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Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

Let μ ( n ) be the Mobius function and x real. In this paper, we investigated the best possible estimates for the sum ∑ n ≤ x μ ( n ) e ( n k θ ) under the weak Generalized Riemann Hypothesis. A similar result also holds for the Liouville function λ ( n ) .

In this paper, we proved an arithmetic Siegel–Weil formula and the modularity of some arithmetic theta function on the modular curve X 0 ( N ) when N is square free. In the process, we also constructed some generalized Delta function for Γ 0 ( N ) and proved some explicit Kronecker limit formula for Eisenstein series on X 0 ( N ) .

In this paper, we study the problem of whether the best approximation to any x in a real Hilbert space X from the closed convex set K ≔ C ∩ D can be characterized by the best approximation to a perturbation x − l of x from the set C for some l in a certain cone in X . The set C is a closed convex subset of X and D ≔ { x ∈ X : g j ( x ) ≤ 0 , ∀ j = 1 , 2 , … , m } , where the functions g j : X ⟶ R ( j = 1 , 2 , … , m ) are continuously Frechet differentiable that are not necessarily convex. We show under suitable conditions that this “perturbation property” is characterized by the strong conical hull intersection property of C and D at the point x 0 ∈ K . We prove this by first establishing a dual cone characterization of a nearly convex set. Our result shows that the convex geometry of K is critical for the characterization rather than the representation of D by convex inequalities, which is commonly assumed for the problems of best approximation from a convex set. In the special case where the set D is convex, we show that the Lagrange multiplier characterization of best approximation holds under the standard Slater’s constraint qualification together with a non-degeneracy condition. The lack of representation of D by convex inequalities is supplemented by the non-degeneracy condition, but the characterization, even in this special case, allows applications to problems with quasi-convex functions g j , j = 1 , 2 , … , m , as they guarantee the convexity of D . Simple numerical examples illustrate the nature of our assumptions.

In this paper we prove the existence of periodic solutions for gradient systems in finite and infinite dimensional spaces. The techniques of the proofs are based on the application of a global inverse functions theorem, the Schaefer fixed point theorem and the Faedou–Galerkin method. We apply our results in order to solve nonlinear reaction–diffusion equations with Dirichlet and Neumann boundary conditions.

In this paper, we studied a diffusive predator-prey model with competitive interference and Crowley-Martin type functional response. The conditions for local stability of coexisting equilibrium are given by analyzing the eigenvalue spectrum. By using delay as bifurcation parameter, conditions for occurrence of Hopf bifurcation are also given. The property of bifurcating period solutions is investigated by calculating the normal form. Some numerical simulations are performed to support our theoretical result. Our conclusions show that diffusion and delay are two factors that should be considered in establishing the predator-prey model, since they can induce spatially bifurcating period solutions.

In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained.

In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials., 40 pages, 1 figure

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group., 32 pages, 19 figures

The aim of this paper is to bring together two approaches, Heglotz variational principle and fractional calculus, to deal with non-conservative systems in phase space. Namely, we study the functional of Herglotz type whose extremum is sought, by the differential equation that involves Caputo fractional derivatives in phase space. Firstly, Herglotz variational principle under fractional Hamilton action in phase space is presented, and its Hamilton canonical equations are derived. Secondly, two basic formulae for the variation of the fractional Hamilton–Herglotz action in phase space are obtained. Furthermore, the definition and the criterion of Noether symmetry for fractional Herglotz variational principle are given, and the corresponding Noether’s theorem is established. Under appropriate conditions, the Noether’s theorem can reduce to the classical one of Herglotz type in phase space. Finally, two examples are given to illustrate the application of the results.

We discuss the stability and bifurcation analysis of predator-prey model in the presence of group defense and non-linear harvesting in prey. Mathematically, we analyze the dynamics of the system such as boundedness of the solutions, existence and stability conditions of the equilibrium points. The model undergoes saddle-node, transcritical and Hopf-Andronov bifurcations. The direction of Hopf bifurcation by calculating the first Lyapunov number is examined. The effects of prey harvesting rate and death rate of predator on the model by considering them as bifurcation parameters are analyzed. In this paper, we dedicate ourselves to the investigation of the complex dynamics of the model to maintain the coexistence of the species which is important for ecological balance in the real environment. Several numerical simulations are performed to substantiate our analytical findings.

