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A graph is equitably k-colorable if G has a proper vertex k-coloring such that the sizes of any two color classes differ by at most one. Chen, Lih and Wu conjectured that any connected graph G with maximum degree $$\Delta $$ distinct from the odd cycle, the complete graph $$K_{\Delta +1}$$ and the complete bipartite graph $$K_{\Delta ,\Delta }$$ are equitably m-colorable for every $$m\ge \Delta $$ . Let $${\mathcal {G}}_k$$ be the class of graphs G such that $$e(G')\le k (v(G')-2)$$ for every subgraph $$G'$$ of G with order at least 3. In this paper, it is proved that any graph in $${\mathcal {G}}_4$$ with maximum degree $$\Delta \ge 17$$ is equitably m-colorable for every $$m\ge \Delta $$ . As corollaries, we confirm Chen–Lih–Wu Conjecture for 1-planar graphs, 3-degenerate graphs and graphs with maximum average degree less than 6, provided that $$\Delta \ge 17$$ .

The harmonic index of a graph G is defined as the sum of weights $$\frac{2}{d(u)+d(v)}$$ over all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. Let $$\mathscr {T}(n,\gamma )$$ and $$\mathscr {C}_n^\pi $$ be the sets of trees of order n with domination number $$\gamma $$ and connected graphs of order n with degree sequence $$\pi $$ , respectively. In this paper, the tree with minimum harmonic index among trees in $$\mathscr {T}(n,\gamma )$$ is determined, as well as the trees with maximum harmonic index among trees in $$\mathscr {T}(n,\gamma )$$ when $$\gamma =2,\lceil \frac{n}{3}\rceil $$ . Moreover, we show that among graphs in $$\mathscr {C}_n^\pi $$ , the maximum harmonic index is attained by a BFS graph.

This paper is devoted to a random dispersion Schrodinger equation including time-oscillating nonlinearity and dissipation/gain: \(i\mathrm{d}u + \frac{1}{\varepsilon } m\left( \frac{t}{\varepsilon ^{2}}\right) \partial _{xx}u\mathrm{d}t + \phi _{1}\left( \frac{t}{\varepsilon ^{2}}\right) |u|^{2\sigma }u \mathrm{d}t + i\phi _{2}\left( \frac{t}{\varepsilon ^{2}}\right) u\mathrm{d}t = 0\). The main aim is to prove the equation converges to a white noise dispersion Schrodinger equation with the average of time-oscillating nonlinearity and dissipation/gain. Firstly, the existence of the local solution for the limit equation is established for \(0 0\) if \(d = 1\) or 2). Then, the convergence is proved in one dimension for \(\sigma >0\).

In this paper, we present some new results concerning positive solutions for the singular fractional boundary value problem with p-Laplacian. By imposing some suitable conditions on the nonlinear term f, existence results of positive solutions are obtained. The proof is based upon theory of Leray–Schauder degree. The interesting point is the nonlinear term f(t, u) may be singular at \(u=0\).

An edge-coloring of a connected graph is a monochromatically-connecting coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices, which was introduced by Caro and Yuster. Let mc(G) denote the maximum number of colors used in an MC-coloring of a graph G. Note that an MC-coloring does not exist if G is not connected, in which case we simply let \(mc(G)=0\). In this paper, we characterize all connected graphs of size m with \(mc(G)=1, 2, 3, 4\), \(m-1\), \(m-2\) and \(m-3\), respectively. We use G(n, p) to denote the Erdős-Renyi random graph model, in which each of the \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) pairs of vertices appears as an edge with probability p independent from other pairs. For any function f(n) satisfying \(1\le f(n)

In this paper, we firstly consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As applications of the result, the largest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order and girth is determined. Moreover, the largest normalized Laplacian spectral radius among non-bipartite unicyclic graphs with fixed girth and order is also determined. The maximizer is the tadpole graph with the same (odd) girth.

In this paper, Ruspini partitioning is used to represent the reduced form of the fuzzy basis function. Among all the predefined membership functions, Triangular-shaped membership function is used to represent the whole partitioned space of a given fuzzy system. According to the formulation, it shows that the reduced fuzzy basis function using Ruspini partitioning is indeed a strong fuzzy partition and a complete rule base method. By using the Stone–Weierstrass Theorem, for any real continuous function on a compact set, the reduced fuzzy basis function is proven to be a universal approximator at arbitrary accuracy.

