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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.

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.

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.

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.

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 $$\varDelta (SD_{8n})$$ .

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected} if for each pair of distinct vertices of $G$ there exists a proper edge-cut separating them. For a connected graph $G$, the \emph{proper disconnection number} of $G$, denoted by $pd(G)$, is the minimum number of colors that are needed in order to make $G$ proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of $pd(G)$ for a connected graph $G$ of order $n$, i.e, $pd(G)\leq \min\{ \chi'(G)-1, \left \lceil \frac{n}{2} \right \rceil\}$. Finally, we show that for given integers $k$ and $n$, the minimum size of a connected graph $G$ of order $n$ with $pd(G)=k$ is $n-1$ for $k=1$ and $n+2k-4$ for $2\leq k\leq \lceil\frac{n}{2}\rceil$., Comment: 14 pages

In this paper, we obtain error bound for pseudo-binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein’s method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.

This paper is concerned with the regularity criteria in terms of the middle eigenvalue of the deformation (strain) tensor $$\mathcal {D}(u)$$ D ( u ) to the 3D Navier–Stokes equations in Lorentz spaces. It is shown that a Leray–Hopf weak solution is regular on (0, T] provided that the norm $$\Vert \lambda _{2}^{+}\Vert _{L^{p,\infty }(0,T; L ^{q,\infty }(\mathbb {R}^{3}))} $$ ‖ λ 2 + ‖ L p , ∞ ( 0 , T ; L q , ∞ ( R 3 ) ) with $$ {2}/{p}+{3}/{q}=2$$ 2 / p + 3 / q = 2 $$( {3}/{2} ( 3 / 2 < q ≤ ∞ ) is small. This generalizes the corresponding works of Neustupa–Penel and Miller.

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. The authors introduced and studied $\xi$-$\mathcal{G}$projective and $\xi$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $\xi$-$\mathcal{G}$projective dimension by using derived functors on $\mathcal{C}$. Second, let $\mathcal{P}(\xi)$ (resp. $\mathcal{I}(\xi)$) be a generating (resp. cogenerating) subcategory of $\mathcal{C}$. We show that the following equality holds under some assumptions: $$\sup\{\xi\textrm{-}\mathcal{G}{\rm pd}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\}=\sup\{\xi\textrm{-}\mathcal{G}{\rm id}M \ | \ \textrm{for} \ \textrm{any} \ M\in{\mathcal{C}}\},$$ where $\xi\textrm{-}\mathcal{G}{\rm pd}M$ (resp. $\xi\textrm{-}\mathcal{G}{\rm id}M$) denotes $\xi$-$\mathcal{G}$projective (resp. $\xi$-$\mathcal{G}$injective) dimension of $M$. As an application, our main results generalize their work by Bennis-Mahdou and Ren-Liu. Moreover, our proof is not far from the usual module or triangulated case., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1906.10989

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms.

A set of vertices W of a graph G is a resolving set if every vertex of G is uniquely determined by its vector of distances to W. In this paper, the Maker–Breaker resolving game is introduced. The game is played on a graph G by Resolver and Spoiler who alternately select a vertex of G not yet chosen. Resolver wins if at some point the vertices chosen by him form a resolving set of G, whereas Spoiler wins if the Resolver cannot form a resolving set of G. The outcome of the game is denoted by o(G), and $$R_{\mathrm{MB}}(G)$$ (resp. $$S_{\mathrm{MB}}(G)$$ ) denotes the minimum number of moves of Resolver (resp. Spoiler) to win when Resolver has the first move. The corresponding invariants for the game when Spoiler has the first move are denoted by $$R'_{\mathrm{MB}}(G)$$ and $$S'_\mathrm{MB}(G)$$ . Invariants $$R_{\mathrm{MB}}(G)$$ , $$R'_{\mathrm{MB}}(G)$$ , $$S_\mathrm{MB}(G)$$ , and $$S'_{\mathrm{MB}}(G)$$ are compared among themselves and with the metric dimension $$\mathrm{dim}(G)$$ . A large class of graphs G is constructed for which $$R_{\mathrm{MB}}(G) > \mathrm{dim}(G)$$ holds. The effect of twin equivalence classes and pairing resolving sets on the Maker–Breaker resolving game is described. As an application, o(G), as well as $$R_{\mathrm{MB}}(G)$$ and $$R'_{\mathrm{MB}}(G)$$ (or $$S_{\mathrm{MB}}(G)$$ and $$S'_{\mathrm{MB}}(G)$$ ), is determined for several graph classes, including trees, complete multi-partite graphs, grid graphs, and torus grid graphs.

