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Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ - Δ Φ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ Δ Φ u = div ( φ ( x , | ∇ u | ) ∇ u ) and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ Φ ( x , t ) = ∫ 0 | t | φ ( x , s ) s d s is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ Ω N , Ω p with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ Ω ¯ N ∩ Ω ¯ p = ∅ . The features of this paper are that $$-\Delta _{\Phi }u$$ - Δ Φ u behaves like $$-\Delta _N u $$ - Δ N u on $$\Omega _N$$ Ω N and $$-\Delta _p u $$ - Δ p u on $$\Omega _p$$ Ω p , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ f : Ω × R → R is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ e α | t | N N - 1 on $$\Omega _N$$ Ω N and as $$|t|^{p^{*}-2}t$$ | t | p ∗ - 2 t on $$\Omega _p$$ Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.

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

In this paper, we prove that if a nonconstant finite order meromorphic function f and its n-th order difference operator $$\Delta ^n_{\eta }f$$ share $$a_1,$$ $$a_2,$$ $$a_3$$ CM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, and $$a_1,$$ $$a_2,$$ $$a_3$$ are three distinct finite complex values, then $$f(z)=\Delta ^n_{\eta }f(z)$$ for each $$z\in \mathbb {C}.$$ The main results in this paper improve and extend many known results concerning a conjecture posed by Chen and Yi in 2013.

In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

The first aim of this paper is to investigate the growth of the entire function defined by the Laplace-Stieltjes transform converges on the whole complex plane. By introducing the concept of generalized order, we obtain two equivalence theorems of Laplace-Stieltjes transforms related to the generalized order, A * and λn. The second purpose of this paper is to study the problem on the approximation of this Laplace-Stieltjes transform. We also obtain some theorems about the generalized order, the error, and the coefficients of Laplace-Stieltjes transforms, which are generalization and improvement of the previous results.

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

In this paper we deal with the martingales in variable Lebesgue space over a probability space. We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space. The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent. In particular, we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities. Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p > 1.

This paper aims to discuss the mathematical details in Lewis’ model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Levy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis’ model and validate his results for particular Levy processes. We also simply give the analyticity conditions for the characteristic function of the Carr–Geman–Madan–Yor model and a simple derivation of the characteristic function of Kou’s double exponential model.

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator $${\cal L} = \Delta + {\bf{V}}$$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.

In the present paper, we obtain a Halanay inequality on time scales with unbounded coefficient for a dynamic problem, which extends a result of Wen et al. (J. Math. Anal. Appl., 347 (2008), 169–178.) to the inequality of integral type on time scales. Moreover, we list two dynamic problems to which the Halanay inequality obtained above can be applied and prove the zero solution of two delay dynamic problems are asymptotically stable. Moreover, it is worth mentioning that the Halanay inequality obtained in the present paper is more precise than the results in [3, 14, 17].

In this paper, we introduce a new family of generalized blending-type bivariate Bernstein operators which depends on four parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ . Approximation properties of these operators are studied, and we obtain the rate of convergence in terms of mixed and partial modulus of continuities. Moreover, we prove a Korovkin- and a Voronovskaja-type theorems for these operators. The last part of the paper is devoted to the associated GBS operators. In this part, we study degree of approximation of the GBS operators in terms of mixed modulus of continuity. GBS operators obtained here give better approximation than the original operators to the function f(x, y). Finally, approximation properties of the suggested operators and their associated GBS operators are discussed on graphs, for some numerical examples to show how GBS operator gives better approximation to f(x, y). Also, approximation properties of the suggested operators for different values of parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ are illustrated on graphs. It should be mentioned that any increase in $$\alpha _{i}$$ values or any decrease in $$s_{i}$$ values gives better approximation of the suggested operators to f(x, y).

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

In the present paper we study the so-called sampling Kantorovich operators in the very general setting of modular spaces. Here, modular convergence theorems are proved under suitable assumptions, together with a modular inequality for the above operators. Further, we study applications of such approximation results in several concrete cases, such as Musielak–Orlicz and Orlicz spaces. As a consequence of these results we obtain convergence theorems in the classical and weighted versions of the $$L^p$$ L p and Zygmund (or interpolation) spaces. At the end of the paper examples of kernels for the above operators are presented.

In their remarkable paper, Fazli et al. (Int J Comput Math, 2019. https://doi.org/10.1080/00207160.2019.1658870 ) have recently established existence results for fractional Basset–Boussinesq–Oseen equation in an appropriate partially ordered Banach space. In this paper, we improve this study by proposing different approach that provides existence criterion for the addressed equation. The main assumptions are less restrictive and easily verifiable. Examples with graphical representations are constructed to demonstrate consistency to theoretical findings. Besides, we construct an iterative sequence that converges to the unique solution. We end the paper by a conclusion that demonstrates the advantage of our theorem compared to the previous results.