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, we consider Hilbert \(C^*\)-module-valued random variables and discuss about their expectations, covariance operators and correlation operators by introducing some adjointable operators on Hilbert \(C^*\)-modules. We also study Hilbert \(C^*\)-module-valued stochastic processes instead of Hilbert space-valued processes as a generalization. Using the characterization of \(C^*\)-algebra-valued bimeasures, we prove the equivalence between V-boundedness and harmonizability of Hilbert \(C^*\)-module-valued stochastic processes. Finally, we consider Hilbert \(C^*\)-module operator-valued stochastic processes and construct spectral distributions associated with Hilbert \(C^*\)-module operator-valued stationary stochastic processes.

In this paper we prove that the group of all biholomorphic maps from the open unit ball in $$\mathbb {C}^n$$ onto itself endowed with the compact-open topology is a root-approximable topological group.

In this paper, we show the blow-up phenomenon of smooth solutions to the compressible Navier–Stokes–Poisson (N-S-P) equations in \(\mathbb {R}^{2}\), under the assumption that the initial density has compact support. The proof is based on some useful physical quantities. In particular, our result is valid for both isentropic and isothermal cases.

In this paper, a system of quasilinear elliptic equations is investigated, which involves critical exponents and multiple Hardy-type terms. By variational methods and analytic techniques, the existence of positive solutions to the system is established. The conclusions are new even when the Hardy-type terms disappear.

In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform.

In this paper, a delayed diffusive modified Leslie–Gower-type predator–prey model with Beddington–DeAngelis functional response subject to Neumann boundary condition is considered. The stability/instability of the non-negative equilibria and a detailed Hopf bifurcation analysis are investigated. And explicit formulas for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived by the theory of normal form and center manifold. In addition, some numerical simulations are carried out.

This paper is concerned with a class of nonlinear diffusion equations coupled by the nonlinear boundary sources on the exterior domain of the unit ball in \(\mathbb R^N\). We are interested in the critical curves which can describe the large time behavior of the solutions. It is shown that the critical global existence curve coincides with the critical Fujita curve for the system we considered.

The finite Hilbert transform is a helpful tool in fields like aerodynamics, the theory of elasticity, and other areas of the engineering sciences. In this paper, by using some companions of Ostrowski’s inequalities for absolutely continuous functions due to Dragomir, we give some new inequalities and approximations for the finite Hilbert transform, which may have the better error bounds than the known results.

In this paper, we consider regularity criteria for the 3D generalized MHD and Hall-MHD systems with fractional dissipative terms. Some scaling invariant regularity criteria are established for the two systems. Global regularity for the Hall-MHD equation is also proved for the case $$\alpha \ge \frac{5}{4}, \beta \ge \frac{7}{4}$$ .

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.

This paper is concerned with the distribution of zeros of solutions of first-order linear advanced dynamic equations. By introducing a class of sequences involving iterates of the argument, some estimates for the distances between consecutive zeros of solutions are obtained on arbitrary time scales. Our results improve those for differential equations, and provide new results on the distribution of zeros of solutions for arbitrary time scales. Some examples are given to illustrate our results.

In this paper, we discuss a new class of nonlocal boundary value problems of fractional differential equations and inclusions with a new integral boundary condition. This new boundary condition states that the value of the unknown function at an arbitrary (local) point $$\xi $$ is proportional to the contribution due to a sub-strip of arbitrary length $$(1-\eta ),$$ that is, $$x(\xi )=a \int _{\eta }^{1}x(s)\mathrm{d}s,$$ where $$0< \xi < \eta

In this paper, we study \(D(-1)\)-triples of the form \(\{1,b,c\}\) in the ring \({\mathbb {Z}}[\sqrt{-t}]\), \(t>0\), for positive integer b such that b is a prime, twice prime, and twice prime squared. We prove that in those cases, c has to be an integer. In cases of \(b=26,37\), or 50, we prove that \(D(-1)\)-triples of the form \(\{1,b,c\}\) cannot be extended to a \(D(-1)\)-quadruple in the ring \({\mathbb {Z}}[\sqrt{-t}\,], t>0\), except in cases \(t\in \{1,4,9,25,36,49\}\). For those exceptional cases of t we show that there exist infinitely many \(D(-1)\)-quadruples of the form \(\{1,b,-c,d\}\), \(c,d>0\) in \({\mathbb {Z}}[\sqrt{-t}\,]\).

In this paper, first, a sufficient condition for almost convex mappings of order \(\alpha \) defined on the unit ball of complex Hilbert spaces and another sufficient condition for almost quasi-convex mappings of order \(\alpha \) defined on the unit ball of complex Banach spaces are given. Second, the distortion theorem of the Frechet derivative for almost convex mappings of order \(\alpha \) on the unit ball of complex Banach spaces, the homogeneous ball of complex Banach spaces, and the unit ball of complex Hilbert spaces are established respectively. Finally, the distortion theorem of the Jacobi determinant for almost convex mappings of order \(\alpha \) on the Euclidean unit ball in \(\mathbb {C}^n\) is obtained. Our results generalize many known results.