Let $k$ be an positive integer. For any two integers $i$ and $j$ in $\{0,1,\dots,k-1\}$, let $|i-j|_k=\min\{|i-j|,k-|i-j|\}$ be the circular distance between $i$ and $j$. Let $t$ be a nonnegative integer. Suppose $f$ is a mapping from $V(G)$ to $\{0,1,\dots,k-1\}$. If adjacent vertices receive different integers, and for each vertex $u$ of $G$, the number of neighbors $v$ of $u$ with $|f(u)-f(v)|_k=1$ is at most $t$, then $f$ is called a $t$-relaxed 2-distant circular $k$-coloring, or simply a $(\frac{k}{2},t)^*$-coloring of $G$. If $G$ has a $(\frac{k}{2},t)^*$-coloring, then $G$ is called $(\frac{k}{2},t)^*$-colorable. In this paper, we prove that, for any two fixed integers $k$ and $t$ with $k\geq2$ and $t\geq1$, deciding whether $G$ is $(\frac{k}{2},t)^*$-colorable is NP-complete expect the case $k=2$ and the case $k=3$ and $t\leq3$, which are polynomially solvable. For any outerplanar graph $G$, e show that all outerplanar graphs are $(\frac{5}{2},4)^*$-colorable, we prove that there is no fixed positive integer $t$ such that all outerplanar graphs are $(\frac{4}{2},t)^*$-colorable., Comment: 19 pages, 8 figures

In this paper, we study the fattening effect of points over the complex numbers for del Pezzo surfaces $$\mathbb {S}_r$$ S r arising by blowing-up of $$\mathbb {P}^2$$ P 2 at r general points, with $$ r \in \{1, \dots , 8 \}$$ r ∈ { 1 , ⋯ , 8 } . Basic questions when studying the problem of points fattening on an arbitrary variety are what is the minimal growth of the initial sequence and how are the sets on which this minimal growth happens characterized geometrically. We provide a complete answer for del Pezzo surfaces.

For a simple connected graph G, the Q-generating function of the numbers $$N_k$$ of semi-edge walks of length k in G is defined by $$W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$$ . This paper reveals that the Q-generating function $$W_Q(t)$$ may be expressed in terms of the Q-polynomials of the graph G and its complement $$\overline{G}$$ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $$W_Q(t)$$ , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $$(\lambda I_n-Q(G))^{-1}$$ . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.

Let $$n >3$$ and $$ 0< k < \frac{n}{2} $$ be integers. In this paper, we investigate some algebraic properties of the line graph of the graph $$ {Q_n}(k,k+1) $$ where $$ {Q_n}(k,k+1) $$ is the subgraph of the hypercube $$Q_n$$ which is induced by the set of vertices of weights k and $$k+1$$ . The graph $$ {Q_n}(k,k+1) $$ has a close relation to Johnson graph $$J(n+1,k+1)$$ . In fact, it is the square root of the graph $$J(n+1,k+1)$$ . We will see that when $$ n\ne 2k+1$$ , then the graph $$ {Q_n}(k,k+1) $$ is a non-regular edge-transitive graph; hence, its line graph is a vertex-transitive graph. In the first step, we determine the automorphism groups of these graphs for all values of n, k. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if $$k\ge 3$$ and $$ n \ne 2k+1$$ , then except for the cases $$(k,n) \ne (3,9)$$ and $$(k,n) \ne (3,33)$$ , the line graph of the graph $$ {Q_n}(k,k+1) $$ is a vertex-transitive non-Cayley graph. Also, we show that the line graph of the graph $$ {Q_n}(1,2) $$ is a Cayley graph if and only if n is a power of a prime p. Moreover, we show that for ‘almost all’ even values of k, the line graph of the graph $$ {Q_{2k+1}}(k,k+1) $$ is a vertex-transitive non-Cayley graph.