In this paper, we introduce the notion of FI-t-lifting modules which is a proper generalization of both the concepts of t-lifting modules and FI-lifting modules. We show that a direct sum of FI-t-lifting modules is not FI-t-lifting, in general. It is also shown that if M is an FI-t-lifting module, then $${\overline{Z}}^2(M)$$ is a direct summand of M and $${\overline{Z}}^2(M)$$ is a noncosingular FI-lifting module. The last part of the paper is devoted to the study of amply supplemented FI-t-lifting modules.

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution (Y, Z) but also on the law PY of Y. The first part of the paper is devoted to the existence and uniqueness of solutions in Lp, 1 < p ≤ 2, where the monotonicity conditions are satisfied. Next, we show that if the generator f is uniformly continuous in (μ, y, z), uniformly with respect to (t, ω), and if the terminal value ξ belongs to Lp(Ω, F, P) with 1 < p ≤ 2, the mean-field BSDE has a unique Lp solution.

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach’s fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation.

This paper deals with normal submanifolds immersed in a Riemannian manifold $${\overline{M}}$$ . We generalized some recent results of surfaces in space forms obtained by Hernandez-Lamoneda and Ruiz-Hernandez (Bull Braz Marh Soc (NS) 49:447–462, 2018) to arbitrary submanifolds. More precisely, given a submanifold M in $${\overline{M}}$$ , we study the submanifolds formed by orthogonal geodesics to M, and call it a ruled normal submanifold to M. In the first part of this paper, we analyze these submanifolds and establish some geometric properties of them. Furthermore, we extend some properties about the lines of curvature and using the ideas of [3] also give an extension of the classical Theorem of Bonnet to hypersurfaces of $${\overline{M}}$$ .

In this paper we consider a link $$B_{n,\rho }$$Bn,ρ between Baskakov type operators $$B_{n,\infty }$$Bn,∞ and genuine Baskakov–Durrmeyer type operators $$ B_{n,1}$$Bn,1 depending on a positive real parameter $$\rho $$ρ. The topic of the present paper is the pointwise limit relation $$\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) $$Bn,ρfx→Bn,∞fx as $$\rho \rightarrow \infty $$ρ→∞ for $$x\ge 0.$$x≥0. As a main result we derive uniform convergence on each compact subinterval of the positive real axis for all continuous functions f of polynomial growth.

This is a follow-up of the paper J. Fernandez-Bonder, A. Ritorto and A. Salort, H-convergence result for nonlocal elliptic-type problems via Tartar’s method, SIAM J. Math. Anal., 49 (2017), pp. 2387–2408, where the classical concept of H-convergence was extended to fractional $$p$$ -Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak- $$*$$ convergence of the coefficients is an equivalent condition for H-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local $$p$$ -Laplacian case.

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation, $${\rm{i}}u_t-u_{xx}+{\rm{i}}(|u|^4u)_x=0, \;\; x\in\mathbb{T}.$$ The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)].

This paper is based mainly on Titchmarsh’s theorem (Introduction to the theory of Fourier integrals. Clarendon Press, Oxford, 1937, Theorem 84) in the one-dimensional case. Abilov et al. (Comput Math Math Phys 48:2146, 2008) proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilovs for only two variables, using a translation operator. The purpose of this paper is to study these estimates for Quaternion Fourier transforms, also the functions satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space $$L^r({\mathbb {R}}^{2},{\mathcal {H}})$$, where $${\mathcal {H}}$$ a quaternion algebra which will be specified in due course.

In this paper, by the affine analogue of the fundamental theorem for Euclidean planar curves, we classify the affine curves with constant affine curvatures. Note that we use the fully affine group and not the equi-affine subgroup consisting of area-preserving affine transformations. (Caution: much of the literature omits the “equi-” in their treatment.) According to the equivariant method of moving frames, explicit formulas for the generating affine differential invariants and invariant differential operators are constructed. At the same time, by using the fact that the affine transformation group GA$$(2,\mathbb {R})$$ can factor as a product of two subgroup $$B\cdot \mathrm{SE}(2,\mathbb {R})$$ and the moving frame of the subgroup SE$$(2,\mathbb {R})$$, we build the moving frame of GA$$(2,\mathbb {R})$$ and obtain the relations among invariants of group GA$$(2,\mathbb {R})$$ and its subgroup SE$$(2,\mathbb {R})$$. Applying the affine curvature to recognize affine equivalent objects is considered in the last part of this paper.

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

In this paper, we present some fixed point theorems of Krasnosel’skii’s type in Banach spaces. The involved operators need not to be compact nor weakly continuous. The results are obtained and formulated with the use of the measures of weak noncompactness and a large classes of contractions (strict contractions, nonlinear contractions, as well as nonexpansive or pseudocontractive mappings). Throughout the paper, we use the hypothesis $$\mathsf {(H1)}$$ and $$\mathsf {(H2)}$$ , which are one of the main ingredients of the proofs. Finally, with the obtained fixed point results, we discuss the existence of solutions to a stationary transport equation with delayed neutrons.

