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

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

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

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

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

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

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

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

In this paper, the products of modified Bessel functions of the first kind are considered. Our main aim is to provide various conditions so that these products have certain geometric properties such as starlikeness, convexity, and close to convexity (univalency) with respect to certain functions in the unit disk. Subordinating factor sequences for these products are also obtained. Interesting consequences are provided. The results obtained are new in the literature.

In 2011, Gross derived an $$O(n^2)$$ -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of the overlap matrix, we obtain a recurrence relation for the Euler-genus polynomials of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of the cubic caterpillar-Halin graphs into surfaces with Euler-genus 0, 1 and 2 are also obtained.

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.

In this paper, we prove an n-dimensional radially flat gradient shrinking Ricci solitons with $$div^2W(\nabla f,\nabla f)=0$$ is rigid. Moreover, we show that a four-dimensional radially flat gradient shrinking Ricci soliton with $$\text {div}^2W^\pm (\nabla f,\nabla f)=0$$ is either Einstein or a finite quotient of $${\mathbb {R}}^4$$ , $${\mathbb {S}}^2\times {\mathbb {R}}^2$$ or $${\mathbb {S}}^3\times {\mathbb {R}}$$ .

In this paper, the initial-boundary value problem of the original three-dimensional compressible Euler equations with (or without) time-dependent damping is considered. By considering a functional $$F(t,\alpha ,f)$$ weighted by a general time-dependent parameter function $$\alpha $$ and a general radius-dependent parameter function f, we show that if the initial value $$F|_{t=0}$$ is sufficiently large, then the lifespan of the system is finite. Here, f can be any $$C^1$$ strictly increasing function such that the sum of initial values of f and $$\alpha $$ is non-negative. It follows that a class of conditions for non-existence of global classical solutions is established. Moreover, the conditions imply that a strong $$\alpha $$ will lead to a more unrestrained necessary condition for classical solutions of the system to exist globally in time.

In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in $${\mathbb {R}}^n, n\ge 2$$ to attain the desire result.

A graph G is $$(d_{1},d_{2},\ldots ,d_{k})$$ -colorable if the vertex set of G can be partitioned into subsets $$V_{1}, V_{2},\ldots , V_{k}$$ such that the subgraph $$G[V_{i}]$$ induced by $$V_{i}$$ has maximum degree at most $$d_{i}$$ for $$i = 1, 2, \ldots , k$$ . Novosibirsk’s Conjecture (Sib lektron Mat Izv 3:428–440, 2006) says that every planar graph without 3-cycles adjacent to cycles of length 3 or 5 is 3-colorable. Borodin et al. (Discrete Math 310: 167–173, 2010) asked whether every planar graph without adjacent cycles of length at most 5 is 3-colorable. Cohen-Addad et al. (J Comb Theory Ser B 122:452–456, 2017) gave a negative answer to both Novosibirsk’s conjecture and Borodin et al.’s problem. Zhang et al. (Discrete Math 339:3032–3042, 2016) asked whether every planar graph without adjacent cycles of length at most 5 is (1, 0, 0)-colorable. In this paper, we show that every planar graph without adjacent cycles of length at most five is (2, 0, 0)-colorable, which improves the result of Chen et al. (Discrete Math 339:886–905, 2016) who proved that every planar graph without cycles of length 4 and 5 is (2, 0, 0)-colorable.

In this paper, the split common fixed point problem for quasi-pseudo-contractive mappings is studied in Hilbert spaces. By using the hybrid projection method, a new algorithm and some strong convergence theorems are established under suitable assumptions. Our results not only improve and generalize some recent results but also give an affirmative answer to an open question.

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.

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 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 $$\lambda \rightarrow 0$$ .

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.

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 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.

The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton–Watson process with a Poisson process as its time index. Harmonic moments and Shroder index play an important role in the proofs.

The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that the edge coloring set on any pair of adjacent vertices is distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of G is denoted by $$\chi _{a}'(G)$$. It is observed that $$\chi _a'(G)\ge \Delta (G)+1$$ when G contains two adjacent vertices of degree $$\Delta (G)$$. In this paper, we prove that if G is a planar graph without 4-cycles, then $$\chi _a'(G)\le \max \{9,\Delta (G)+1\}$$.

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.

This paper is concerned with the global well-posedness of the n-dimensional generalized incompressible Navier–Stokes equations with initial data in the critical Besov space. By fully using the advantage of suitable weighted functions and the Fourier localization technique, the global well-posedness is obtained without the hypothesis of smallness on initial data, but for some nonlinear smallness of the first iterate. We also give an example of initial data satisfying the nonlinear smallness condition, but whose norm is arbitrarily large.

