a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

viscosity solutions for weakly coupled systems of first order partial differential equations abstract we consider viscosity solutions for weakly coupled systems of first order partial differential equations and apply these existence and uniqueness results to examples involving random evolutions

local automorphisms of finite dimensional simple lie algebras abstract let g be a finite dimensional simple lie algebra over an algebraically closed field k of characteristic 0 a linear map g g is called a local automorphism if for every x in g there is an automorphism x of g such that x x x we prove that a linear map g g is local automorphism if and only if it is an automorphism or an anti automorphism

schlomilch series that arise in diffraction theory and their efficient computation we are concerned with a certain class of schlomilch series that arise naturally in the study of diffraction problems when the scatterer is a semi infinite periodic structure by combining new results derived from integral representations and the poisson summation formula with known identities we obtain expressions which enable the series to be computed accurately and efficiently many of the technical details of the derivations are omitted they can however be obtained from linton 2005 schlomilch series that arise in diffraction theory and their efficient computation technical report loughborough university available online at http www staff lboro ac uk macml1 schlomilch techreport pdf

minimax control problems of periodic competing parabolic systems with logistic growth terms in this article we consider a bioeconomic model for minimax control problems which are governed by periodic competing parabolic lotka volterra equations firstly the existence and uniqueness results to the state equations are proved as well as stability under mild assumptions afterwards we formulate the minimax control problem the optimality criteria are to minimize damage and trapping costs of the first species nuisance and to maximize the difference between economic revenue and cost of the second species the existence of an optimal solution is obtained also necessary conditions for a saddle point optimality are derived

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

on hopf s formulas for solutions of hamilton jacobi equations soit h r n r on considere l equation de hamilton jacobi u t h du 0 dans r n 1 r n 0 on etudie les proprietes des formules de hopf pour les solutions

a viscosity solution approach to the asymptotic analysis of queueing systems we consider a system of several interconnected queues with a single class of customers and model the state xt of the system as a jump markov process the problem of interest is to estimate the large deviations behavior of the rescaled system xt ext s corresponding to large time and large excursions of the original unscaled system the techniques employed are those of the theory of viscosity solutions to hamilton jacobi equations from the point of view of large deviation theory the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty since an abrupt change in the jump measure occurs from the point of view of viscosity solutions the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations much of the paper is devoted to proving uniqueness of viscosity solutions for these equations and these sections are of independent interest while our use of test functions in proving the uniqueness is an adaptation of the usual technique the construction of the test functions themselves via the legendre transform is new we obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem which suggests the correct integrand in the large deviation rate functional since it is the treatment of the effects due to the boundaries that is novel we devote the majority of the paper to the detailed development of a simple two dimensional system that exhibits all the essential new features however the arguments may be applied to nueueing systems that are considerably more general and we attempt to indicate this generality as well

2 local automorphisms on finite dimensional lie algebras abstract we prove that every 2 local automorphism on a finite dimensional semi simple lie algebra l over an algebraically closed field of characteristic zero is an automorphism we also show that each finite dimensional nilpotent lie algebra l with dim l 2 admits a 2 local automorphism which is not an automorphism

a stochastic control approach to the pricing of options we construct a stochastic control model of a portfolio in which the investor may invest in stock call options on the stock or risk free bonds he may also borrow funds we assume that the investor is attempting to maximize expected utility of total wealth at expiration of the options where the utility function is given as wealth to the power 0 1 this includes both the risk averse and risk neutral cases we allow the borrowing interest rate r to be distinct from the lending interest rate r we make two fundamental assumptions i the investor cannot achieve infinite expected utility of wealth in finite time stable markets and ii the market will set the option price to minimize the investor s maximal expected utility of wealth i e options are priced using a minimax strategy the market is stable only when r r we prove in the case r r and 1 that the minimax european call option price ft s is given as the solution of the black scholes equation ft i i 2s2fss i s fs i f 0 where i r when r i 2 1 i r when i 2 1 r and i i 2 1 if r i 2 1 r the parameter is the stock return drift and i is the stock return diffusion if r r then the call option price must by stable markets always be the black scholes price with rate r in the risk neutral 1 case we must have r r for a stable market and the option price given by the black scholes equation but with i explicit solutions are presented and the optimal portfolios and returns are also derived comparable results are obtained for american call options which can be exercised at any time up to and including the expiration date in this case the option pricing function is shown to be the solution of an obstacle problem

