a stability theorem for identification and adaptive control establishes a globally convergent algorithm for discrete time model reference adaptive systems an extension of the classical decreasing gain algorithm using this approach the process in adaptive control may contain zeros outside the unit circle if the external reference signal is a solution of a linear homogeneous difference equation

behaviors of solutions for a singular diffusion equation abstract this paper is devoted to some behaviors of solutions of the initial boundary problem for a singular diffusion equation namely localization and large time behavior after given some special explicit solutions it is proved that solutions of the problem possess the localization property next l 2 decay estimate as t is proved by a rather standard energy method finally by comparison with a special solution the expected l decay estimate is derived

exponential stability of general tracking algorithms tracking and adaptation algorithms are from a formal point of view nonlinear systems which depend on stochastic variables in a fairly complicated way the analysis of such algorithms is thus quite complicated a first step is to establish the exponential stability of these systems this is of interest in its own right and a prerequisite for the practical use of the algorithm it is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time varying true ones in this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms this includes least mean squares recursive least squares and kalman filter based adaptation algorithms we show how stability of an averaged linear and deterministic equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes we also give explicit conditions for exponential stability of the most common algorithms the tracking performance of the algorithms is studied in a companion paper

positivity properties of viscosity solutions of a degenerate parabolic equation we consider the problem i u t u u u 2 in r n r u x 0 u 0 x in r n where is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on r n

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

computing the region of convergence for power series in many real variables a ratio like test abstract we give an elementary proof that the region of convergence for a power series in many real variables is a star convex domain but not in general a convex domain in doing so we deduce a natural higher dimensional analog of the so called ratio test from univariate power series from the constructive proof of this result we arrive at a method to approximate the region of convergence up to a desired accuracy while most results in the literature are for rather specialized classes of multivariate power series the method devised here is general as far as applications are concerned note that while theorems such as the cauchy kowalevski theorem and its generalizations to many variables grant the existence of a region of convergence for a multivariate taylor series to certain pdes under appropriate restrictions they do not give the actual region of convergence the determination of the maximal region of convergence for such a series solution to a pde is one application of our result

on the cauchy problem for the fast diffusion equation abstract for u 0 1 u 0 l loc 1 r n the author studies the existence of a kind of weak solution to the cauchy problem u t div u m 1 d u in r n 0 t u x 0 u 0 x 0 in r n where m 0 is a constant the uniqueness and regularity of solutions are also discussed

life span and a new critical exponent for a degenerate parabolic equation abstract in this paper we consider the positive solution of the cauchy problem for the equation u t u p u u q p 1 q 1 and give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the cauchy problem furthermore the life span of solutions are also studied

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

regularity of viscosity solutions to a degenerate nonlinear cauchy problem abstract this paper is concerned with the regularity of certain weak solutions to the cauchy problem in r n for a degenerate nonlinear parabolic equation we get the holder regularity of the weak solution when the parameter n 1 and improve the parameter s region from 2 n 1 to n 1 since n 1 2 n 1

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

viscosity solutions of a class of degenerate quasilinear parabolic equations with dirichlet boundary condition abstract this paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous dirichlet boundary condition the equation under consideration arises from a number of practical model problems including reaction diffusion processes in a porous medium the degeneracy of the problem appears on the boundary and possibly in the interior of the domain the goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them an application of the above results is given to a porous medium type of reaction diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem

some degenerate and quasilinear parabolic systems not in divergence form abstract this paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form u t u p u av v t v q v bu with null dirichlet boundary conditions and positive initial conditions where p q a and b are all positive constants the local existence and uniqueness of classical solution are proved moreover it will be proved that all solutions exist globally if and only if ab 1 2 where 1 is the first eigenvalue of in with homogeneous dirichlet boundary condition