A graph is $$(d_1,\ldots ,d_r)$$-colorable if the vertex set can be partitioned into r sets $$V_{1},\ldots ,V_{r}$$ where the maximum degree of the subgraph induced by $$V_{i}$$ is at most $$d_{i}$$ for each $$i\in \{1,\ldots ,r\}$$. In this paper, we prove that every planar graph without 4-cycles and 5-cycles is (2, 6)-colorable, which improves the result of Sittitrai and Nakprasit, who proved that every planar graph without 4-cycles and 5-cycles is (2, 9)-colorable.

This paper mainly deals with extinction and non-extinction of weak solutions for a non-Newton polytropic filtration system coupled with non-local source terms under homogeneous Dirichlet boundary condition by comparing the exponents of nonlinear terms. Particularly, precise decay estimates of extinction solutions in the whole dimensional space are also established.

In this paper, we consider the following quasilinear Schrodinger equation $$\begin{aligned} -\Delta u+V(x)u+\frac{\kappa }{2}\Delta (u^2)u=\lambda f(u)+h(u),\,\, x\in {\mathbb {R}}^N, \end{aligned}$$where $$\lambda ,\kappa >0$$, $$N\ge 3$$ and $$2^*=\frac{2N}{N-2}$$. By using a change of variable, we obtain the existence of positive solutions for this problem with subcritical nonlinearities by using the mountain pass theorem and Moser iterative method. Our results extend and supplement some other related literatures.

In this paper, we study the Berezin number inequalities by using the transform $$C_{\alpha ,\beta }\left( A\right) $$ on reproducing kernel Hilbert spaces (RKHS). Moreover, we give Gruss-type inequalities for selfadjoint operators in RKHS.

DP-coloring is a generalization of list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that every planar graph without k-cycles where $$k=3,4,5,$$ or 6 is DP-4-colorable. Recently, Kim and Yu extended the result on 3- and 4-cycles by showing that every planar graph without triangles adjacent to 4-cycles are DP-4-colorable. Xu and Wu showed that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. In this paper, we extend the results on 3-, 4-, and 5-cycles as follows. Let G be a planar graph without pairwise adjacent 3-, 4-, and 5-cycle. We prove that each precoloring of a 3-cycle of G can be extended to a DP-4-coloring of G. As a consequence, each planar graph without pairwise adjacent 3-, 4-, and 5-cycle is DP-4-colorable.

In this paper, we consider the boundary value problem for the one-dimensional Dirac system with spectral parameter in the boundary condition. We completely study the oscillatory properties of components of the eigenvector functions of this problem.

This paper focuses on the multidimensional spherically symmetric full compressible Euler equations and constructs rigorously a family of global self-similar bounded weak solutions for all positive time to its initial value problem with constant initial data. The main approach is to reduce the full compressible Euler equations to an autonomous system of ordinary differential equations under the spherically symmetric and self-similar assumptions. We establish the detailed structures of solutions as well as their existence by analyzing carefully the properties of the integral curves of the autonomous ODE system.

In the present paper, we introduce a new kind of Riemannian submersion such that the fibers of such submersion are generic submanifolds in the sense of Ronsse that we call generic submersion. Some examples are given for generic submersion. Necessary and sufficient conditions are found for the integrability and totally geodesicness of the distributions which are mentioned in the definition. The geometry of the fibers is investigated. New results are obtained by considering the parallelism condition of canonical structures.

In this paper, nonlinear Fredholm integral equation of the second kind is solved by using parameter continuation method. Then, we propose parameter continuation method to solve perturbed nonlinear Fredholm integral equation of the second kind, which appear as an extension of the method of contractive mapping and parameter continuation method for solving nonlinear Fredholm integral equation of the second kind. Illustrative examples are presented to show the effectiveness and convenience of parameter continuation method.

Prime partitions are partitions of integers into prime parts. In this paper, we first consider prime partitions with distinct parts. By using generating functions, we obtain some inductive formulas to calculate the number of prime partitions with distinct parts. Our formulas give two generalizations of the Euler’s formula for the integer partition case. Then, we consider general prime partitions with not necessarily distinct parts. By keeping track of the recurrence of primes in a partition and finding bijections between different prime partitions, we get some inductive formulas to calculate the number of general prime partitions. Finally, by numerical experimentation we find an approximation of some analytical formulas for the number of general prime partitions.