Let ( X n , g + ) ( n ≥ 3 ) be a Poincare–Einstein manifold which is C 3 , α conformally compact with conformal infinity ( ∂ X , [ g ˆ ] ) . On the conformal compactification ( X ‾ , g ¯ = ρ 2 g + ) via a boundary defining function ρ, there are two types of Yamabe constants: Y ( X ‾ , ∂ X , [ g ¯ ] ) and Q ( X ‾ , ∂ X , [ g ¯ ] ) . (See definitions (1) and (2) .) In [13] , Gursky and Han obtained an inequality between Y ( X ‾ , ∂ X , [ g ¯ ] ) and Y ( ∂ X , [ g ˆ ] ) . In this paper, we first show that the equality holds in Gursky–Han's inequality if and only if ( X n , g + ) is isometric to the standard hyperbolic space ( H n , g H ) . Secondly, we derive a conformal invariant inequality between Q ( X ‾ , ∂ X , [ g ¯ ] ) and Y ( ∂ X , [ g ˆ ] ) , and show the equality holds if and only if ( X n , g + ) is isometric to ( H n , g H ) . Based on this, we give a simple proof of the rigidity theorem for Poincare–Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.

In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan.

Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group H ( G ) to be frames and Riesz bases in terms of the group Fourier transform.

In this paper, we present an upwind finite volume method to solve the convection-diffusion equations with Dirichlet boundary on rectangular mesh. By utilizing the technique of element-by-element analysis, the stability of the method has been proved and the H1-norm error estimate is presented. Furthermore, we provide the proofs of the maximum principle and L∞-norm error estimate. Finally, some numerical experiments are provided to confirm our theoretical results.

Breaking of ensemble equivalence between the microcanonical ensemble and the canonical ensemble may occur for random graphs whose size tends to infinity, and is signaled by a non-zero specific relative entropy of the two ensembles. In [3] and [4] it was shown that breaking occurs when the constraint is put on the degree sequence (configuration model). It is not known what is the effect on the relative entropy when the number of constraints is reduced, i.e., when only part of the nodes are constrained in their degree (and the remaining nodes are left unconstrained). Intuitively, the relative entropy is expected to decrease. However, this is not a trivial issue because when constraints are removed both the microcanonical ensemble and the canonical ensemble change. In this paper a formula for the relative entropy valid for generic discrete random structures, recently formulated by Squartini and Garlaschelli, is used to prove that the relative entropy is monotone in the number of constraints when the constraint is on the degrees of the nodes. It is further shown that the expression for the relative entropy corresponds, in the dense regime, to the degrees in the microcanonical ensemble being asymptotically multivariate Dirac and in the canonical ensemble being asymptotically Gaussian., arXiv admin note: substantial text overlap with arXiv:1711.04273

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions., Comment: 21 pages

In this paper we construct the higher order Heisenberg supermagnet models in (2+1)-dimensions by introducing different auxiliary matrix variables. We investigate their structure and integrability with respect to two constraints. By virtue of the gauge transformation, we derive their gauge equivalent counterparts which are the super and fermionic higher order nonlinear Schrodinger equations in (2+1)-dimensions. Furthermore, we obtain their corresponding Backlund transformations.

This paper considers backward stochastic differential equations (BSDEs for short) on the infinite horizon under consistent nonlinear expectations which are dominated by consistent sublinear expectations. We derive the existence and uniqueness result by discretization and approximation method. Besides, we show the existence of the solution of Markovian ergodic BSDEs under G ˜ -expectations and state some applications.

In this paper we study pseudo-differential operators with operator valued symbols on the Similitude group, SIM ( 2 ) . We characterize the unique infinite-dimensional, unitary, irreducible representations of SIM ( 2 ) and using that the Fourier inversion formula is given. The operator valued Wigner transform and the Weyl transform are also studied here. Furthermore we also obtain a necessary and sufficient condition on the symbols for which the pseudo-differential operators SIM ( 2 ) are Hilbert-Schmidt and trace class operators. Finally, we also give the trace formula for these operators.

The dynamics of the language competition model is considered in this paper. The classical system is converted to non-integer order case by replacing the second-order partial derivative with the Riesz fractional derivative. A well-known numerical approximation methods based on the Fourier spectral algorithm in space and the third-order exponential time-differencing scheme are formulated to numerically simulate the three component fractional-in-space reaction-diffusion system in one and high dimensions for different values of α. Numerical results indicate α ∈ (1, 1.5] as the key control parameter that can influence the coexistence of various speakers over a period of time.

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^3, [\hat{g}])$ is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is a Berger metric on $S^3$ with $Y(S^3, [\hat{g}])$ close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries., Some changes are made in the references. Comments are welcome!

In the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the Hardy space on the unit polydisc. It turns out that our approach also applies to quotients of weighted Bergman modules.

In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds., to appear in Advances in Mathematics

We develop a new and further generalized form of the fractional kinetic equation involving generalized k-Bessel function. The manifold generality of the generalized k-Bessel function is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results. MSC: 26A33, 33C45, 33C60, 33C70, Keywords: Pochhammer symbol, Fractional calculus, k-Bessel function, Laplace transform, Fractional derivative, Fractional integration

In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.

