modica type gradient estimates for an inhomogeneous variant of the normalized p laplacian evolution abstract in this paper we study an inhomogeneous variant of the normalized p laplacian evolution which has been recently treated in banerjee and garofalo 2013 does 2011 manfredi et al 2010 and juutinen 2014 we show that if the initial datum satisfies the pointwise gradient estimate 1 6 a e then the unique solution to the cauchy problem 1 2 satisfies the same gradient estimate a e for all later times see 1 7 a general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of 1 2 was first obtained in caffarelli et al 1994 such estimate generalizes one obtained by l modica for the laplacian and it has connections to a famous conjecture of de giorgi

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

a fast approximation of the bilateral filter using a signal processing approach the bilateral filter is a nonlinear filter that smoothes a signal while preserving strong edges it has demonstrated great effectiveness for a variety of problems in computer vision and computer graphics and fast versions have been proposed unfortunately little is known about the accuracy of such accelerations in this paper we propose a new signal processing analysis of the bilateral filter which complements the recent studies that analyzed it as a pde or as a robust statistical estimator the key to our analysis is to express the filter in a higher dimensional space where the signal intensity is added to the original domain dimensions importantly this signal processing perspective allows us to develop a novel bilateral filtering acceleration using downsampling in space and intensity this affords a principled expression of accuracy in terms of bandwidth and sampling the bilateral filter can be expressed as linear convolutions in this augmented space followed by two simple nonlinearities this allows us to derive criteria for downsampling the key operations and achieving important acceleration of the bilateral filter we show that for the same running time our method is more accurate than previous acceleration techniques typically we are able to process a 2 megapixel image using our acceleration technique in less than a second and have the result be visually similar to the exact computation that takes several tens of minutes the acceleration is most effective with large spatial kernels furthermore this approach extends naturally to color images and cross bilateral filtering

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

transfer function approach to quantum control part i dynamics of quantum feedback systems this paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators a quantum optical closed loop including a plant and controller is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view in order to facilitate the use of control theory in microscopic world described by quantum theory in particular an input output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed this would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics this paper is divided into two parts the first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device cavity that can be regarded as a unit of quantum dynamics of bosonic modes the formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations the input output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems in the follow up paper the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory uncertainty principle e g and some applications are presented

nonlinear evolution equations that are non local in space and time abstract we deal with a nonlocal nonlinear evolution problem of the form r n r j x y t s v y s v x t p 2 v y s v x t d y d s 0 for x t r n 0 here p 2 j r n 1 r is a nonnegative kernel compactly supported inside the set x t r n 1 t 0 with r n r j x t d x d t 1 and v stands for an extension of a given initial value f that is v x t v x t t 0 f x t t 0 for this problem we prove existence and uniqueness of a solution in addition we show that the solutions approximate viscosity solutions to the local nonlinear pde u p 2 u t p u when the kernel is rescaled in a suitable way

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

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 flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

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

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

coherent quantum lqg control based on a recently developed notion of physical realizability for quantum linear stochastic systems we formulate a quantum lqg optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum such a control scheme is often referred to in the quantum control literature as coherent feedback control it distinguishes the present work from previous works on the quantum lqg problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom the difference in our formulation is the presence of additional non linear and linear constraints on the coefficients of the sought after controller rendering the problem as a type of constrained controller design problem due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard lqg problem we propose a numerical procedure for solving this problem based on an alternating projections algorithm and as an initial demonstration of the feasibility of this approach we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained for comparison we also consider the case of classical linear controllers that use direct or indirect measurements and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones

linear quantum systems with diagonal passive hamiltonian and a single dissipative channel abstract given any covariance matrix corresponding to a so called pure gaussian state a linear quantum system can be designed to achieve the assigned covariance matrix in most cases however one might obtain a system that is difficult to realize in practice in this paper we restrict our attention to a special class of linear quantum systems i e systems with diagonal passive hamiltonian and a single dissipative channel the practical implementation of such a system would be relatively simple we then parametrize the class of pure gaussian state covariance matrices that can be achieved by this particular type of linear quantum system

cascade and locally dissipative realizations of linear quantum systems for pure gaussian state covariance assignment this paper presents two realizations of linear quantum systems for covariance assignment corresponding to pure gaussian states the first one is called a cascade realization given any covariance matrix corresponding to a pure gaussian state we can construct a cascaded quantum system generating that state the second one is called a locally dissipative realization given a covariance matrix corresponding to a pure gaussian state if it satisfies certain conditions we can construct a linear quantum system that has only local interactions with its environment and achieves the assigned covariance matrix both realizations are illustrated by examples from quantum optics