In this paper, we consider two kinds of nonlinear matrix equations $$X+ \sum _{i=1}^{m}B_{i}^*X^{t_{i}}B_{i}=I\;(01,\;s\ge 1)$$ . By means of the integral representation of matrix functions, properties of Kronecker product and the monotonic p-concave operator fixed point theorem, we derive necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equations. We also obtain some properties of the Hermitian positive definite solutions, the bounds of the determinant’s sum for $$A_{i}^{*}A_{i}$$ and the spectral radius of $$A_{i}$$ .

In this paper, we present some inequalities for Riemann–Liouville and Caputo q-fractional derivatives related to the q-Leibniz rule. We used one of the inequalities to prove the maximum principle for a time fractional q-Cauchy differential equation.

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrodinger–Poisson problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\phi u=\lambda Q(x)|u|^{q-2}u+ K(x)|u|^4u, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ -\Delta \phi =u^2, \quad &{}\hbox {in} \ \mathbb {R}^3,\\ \end{array}\right. \end{aligned}$$ where $$40$$ is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small $$\lambda $$ .

In this paper, we prove Doob’s inequality and Burkholder–Gundy inequalities for quasi-martingales in noncommutative symmetric spaces. We also consider the inclusion relations between $$\widetilde{\mathcal {H}}_E(\mathcal {M})$$ and $$\widetilde{h}_{E}(\mathcal {M})$$ , where $$\widetilde{\mathcal {H}}_E(\mathcal {M})$$ and $$\widetilde{h}_{E}(\mathcal {M})$$ are the Hardy spaces of quasi-martingales in noncommutative symmetric spaces.

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature.

Let H be a complex Hilbert space with dim H $$\ge $$ 3, $$\mathcal {B}_s(H)$$ be the Jordan algebra of all bounded self-adjoint operators on H, and let $$F:B(H)\rightarrow [d,\infty ]$$ with $$d\ge 0$$ be a radial unitary similarity invariant function. In this paper, a characterization is obtained for maps $$\phi $$ on $$\mathcal {B}_s(H)$$ satisfying $$F(\phi (A)\phi (B)\phi (A))=F(ABA)\ \ (A,B\in \mathcal {B}_s(H))$$ . As applications, a general form of the maps on $$\mathcal {B}_s(H)$$ preserving the p-norm, the maps preserving the pseudo-spectral radius and the maps preserving the pseudo-spectrum are also described.

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 investigate the Hardy–Lorentz–Karamata spaces for Banach space-valued martingales. Relying on the geometrical properties of the underlying Banach spaces, we establish the atomic decompositions and characterize the dual spaces of these spaces. We also obtain some martingale inequalities in the setting of Hardy–Lorentz–Karamata spaces.

In this paper, we concern with the following Schrodinger–Poisson system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\phi u = f(x,u)+u^5 ,\quad &{} x\in \Omega ,\\ -\Delta \phi =u^2,\quad &{} x\in \Omega ,\\ u=\phi =0, \quad &{} x \in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $$\mathbb {R}^{3}$$ and $$u^5$$ reaches the Sobolev critical exponent since $$2^*=6$$ in dimension 3. Under some appropriate assumptions on f, a new result on the existence of nontrivial solutions is obtained via variational methods.

In this paper, we develop a formula for spacelike surfaces in a four-dimensional Lorentzian space form which involves its mean curvature vector field, the Gauss curvature of the induced metric and the Gauss curvature of the second fundamental form associated to a non-degenerate null normal section. By means of this formula, we establish several sufficient conditions for a compact spacelike surface in a four-dimensional Lorentzian space form which has a null umbilical normal direction. As another application, we give a new proof of Liebmann rigidity theorems in Euclidean, hemispherical, hyperbolic spaces and in the De Sitter spacetime.

In this paper, we study the boundedness of multilinear commutators of Hardy–Littlewood maximal operators in variable exponent Herz and Herz–Morrey spaces, which in turn are used to obtain the boundedness for a large class of the multilinear commutators related to sublinear operators. Moreover, based on the atomic decomposition and on generalization of the BMO norm, we study the boundedness of multilinear commutators of singular integral operators with Calderon–Zygmund kernels in variable exponent Herz-type Hardy spaces.

This paper deals with the asymptotic behavior, regularity criterion and global existence for the generalized Navier–Stokes equations. Firstly, an upper bound for the difference between the solution of our equation and the generalized heat equation in $$L^2$$ space is proved. We optimize the upper bound of decay for the solutions and obtain the algebraic lower bound by using Fourier splitting method. Then, a new scaling invariant regularity criterion on the fractional derivative is established. Finally, global existence is obtained provided that the initial data are small enough.