic delayed system. Second, we give the dynamical behavior of the solution of stochastic delay system around the washout equilibrium of deterministic system; moreover, we discuss the competition exclusion and coexistence of microorganisms $$x_1$$ and $$x_2$$ . Finally, computer simulations are carried out to illustrate the obtained results; in addition, results show that time delay has critical effects on the survival of the microorganisms.

We study the common singular direction problem of meromorphic function for q-difference version operator; some criterions of the existence of common singular direction have been established. Further, the common singular direction of solutions of q-difference equations is also discussed in this paper.

In this paper, we use Schauder’s fixed point theorem to establish the existence of invasion waves in a stage-structured competitive system on higher-dimensional lattices. To illustrate our results, we construct a pair of upper and lower solutions.

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

Generalized restriction P-restriction semigroups are common generalizations of restriction semigroups and generalized inverse $$*$$ -semigroups. Gomes and Szendrei (resp. Ohta and Imaoka) have shown that every restriction semigroup (every generalized inverse $$*$$ -semigroup) can be embedded in a complete, infinitely distributive restriction semigroup (resp. a $$*$$ -complete, infinitely distributive generalized inverse $$*$$ -semigroup). The main aim of this paper is to obtain an entirely corresponding result for generalized restriction P-restriction semigroups. Specifically, among other things, we show that every generalized restriction P-restriction semigroup can be (2,1,1)-embedded in a complete, infinitely distributive generalized restriction P-restriction semigroup. Our results generalize and enrich the corresponding results of Gomes, Szendrei, Ohta and Imaoka.

In this paper, we introduce the theory of multiplication alteration by two-cocycles for non-associative structures like non-associative bimonoids with left (right) division. Also, we explore the connections between Yetter–Drinfeld modules for Hopf quasigroups, projections of Hopf quasigroups, skew pairings and quasitriangular structures, obtaining the non-associative version of the main results proved by Doi and Takeuchi in the Hopf algebra setting.