approximate dynamic programming based approaches for input output data driven control of nonlinear processes we propose two approximate dynamic programming adp based strategies for control of nonlinear processes using input output data in the first strategy which we term j learning one builds an empirical nonlinear model using closed loop test data and performs dynamic programming with it to derive an improved control policy in the second strategy called q learning one tries to learn an improved control policy in a model less manner compared to the conventional model predictive control approach the new approach offers some practical advantages in using nonlinear empirical models for process control besides the potential reduction in the on line computational burden it offers a convenient way to control the degree of model extrapolation in the calculation of optimal control moves one major difficulty associated with using an empirical model within the multi step predictive control setting is that the model can be excessively extrapolated into regions of the state space where identification data were scarce or nonexistent leading to performances far worse than predicted by the model within the proposed adp based strategies this problem is handled by imposing a penalty term designed on the basis of local data distribution a cstr example is provided to illustrate the proposed approaches

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

evolution equations for nonlinear degenerate parabolic pde abstract we define a convex function on h 1 and characterize its subdifferential by which we introduce an evolution equation as a weak formulation of a class of nonlinear degenerate singular and noncoercive parabolic pdes associated with an arbitrary maximal monotone graph and with the dirichlet boundary condition

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

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

instantaneous shrinking and single point extinction for viscous hamilton jacobi equations with fast diffusion for a large class of non negative initial data the solutions to the quasilinear viscous hamilton jacobi equation partial t u delta p u nabla u q 0 in 0 infty times real n are known to vanish identically after a finite time when 2n n 1 0 the positivity set of u t is a bounded subset of real n even if u 0 0 in real n this decay condition on u 0 is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as x to infty is the whole real n for all times the time evolution of the positivity set is also studied on the one hand it is included in a fixed ball for all times if it is initially bounded emph localization on the other hand it converges to a single point at the extinction time for a class of radially symmetric initial data a phenomenon referred to as emph single point extinction this behavior is in sharp contrast with what happens when q ranges in p 1 p 2 and p in 2n n 1 2 for which we show emph complete extinction instantaneous shrinking and single point extinction take place in particular for the semilinear viscous hamilton jacobi equation when p 2 and q in 0 1 and seem to have remained unnoticed

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

a new result for the porous medium equation derived from the ricci flow given r 2 with a good complete metric we show that the unique solution of the ricci flow approaches a soliton at time infinity solitons are solutions of the ricci flow which move only by diffeomorphism the ricci flow on r 2 is the limiting case of the porous medium equation when m is zero the results in the ricci flow may therefore be interpreted as sufficient conditions on the initial data which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a soliton solution

nonuniqueness in the quenching problem this paper deals with nonnegative solutions of for u t delta u u q chi u 0 qquad rm in omega times 0 infty qquad qquad rm q with q in 0 1 and prescribed continuous dirichlet data b b x on it is proved that for n 6 there is a critical parameter q c in 0 1 with the following property if q q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w one that coincides with w and another one that satisfies u w but u not equiv w for n 6 and q q c or n 7 however such a second solution above w is impossible moreover it is shown that for n 6 q q c and any sufficiently small nonnegative boundary data b there exist initial values admitting at least two continuous weak solutions of q the final result asserts that for any n and q nonuniqueness for q holds at least for some boundary and initial data

nonlinear heat conduction with absorption space localization and extinction in finite time additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions the statements answer the questions when the support of the solution is bounded in space and when the solution dies in finite time

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

asymptotic properties of a semilinear heat equation with strong absorption and small diffusion in this paper the authors study the asymptotic behaviour of solutions ue x t of the cauchy problems as e goes to zero ut e u up 0 x rn t 0 u x 0 u0 x x rn 0 0 u x 0 u0 x x rn it is proved under certain assumptions that ue x t u x t as e 0 uniformly on compact subsets of rn 0 and moreover a precise estimate is given local and global estimates for extinction time are also given the proofs are somewhat technical

