time discretization of nonlinear control systems with state delay via taylor lie series abstract the state delay is always existent in the practical systems analysis of the delay phenomenon in a continuous time domain is sophisticated it is appropriate to obtain its corresponding discrete time model for implementation via digital computer in this paper we propose a new scheme for the discretization of nonlinear systems using taylor series expansion and the zero order hold assumption this scheme is applied to the sample data representation of a nonlinear system with constant state time delay the mathematical expressions of the discretization scheme are presented and the effect of the time discretization method on equilibrium properties of nonlinear control system with state time delay is examined the proposed scheme provides a finite dimensional representation for nonlinear systems with state time delay enabling existing controller design techniques to be applied to them the performance of the proposed discretization procedure is evaluated using a nonlinear system for this nonlinear system various sampling rates and time delay values are considered

a numerical study of the spectral gap we present a numerical study of the spectral gap of the dirichlet laplacian k 2 k 1 k of a planar convex region k besides providing supporting numerical evidence for the long standing gap conjecture that k 3 2 d2 k where d k denotes the diameter of k our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter but also on the perimeter and the area one of these conjectures is a stronger version of the gap conjecture mentioned above a similar study is carried out for the quotient of the first two dirichlet eigenvalues

on the solution of a mixed nonlinear integral equation in this paper we consider a mixed nonlinear integral equation of the second kind in position and time the existence of a unique solution of this equation is discussed and proved a numerical method is used to obtain a system of harmmerstein integral equations of the second kind in position then the modified toeplitz matrix method as a numerical method is used to obtain a nonlinear algebraic system many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived the rate of convergence of the total error is discussed finally numerical examples when the kernel of position takes a logarithmic and carleman forms are presented and the error estimate in each case is calculated

proof of the fundamental gap conjecture we prove the fundamental gap conjecture which states that the difference between the first two dirichlet eigenvalues the spec tral gap of a schrodinger operator with convex potential and dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential more generally for an arbitrary smooth potential in higher dimensions our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction in terms of the diameter of the domain and a modulus of convexity for the potential

monotonic positive solution of a nonlinear quadratic functional integral equation abstract the existence of positive monotonic solutions in the class of continuous functions for some nonlinear quadratic integral equation have been studied in 4 5 6 7 8 here we are concerning with a nonlinear quadratic integral equation of volterra type and we shall prove the existence of at least one l 1 positive monotonic solution for that equation under caratheodory condition

integrable solutions of a nonlinear functional integral equation on an unbounded interval abstract in this paper we prove the existence of integrable solutions of a generalized functional integral equation which includes many key integral and functional equations that arise in nonlinear analysis and its applications this is achieved by means of an improvement of a krasnosel skii type fixed point theorem recently proved by k latrach and the author the result presented in this paper extends the corresponding result of j banas a chlebowicz on existence of integrable solutions of a functional integral equation under caratheodory condition nonlinear anal 2008 doi 10 1016 j na 2008 04 020 an example which shows the importance and the applicability of our result is also included

lipschitz bounds for solutions of quasilinear parabolic equations in one space variable abstract we bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time in particular we characterize those equations for which the lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time

the fundamental gap of simplices the gap function of a domain omega subset mathbb r n is xi omega d 2 lambda 2 lambda 1 where d is the diameter of and 1 and 2 are the first two positive dirichlet eigenvalues of the euclidean laplacian on it was recently shown by andrews and clutterbuck j amer math soc 24 899 916 2011 that for any convex omega subset mathbb r n xi omega geq 3 pi 2 where the infimum occurs for n 1 on the other hand the gap function on the moduli space of n simplices behaves differently our first theorem is a compactness result for the gap function on the moduli space of n simplices next specializing to n 2 our second main result proves the recent conjecture of antunes freitas j phys a math theor 41 5 055201 2008 for any triangle t subset mathbb r 2 xi t geq frac 64 pi 2 9 with equality if and only if t is equilateral

integral solution of a class of nonlinear integral equations this paper is concerned with the existence of integral solutions to a general nonlinear integral equationx t f 1 t x f 1 t f 2t 0 f 2 t k t s f 3 s x f 3 s ds t r with the help of krasnoselskii s fixed point theorem and the theory of measure of weak noncompactness we establish a new and general existence theorem for the nonlinear functional integral equation moreover an example which can not be treated by the related theorems in 5 20 22 is given to illustrate the new existence theorem