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

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

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 $$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$ .

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.

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.

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

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 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.

In this paper, we introduce q,{\omega}-Dirac system. We investigate the existence and uniqueness of solutions for this system and obtain some spectral properties based on the Hahn difference operator. Also we give two examples, which indicate that asymptotic formulas for eigenvalues., Comment: 12 pages

Let G be a locally compact abelian group and $$\Lambda $$ be a closed subgroup of the dual group $${\widehat{G}}$$. In this paper we investigate modulation preserving operators with respect to $$\Lambda $$ and give a characterization of them in terms of modulation-range operators.

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Serbanuţa. Recently, a decade-old conjecture by Serbǎnuţǎ on the M-ambiguity of words has disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of each of those words is then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word result in an M-ambiguity sequence that is ultimately periodic.

The strong geodetic number, $$\text {sg}(G),$$ of a graph G is the smallest number of vertices such that by fixing a suitable geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the formula for $$\text {sg}(K_{n,m})$$ is given, as well as a formula for the crown graphs $$S_n^0$$. Bounds on the strong geodetic number of the hypercube $$Q_n$$ are also discussed.

Let $$P_n^{(\alpha ,\beta )}$$ be the n-th monic Jacobi polynomial with $$\alpha ,\beta >-1$$. Given m numbers $$\omega _1, \ldots , \omega _m \in \mathbb {C} \setminus [-1,1]$$, let $$\varOmega _m=( \omega _1, \ldots , \omega _m)$$ and $$\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}$$ be the m-th iterated integral of $$\frac{(n+m)!}{n!}\;P^{(\alpha ,\beta )}_{n}$$ normalized by the conditions $$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}}{\mathrm{d}z^k}(\omega _{m-k})=0, \; \text { for } \; k=0,1,\ldots , m-1. \end{aligned}$$The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials $$\{ \mathscr {P}_{n,m,\varOmega _m}^{(\alpha , \beta )}\}_n$$. In particular, we obtain the relative asymptotic for the ratio of the sequences $$\{ \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\}_n$$ and $$\{P_n^{(\alpha ,\beta )}\}_n$$. We prove that the zeros of these polynomials accumulate on a suitable ellipse.

The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further we consider automorphisms of simple Leibniz algebras. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show that nilpotent Leibniz algebras admit 2-local automorphisms which are not automorphisms. A similar problem concerning local automorphism on simple Leibniz algebras is reduced to the case of simple Lie algebras., 28 pages

In this paper, we study Triebel–Lizorkin-type spaces with variable smoothness and integrability. We show that our space is well-defined, i.e., independent of the choice of basis functions and we obtain their atomic characterization. Moreover, the Sobolev embeddings for these function spaces are obtained.

The total-coloring of a graph is a coloring of the edge set and vertex set. A path in a total-colored graph is a total proper path if the coloring of the edges and internal vertices is proper, that is, any two adjacent or incident elements of edges and internal vertices on the path differ in color. For a connected graph G, the total proper connection number of G, denoted by tpc(G), is defined as the smallest number of colors such that any two vertices of the graph are connected by a total proper path of G. These concepts are inspired by the concepts of total chromatic number $$\chi ''(G)$$ , proper connection number pc(G) and proper vertex connection number pvc(G) of a connected graph G. In this paper, we first determine the value of the total proper connection number tpc(G) for some special graphs G. Secondly, we obtain that $$tpc(G)\le 4$$ for any 2-(edge-)connected graph G and give examples to show that the upper bound 4 is sharp. For general graphs, we also obtain sharp upper bounds for tpc(G) in terms of the maximum degree of a vertex incident with a bridge in G and the minimum degree, respectively. Finally, we compare tpc(G) with pvc(G) and pc(G), respectively, and obtain that $$tpc(G)>pvc(G)$$ for any nontrivial connected graph G and that there exist graphs G such that $$tpc(G)=pc(G)+t$$ for $$0\le t\le 2$$ .