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

behaviour of the velocity of one dimensional flows in porous media the one dimensional flow of gas of density u through a porous medium obeys the equation u um x where m 1 x e r and t 0 we prove that the local velocity of the gas given by v mu 2u1 not only is bounded for t t 0 but approaches an n wave profile as t oo n waves are the typical asymptotic profiles for some first order conservation laws a class of nonlinear hiperbolic equations the case m 0 0 2 u x 0 u0 x forx e r where m is a positive constant and u0 satisfies 0 3 u0 e l r u0 0 a e u0 0 apart from its mathematical interest as a simple model of nonlinear evolution equation of possibly degenerate or singular parabolic type equation 0 1 and its n dimensional analogue appear in a number of applications cf 18 for a reference in the typical application to the flow of gas through a porous medium we have m 1 and u stands for the density of the gas writing 0 1 as a conservation law for the mass

concavity of solutions of nonlinear ordinary differential equations in this work we study concavity properties of solutions of u n 1 x x2 u u x 0 as a by product we establish the nonexistence of global solutions in c2 0 some applications to nonlinear parabolic equations are mentioned

cascade lattice iir adaptive filters the feedback lattice filter forms including the two multiplier form and the normalized form are examined with respect to their relationships to the feedback direct form filter specifically the transformation matrix between the lattice forms and the direct form is derived parameter and state relationships between the lattice forms and the direct form are therefore obtained an iir filter structure the cascade lattice iir structure is constructed based on this structure three iir adaptive filtering algorithms in the two multiplier form can then be developed following the gradient approach the steiglitz mcbride approach and the hyperstability approach convergence of these algorithms is theoretically analyzed using either the ode approach or the hyperstability theorem these algorithms are then simplified into forms computationally as efficient as their corresponding direct form algorithms relationships of the simplified algorithms to the direct form algorithms are also studied which disclose a consistency in algorithm structure regardless of the filter form three normalized lattice algorithms can also be derived from the two multiplier lattice algorithms experimental results show much improved performance of the normalized lattice algorithms over the two multiplier lattice algorithms and the direct form algorithms

localization and boundedness of the solutions of the neumann problem for a filtration equation on etudie l existence et l unicite des solutions faibles du probleme de neumann u t u xx dans d t u x 0 u 0 x sur 0 u x 0 t h t sur 0 t ici c 1 r c 2 r 0 avec s 0 si s 0 pour t 0 d t 0 0 t on suppose que h 0 t r est bornee sur des sous ensembles compacts de 0 t et u 0 l 1 0 est telle que u 0 x est bornee sur 0

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

well posedness of initial boundary value problem of degenerate parabolic equations abstract we study a nonlinear degenerate parabolic equation associated with an arbitrary maximal monotone graph we impose initial and dirichlet boundary conditions and prove that the problem is well posed

existence and non existence of a fast diffusion equation inrn in this paper we study the global existence and asymptotic behaviour of solutions tout log ufor the cauchy initial value problem inrn we prove that ifn 3 then every solution satisfies rn up x t dx for any 1 n 2 hence we extend a previous result of vazquez 19 which claims that rn u dx for 0 t tmax

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

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

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

acoustic echo cancellation do iir models offer better modeling capabilities than their fir counterparts the adequateness of iir models for acoustic echo cancellation is a long standing question and the answers found in the literature are conflicting we use the results from rational hankel norm and least squares approximation and we recall a test that provides a priori performance levels for fir and iir models we apply this test to the measured acoustic impulse responses upon comparing the performance levels of fir and iir models with the same number of free parameters we do not observe any significant gain from the use of iir models we attribute this phenomenon to the shape of the energy spectra of the acoustic impulse responses so tested which possess many strong and sharp peaks faithful modeling of these peaks requires many parameters irrespective of the type of the model

bounded speed of propagation for solutions to radiative transfer equations the radiative transfer equation is the nonlinear transport equation partial t f frac 1 varepsilon v cdot nabla x f frac 1 varepsilon 2 sigma tilde f f tilde f 0 rte where tilde f x mathbf t ff x mathbf v t dv denotes the average off x t on the unit sphere v 1 it describes the absorption and emission of photons in a hot medium as the mean free path goes to 0 f converges to a solution of the porous medium equation t u f u withf u n u 1 since blows up atu 0 solutions to the pme propagate with finite speed specifically ifu 0 has compact support inr n so doesu t for everyt 0 and the sets t x u x t 0 t 0 form an expanding family ast increases andu t r n we show in this paper that these propagation properties hold for the solutionsf of the rte for all small moreover the growth of the support off is uniform in our proofs rely on the construction of explicit solutions of the travelling wave type and subsolutions to the rte to our knowledge this is the first example of a kinetic equation with high velocities where localized data propagate always with bounded speed for vlasov poisson equations this arises only for particular initial data