In this paper, we present a nonlinear fractional order model in the Caputo sense to explore and simulate the TB dynamics. Using the TB confirmed notified cases from the year 2002 to 2017 in Khyber Pakhtunkhwa, Pakistan, we estimate the model parameters and demonstrate that the proposed fractional model provides a good fit to the real data. Initially, we compute the basic reproduction number and the model equilibria. Then, the existence and uniqueness of the model are shown via generalized mean value theorem. Further, we explore the local and global stability of the disease free equilibria in fractional environment. Finally, numerical results are obtained in order to validate the importance of the arbitrary order derivative and theoretical results. We conclude that the fractional epidemic model is more generalized than the classical model, which give more information about the disease dynamics and give a good agreement to the real data of TB infection.

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called 'zero product preserving', and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford-Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert Schmidt operators. (C) 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

In this paper, we consider the mathematical analysis and numerical simulation of time-fractional multicomponent systems. Here, the classical time derivatives in such systems are replace with the Atangana–Baleanu fractional derivative in the sense of Caputo. This derivative is found useful in the sense that it combines both the non-local and nonsingular kernels in its formulation. A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu derivative. Numerical experiments presented for different instances of α, 0

Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results.

A general theory of partial balayage on Riemannian manifolds is developed, with emphasis on compact manifolds. Partial balayage is an operation of sweeping measures, or charge distributions, to a prescribed density, and it is closely related to (construction of) quadrature domains for subharmonic functions, growth processes such as Laplacian growth and to weighted equilibrium distributions. Several examples are given in the paper, as well as some specific results. For instance, it is proved that, in two dimensions, harmonic and geodesic balls are the same if and only if the Gaussian curvature of the manifold is constant., Comment: 68 pages

In this paper, we characterize the closures of Dirichlet type spaces D α in the logarithmic Bloch space. Moreover, we study the boundedness and compactness of composition operators from the logarithmic Bloch space to closures of Dirichlet type spaces D α in the logarithmic Bloch space.

In this paper we show sharp lower bounds for norms of even homogeneous Fourier multipliers in L ( L p ( R d ; X ) ) for 1 p ∞ and for a UMD Banach space X in terms of the range of the corresponding symbol. For example, if the range contains a 1 , … , a N ∈ C , then the norm of the multiplier exceeds ‖ a 1 R 1 2 + ⋯ + a N R N 2 ‖ L ( L p ( R N ; X ) ) , where R n is the corresponding Riesz transform. We also provide sharp upper bounds of norms of Banuelos–Bogdan type multipliers in terms of the range of the functions involved. The main tools that we exploit are A -weak differential subordination of martingales and UMD p A constants, which are introduced here.

In this paper we give an explicit construction of a representing system generated by the Szego kernel for the Hardy space. Thus we answer an open question posed by Fricain, Khoi and Lefevre. We use frame theory to prove the main result.

In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space.

Substantial evidence supports that financial returns time series exhibit abnormal properties including leptokurtosis, volatility clustering as well as intermittent jumps and leverage effects between returns and volatility processes. This paper studies a heavy-tailed stochastic volatility (SV) model with jumps components and leverage effects, and the Student's-t distribution is employed to describe the error innovations (SVJLt). Since the existence of high-dimensionality of the latent variables and the special structure of Hessian matrix of the stochastic volatility density, we develop an efficient Markov chain Monte Carlo (MCMC) posterior simulator exploiting the adaptive importance sampling technique based on band and sparse matrix routine rather than the conventional Kalman filter to estimate the new model. And the precision sampler is exploited due to the band structure of the inverse covariance matrix of the state variables. The model comparisons of returns volatility are conducted utilizing the observed-data based deviance information criterion (DIC) and the cross-entropy (CE) based marginal likelihood estimation. The effectiveness of the proposed model and the methodology are illustrated with applications in stock returns volatility forecast. Through employing several loss functions for evaluation, the empirical studies suggest strong evidence in heavy tailed distributions, jumps features and leverage effects simultaneously.

In this paper we study transformed trigonometric series with Hausdorff averages of Fourier coefficients. We prove Hardy–Littlewood and Pitt's inequalities for such series. The corresponding results for the Hausdorff averages of the Fourier transforms are also obtained.

In this paper, we are interested in the study of a stochastic viral infection model with immune impairment driven by Levy noise. First we prove the existence of a unique global solution to the model. By means of the Lyapunov method we study the stability of the equilibria. We present sufficient conditions for the extinction and persistence in mean. Furthermore, we present some numerical results to support the theoretical work.