A subgroup H of a finite group G is called to be a CAP\(^*\)-subgroup of G if H either covers or avoids every non-Frattini chief factor of G. In this paper, we study the influence of the CAP\(^*\)-subgroups of a finite group G on the structure of G, and some recent results were extended.

In this paper, the author studies the long-time behavior of solutions to the p(x)-Laplace equation \(u_t=\mathrm {div}(|\nabla u|^{p(x)-2}\nabla u)+f(u)\), with homogeneous Dirichlet boundary condition in a bounded domain. As for the blow-up results, it is shown, by using the energy method, that the solutions of this problem blow up in finite time for nonpositive initial energy, or even for small positive initial energy. As for the extinction results, we give some sufficient conditions for the solutions to vanish in finite time. All these results generalize the ones when p(x) is a constant.

In this paper, we consider the boundedness properties of commutator operators generated by $$BMO(\mathbb {R}^n)$$ function and intrinsic square function including the Lusin area integral, Littlewood-Paley g-function and $$g^*_{\uplambda }$$ -function on weighted Herz spaces $$\dot{K}_q^{\alpha ,p}(\omega _1,\omega _2)(\mathbb {R}^n)$$ with general Muckenhoupt weights.

In this paper, we extend the notion of generalized Sasakian space form to the semi-Riemannian setting. We consider several interesting cases and we give examples of them all. We also study their structures.

In this paper, first, we make the uniform estimates for a class of fourth-order elliptic system in bounded and smooth domains. Second, we study the qualitative properties of solutions with prescribed integration in $$R^4$$ . Finally, we also will obtain some radially symmetric results by using moving planes methods.

In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and a new formula of products of the Apostol–Bernoulli and Apostol–Euler polynomials is established by applying the generating function methods and some summation transform techniques. It turns out that some known results are obtained as special cases.

In the paper we study groups in which the factor-group by k-th hypercenter is locally finite and has finite exponent. We proved that in these groups the (k+1)-th term of lower central series is locally finite and has finite exponent. Moreover we are able to find bounds for the exponent of \(\gamma _{k+1}(G)\) and for the exponent of the locally nilpotent residual of G.

In this paper the notion of Lie-Codazzi type of a tensor field T of type (1,1) on real hypersurfaces, which generalizes the notion of Lie parallel, is introduced. Real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator is of Lie-Codazzi type, are studied. More precisely, the non-existence of such real hypersurfaces is proved.

In this paper, a non-autonomous SIR epidemic model with almost periodic transmission rate and a constant removal rate is considered. By means of Mawhin’s continuous theorem of coincidence degree, some new sufficient conditions for the existence and multiplicity of positive almost periodic solutions to the model are established. Further, the global asymptotical stability of positive almost periodic solution of the model is also investigated by constructing a suitable Lyapunov functional. Finally, some examples and numerical simulations are given to illustrate the main results.

In this paper, we present a modular version of KKM and generalized KKM mappings and then we establish a characterization of generalized KKM mappings in modular spaces. Also we prove an analogue to KKM principle in modular spaces. Moreover, as an application, we give some sufficient conditions which guarantee existence of solutions of minimax problems in which we get Fan’s minimax inequality in modular spaces.

The purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let \(D:R\rightarrow R\) be an additive mapping satisfying the relation \(2D(x^{3})=D(x^{2})x+x^{2}D(x)+D(x)x^{2}+xD(x^{2})\) for all \(x\in R.\) In this case D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring of characteristic different from two is a derivation.

In this paper we prove, using inequalities between infinite cardinals, that, if R is an hereditary ring, the contravariant derived functor $$\mathrm {Ext}^{1}_{R}(-,G)$$ commutes with direct products if and only if G is an injective R-module.

In this paper, we study the existence of asymptotic almost automorphic solution of fractional neutral integro-differential equation. We prove the result using fixed-point theorems. We show the result with Lipschitz condition and without Lipschitz condition on the forcing term. Finally, examples are given to illustrate the analytical findings.

The object of this paper is to obtain some results on a weaker separation axiom along with preservation theorems under some weaker form of continuity and openness of a function on spaces weaker than pairwise compactness. We also obtain some results concerning subspaces of a bitopological space.