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

self similar solutions satisfying or not the equation of the interface abstract we study the existence of self similar solutions for the porous medium equation with reaction and convection u t u m 1 u x x u n 1 u x ku p in r 0 where m n 1 0 0 we are in particular interested in compactly supported self similar solutions satisfying some good equation at the interface the same equation appearing in the pure diffusion case we prove that there exist such self similar solutions only if k 1 4n an infinity of solutions with bad behaviour at least at one interface also exist there exist no self similar solutions with support arbitrarily small we complete the study by considering the case 0

approaching an extinction point in one dimensional semilinear heat equations with strong absorption abstract this paper deals with the cauchy problem ut uxx up 0 0 u x 0 u0 x 0 and u x t 0 appear when t is close to t we describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached we also give conditions under which some of these behaviours actually occur

global travelling waves in reaction convection diffusion equations we study the existence and properties of solutions in travelling wave form u x t x st defined for every z x st r for the reaction convection diffusion equationut a um xx b un x kupforx r t 0 with a m n 0 b k p r in the reaction case k 0 we prove that there exist travelling waves vanishing for z if and only if b 0 and min 1 n m p2 max 1 n p max 1 n moreover if m p 2n there exists a minimal velocity s a b k m n p 0 for which there are travelling waves only with s s while in the case m p 2n there are travelling waves only when 4amk b2n and for every velocity s 0 some properties of the function s are established all the waves are decreasing in their support and waves having bounded support from the right exist if and only if m min 1 n also the absorption case k 0 is treated where we find that for different values of the parameters there exists a unique travelling wave for every velocity s r but for some case where only negative velocities exist the cases b 0 or k 0 are well known in the literature

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

extinction and positivity of the solutions of the heat equations with absorption on networks in this paper we propose a discrete version of the following semilinear heat equation with absorption ut u uq with q 1 which is said to be the heat equation with absorption on a network using the discrete laplacian operator on a weighted graph we define the heat equations with absorption on networks and give their physical interpretations the main concern is to investigate the large time behaviors of nontrivial solutions of the equations whose initial data are nonnegative and the boundary data vanish it is proved that the asymptotic behaviors of the solutions u x t as t tends to strongly depend on the sign of q 1

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

data based analysis of discrete time linear systems in noisy environment in this paper a data based method is developed for analyzing the controllability and observability of discrete time linear systems in noisy environment this method uses measured data to estimate the controllability matrix and the observability matrix without identifying system models the unbiasedness and consistency of this estimate with measurement noise and system noise are proven respectively as the estimated error of system parameters will not accumulate in calculating the controllability matrix and observability matrix this method has a higher precision than traditional methods especially in high dimensional state space in the simulation the advantages of the data based method in accuracy and convergence are illustrated

obstructions to existence in fast diffusion equations abstract the study of nonlinear diffusion equations produces a number of peculiar phenomena not present in the standard linear theory thus in the sub field of very fast diffusion it is known that the cauchy problem can be ill posed either because of non uniqueness or because of non existence of solutions with small data the equations we consider take the general form u t d u u x u x x or its several dimension analogue fast diffusion means that d at some values of the arguments typically as u 0 or u x 0 here we describe two different types of non existence phenomena some fast diffusion equations with very singular d do not allow for solutions with sign changes while other equations admit only monotone solutions no oscillations being allowed the examples we give for both types of anomaly are closely related the most typical examples are v t v x v x and u t u xx u x for these equations we investigate what happens to the cauchy problem when we take incompatible initial data and perform a standard regularization it is shown that the limit gives rise to an initial layer where the data become admissible positive or monotone respectively followed by a standard evolution for all t 0 once the obstruction has been removed

extinction and non extinction for a p laplacian equation with nonlinear source abstract in this paper we deal with the extinction of solutions of the initial boundary value problem of the p laplacian equation u t div u p 2 u u q in a bounded domain of r n with n 2 for 1 p 2 we show that q p 1 is the critical exponent of extinction for the weak solution furthermore for 1 p 2 and q p 1 we prove that the extinction and non extinction of the solution depends strongly on the first eigenvalue of the problem div p 2 p 2 in 0

on the existence of stationary points for the steiglitz mcbride algorithm most convergence results for adaptive identification algorithms have been developed in sufficient order settings involving an unknown system with known degree reduced order settings in which the degree of the unknown system is underestimated are more common but more difficult to analyze deducing stationary points in these cases typically involves solving nonlinear equations hence the sparseness of results for reduced order cases if we allow ourselves the tractable case in which the input to an identification experiment is white noise we shall show that the steiglitz mcbride method 1965 indeed admits a stationary point in reduced order settings for which the resulting model is stable our interest in this study stems from a previous result showing an attractive a priori bound on the mismodeling error at any such stationary point

an overdetermined initial and boundary value problem for a reaction diffusion equation with m 1 p 1 and p 0 for p 1 2 m the minimum speed is 0 here we are interested in understanding the phenomenon of nonuniqueness therefore we restrict ourselves to the exponent range 2 m i p 1 we remind the reader that negative exponents p i 0 are also allowed with the convention that up 0 wherever u 0 we will consider initial data u0 such that problem 1 1 admits more than one solution and we will try to