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

lattice sums for the helmholtz equation a survey of different representations for lattice sums for the helmholtz equation is made these sums arise naturally when dealing with wave scattering by periodic structures one of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d lambda lattice sums are related to and can be calculated from the quasi periodic green s function and this object serves as the starting point of the analysis

optimal control for elliptic volterra lotka type equations abstract an optimal control problem for elliptic volterra lotka type equations is considered the control problem simulates optimal harvesting of growing species under certain assumptions an optimality system is derived characterizing optimal control constructive analytic procedure is designed providing a complete solution of the optimal control problem in some cases

2 local automorphisms of finite dimensional simple lie algebras abstract let f be an algebraically closed field of characteristic 0 l be a finite dimensional simple lie algebra of type a l l 1 d l l 4 e k k 6 7 8 over f a not necessarily linear map l l is called a 2 local automorphism if for every x y l there is an automorphism x y l l depending on x and y such that x x y x y x y y in this paper we prove that any 2 local automorphism of l is an automorphism

2 local derivations on finite dimensional lie algebras abstract we prove that every 2 local derivation on a finite dimensional semi simple lie algebra l over an algebraically closed field of characteristic zero is a derivation we also show that a finite dimensional nilpotent lie algebra l with dim l 2 admits a 2 local derivation which is not a derivation

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

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

fully nonlinear neumann type boundary conditions for first order hamilton jacobi equations we consider problems of the following type h x u u 0 in where is a smooth bounded open set in r n u is a scalar function from into r u denotes its frechet differential at x and h is a given continuous function on xr r n this equation is nothing but a general form of the classical first order hamilton jacobi equation in order to investigate existence and uniqueness questions the equation has to be complemented with a boundary condition on or on a part of we consider here for reasons described below a fully nonlinear boundary condition of the form f x u u 0 on that we describe as being of the neumann type

2 local derivations and automorphisms on b h abstract this paper is devoted to 2 local derivations and 2 local automorphisms on the algebra b h of all bounded linear operators on a hilbert space h we prove that every 2 local derivation on b h is a derivation a similar result is obtained for automorphisms

nonlinear optimal control problems of degenerate parabolic equations with logistic time varying delays of convolution type in this article we consider a bioeconomic model for optimal control problems which are governed by degenerate parabolic equations governing diffusive biological species with logistic growth terms and multiple time varying delays the time varying delays are given in a convolution form the existence uniqueness and regularity results to the state equations with homogeneous dirichlet and neumann boundary conditions are established the vanishing viscosity method is used to obtain the existence result afterwards we formulate the optimal control problem in two cases firstly we suppose that this biological species causes damage to environment e g forest agriculture the optimal control is the trapping rate and the cost functional is a combination of damage and trapping costs secondly an optimal harvesting control of a biological species is considered the optimal control is a distribution of harvesting effort on the biological species and the cost functional measure the difference between economic revenue and cost the existence and the condition of uniqueness of the optimal solution are obtained a nonlinear optimality system is derived characterizing the optimal control

frequency domain green s function for a planar periodic semi infinite phased array i truncated floquet wave formulation this two part sequence deals with the derivation and physical interpretation of a uniform high frequency solution for the field radiated at finite distance by a planar semi infinite phased array of parallel elementary electric dipoles the field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent floquet waves fws together with corresponding diffracted rays which arise from scattering of the fw at the edge of the array the formal aspects of the solution are treated in the present paper they involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent different species of spectral poles define the various species of propagating and evanescent fw their interception by the steepest descent path sdp determines the variety of shadow boundaries for the edge truncated fw the uniform asymptotic reduction of the sdp integrals performed by the van der waerden 1951 procedure and yielding a variety of edge diffracted fields completes the formal treatment

the effect on bending waves by defects in pinned elastic plates abstract this paper presents solutions to a number of problems posed for the out of plane displacement of infinite thin elastic plates that are rigidly pinned in periodic configurations but that possess a finite number of defects we begin by considering a single one dimensional periodic array of pins we derive an analytic solution for the displacement produced by the forced oscillation of the central pin in the array and this solution is shown to be closely connected to the problem of scattering of plane waves by an array when a finite number of pins are removed attention then focuses on doubly periodic rectangular arrays of pinned points possessing defects central to approaching such problems is an understanding of bloch floquet waves in periodic arrays in the absence of defects and a simple method is described for computing the associated dispersion surfaces the solutions to three problems are then sought the trapping of localised waves by a finite number of missing pins trapping of waves by entire rows of missing pins and the wave radiation pattern due to the forcing of a single pin all problems are treated analytically using bounded green s functions for thin elastic plates a discrete fourier transform solution method and simple explicit and rapidly convergent evaluations of the one and two dimensional lattice sums that arise

periodic optimal control for competing parabolic volterra lotka type systems in this paper we study the optimal harvesting control problem governed by a time periodic competing parabolic volterra lotka system we show the existence of an optimal control and we also find some conditions which enable the characterization of the optimal control in terms of a large parabolic optimality system we further construct monotone sequences closing in to all appropriate solutions of the periodic optimality system

optimal control strategies for a new ecosystem governed by reaction diffusion equations abstract an optimal control problem for a new ecosystem described by reaction diffusion equations is studied in this contribution different from the classical prey predator law the phenomenon that the growth of predator population gradually reduces due to the after effects of prey population is ubiquitous in the real world such as allelopathy phenomenon in planktonic ecosystems to model this relationship between predator and prey populations together with diffusion behaviors a novel model with general holling type functional response is presented after that the existence and uniqueness of the global positive strong solution to the ecosystem is obtained by the method of semigroups of operators on this basis an optimal control problem of the ecosystem is considered the existence of optimal control strategy is proved by using minimal sequence then the first order necessary optimality condition is obtained and the corresponding optimal control is found to be of on off form furthermore the second order necessary condition for optimal control and sufficient condition for local optimal control are all established finally a numerical example is offered to demonstrate the applications of the results obtained in this work

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

remarks on hamilton jacobi equations with measurable time dependent hamiltonians on considere les equations de hamilton jacobi de la forme suivante u t h x t u u 0 dans r n 0 t ou h x t s p est continu en x s p et mesurable en t on considere une extension de la notion de solution viscosite au cas de h mesurable dependant du temps

2 bilocal derivations abstract let x be a banach space and a be a unital subalgebra of b x containing all finite rank operators in b x a map a b x is called a 2 bilocal derivation if for a b a x x there exists a derivation a b x such that a a b x a x and b x a b x b x in this paper we show that each 2 bilocal derivation on a is a derivation

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

a remark on a system of inequalities with bilateral obstacles up x k s up x i up l x k x e sz apup fp if up x k up x up x k x e q apup 5 fp if up x up l x k x e l2 1 1 apup l f if up x up x k x e q u x 0 onaq p 1 2 m where p ranges over 1 m and urn represents u is dealt with by the second author in 5 and the existence of a weak solution of 1 1 is established under the assumption ap s have sufficiently large zeroth order coefficients 1 2 the proof of the existence result in 5 is based on the iv estimate independent of e for the solution u 124 u of the following approximate system apu u p uf l k u k uf f in r 1 3 u 0 0naq p 1 2 m where p ranges over 1 ml p is the so called penalty function and uf represents u the gradient estimate for solutions u of 1 3 in 5 is obtained by using bernstein s method and assumption 1 2 is solely employed to make it possible in addition it should be remarked that standard variants of the bernstein trick like in 2 and 4 which are useful to obtain the iv estimate for solutions of the system of variational inequalities with unilateral constraints without 1 2 do not work in our case

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

representation of solutions of hamilton jacobi equations on etudie le probleme de cauchy pour l equation de hamilton jacobi u t h t x du 0 dans o t r n avec la condition initiale u o x u o x pour x r n on a t 0 hi o t r n r n r et u o r n r sont donnes on represente la solution viscosite comme la fonction valeur d un jeu differentiel approprie

a regularity result for viscosity solutions of hamilton jacobi equations in one space dimension on demontre que dans le cas particulier d un probleme stationnaire a une dimension sous des hypotheses tres generales les solutions viscosite continues de lipschitz ont des derivees a droite et a gauche en chaque point

edge effects in axial arrays of circular rings a boundary value problem is solved for the current induced by any source in particular for an incident plane wave on any number of identical rings that are arranged to form an axisymmetric linear array the net aperture distribution along the finite array is approximated by a combination of three physically identifiable components a term from the related infinite array plus contributions from two independent edge effect waves that originate from each of the two ends however if individually resonant loops are closer than a critical spacing the entire structure exhibits a wild resonance and the simple approximation becomes invalid

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

analysis of finite arrays a new approach a novel method for the analysis of finite arrays is presented the method is based on a global array concept where the array problem for single mode elements is reduced to a solution of a single fredholm integral equation of the second kind this formulation offers several types of solutions not all explored yet with illuminating results the approximate solution of this integral equation for example yields finite array characteristics in terms of equivalent infinite array scattering parameters and mutual admittances the method is general i e applicable to any element type and periodic array geometry presently the method applies to single mode elements one unknown per element however it can be extended to a multimode analysis

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays i basic principles and 2 d cases this two part sequence deals with the formulation of an efficient method for the full wave analysis of large phased array antennas this is based on the method of moments mom solution of a fringe integral equation ie in which the unknown function is the difference between the exact solution of the finite array and that of the associated infinite array the unknown currents can be interpreted as produced by the field diffracted at the array edge which is excited by the floquet waves fws pertinent to the infinite configuration following this physical interpretation the unknown in the ie is efficiently represented by a very small number of basis functions with domain on the entire array aperture in order to illustrate the basic concepts the first part of this sequence deals with the two dimensional example of a linearly phased slit array it is shown that the dominant phenomenon fur describing the current perturbation with respect to the infinite array is accurately represented in most cases by only three diffracted ray shaped unknown functions this also permits a simple interpretation of the element by element current oscillation which was described by other authors

weak 2 local symmetric maps on c algebras abstract we introduce and study weak 2 local symmetric maps between c algebras a and b as not necessarily linear nor continuous maps a b such that for each a b a and b there exists a symmetric linear map t a b a b depending on a b and satisfying a t a b a and b t a b b we prove that every weak 2 local symmetric map between c algebras is a linear map among the consequences we show that every weak 2 local derivation on a general c algebra is a linear derivation we also establish a 2 local version of the kowalski slodkowski theorem for general c algebras by proving that every 2 local homomorphism between c algebras is a linear homomorphism

local derivations on finite dimensional lie algebras abstract we prove that every local derivation on a finite dimensional semisimple lie algebra over an algebraically closed field of characteristic zero is a derivation we also give examples of finite dimensional nilpotent lie algebras l with dim l 3 which admit local derivations which are not derivations

an asymptotic formula for solutions of hamilton jacobi bellman equations on demontre que la solution de l equation de hamilton jacobi bellman converge exponentiellement vers 0 quand e 0 avec un taux de decroissance i x 0 1 e et on donne une formule de representation pour i x

2 local derivations on associative and jordan matrix rings over commutative rings abstract in the present paper we prove that every 2 local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring we also develop a jordan analog of the above method and prove that every 2 local inner derivation on the jordan matrix ring over a commutative ring is a derivation

on acoustic and electric faraday cages two dimensional problems involving many identical small circles are considered the circles are the cross sections of parallel wires modelling a cage or a grating both electrostatic and acoustic fields are considered the main emphasis is on periodic configurations of n circles distributed evenly around a large circle a ring foldy s theory is used for acoustic problems and then adapted for electrostatic problems in both cases circulant matrices are encountered the fields can be calculated explicitly then the limit n is studied a connection between the n circle problem and the limiting problem fields exterior to the ring is established using known results on the convergence of a defective form of the trapezoidal rule defective in that endpoint contributions are ignored because the integrand has logarithmic singularities at those points this shows that the solution of the limiting problem is approached very slowly as n 1 log n as n

on existence and uniqueness of solutions of hamilton jacobi equations abstract the theory of scalar first order nonlinear partial differential equations has been enjoying a rapid development in the last few years this development occurred because the authors established uniqueness criteria for generalized solutions called viscosity solutions which correctly identify the solutions sought in areas of application including control theory differential games and the calculus of variations the concept of viscosity solutions is relatively easy to work with and many formally heuristic or difficult proofs have been made rigorous or simple using this concept a feedback process has begun and the experience recently gained in working with viscosity solution has suggested new existence and uniqueness results the current paper continues this interaction by establishing new existence ane uniqueness results in a natural generality suggested by earlier proofs it is also felt that the presentation of the comparison results which imply uniqueness continuous dependence and are used to estimate moduli of continuity has something to offer over earlier presentations in special cases author

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

the pontryagin maximum principle from dynamic programming and viscosity solutions to first order partial differential equations on demontre le principe de maximum de pontryagin pour le probleme de lagrange de commande optimale en utilisant le fait que la fonction valeur du probleme est la solution viscosite de l equation de hamilton jacobi bellman associee

perturbed dynamical systems with an attracting singularity and weak viscosity limits in hamilton jacobi equations nous donnons une nouvelle preuve par des methodes edp du theoreme de wentzell freidlin concernant les petites perturbations d un systeme dynamique l e u e e 2 u e b u e 0 dans u e sur nous prouvons que si b a un seul point singulier attractif alors u e converge vers une constante uniformement sur tout compact et avec une vitesse exponentielle nous determinons nous traitons aussi le cas de conditions aux limites de neumann pour cela nous faisons l analyse asymptotique d une mesure ergodique intervenant naturellement dans le probleme ce qui revient a etudier la limite par viscosite evanescente dans une equation de hamilton jacobi

semi infinite arrays of isotropic point scatterers a unified approach we solve the two dimensional problem of acoustic scattering by a semi infinite pe riodic array of identical isotropic point scatterers i e objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions this model is appropriate for scatterers on which dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity the problem is also relevant to the scattering of an e polarized electromagnetic wave by an array of highly conducting wires the actual geometry of each scatterer is characterized by a single parameter in the equations related to the single body scattering problem and determined from a harmonic boundary value problem using a mixture of analytical and numerical techniques we confirm that a number of phenomena reported for specific geometries are in fact present in the general case such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions we show that the semi infinite array problem is equivalent to that of inverting an infinite toeplitz matrix which in turn can be formulated as a discrete wiener hopf problem numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes

wave diffraction by semi infinite venetian blind type grating the h polarized electromagnetic wave diffraction by semi infinite periodic venetian blind type grating is considered the grating consists of strips placed in the parallel planes the problem is reduced to a nonlinear operator equation with respect to the unknown reflection operator of a structure for its solution a complex approach which consists in using the successive over relaxation method and the gauss seidel method is suggested the near and far fields scattered by the grating are calculated the dependences of the reflection coefficient and diffraction patterns are analyzed for different incidence angles width of strips and distance between them