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

constant time median and bilateral filtering this paper formulates both the median filter and bilateral filter as a cost volume aggregation problem whose computational complexity is independent of the filter kernel size unlike most of the previous works the proposed framework results in a general bilateral filter that can have arbitrary spatial 1 1 and arbitrary range filter kernels this bilateral filter takes about 3 5 s to exactly filter a one megapixel 8 bit grayscale image on a 3 2 ghz intel core i7 cpu in practice the intensity range and spatial domain can be downsampled to improve the efficiency this compression can maintain very high accuracy e g 40 db but over 100 times 100 faster

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

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

multiscale image segmentation by integrated edge and region detection this paper is concerned with the detection of low level structure in images it describes an algorithm for image segmentation at multiple scales the detected regions are homogeneous and surrounded by closed edge contours previous approaches to multiscale segmentation represent an image at different scales using a scale space however structure is only represented implicitly in this representation structures at coarser scales are inherently smoothed and the problem of structure extraction is unaddressed this paper argues that the issues of scale selection and structure detection cannot be treated separately a new concept of scale is presented that represents image structures at different scales and not the image itself this scale is integrated into a nonlinear transform which makes structure explicit in the transformed domain structures that are stable locally invariant to changes in scale are identified as being perceptually relevant the transform can be viewed as collecting spatially distributed evidence for edges and regions and making it available at contour locations thereby facilitating integrated detection of edges and regions without restrictive models of geometry or homogeneity in this sense it performs gestalt analysis all scale parameters of the transform are automatically determined and the structure of any arbitrary geometry can be identified without any smoothing even at coarse scales

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

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

a pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature abstract we prove a pointwise gradient bound for bounded solutions of u f u 0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature we also obtain some rigidity results when equality in the bound holds at some point

median filtering in constant time the median filter is one of the basic building blocks in many image processing situations however its use has long been hampered by its algorithmic complexity o tau of in the kernel radius with the trend toward larger images and proportionally larger filter kernels the need for a more efficient median filtering algorithm becomes pressing in this correspondence a new simple yet much faster algorithm exhibiting o 1 runtime complexity is described and analyzed it is compared and benchmarked against previous algorithms extensions to higher dimensional or higher precision data and an approximation to a circular kernel are presented as well

tug of war games and pdes in this talk we review some recent results concerning tug of war games and their relation to some well known pdes see 1 2 4 3 in particular we will show that solutions to certain pdes like the p laplacian and the parabolic p laplacian can be obtained as limits of values of tug of war games when the parameter that controls the length of the possible movements goes to zero since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions seccion en el cedya 2011 edp

on the evolution governed by the infinity laplacian we investigate the basic properties of the degenerate and singular evolution equation ut d 2 u du du du du which is a parabolic version of the increasingly popular infinity laplace equa tion we prove existence and uniqueness results for both dirichlet and cauchy problems establish interior and boundary lipschitz estimates and a harnack inequality and also provide numerous explicit solutions

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

low frequency approximation for a class of linear quantum systems using cascade cavity realization abstract this paper presents a method for approximating a class of complex transfer function matrices corresponding to physically realizable complex linear quantum systems the class of linear quantum systems under consideration includes interconnections of passive optical components such as cavities beam splitters phase shifters and interferometers this approximation method builds on a previous result for cascade realization and gives good approximations at low frequencies

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

the transfer function of generic linear quantum stochastic systems has a pure cascade realization this paper establishes that generic linear quantum stochastic systems have a pure cascade realization of their transfer function generalizing an earlier result established only for the special class of completely passive linear quantum stochastic systems in particular a cascade realization therefore exists for generic active linear quantum stochastic systems that require an external source of quanta to operate the results facilitate a simplified realization of generic linear quantum stochastic systems for applications such as coherent feedback control and optical filtering the key tools that are developed are algorithms for symplectic qr and schur decompositions it is shown that generic real square matrices of even dimension can be transformed into a lower 2 2 block triangular form by a symplectic similarity transformation the linear algebraic results herein may be of independent interest for applications beyond the problem of transfer function realization for quantum systems numerical examples are included to illustrate the main results in particular one example describes an equivalent realization of the transfer function of a nondegenerate parametric amplifier as the cascade interconnection of two degenerate parametric amplifiers with an additional outcoupling mirror

h infty control of linear quantum stochastic systems the purpose of this paper is to formulate and solve a h infin controller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology the paper includes results on the class of such systems for which the quantum commutation relations are preserved such a requirement must be satisfied in a physical quantum system a quantum version of standard classical dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived this enables a quantum version of the two riccati solution to the h infin control problem to be presented this result leads to controllers which may be realized using purely quantum purely classical or a mixture of quantum and classical elements this issue of physical realizability of the controller is examined in detail and necessary and sufficient conditions are given our results are constructive in the sense that we provide explicit formulas for the hamiltonian function and coupling operator corresponding to the controller