In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

In this paper, we study the uniqueness problem of meromorphic functions sharing two finite sets ignoring multiplicities. This paper is inspired from an article due to Chen (Open Math 15:1244–1250, 2017).

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 prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extend the works of different authors carried out in recent years, for example Charak and Sharma (Turk J Math 41: 404–411, 2017), Fang and Yuan (J Aust Math Soc Ser A 68(3): 321–333, 2000) Jiang and Gao (Acta Math Sci 32(4): 1503–1512, 2012), Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015), Sun (Kragujev J Math 38(2):273–282, 2014), Xiao et al. (Abstr Appl Anal Article ID 373910, 1–8, 2011), Yuan (Arch Math 96(5): 435–444 2011).

In this paper, a general five-step discrete-time Zhang neural network is proposed for time-varying nonlinear optimization (TVNO). First, based on the bilinear transform and Routh–Hurwitz stability criterion, we propose a general five-step third-order Zhang dynamic (ZeaD) formula, which is shown to be convergent with the truncation error $$O(\tau ^3)$$ with $$\tau >0$$ the sampling gap. Second, based on the designed ZeaD formula and the continuous-time Zhang dynamic design formula, a general five-step discrete-time Zhang neural network (DTZNN) model with quartic steady-state error pattern is presented for TVNO, which includes many multi-step DTZNN models as special cases. Third, based on the Jury criterion, we derive the effective domain of step size in the DTZNN model, which determines its convergence and convergence speed. Fourth, a nonlinear programming is established to determine the optimal step size. Finally, simulation results and discussions are given to validate the theoretical results obtained in this paper.

In this paper, we consider the following logarithmic Choquard equation $$\begin{aligned} - \,\Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~\ \text{ in }~\ {\mathbb {R}}^n, \end{aligned}$$where $$n\ge 3, \lambda >0, 2 2$$, this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for $$n\ge 3$$ by using the variational method.

We consider the Siegel upper half space $$H_{2m}$$ of degree 2m and a subset $$H_m\times H_m$$ of $$H_{2m}$$ consisting of two $$m\times m$$ diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ , one is the action on holomorphic functions on $$H_{2m}$$ defined by the automorphy factor of weight k on $$H_{2m}$$ and the other is the action on vector valued holomorphic functions on $$H_m\times H_m$$ defined on each component by automorphy factors obtained by $$det^k \otimes \rho $$ , where $$\rho $$ is a polynomial representation of $$GL(n,{\mathbb C})$$ . We consider vector valued linear holomorphic differential operators with constant coefficients on holomorphic functions on $$H_{2m}$$ which give an equivariant map with respect to the above two actions under the restriction to $$H_m\times H_m$$ . In a previous paper, we have already shown that all such operators can be obtained either by a projection of the universal automorphic differential operator or alternatively by a vector of monomial basis corresponding to the partition $$2m=m+m$$ . Here in this paper, based on a completely different idea, we give much simpler looking one-line formula for such operators. This is obtained independently from our previous results. The proofs also provide more algorithmic approach to our operators.

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

In the present paper, we apply Hopf’s lemma to the contact CR-warped product submanifolds of generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. The indicated inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iranian Math. Soc., 39, No. 3, 415–429 (2013), M. Atceken, Collect. Math., 62, No. 1, 17–26 (2011), and S. Sular and C. Ozgur, Turkish J. Math., 36, 485–497 (2012)]. Moreover, we also deduce another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities obtained in [I. Mihai, Geom. Dedicata, 109, 165–173 (2004) and K. Arslan, R. Ezentas, I. Mihai, and C. Murathan, J. Korean Math. Soc., 42, No. 5, 1101–1110 (2005)]. The inequalities deduced in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for the contact CR-warped product submanifolds of any almost contact metric manifold.

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M G , k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for M G k were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from M G . We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Polubinski.

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl-Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

International audience; In this paper, we first introduce the concept and properties of ω-periodic limit process. Then we apply specific criteria obtained to investigate asymptotically ω-periodic mild solutions of a Stochastic Differential Equation driven by a Brownian motion. Finally, we give an example to show usefulness of the theoritical results that we obtain in the paper. MSC : 34C25, 34C27, 60H30, 34 F05.

In the present paper, our boundary value problem is Sturm–Liouville problem with retarded argument and transmission conditions at the finitely many points of discontinuity. This is the first work containing several points of discontinuity in the theory of differential equations with retarded argument. The main purpose of this paper is to obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions of this boundary value problem. The simplicity of the eigenvalues is proved. In special case when our problem with retarded argument and transmission conditions at one point of discontinuity, the obtained results coincide with the corresponding results in Bayramov et al. (Appl Math Comput 191, 592–600, 2007).

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.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

The aim of this paper is to consider the relation between $$\mathsf {Lie}$$-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional Lie algebras, these concepts are not identical, even if the pairs of Leibniz algebras are $$\mathsf {Lie}$$-stem. Moreover, throughout the paper, we provide some conditions under which $$\mathsf {Lie}$$-isoclinism and isomorphism of $$\mathsf {Lie}$$-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well.