a note on solving cauchy integral equations of the first kind by iterations abstract the successive approximation method was applied for the first time by n i ioakimidis to solve practical cases of a cauchy singular integral equation the airfoil one in this paper we study a more general case we prove the convergence of the method in this general case the proposed method has been tested for two kernels which are particularly important in practice finally some numerical examples illustrate the accuracy of the method

semilinear integrodifferential equations with nonlocal initial conditions abstract of concern is the cauchy problem for semilinear integrodifferential equations with nonlocal initial conditions under general and natural hypotheses we establish some new theorems about the existence and uniqueness of solutions for the cauchy problem as a consequence we unify and extend the corresponding theorems given previously for the cauchy problem for differential equations or integrodifferential equations with nonlocal initial conditions moreover we present two examples one of which comes from heat conduction in materials with memory to show that the existing results are not applicable to them in contrast with ours

on the inverse spectral problem for euclidean triangles we consider the inverse spectral problem for the laplace operator on triangles with dirichlet boundary conditions providing numerical evidence to the effect that the eigenvalue triplet 1 2 3 is sufficient to determine a triangle uniquely on the other hand we show that other combinations such as 1 2 4 will not be enough and that there will exist at least two triangles with the same values on these triplets

power series solutions to volterra integral equations abstract power series type solutions are given for a wide class of linear and q dimensional nonlinear volterra equations on rp the basic assumption on the kernel k x y is that k x xt has a power series in x for example this holds for any analytic kernel the kernel may be strongly singular provided certain constants are finite one and only one such power series solution exists its coefficients are given by a simple iterative formula in many cases this may be solved explicitly in particular an explicit formula for the resolvent is given

gradient estimates for the ground state schrodinger eigenfunction and applications let be a planar convex domain which is symmetric with respect to each coordinate axes in this paper we give a simple and very short proof based on the maximum principle of the sharp lower bound for the spectral gap of the dirichlet laplacian in

lower bounds of the gap between the first and second eigenvalues of the schrodinger operator soit un domaine regulier borne strictement convexe dans r n et v r une fonction convexe non negative soient 1 et 2 les 2 premieres valeurs propres non nulles de l equation f vf f f o alors 2 1 2 d 2 ou d est le diametre de

local existence for retarded volterra integrodifferential equations with hille yosida operators abstract this paper is devoted to finding some existence and uniqueness theorems for classical solutions of the integrodifferential equations with infinite delay we give some sufficient conditions ensuring the existence and uniqueness of solutions we assume the linear part is not necessary to be densely defined and satisfies a hille yosida condition so that it is the generator of a nondegenerated locally lipschitz continuous integrated semigroup by using matrix operators and fixed point theorems we obtain new results for the retarded integrodifferential equations

system theoretic properties of sampled data representations of nonlinear systems obtained via taylor lie series it is possible to obtain a sampled data representation of a nonlinear dynamic system under zero order hold using taylor lie series within the context of taylor lie series theory explicit formulae for the resulting discrete time system are derived in terms of the system lie derivatives the effect of sampling on systemtheoretic properties such as equilibrium properties relative order stability zero dynamics and minimum phase characteristics is examined revealing the natural and transparent way in which the use of taylor lie series permeates the relevant theoretical aspects the results derived are also illustrated in two chemical engineering examples

a delay dependent approach to robust h sub spl infin filtering for uncertain discrete time state delayed systems a delay dependent approach to robust h sub spl infin filtering is proposed for linear discrete time uncertain systems with multiple delays in the state the uncertain parameters are supposed to reside in a polytope and the attention is focused on the design of robust filters guaranteeing a prescribed h sub spl infin noise attenuation level the proposed filter design methodology incorporates some recently appeared results such as moon s new version of the upper bound for the inner product of two vectors and de oliveira s idea of parameter dependent stability which greatly reduce the overdesign introduced in the derivation process in addition to the full order filtering problem the challenging reduced order case is also addressed by using different linearization procedures both full and reduced order filters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities which can be solved via efficient interior point algorithms numerical examples have been presented to illustrate the feasibility and advantages of the proposed methodologies

sharp inequalities for heat kernels of schrodinger operators and applications to spectral gaps abstract this paper derives inequalities for multiple integrals from which sharp inequalities for ratios of heat kernels and integrals of heat kernels of certain schrodinger operators follow such ratio inequalities imply sharp inequalities for spectral gaps the multiple integral inequalities although very different are motivated by the now classical brascamp lieb luttinger rearrangement inequalities