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

transfer function approach to quantum control part ii control concepts and applications based on the stochastic differential equation of quantum mechanical feedback obtained in the first part of this paper detailed control concepts and applications are discussed for quantum systems interacting with a noncommutative noise source a feedback system in our framework is purely nonclassical in the sense that feedback control is performed via local operation and quantum communication through a quantum channel the role of the controller is to alter the quantum dynamic characteristics of the plant through entanglement shared between the plant and controller by sending quantum states that is modulated by the hamiltonian on the controller the input output relation of quantum systems provides a natural extension of control theory to the quantum domain this enables one to present a control theoretical interpretation of some fundamental quantum mechanical notions such as the uncertainty principle but also to find applications of ideas and tools of control theory one of the most important applications is the production of squeezed states which has been an important issue of quantum theory in relation to quantum computation and quantum communication the method proposed here reduces the application to a conventional noise reduction problem with feedback the h sub spl infin control then leads to complete squeezing

a transform for multiscale image segmentation by integrated edge and region detection describes a transform to extract image regions at all geometric and photometric scales it is argued that linear approaches have the shortcoming that they require a priori models of region shape the proposed transform avoids this by letting the structure emerge bottom up from interactions among pixels the transform involves global computations on pairs of pixels followed by vector integration of the results an attraction force field is computed over the image in which pixels belonging to the same region are mutually attracted and the region is characterized by a convergent flow it is shown that the transform possesses properties that allow multiscale segmentation or extraction of original unblurred structure at all different geometric and photometric scales present this is in contrast with much previous work wherein multiscale structure is viewed as the smoothed structure in a multiscale signal decimation scale is an integral parameter of the force computation and the number and values of scale parameters associated with the image can be estimated automatically regions are detected at all a priori unknown scales resulting in automatic construction of a segmentation tree in which each pixel is annotated with descriptions of all the regions it belongs to transform properties are presented for piecewise constant images but hold for more general ones thus the proposed method is intended as a solution to the problem of multiscale integrated edge and region detection or low level image segmentation experimental results are given

an introduction to quantum filtering this paper provides an introduction to quantum filtering theory an introduction to quantum probability theory is given focusing on the spectral theorem and the conditional expectation as a least squares estimate and culminating in the construction of wiener and poisson processes on the fock space we describe the quantum ito calculus and its use in the modeling of physical systems we use both reference probability and innovations methods to obtain quantum filtering equations for system probe models from quantum optics

brief paper cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control this paper presents a realization algorithm for a class of complex transfer function matrices corresponding to physically realizable linear quantum systems the aim of the realization algorithm is to enable a coherent quantum feedback controller which has been synthesized using methods such as quantum h control or quantum lqg control to be constructed using optical components such as cavities and phase shifters the class of linear quantum systems under consideration are passive linear quantum systems which can be described purely in terms of annihilation operators the proposed algorithm enables a complex transfer function matrix to be realized as a pure cascade connection involving only cavities and phase shifters

on the quasi balanceable class of linear quantum stochastic systems abstract this paper is concerned with the characterization of the quasi balanceable class of linear quantum stochastic systems i e systems that can be approximated via the recently proposed quasi balanced truncation method it has previously been shown that the quasi balanceable class of systems includes the class of completely passive systems in this work we refine the previously established characterization of quasi balanceable systems and show that the class of quasi balanceable systems is strictly larger than the class of completely passive systems we derive a novel characterization of completely passive linear quantum stochastic systems solely in terms of the controllability gramian of such systems exploiting this result we prove that all linear quantum stochastic systems with a pure gaussian steady state active systems included are all quasi balanceable and establish a new complete parameterization for this important class of systems examples are provided to illustrate our results

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9

a gradient bound and a liouville theorem for nonlinear poisson equations soit f c 2 r une fonction non negative et u c 3 r n une solution dans tout r n de l equation u f u ou f f est la derivee 1ere de f si u est borne sur r n et s il existe x 0 r n tel que f u x 0 0 alors u est constante