In this paper, we characterize the graded post-Lie algebra structures and a class of shifting post-Lie algebra structures on the Witt algebra. We obtain some new Lie algebras and give a class of their modules. As an application, the homogeneous Rota-Baxter operators and a class of non-homogeneous Rota-Baxter operators of weight $1$ on the Witt algebra are studied., Comment: 24 pages

In the present paper, the estimate of the third Hankel determinant $$\begin{aligned} \begin{aligned} H_{3,1}(f)&= \begin{vmatrix} a_{1}&a_{2}&a_{3} \\ a_{2}&a_{3}&a_{4} \\ a_{3}&a_{4}&a_{5} \end{vmatrix} \end{aligned} \end{aligned}$$ for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that $${{\mathrm{Re}}}(zf'(z)/f(z)) > 0,\ z\in {{\mathbb {D}}}:=\{z \in {\mathbb {C}} : |z

In this paper, we prove that the equation $$x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$$ , $$l \in \{0,1,\dots ,k\}, k \ge 0$$ , where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some $$D(-1)$$ -pairs to quadruples in the ring $${\mathbb {Z}}[\sqrt{-t}], t>0$$ .

The aim of this paper is to introduce a Dunkl generalization of the operators including two variable Hermite polynomials which are defined by Krech [14](Krech, G. A note on some positive linear operators associated with the Hermite polynomials, Carpathian J. Math., 32 (1) (2016), 71--77) and to investigate approximating properties for these operators by means of the classical modulus of continuity, second modulus of continuity and Peetre's K-functional.

Let $$\text {PL}(S)$$ be the group of piecewise linear homeomorphisms of the circle S which are locally affine at all but a finite number of points of S (called break points) and $$\text {PL}^+(S)$$ be the subgroup of $$\text {PL}(S)$$ whose elements are order preserving. An element in $$\text {PL}^+(S)$$ is called reversible in $$\text {PL}^+(S)$$ if it is conjugate to its inverse in $$\text {PL}^+(S)$$ . In this paper, we answer partially one of O’Farrell and Short’s problems in their recent book (O’Farrell and Short in Reversibility in dynamics and group theory. London Mathematical Society Lecture Note Series: 416. Cambridge University Press, Cambridge, 2014). We characterize the reversible elements in $$\text {PL}^+(S)$$ and also perform a similar characterization in the full group $$\text {PL}(S)$$ . In particular, we give a complete characterization of reversibility in $$\text {PL}^+(S)$$ and $$\text {PL}(S)$$ of elements f of $$\text {PL}^+(S)$$ having the (D)-property (i.e. for which the product of the f-jumps along the full orbit of any break point is one).

The metric dimension is quite a well-studied graph parameter. Recently, the local metric dimension and the adjacency dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $$G \circ \mathcal {H}$$ of a connected graph G of order n and a family $$\mathcal {H}$$ composed of n graphs. We show that the local metric dimension of $$G \circ \mathcal {H}$$ can be expressed in terms of the numbers of vertices in the true twin equivalence classes of G, and the local adjacency dimension of the graphs in $$\mathcal {H}$$ .

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then $$\mathrm{sg}(G)$$ is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair $$\{x,y\}\subseteq S$$ so that these $$\left( {\begin{array}{c}|S|\\ 2\end{array}}\right) $$ geodesics cover all the vertices of G. In this paper, the strong geodetic problem is studied on Cartesian product graphs. A general upper bound is proved on the Cartesian product of a path with an arbitrary graph and showed that the bound is tight on thin grids and thin cylinders.

