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In this paper, we established a mathematical model of an S I 1 I 2 R epidemic disease with saturated incidence and general recovery functions of the first disease I 1 . Considering the basic reproduction number, we obtained conditions for both disease-free and co-existing cases. The equilibrium points local stability is verified by using the Routh-Hurwitz criterion, while for the global stability, we used a suitable Lyapunov function to analyze the endemic spread of the positive equilibrium point. Moreover, we carried out the local bifurcation around both equilibrium points (disease-free and co-existing), where we obtained that the disease-free equilibrium point undergoes a transcritical bifurcation. We conduct numerical simulations that supported our theoretical findings.

In this paper we study approximation theorems for L 2 -space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we prove equivalence of the K -functional and modulus of smoothness for Damek–Ricci spaces.

This paper is concerned with a solution of backward pseudo-Parabolic equation u t − α Δ u t + G ( ∥ ∇ u ∥ L 2 ( Ω ) ) ( − Δ ) β u = K ( t , x ) subject to the final data which is blurred by random Gaussian white noise. The primary aim of this work is to obtain the solution u . This problem is severely ill-posed (the solution’s behavior does not change continuously with the final condition). By using a statistical method, we estimate the value data from observation data and the Fourier truncation method is used to propose a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution. Finally, a numerical example is given to illuminate the effect of our method.

In real space forms, any submanifold must satisfy the so-called DDVV inequality which relates the scalar curvature, the mean curvature, and the scalar normal curvature pointwise. When the equality is attained at each point, it is called a Wintgen ideal submanifold. This property is invariant under the conformal transformations. So we try to give a complete classification of this class of submanifolds. This is done under the additional assumption of Mobius homogeneity in this paper. Some new interesting examples are constructed using the real representation of SU(2), which turn out to constitute the majority of Mobius homogeneous Wintgen ideal submanifolds. The classification follows by constructing a frame sequence. This reveals a wonderful connection with the classical harmonic sequence of Riemann surfaces.

In this paper we make use of energy methods to study the Navier–Stokes equations with time-fractional derivative. Such equations can be used to simulate anomalous diffusion in fractal media. In the first step, we construct a regularized equation by using a smoothing process to transform unbounded differential operators into bounded operators and then obtain the approximate solutions. The second part describes a procedure to take a limit in the approximation program to present a global solution to the objective equation.

In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.

This paper investigates chaotic behavior and stability of fractional differential equations within a new generalized Caputo derivative. A semi–analytical method is proposed based on Adomian polynomials and a fractional Taylor series. Furthermore, chaotic behavior of a fractional Lorenz equation are numerically discussed. Since the fractional derivative includes two fractional parameters, chaos becomes more complicated than the one in Caputo fractional differential equations. Finally, Lyapunov stability is defined for the generalized fractional system. A sufficient condition of asymptotic stability is given and numerical results support the theoretical analysis.

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals., Comment: This is a preprint of a paper whose final and definite form is with 'Chaos, Solitons & Fractals', ISSN: 0960-0779. Submitted 1 Dec 2016; Article revised 17 Apr 2017; Article accepted for publication 19 Apr 2017; see [http://dx.doi.org/10.1016/j.chaos.2017.04.034]

This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming (CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone (SOC), we reformulate the CCP problem as the second-order cone problem (SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.

The nature is very complex to model with mathematical equations. Some physical problems found in nature could follow the power law; other could follow the Mittag–Leffler law and other the exponential decay law. On the other hand one could observe in nature a physical problem that combines both, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator exponential-Mittag–Leffler kernel with two fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed.

Let Ω ∈ L 2 ( S n − 1 ) be a homogeneous function of degree zero and b be a BMO or Lipschitz function. In this paper, we obtain some boundedness of the parametrized Littlewood–Paley operators and their high-order commutators on Herz spaces with variable exponent. Keywords: Herz space, Variable exponent, Commutator, Parametrized area integral, Parametrized Littlewood–Paley g λ∗ function

In the paper, the authors present explicit formulas, nonlinear ordinary differential equations, and recurrence relations for Eulerian polynomials, higher order Eulerian polynomials, and their generating functions in terms of the Stirling numbers of the second kind.

In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem., 21 pages, 1 figure. Revised version, minor changes and references added

This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green's function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.

We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nystr\"om, and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of "flat" blowups for the parabolic problem., Comment: 91 pages. Comments welcome

This paper deals with the singularity and global regularity for a class of nonlinear porous medium system with time-dependent coefficients under homogeneous Dirichlet boundary conditions. First, by comparison principle, some global regularity results are established. Secondly, using some differential inequality technique, we investigate the blow-up solution to the initial-boundary value problem. Furthermore, upper and lower bounds for the maximum blow-up time under some appropriate hypotheses are derived as long as blow-up occurs.

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given., 19 pages, to appear in Advances in Mathematics

Sarason's Toeplitz product problem asks when the operator T u T v ‾ is bounded on various Hilbert spaces of analytic functions, where u and v are analytic. The problem is highly nontrivial for Toeplitz operators on the Hardy space and the Bergman space (even in the case of the unit disk). In this paper, we provide a complete solution to the problem for a class of Fock spaces on the complex plane. In particular, this generalizes an earlier result of Cho, Park, and Zhu.

Defense mechanisms are very important to all animal life. Predators in every biome must eat to survive. With predators being top on the food chain and always on the lookout for a meal, prey must constantly avoid being eaten. In this paper, we have proposed and analyzed a tri-trophic predator–prey model of one prey and two predator exhibiting group defense mechanism. We have assumed Monod-Haldane functional response for interaction between species due to group defense ability of prey and middle predator. We have performed Kolmogorov and Hopf bifurcation analysis for the model system. Linear and global stability of the model system have been analyzed. Lyapunov exponents are computed numerically and 2D scan for different parameters of the model have performed to characterize the complex behavior of the model system. The numerical simulations shows the chaotic and periodic oscillations of the model system for certain range of parameter. We have drawn bifurcation diagrams for different parameter values which shows the complex dynamical behavior of model system. This work is an attempt to study the effect of group defense mechanism of prey in predator population and effect of immigration within top predator population is investigated. It is also observed that in the presence of group defense, the model system stabilizes after adding a small amount of constant immigration within top predator population.

In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound estimates for the first eigenvalue of the mean Laplacian on Berwald manifolds, which generalize some results in Riemannian geometry.

This paper is concerned with a stochastic HBV infection model with logistic growth. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HBV infection model. Then we obtain sufficient conditions for extinction of the disease. The stationary distribution shows that the disease can become persistent in vivo.

In this paper, we construct a new integrable equation called Mobius-Toda equation which is a generalization of q -Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mobius-Toda equation and a whole integrable Mobius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Mobius-Toda hierarchy are given and this leads to the existence of the tau function.

This paper is concerned with obtaining theapproximate solution for Volterra-Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function ω( x )=1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L 2 norm and L ∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.

The smooth bifurcation and non-smooth grazing bifurcation of periodic solution of three-degree-of-freedom vibro-impact systems with clearance are studied in this paper. Firstly, six-dimensional Poincare maps are established through choosing suitable Poincare section and solving periodic solutions of vibro-impact system. Then, as the analytic expressions of all eigenvalues of Jacobi matrix of six-dimensional map are unavailable, the numerical calculations to search for the critical bifurcation values point by point is a laborious job based on the classical critical criterion described by the properties of eigenvalues. To overcome the difficulty from the classical bifurcation criteria, the explicit critical criterion without using eigenvalues calculation of high-dimensional map is applied to determine bifurcation points of Co-dimension-one bifurcations and Co-dimension-two bifurcations, and then local dynamical behaviors of these bifurcations are further analyzed. Finally, the existence of the grazing periodic solution of the vibro-impact system and grazing bifurcation point are analyzed, the discontinuous grazing bifurcation behavior is studied based on the compound normal form map near the grazing point, the discontinuous jumping phenomenon and the co-existing multiple solutions near the grazing bifurcation point are revealed.

Based on the generalized Hamiltonian system, a new method for constructing a class of three-dimensional (3-D) chaotic systems is presented in this paper. After choosing the proper parameters of skew-symmetric matrix, dissipative matrix and external input, one smooth 3-D chaotic system is proposed to show the effectiveness of the proposed method. Numerical simulation techniques, including phase portraits, Poincare sections, Lyapunov exponents and bifurcation diagram, illustrate that the proposed 3-D system has periodic, quasi-periodic and chaotic flows under the conditions of different parameters.

In this paper, we make use of the Lagrange interpolation to obtain some algebraic identities, involving one or two infinite sets of variables.

Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer’s G function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke’s inequalities for the Kummer and Gauss type functions.

This paper mainly focuses on the analysis of a hyperchaotic financial system as well as its chaos control and synchronization. The phase diagrams of the above system are plotted and its dynamical behaviours like equilibrium points, stability, hyperchaotic attractors and Lyapunov exponents are investigated. In order to control the hyperchaos, an efficient optimal controller based on the Pontryagin’s maximum principle is designed and an adaptive controller established by the Lyapunov stability theory is also implemented. Furthermore, two identical financial models are globally synchronized by using an interesting adaptive control scheme. Finally, a fractional economic model is introduced which can also generate hyperchaotic attractors. In this case, a linear state feedback controller together with an active control technique are used in order to control the hyperchaos and realize the synchronization, respectively. Numerical simulations verifying the theoretical analysis are included.

This paper is purported to investigate a cross-diffusion system arising in a predator-prey population model including Holling type-II functional response in a bounded domain with Dirichlet boundary condition. The asymptotical stabilities are investigated to this system by using the method of eigenvalue. Moreover, the existence of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function.

Fractional order dynamical systems admit chaotic solutions and the chaos disappears when the fractional order is reduced below a threshold value [1]. Thus the order of the dynamical system acts as a chaos controlling parameter. Hence it is important to study the fractional order dynamical systems and chaos. Study of fractional order dynamical systems is still in its infancy and many aspects are yet to be explored. In pursuance to this in the present paper we prove the existence of fractional Hopf bifurcation in case of fractional version of a chaotic system introduced by Bhalekar and Daftardar-Gejji [2]. We numerically explore the (A, B, α) parameter space and identify the regions in which the system is chaotic. Further we find (global) threshold value of fractional order α below which the chaos in the system disappears regardless of values of system parameters A and B.

In the paper, the stochastic stability of the binary airfoil subject to the effect of a bounded noise is studied through the determination of moment Lyapunov exponents. The noise excitation here is often used to model a realistic model of noise in many engineering application. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. Via the Feller boundary classification, the types of singular points are discussed here, and for the system discussed, the singular points only exist in end points. The fundamental methods used are the perturbation approach and the Green's functions method. With these methods, the second-order expansions of the moment Lyapunov exponents are obtained, which are shown to be in good agreement with those obtained using Monte Carlo simulation. The effects of noise and system parameters on the moment Lyapunov exponent and the stochastic stability of the binary airfoil system are discussed.

In the present paper, the synchronization by the Ge-Yao-Chen (GYC) partial region stability theory of chaotic Mathieu-Van der Pol and chaotic Duffing-Van der Pol systems with fractional order-derivative is proposed. Numerical simulations show that this synchronization technique is very effective and it turns out that the fractional order-derivative induces quick synchronization compared to integer order-derivative of these systems. In order to bring out the chaotic behavior of these systems either with fractional or with integer order-derivative, we simulate their phase portraits and the Lyapunov exponent. Moreover, we provide in this work an approximated solution to both systems to show that the solution of such a system can be represented as a simple power-series function. Furthermore, the representation of the error dynamics with respect to the time before and after the control action approves the effectiveness of the control method and proves the possibility of stabilization and controllability of chaotic systems with an appropriate. Furthermore, the synchronization of the fractional Mathieu-Van der Pol system using the fractional Duffing-Van der Pol system is simulated.

In this paper we define a more general convolution product of functionals on Wiener space and develop the fundamental relationships between the generalized Fourier–Feynman transform and the convolution product.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map., Comment: 35 pages, 9 figures. Minor corrections, new Figures 3, 5, 6, and 7

In this paper we introduce the weighted version of fully measurable grand Lebesgue spaces and obtain characterizations for the boundedness of maximal operator, Hilbert transform and the Hardy averaging operator on these spaces.

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets—a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2 . For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2 -weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion.

Let b p and h p ( 0 p ∞ ) be the harmonic Bergman space and harmonic Hardy space of harmonic functions on the open unit disk U , respectively. Given 1 ≤ p ∞ , denote by ‖ ⋅ ‖ b p the norm in the space b p and by ‖ ⋅ ‖ h p the norm in the space h p . In this paper we establish some improvements of the constant a p appearing in the inequality ‖ f ‖ b 2 p ≤ a p ‖ f ‖ h p , given on Kalaj and Mestrovic (2011) [Theorem 1.1], for p = 4 and f real harmonic, as ‖ f ‖ b 8 ≤ 28 + 35 2 + 24 2 ( 2 + 2 ) + 4 4 ( 2 + 2 ) 64 8 ‖ f ‖ h 4 .

The aim of this paper is to enlarge some known results from Fredholm and perturbation theory in the Banach algebra of bounded operators on a Banach space to the Fredholm theory in Banach algebra with respect to a subalgebra. The obtained results allow us to characterize the stability of some essential spectra with respect to a subalgebra.

This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the 2 n roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained.

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.

We obtain integral representations for the resolvent of $\psi(A)$, where $\psi$ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and $A$ is a sectorial operator on a Banach space. As a corollary, for a wide class of functions $\psi$, we show that the operator $-\psi(A)$ generates a sectorially bounded holomorphic $C_0$-semigroup on a Banach space whenever $-A$ does, and the sectorial angle of $A$ is preserved. When $\psi$ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for $A$ can be described, at least on Hilbert spaces, in terms of the existence of a bounded $H^{\infty}$-calculus for $A$. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators., Comment: The paper has been accepted for publication in Advances in Mathematics. This is the authors' accepted version

In this paper, we derive the non Markovian master equation (NMME) that correspond to position non Markovian stochastic Schrodinger equation (PNMSSE) in linear and non linear cases. In this case, using Nivokov property we derive four formulas of (NMME) for linear and non linear cases respectively. The functional derivative operator may depend on time and independent with respect to noise. Here, we determine the functional derivative of statistical operator. When the functional derivative operator depends on time and noise, one can calculate the perturbation and post Markovian perturbation for the functional operator, which exists in position non Markovian equation of motion (PNMEM). In order to explain our theory, we present a simple non Markovian example. Finally, we give the conclusion and the plan for future works.

In this paper, using a new method provided in [4] we characterize all algebraic traveling wave solutions of the fourth order Boussinesq equation, the nonlinear Klein–Gordon equation and a generalized Benjamin–Bona–Mahony equation.

Siedlecki and Weimar (2015) defined the notion of ( s , t ) -weak tractability for linear multivariate problems, which holds if the information complexity of the multivariate problem is not exponential in d t and e − s , where d is the number of variables and e is the error threshold with positive s and t . For Hilbert spaces, they were able to characterize ( s , t ) -weak tractability in terms of how quickly the corresponding ordered singular values decay. Using this result, they studied the embedding of H r ( T d ) into L 2 ( T d ) , where T d is the d -dimensional torus, determining precisely when this problem is ( s , t ) -tractable for a given d and r . Their proof is based on deep results of Kuhn et al. (2014), which are complicated by the difficulty of ordering the singular values. In this paper, we provide a new characterization of ( s , t ) -weak tractability of multivariate problems over Hilbert spaces, which does not require us to order the singular values. This allows us to obtain a new, and somewhat simpler, proof of the Siedlecki and Weimar (2015) result that does not need to use the results of Kuhn et al. (2014).

Recently, Abdon Atangana and Dumitru Baleanu suggested a novel fractional operator based in the Mittag-Leffler function with non-singular and nonlocal kernel. In this paper using the newly established fractional operator, an alternative representation of the Irving–Mullineux oscillator via Atangana–Baleanu fractional derivative in Liouville–Caputo sense is presented. Numerical simulations are obtained using an iterative scheme via Sumudu-Picard iterative method. The existence and uniqueness of the solutions are studied in detail using the fixed-point theorem and some properties of the inner product and the Hilbert space. Numerical simulations of the special solutions were done and new chaotic behaviors are obtained.

In this paper we prove global existence and uniqueness of solutions to the stochastic logarithmic Schrodinger equation with linear multiplicative noise. Our approach is mainly based on the rescaling approach and the method of maximal monotone operators. In addition, uniform estimates of solutions in the energy space H-1(R-d) and in an appropriate Orlicz space are also obtained here. (C) 2016 Elsevier Masson SAS. All rights reserved.

In this paper, we investigate the dynamics of a diffusive predator–prey system with Crowley–Martin functional response and delay subject to Neumann boundary condition. More precisely, we study the stability and Turing instability of positive equilibrium for non-delay system, instability and Hopf bifurcation induced by time delay for delay system. In addition, by the theory of normal form and center manifold method, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincare series introduced by Petersson and a new family of Poincare series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.

In 1969 R.H. Bing asked the following question: Is there a compact two-dimensional polyhedron with the fixed point property which has even Euler characteristic? In this paper we prove that there are no spaces with these properties and abelian fundamental group. We also show that the fundamental group of an example cannot have trivial Schur multiplier., Comment: 9 pages, 1 figure