positivity local smoothing and harnack inequalities for very fast diffusion equations abstract we investigate qualitative properties of local solutions u t x 0 to the fast diffusion equation t u u m m with m 1 corresponding to general nonnegative initial data our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form 0 t with r d they combine into forms of new harnack inequalities that are typical of fast diffusion equations such results are new for low m in the so called very fast diffusion range precisely for all m m c d 2 d the boundedness statements are true even for m 0 while the positivity ones cannot be true in that range

existence of positive solutions for the one dimensional singular p laplacian this paper concerns the positive solutions of boundary value problems for the one dimensional singular p laplacian by the classical method of elliptic regularization we obtain some existence results which generalize some results of w zhou x wei positive solutions to bvps for a singular differential equation nonlinear anal 2006 doi 10 1016 j na 2006 06 015

discontinuous viscosity solutions of a degenerate parabolic equation we study a nonlinear degenerate parabolic equation of the second order regularizing the equation by adding some artificial viscosity we construct a generalized solution we show that this solution is not necessarily continuous at all points

solution of porous medium type systems by linear approximation schemes the convergence of semi discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type the enthalpy formulation and variational technique are used the semi discretization used reduces the original parabolic p d e to linear elliptic p d e the algebraic correction arising from nonlinearities is treated by newton like iterations in finite steps some numerical experiments are discussed and compared with the analytical solutions

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

radial equivalence and study of self similarity for two very fast diffusion equations abstract in this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p laplacian equation begun in a previous work we treat the cases m 0 and p 1 we perform an exhaustive study of self similar solutions for both equations based on a phase plane analysis and the correspondences we discover we also obtain special correspondence relations and self maps for the limit case m 1 p 0 which is particularly important in applications in image processing we also find self similar solutions for the very fast p laplacian equation that have finite mass and in particular some of them that conserve mass while this phenomenon is not true for the very fast diffusion equation

null controllability of a class of newtonian filtration equations abstract in this paper we study the null controllability of a class of newtonian filtration equations using the properties of finite extinction time and finite speed of propagation we construct a control function in feedback form such that system is exactly null controllable at any time t 0 if we put a nonnegative constraint on the control function this system is not exactly null controllable any more but approximately null controllable for a long time

self similar subsolutions and blowup for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the formut u f u u we prove the nonexistence of global solutions for large initial data we also estimate the maximal existence time to do so we use a method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results onut u up on generalized burgers equations and on other semilinear equations this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

some results on controllability of a nonlinear degenerate parabolic system by bilinear control in this paper we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control based on the shrinking property of the solutions we prove that the system is not globally approximately controllable furthermore we give an approximate null controllability result we also prove that the system is not globally exactly null controllable by a comparison principle

second order convergence analysis of stochastic adaptive linear filtering the convergence of an adaptive filtering vector is studied when it is governed by the mean square error gradient algorithm with constant step size we consider the mean square deviation between the optimal filter and the actual one during the steady state this quantity is known to be essentially proportional to the step size of the algorithm however previous analyses were either heuristic or based upon the assumption that successive observations were independent which is far from being realistic actually in most applications two successive observation vectors share a large number of components and thus they are strongly correlated in this work we deal with the case of correlated observations and prove that the mean square deviation is actually of the same order or less than the step size of the algorithm this result is proved without any boundedness or barrier assumption for the algorithm as it has been done previously in the literature to ensure the nondivergence our assumptions are reduced to the finite strong memory assumption and the finite moments assumption for the observation they are satisfied in a very wide class of practical applications

directions for computing truncated multivariate taylor series efficient recurrence relations for computing arbitrary order taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in r n after a sequence of directions the multivariate taylor coefficients or partial derivatives can be reconstructed or interpolated the sequence of univariate calculations is more efficient than multivariate methods although previous work indicates a space cost for this savings and significant cost for the reconstruction we completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction by appropriate choice of directions the reconstruction reduces to a sequence of lagrange polynomial interpolation problems in r n 1 for which a divided difference algorithm computes the coefficients of a newton form another algorithm collects like terms from the newton form and returns the desired multivariate coefficients