In this paper, we study Rota–Baxter operators and super $$\mathcal {O}$$ -operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L-dendriform superalgebras. Moreover, we provide all Rota–Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3.

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A path P in a total-colored graph G is called a total-proper path if (1) any two adjacent edges of P are assigned distinct colors; (2) any two adjacent internal vertices of P are assigned distinct colors; and (3) any internal vertex of P is assigned a distinct color from its incident edges of P. The total-colored graph G is total-proper connected if any two distinct vertices of G are connected by a total-proper path. The total-proper connection number of a connected graph G, denoted by tpc(G), is the minimum number of colors that are required to make G total-proper connected. In this paper, we first characterize the graphs G on n vertices with $$tpc(G)=n-1$$ . Based on this, we obtain a Nordhaus–Gaddum-type result for total-proper connection number. We prove that if G and $$\overline{G}$$ are connected complementary graphs on n vertices, then $$6\le tpc(G)+tpc(\overline{G})\le n+2$$ . Examples are given to show that the lower bound is sharp for $$n\ge 4$$ . The upper bound is reached for $$n\ge 4$$ if and only if G or $$\overline{G}$$ is the tree with maximum degree $$n-2$$ .

In this paper, we study the non-monotonic behavior of the solutions of second-order delay differential equations. We establish criteria such that every positive solution is non- monotonic, on a bounded interval. While there has been previous studies on non-monotone solutions, the results are the first to be reported, for functional differential equations. In the final section, examples illustrating the significance of our results are also given.

The subdivision graph $\mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The \emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1\dot{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The \emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1\underline{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $\mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$., Comment: The Bulletin of the Malaysian Mathematical Society (2017)

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $$sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,$$ where R is a solvable radical. The classifications of such Leibniz algebras in the cases $$\mathrm{dim} R=2, 3$$ and $$\mathrm{dim} I\ne 3$$ have been obtained. Moreover, we classify Leibniz algebras with $$L/I\cong sl_2^1\oplus sl_2^2$$ and some conditions on ideal $$I=\mathrm{id} $$ .

A path $P$ in an edge-colored graph $G$ is called a proper path if no two adjacent edges of $P$ are colored the same, and $G$ is proper connected if every two vertices of $G$ are connected by a proper path in $G$. The proper connection number of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we investigate the proper connection number of the complement of graph $G$ according to some constraints of $G$ itself. Also, we characterize the graphs on $n$ vertices that have proper connection number $n-2$. Using this result, we give a Nordhaus-Gaddum-type theorem for the proper connection number. We prove that if $G$ and $\overline{G}$ are both connected, then $4\le pc(G)+pc(\overline{G})\le n$, and the only graph attaining the upper bound is the tree with maximum degree $\Delta=n-2$., Comment: 12 pages

Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are $\mathbb{Z}_{2}$-graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcev-admissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the $\mathbb{Z}_{2}$-graded Bruck-Kleinfled function are obtained., arXiv admin note: substantial text overlap with arXiv:1002.3944 by other authors

This paper is devoted to the study of the problem of prevalence in the class of quadratic stochastic operators acting on the L1 space for the uniform topology. We obtain that the set of norm quasi-mixing quadratic stochastic operators is a dense and open set in the topology induced by a very natural metric. This shows the typical long-term behaviour of iterates of quadratic stochastic operators.

In this paper, a full incompressible magneto-micropolar system is investigated. We prove the local well-posedness of strong solutions to the full system with vacuum. Moreover, previous compatibility conditions on the initial data are also moved.

Dorff, proved in [2] that the convolution of two harmonic right-half plane mappings is convex in the direction of real axis provided that the convolution is locally univalent and sense preserving. Later, it was shown in [3] that the condition of locally univalent and sense preserving can be dropped in some special cases. In this paper, we generalize the main result from [3]., The manuscript includes 14 pages and 2 figures