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

extinction properties of semilinear heat equations with strong absorption consider the initial boundary value problem for ut u uq with 0 0 q 1 the initial data are nonnegative and the boundary data vanish it is well known that the solution becomes extinct in finite time t i e u x t becomes identically zero for t t where t is some positive number in this paper we study the profile of x u x t as t t

c regularity for the porous medium equation on etablit un resultat de regularite optimale pour l equation d un milieu poreux u t u m xx 0 x r t 0 u 0 u 0 pour m 1 et la donnee initiale u 0 positive continue a support compact connexe

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

extinction and positivity for the evolution p laplacian equations with absorption on networks in this paper we discuss the extinction and positivity of solutions to evolution p laplacian equations u t p u u q 1 u 0 with absorption on finite networks with p 1 q 0 and 0 we present necessary and sufficient conditions for parameters p q and in order that solutions to the equation are extinctive or positive respectively in addition when a solution is positive we provide its decay rate and when a solution becomes extinctive in finite time we give estimate for the extinction time

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

lattice form adaptive infinite impulse response filtering algorithm for active noise control in some situations of active noise control infinite impulse response iir filters are more suitable than finite impulse response fir filters owing to the poles in the transfer function a number of algorithms have been derived for applying iir filters in active noise control however most of them use the direct form iir filter structure which faces the difficulties of checking stability and relatively slow convergence speed for noise composed of narrow band components with large power disparity to overcome these difficulties along with using the direct form iir filters a new adaptive algorithm is proposed in this paper which uses and updates the lattice form adaptive iir filter in an active noise control system full mathematical derivations of the proposed algorithm are presented and the comparison between the proposed algorithm and the commonly used filtered u lms and filtered v lms algorithms shows the superiority of the proposed algorithm

extinction and positivity for a system of semilinear parabolic variational inequalities abstract a simple model of chemical kinetics with two concentrations u and v can be formulated as a system of two parabolic variational inequalities with reaction rates v p and u q for te diffusion processes of u and v respectively it is shown that if pq u and v are comparable then at least one of the concentrations becomes extinct in finite time on the other hand for any p q 0 there are initial values for which both concentrations do not become extinct in any finite time

on the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium the authors consider the fokker planck equation ut um xx b u x x 0 t 0 with initial and boundary data u x 0 u0 x x 0 u 0 t u1 t t 0 u0 having its support in a bounded interval they concentrate on the case 0 0 it is shown that if u1 tends to zero as t then the free boundary tends to zero if u1 vanishes in a finite time so does the free boundary the possibility that the free boundary tends to infinity is also discussed moreover conditions are found on m and on u1 such that the free boundary can be estimated from above localization and from below by a positive constant when b 1 the free boundary is known to start from the right endpoint of suppu0

nonlinear degenerate parabolic equations with neumann boundary condition abstract we study neumann problem for a class of nonlinear degenerate parabolic pde a typical nonlinearity we have in mind is for instance u 1 u u 0 we establish a necessary and sufficient condition on given data for existence of solution

thermal waves in absorbing media abstract we discuss the existence of travelling wave solutions with interfaces for the nonlinear heat equation with absorption u t a u m xx bu n with a b 0 and m n r several situations occur depending on the relative strength of the diffusion and absorption terms reflected by their exponents m and n we characterize the existence of finite travelling waves in terms of m and n show their uniqueness up to translations in space and time and derive their velocity from the wave profile near the interface or front

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

the lipschitz continuity of the interface in the heat equation with strong absorption for one spatial dimension with cauchy data the continuity of the interface was proved recently by chen et al 2 kennington 3 proved that every level set of a solution to 1 l 1 3 is convex if the domain sz is convex in his proof the convexity of the domain plays a crucial role here we shall use a very simple argument to show the lipschitz continuity of the interface without the convexity condition by constructing an approximate problem we use the implicit function theorem to show that each level set of the approximate solution consists of the graph of a differentiable surface then we show that these approximate surfaces are compact in certain banach spaces the argument was previously used to study the regularity of the free boundary in the stefan problem cf 4 61 it is worthy pointing out that the argument can be applied to the following nonlinear porous medium equation with absorption see remark 1 in section 3 24 v a u vu p u

on nonexistence of similarity solutions on considere une famille de solutions non negatives semblables de l equation u t u up x r n t 0 avec n 1 et 0 p 1