Very little research is available in the field of 4-D autonomous conservative chaotic systems. This paper presents five new 4-D autonomous conservative chaotic systems having non-hyperbolic equilibria with various characteristics. The proposed systems have different numbers of non-hyperbolic equilibrium points. One of the new systems has four non-hyperbolic equilibria points along with lines of equilibria. Hence, this system may belong to the category of hidden attractors chaotic system. The first, second, fourth and fifth type of the systems exhibit coexistence of chaotic flow, whereas the third type of the system exhibits coexistence of chaotic flows with quasi-periodic behaviour. The chaotic behaviours of the proposed systems are verified by using phase portrait plot, Poincare map, local Lyapunov spectrum, bifurcation diagram and frequency spectrum plots. The conservative nature of the proposed systems is proved by finding the sum of finite-time local Lyapunov exponents, finite-time local Lyapunov dimensions and divergence of the vector field. The sum of the finite-time local Lyapunov exponents and divergence of the vector field are equal to zero, and local Lyapunov dimension is equal to the order of the system confirm the conservative nature of the new chaotic systems.

In this paper we have taken a delayed predator-prey system with stage structure among prey species. We assume that there is cooperation among the mature and immature preys to ensure immature prey’s existence and there is a maturation delay for immature to be mature for the preys. With this model the local stability has been discussed in presence of delay. By taking maturation delay as the key parameter, the condition for local stability has been obtained by constructing a Lyapunov functional. By taking the maturation delay as the bifurcation parameter the necessary conditions for Hopf-bifurcation have been discussed both analytically and numerically. The length of delay has been estimated to preserve the stability. Also, Lyapunov exponents have been evaluated numerically for different cases.

Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

In this paper, we use fast oscillating integrated Ornstein–Uhlenbeck (abbreviated as O-U) processes to pathwisely approximate Wiener processes. In physics, such approximation process is known as a colored noise approximation, and is suitable for dealing with stochastic flow problems. Our first result shows that if the drift term of a stochastic differential equation (abbreviated as SDE) satisfies usual Lipschitz constrains and a linear growth condition, then the solution of the SDE can be almost surely approximated with a polynomial rate. Next, we explore the O-U process approximation on the random manifold of stochastic evolution equations with linear multiplicative noise. Our second result shows that if the stochastic evolution equation further satisfies a uniformly hyperbolic condition, then the corresponding random manifold approximation also converges almost surely, with a polynomial rate.

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+\lambda\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^\lambda$ contain an interval for all $x\in(0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017)., Comment: 21 pages, 1 figure. To appear in Indag. Math

This paper focuses on the dynamic behavior of phytoplankton–zooplankton system with cell size and time delay. Remarkably, the existence of cell size and time delay make the dynamic behavior of the system more close to the real-world situation, essentially different from those in the existing related literature. To analyze the dynamic behavior of the system, the positiveness and boundedness of the solution are first derived. Then, the asymptotic stability of the coexistent equilibrium and Hopf bifurcation are studied by analyzing the associated characteristic equation. Once more, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are determined. Finally, the theoretical results are illustrated by a numerical example.

In this paper, we study the iterative method of Aitken type for solving the non-linear equations, in which the interpolation nodes are controlled by variant of Newton method or by a general method of order p.By combining such methods with a generalized secant method, it is shown that the order of convergence can be increased to as high as desired and also in the limiting case efficiency of the method is 2. Several numerical examples are provided in support of the theoretical results. Keywords: Non-linear equations, Newton method, Aitken type method, Generalized secant method

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampere equation with Dirichlet boundary conditions. We formulate the Monge-Ampere equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.

In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is deduced using Benssoussan-Lions' algorithm. Finally, the results of some numerical experiments are presented to support the theory.

Recently, the second and third authors established the sharp Hardy–Adams inequalities of second order derivatives with the best constants on the hyperbolic space B 4 of dimension four ( [40] , Adv. Math. (2017)). A key method used in dimension four in [40] is that the following Hardy operator when n = 4 and k = 2 ∫ B n | ∇ k u | 2 d x − ∏ i = 1 k ( 2 i − 1 ) 2 ∫ B n u 2 ( 1 − | x | 2 ) 2 k d x can be decomposed exactly as the product of fractional Laplacians. However, such a decomposition in dimension four is no longer possible in the case n > 4 and 2 ≤ k n 2 and the situation here is thus considerably more difficult and complicated. Therefore, it remains open as whether a sharp Hardy–Adams inequality still holds on higher dimensional hyperbolic space B n for n > 4 . The main purpose of this paper is to use a substantially new method of estimating the Hardy operator to establish the sharp Hardy–Adams inequalities on hyperbolic spaces B n for all even dimension n and n ≥ 4 . As applications of such inequalities, we will improve substantially the known Adams inequalities on hyperbolic space B n in the literature and also strengthen the classical Adams' inequality and the Hardy inequality on Euclidean balls in any even dimension. The later inequality can be viewed as the borderline case of the sharp Hardy–Sobolev–Maz'ya inequalities for higher order derivatives in high dimensions obtained recently by the second and third authors [41] .

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.