We obtain some results on A-m-isometric operators. The paper consists of two parts. At first, we consider the products of A-m-isometries. It will be proved that the iterates of an A-m-isometry are all of this type. We discuss when an A-m-isometry becomes an A-isometry. Also, we consider the products of A-m-isometries for different values of m. In the second part, we are going to investigate the supercyclicity of these operators and prove that they are never supercyclic.

A graph G is called a fractional $$(g,f,n',m)$$ -critical deleted graph if after deleting any $$n'$$ vertices of G the remaining graph is a fractional (g, f, m)-deleted graph. A graph G is called a fractional ID-(g, f, m)-deleted graph if after deleting any independent set I of G the remaining graph is a fractional (g, f, m)-deleted graph. In this paper, we give some sharp degree conditions for a graph to be a fractional $$(g,f,n',m)$$ -critical deleted graph and a fractional ID-(g, f, m)-deleted graph. The tight degree conditions for fractional $$(a,b,n',m)$$ -critical deleted graphs and fractional ID-(a, b, m)-deleted graphs are also considered.

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

A proper edge-k-coloring of a graph G is an assignment of k colors $$1,2,\ldots ,k$$ to the edges of G such that no two adjacent edges receive the same color. A neighbor sum distinguishing edge-k-coloring of G is a proper edge-k-coloring of G such that for each edge $$uv\in E(G)$$ , the sum of colors taken on the edges incident with u is different from the sum of colors taken on the edges incident with v. By $${ {ndi}}_{\sum }(G)$$ , we denote the smallest value k in such a coloring of G. The maximum average degree of G is $${ {mad}}(G)=\max \{2|E(H)|/|V(H)|\}$$ , where the maximum is taken over all the non-empty subgraphs H of G. In this paper, we obtain that if G is a graph without isolated edges and $${ {mad}}(G)

Let A be a subgroup of a group G and X a non-empty subset of G. A is said to be X-s-semipermutable in G if A has a supplement T in G such that A is X-permutable with every Sylow subgroup of T. In this paper, some new criteria for a finite group G to be p-nilpotent or supersoluble in terms of X-s-semipermutable subgroups are obtained. In particular, a characterization of finite groups all of whose subgroups are G-s-semipermutable is presented.

A function f is called a graceful labeling of a graph G with m edges if f is an injective function from V(G) to $$\{0,1,2,\ldots ,m\}$$ such that when every edge uv is assigned the edge label $$|f(u)-f(v)|$$ , then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular Graceful Tree Conjecture states that every tree is graceful. The Graceful Tree Conjecture remains open for over four decades. Although there are a few general results and techniques on the construction of graceful trees, settling the conjecture seems to be very hard. In this paper, we have introduced a new and different method of constructing graceful trees from a given arbitrary tree. More precisely, we show that every tree can be embedded in a graceful tree with at most km edges, $$k

In this paper, we give a construction of (totally) free quadratic modules of commutative algebras on suitable construction data in terms of free simplicial algebras. Similar freeness results are also explored for reduced quadratic modules and quadratic (chain) complexes of algebras. © 2015, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia.

Our purpose in this paper is to prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

In this paper, we introduce a new concept of \(S^{2}\)-pseudo almost automorphy for stochastic processes. We apply the results obtained to investigate the existence and uniqueness of \(S^{2}\)-pseudo almost automorphic mild solutions to some stochastic differential equations in a real separable Hilbert space. Our main results extend some known ones in the sense of square-mean pseudo almost automorphy or \(S^{2}\)-almost automorphy for stochastic processes.

This paper is concerned with the Lie superalgebra S(n, m) of special type over a field of prime characteristic. We construct the modular Lie superalgebra S(n, m) and discuss some properties of this algebra. Then the derivation superalgebra of S(n, m) is determined.

In this paper, a fourth-order nonlinear p-Laplacian difference equation is considered. Using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Neumann boundary value problem and give some new results. The existing results are generalized and significantly complemented.

Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by $${\mathcal {S}}_{{\mathcal {H}}}$$ . The aim of the paper is to study the properties of a subclass of $${\mathcal {S}}_{{\mathcal {H}}}$$ , such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch’s constant for co-analytic part.

A subgroup H of a group G is called a TI-subgroup if $$H^g\cap H=1$$ or H for all $$g\in G$$ ; and H is called quasi TI if $$\mathcal {C}_G(x)\le \mathcal {N}_G(H)$$ for all non-trivial elements $$x\in H$$ . A group G is called (quasi CTI-group) CTI-group if every cyclic subgroup of G is a (quasi TI-subgroup) TI-subgroup. It is clear that TI subgroups are quasi TI. We first show that finite nilpotent quasi CTI-groups are CTI. In this paper, we classify all finite nilpotent CTI-groups.

In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri.