In this paper we consider the Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 to define powers of Catalan generating function C( T) where T is a linear and bounded operator on a Banach space X. When the operator 4 T is of power-bounded operator, the Catalan generating function is given by the Taylor series C ( T ) : = ∑ n = 0 ∞ C n T n , where c = ( C n ) n ≥ 0 is the Catalan sequence. Note that the operator C( T) is a solution of the quadratic equation T Y 2 − Y + I = 0 . We obtain new formulae which involves Catalan triangle numbers ( B n , k ) n ≥ 1 , 1 ≤ k ≤ n and ( A n , k ) n ≥ 1 , 1 ≤ k ≤ n + 1 . As element in the Banach algebra ℓ 1 ( N 0 , 1 4 n ) , we describe the spectrum of c ∗ j for j≥1, and the expression of c −∗ j in terms of Catalan polynomials. In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.

In this paper, we study two inverse problems for the nonlinear Boussinesq system for incompressible viscoelastic non-isothermal Kelvin-Voigt fluids. The studying inverse problems consist of determining an intensities of density of external forces and heat source under given integral overdetermination conditions. Two types of boundary conditions for the velocity v are considered: sticking and sliding conditions on boundary. In both cases of these boundary conditions, the local and global in time existence and uniqueness of weak and strong solutions are established under suitable assumptions on the data. The large time behavior of weak solutions is also studied.

a) | k ( w ) | ≥ 0 is a piecewise continuous and bounded function in R = ( - ∞ , ∞ ) . The coefficients b ( w ) and q ( w ) are continuous functions in R and can be unbounded at infinity. The operator L admits closure in the space L 2 ( Ω ) and the closure is also denoted by L. Taking into consideration certain constraints on the coefficients b ( w ) q ( w ) , apart from the above-mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two-sided estimates for singular numbers ( s-numbers). Here we note that the estimate of singular numbers ( s-numbers) shows the rate of approximation of the resolvent of the operator L by linear finite-dimensional operators. It is given an example of how the obtained estimates for the s-numbers enable to identify the estimates for the eigenvalues of the operator L. We note that the above results are apparently obtained for the first time for a mixed-type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

In this paper, we propose a unified non-conforming least-squares spectral element ap- proach for solving Stokes equations with various non-standard boundary conditions. Ex- isting least-squares formulations mostly deal with Dirichlet boundary conditions and are formulated using ADN theory based regularity estimates. However, changing boundary conditions lead to a search for parameters satisfying supplementing and complimenting con- ditions [4] which is not easy always. Here we have avoided ADN theory based regularity estimates and proposed a unified approach for dealing with various boundary conditions. Stability estimates and error estimates have been discussed. Numerical results displaying exponential accuracy have been presented for both two and three dimensional cases with various boundary conditions.

This paper presents decoupled second-order accurate algorithms based on Crank-Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. The proposed algorithms deal with the spatial discretization by finite element method and treat the temporal discretization by CNLF method. For the nonlinear term in the Boussinesq equations, the semi-implicit method is used. Unconditional stability and error estimate of the numerical algorithm are proven. Some numerical tests are presented to justify the theoretical analysis.

In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is four and seven, respectively. From their error equations and to increase the convergence order without performing new functional evaluations, memory is introduced in these families of different forms. That allows us to increase from four to seven the convergence order in the first family and from seven to eleven in the second one. We perform some numerical experiments with big size systems for confirming the theoretical results and comparing the proposed methods along other known schemes.

We consider the scattering of a plane wave by a locally perturbed periodic (with respect to x_1) medium. If there is no perturbation it is usually assumed that the scattered wave is quasi-periodic with the same parameter as the incident plane wave. As it is well known, one can show existence under this condition but not necessarily uniqueness. Uniqueness fails for certain incident directions (if the wavenumber is kept fixed), and it is not clear which additional condition has to be assumed in this case. In this paper we will analyze three concepts. For the Limiting Absorption Principle (LAP) we replace the refractive index n=n(x) by n(x)+iε in a layer of finite width and consider the limiting case when ε tends to zero. This will give an unsatisfactory condition. In a second approach we require continuity of the field with respect to the incident direction. This will give the same satisfactory condition as the third approach where we approximate the incident plane wave by an incident point source and let the location of the source tend to infinity.

The arterial vascular system exchanges blood gases using pulmonary capabilities. In humans, hemodynamic flows are subjected to periodic velocity modulations. Fluid mechanics and transport of blood gases play an important role in understanding how constitutive relations of the arterial system contribute to human functioning within physiological limits. Assuming the hemodynamic system as a finite dissipative system, the nonlinear evolution equation is perused to understand dynamical challenges under physiological conditions. The infinitesimal component of the vessel wall with concentric thickening in tunica media is considered as a nonaxisymmetric bulge in an elastic-compliant artery. Using the Lie group of transformations method, we discuss the implications of traveling wave solutions to describe their impact on hemodynamic flow in an elastic-compliant artery. We find that cumulative accretion of potential energy contributes to the creation of bright soliton at the apex of the bulge. The wave speed is maximum at the peak of the bulge and progressively retards with the antegrade flow.

A new fractional difference equation 2D-TFCDM based on Caputo derivative is proposed. Using the bifurcation diagram, the maximum Lyapunov exponent and the phase diagram, the numerical solutions of the fractional difference equations are obtained, and the chaotic behavior is observed numerically. After encrypting the key with elliptic curve cryptosystem, the fractional map is developed as an encryption algorithm and applied to color image encryption. Finally, the proposed encryption system is systematically analyzed from five main aspects, and the results show that the proposed encryption system has a good encryption effect. In respective of application, we apply the proposed discrete fractional map into color image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC). Finally, the image encryption algorithm is analysed in 4 aspects that indicates the proposed algorithm is superior to others.

In this paper, we investigate pattern dynamics in a reaction-diffusion-chemotaxis food chain model with predator-taxis, which enriches previous studies about diffusive food chain models. By virtue of diffusion semigroup theory, we first show the global classical solvability and uniform boundedness of the considered model over a bounded domain Ω ⊂ R N ( N ≥ 1 ) with smooth boundary. Then the linear stability analysis for the considered model shows that chemotaxis can induce the unique positive spatially homogeneous steady state loses its stability via Turing bifurcation and Turing-spatiotemporal Hopf bifurcation, which results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing-spatiotemporal Hopf bifurcation are given explicitly. In addition, the existence and stability of non-constant positive steady state that bifurcates from the positive constant steady state is investigated by the abstract bifurcation theory of Crandall-Rabinowitz and eigenvalue perturbation theory. Finally, numerical simulations are performed to verify our theoretical results, and some interesting non-Turing pattern are found in temporal Hopf parameter space by numerical simulation.

The state-space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time-invariant case has been solved, particular attention was paid to the case of linear periodic systems (see, e.g. 1,2,3,4,5,6,7,8). Recently, such systems have regained importance, for instance, in the context of coding theory (see9), where periodic convolutional encoders play an important role, 10. The majority of the contributions within this area concern the realization of transfer functions as well as impulse responses, thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems (see11,12) suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime, contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. Based on results already obtained in 13,14, we revisit the problem of the realization of linear periodic MIMO behaviors and give further insight into this problem, which allows setting up an algorithm to compute a low-dimensional state-space realization of a periodic behavior. The proposed algorithm is based on a chain decomposition of suitable matrices.

In this paper we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it is a thin layer/quboid of thickness $\eps$ of material whose stiffness is of order $\frac{1}{\eps}$ that in the middle contains another cuboid which is of width and thickness $\eps$ that is made of material with elasticity coefficients of order $\frac{1}{\eps^q}$ for $q>0$. We show that the family of solutions of linearized elasticity problems, when $\eps$ tends to zero, converges to a solution of a problem that is posed only on the unit cube with possibly additional elastic terms on the boundary related to the plate/rod energy of the thin elastic parts. It turns out that there are five different regimes related to different values of $q$ ($q\in \langle 0, 2\rangle, \{2\}, \langle 2, 4\rangle,\{4\},\langle 4, \infty\rangle$) with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when $\eps$ tends to zero as the full 3d problem posed on the union of the unit cube and thin cuboids. This model then can be used as the approximating model in all regimes.

In this paper we prove several optimal inequalities involving the hyperbolicity constant of complementary prisms networks. Moreover, we obtain bounds and closed formulas for the general topological indices A(G)=∑uv∈E(G) a(du,dv) and B(G)=∑u∈V(G)b(du) of complementary prisms networks.

A new approach is proposed to analysis the non-Gaussian random vibration of hyperbolic tangential package. Firstly, the non-Gaussian vibration noise of specified mean, variance, skewness and kurtosis is developed by Hermite polynomial, and verified by Gaussian mixture model (GMM) theory. Secondly, next to analyzing an analytical and numerical response of the two degrees of freedom (TDOF) linear package, the acceleration response of the TDOF hyperbolic tangential package system (HTPS) under non-Gaussian vibration excitation is obtained though the Hermite polynomial and Runge-Kutta method, and the effects of the external excitation and system parameters are investigated. Finally, on the basis of the first-passage probability of the acceleration response, the reliability analysis method is introduced. The analysis provides a guidance to the vibration reliability analysis and packaging optimization design.

A theorem on the maximum regularity of solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space $L_2(R)\,(R=(-\infty,\infty))$ is proved in this paper. Two-sided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm-Liouville equation are also obtained. As is known, the obtained estimates given the opportunity to choose approximation apparatus that guarantees the maximum possible error.

Motivated by important applications in materials science and biology, we consider a non-stationary coagulation equation with in-flux and out-flux of particles (cargo). An exact analytical solution to this integrodifferential equation is found in a parametric form. This solution behaves as a decaying exponent with respect to the particle volume (fluorescence intensity in the case of coagulation of endosomes). We demonstrate that our exact solution substantially depends on coagulation kernels (fusion and fission rates) as well as the in-flux and out-flux processes of cargo.

In this paper, we derive two regularity criteria of solutions to the nematic liquid crystal flows. More precisely, we prove the local smooth solution $(u, d)$ is regular if and only if one of the following two conditions is satisfied: (i) $\nabla_{h} u_{h}\in L^{\frac{2p}{2p-3}}(0,T; L^{p}(\mathbb{R}^{3})),\ \partial_{3} d\in L^{\frac{2q}{q-3}}(0,T; L^{q}(\mathbb{R}^{3})),\ \frac{3}{2}< p\leq\infty,\ 3< q\leq\infty$; and (ii) $\nabla_{h} u_{h}\in L^{q}(0,T; L^{p}(\mathbb{R}^{3})),\ \frac{3}{p}+\frac{2}{q}\leq 1, \ 3

In this paper, Rational Bernstein polynomials on the semi-infinite interval are adapted to solve a Magnetohydrodynamic (MHD) problem. Also, the derivative, product, convert, and Galerkin Exact Operational Matrices (EOMs) of these polynomials are produced. Using the spectral Galerkin method and the Exact Operational Matrices (EOMs) of Rational Bernstein polynomials, we solve the problem with high accuracy and speed. The problem is the flow of MHD micropolar over a moving plate with suction and injection boundary conditions. Comparing the results of the Rational Bernstein Galerkin method with operational matrices and without operational matrices shows that the present method is faster than another method. Also, comparing the results of the Rational Bernstein Galerkin method and Rational Gegenbauer Tau method shows that the present method is more accurate than another method.