positive solutions to bvps for a singular differential equation abstract in this paper we are interested in the positive solutions to a singular differential equation 2 f t 0 0 t 1 subject to the first or all of the following boundary conditions 1 0 0 1 0 0 where is a positive constant f t c 1 0 1 f t 0 on 0 1 by the classical method of elliptic regularization some existence results of the above bvps are obtained moreover the uniqueness and the asymptotic behaviors as 0 and as will also be studied

classical solutions for a class of fully nonlinear degenerate parabolic equations this paper is concerned with the existence and comparison principle of classical solutions for a class of fully nonlinear degenerate parabolic equations

a class of nonlocal and degenerate quasilinear parabolic system not in divergence form abstract in this paper we investigate the positive solution of the nonlocal and degenerate quasilinear parabolic system not in divergence form u t f u u a v d x v t g v v b u d x with null dirichlet boundary conditions the blow up and global existence criteria are obtained furthermore we prove that under certain conditions the solutions have global blow up

a recursive computation scheme for multivariate rational interpolants we derive here a recursive computation scheme for the rational interpolation method introduced in 7 explicit formulas for these multivariate rational interpolants are repeated in 2 while the recursive algorithm is described in 3 a number of interesting special cases such as the univariate rational interpolation problem and the multivariate pade approximants introduced in 6 and 10 are dealt with in 4 for some of these rational approximants other recursive schemes were described previously finally 5 contains the numerical results the multivariate rational interpolants described here are compared with multivariate polynomial interpolants interpolating branched continued fractions introduced by cuyt and verdonk 8 interpolating branched continued fractions introduced by siemaszko 12 and several multivariate pade approximants 6 3 10 besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord

radon measure valued solutions for a class of quasilinear parabolic equations initial value problems for quasilinear parabolic equations having radon measures as initial data have been widely investigated looking for solutions which for positive times take values in some function space in contrast it is the purpose of this paper to define and investigate solutions that for positive times take values in the space of the radon measures of the initial data we call such solutions measure valued in contrast to function valued solutionspreviously considered in the literature we first show that there is a natural notion of measure valued solution of problem p below in spite of its nonlinear character a major consequence of our definition is that if the space dimension is greater than one the concentrated part of the solution with respect to the newtonian capacity is constant in time subsequently we prove that there exists exactly one solution of the problem such that the diffuse part with respect to the newtonian capacity of the singular part of the solution with respect to the lebesgue measure is concentrated for almost every positive time on the set where the regular part with respect to the lebesgue measure is large moreover using a family of entropy inequalities we demonstrate that the singular part of the solution is nonincreasing in time finally the regularity problem is addressed as we give conditions depending on the space dimension the initial data and the rate of convergence at infinity of the nonlinearity to ensure that the measure valued solution of problem p is in fact function valued

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

positive solutions to a singular differential equation of second order abstract this paper concerns the existence of positive solutions to a singular differential equation u u t u 2 u f t 0 0 t 1 with the boundary conditions u 1 u 0 0 where 0 f t c 0 1 and f t 0 on 0 1 by theories of ordinary differential equations bertsch and ughi m bertsch m ughi positivity properties of viscosity solutions of a degenerate parabolic equation nonlinear anal 14 1990 571 592 proved that when n 1 n is a positive integer and f 1 the problem admits a decreasing positive solution in this paper we show by the classical method of elliptical regularization that if 1 2 1 then the above problem admits at least a positive solution which is not decreasing as a by product of the results the problem with n 1 and f 1 admits at least two positive solutions if n 2

some results on a class of degenerate parabolic equations not in divergence form abstract in this paper we first prove the existence of weak solutions of the initial and boundary value problem for u t u div u p 2 u u p with p 2 0 1 and then discuss the asymptotic behavior of weak solutions as 0

nonuniqueness of solutions for a singular diffusion problem in this paper we consider a singular diffusion problem and show by constructing a counterexample that the weak solution to the problem is not unique the proof consists of several steps first we prove that there exists a maximal weak solution to the problem we show that the support of the continuous maximal weak solution cannot decrease in time then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data