the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

sweeping process with unbounded nonconvex perturbation abstract a convex sweeping process with unbounded nonconvex perturbation term of a rather general form is investigated in a separable hilbert space we consider our sweeping process as an evolution inclusion with subdifferential operators this allows to prove the theorem on existence of solutions under conditions generalizing the usual assumptions for convex sweeping processes in particular we show that multivalued mappings whose values are hyperplanes half spaces etc may be chosen as multivalued mappings generating the sweeping process such mappings do not satisfy the conditions under which convex sweeping processes are usually studied

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

coherent structures in strongly interacting many body systems ii classical solutions and quantum fluctuations for pt i see ibid vol 22 p 4877 1989 previously i the author considered a second quantised hamiltonian which can be used to model a large number of important strongly interacting many body systems he has shown that the dynamics of such systems can be described exactly using quantum fields the equations of motion derived in this way are highly nonlinear partial differential equations pde he now uses a standard field theoretical approach where initially the fields are treated as classical functions he studies the solutions of these equations which remarkably may be obtained making full use of very recent mathematical discoveries the symmetry reduction method he finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper these cases may correspond to different physical situations and the last the second order case corresponds to the most general physical situation subsequently he outlines possible quantisation procedures for these fields he then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation abstract an evolution inclusion is considered in a separable hilbert space the right hand side of the inclusion contains the subdifferential of a time dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed not necessarily bounded values along with this inclusion we consider the inclusion with the perturbation term being convexified we prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation in contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets an example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given

group invariant solutions of the 2 1 dimensional cubic schrodinger equation we use lie point symmetries of the 2 1 dimensional cubic schrodinger equation to obtain new analytic solutions in a systematic manner we present an analysis of the reduced odes and in particular show that although the original equation is not integrable they typically can belong to the class of painleve type equations

exact similarity solutions of ideal mhd equations for plane motions for the non linear two dimensional ideal mhd equations we use the known lie point symmetries and for the case of adiabatic exponent 2 we give local lie backlund symmetries we find exact solutions using similarity analysis among the calculated similarity solutions describing plasma expansion into vacuum shock waves and spiral flows with plasma sources

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

are all the equations of the kadomtsev petviashvili hierarchy integrable the kadomtsev petviashvili kp hierarchy is an infinite set of nonlinear partial differential equations in which the number of independent variables increases indefinitely as one proceeds down the hierarchy since these equations were obtained as part of a group theoretical approach to soliton equations it would appear that the kp hierarchy provides integrable scalar equations with an arbitrary number of independent variables it is shown by investigating a specific equation in 3 1 dimensions that the higher equations in the kp hierarchy are only integrable in a conditional sense the equation under study taken in isolation does not pass certain well known and reliable integrability tests thus applying painleve analysis we find that solutions exist allowing movable critical points furthermore solitary wave solutions are shown to exist that do not behave like solitons in multiple collisions on the other hand if the dependence of a solution on the first 2 1 variables is restricted by the fact that it should also satisfy the kp equation itself then the integrability conditions in the other dimensions are satisfied conditional integrability thus means that linear techniques will provide only those solutions of equations in the hierarchy that simultaneously satisfy lower equations in the same hierarchy

multi label moves for mrfs with truncated convex priors optimization with graph cuts became very popular in recent years while exact optimization is possible in a few cases many useful energy functions are np hard to optimize one approach to approximate optimization is the so called move making algorithms at each iteration a move making algorithm makes a proposal move for a pixel p to either keep its old label or switch to a new label two move making algorithms based on graph cuts are in wide use namely the swap and expansion both of these moves are binary in nature that is they give each pixel a choice of only two labels an evaluation of optimization techniques shows that the expansion and swap algorithms perform very well for energies where the underlying mrf has the potts prior however for more general priors the swap and expansion algorithms do not perform as well the main contribution of this paper is to develop multi label moves a multi label move unlike expansion and swap gives each pixel has a choice of more than two labels to switch to in particular we develop several multi label moves for truncated convex priors we evaluate our moves on image restoration inpainting and stereo correspondence we get better results than expansion and swap algorithms both in terms of the energy value and accuracy

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

group analysis of the three wave resonant system in 2 1 dimensions the three wave resonant interaction equations 2d 3wr in two spatial and one temporal dimension within a group framework are analyzed the symmetry algebra of this system which turns out to be an infinite dimensional lie algebra whose subalgebra is of the kac moody type is found the one and two dimensional symmetry subalgebras are classified and the corresponding reduction equations are obtained from these the new invariant and the partially invariant solutions of the original 2d 3wr equations are obtained

optimal control of the sweeping process over polyhedral controlled sets abstract the paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given bolza type functional which depends on control and state variables as well as their velocities besides the highly non lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types all of this creates serious challenges for deriving necessary optimality conditions we develop here the method of discrete approximations and combine it with advanced tools of first order and second order variational analysis and generalized differentiation this approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions as a by product of the developed approach we prove the strong w 1 2 convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous time system and derive necessary optimality conditions for the discrete counterparts the established necessary optimality conditions for the sweeping process are illustrated by several examples

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

group invariant solutions of semilinear schrodinger equations in multi dimensions symmetry group methods are applied to obtain all explicit group invariant radial solutions to a class of semilinear schrodinger equations in dimensions n 1 both focusing and defocusing cases of a power nonlinearity are considered including the special case of the pseudo conformal power p 4 n relevant for critical dynamics the methods involve first reduction of the schrodinger equations to group invariant semilinear complex 2nd order ordinary differential equations odes with respect to an optimal set of one dimensional point symmetry groups and second use of inherited symmetries hidden symmetries and conditional symmetries to solve each ode by quadratures through noether s theorem all conservation laws arising from these point symmetry groups are listed some group invariant solutions are found to exist for values of n other than just positive integers and in such cases an alternative two dimensional form of the schrodinger equations involving an extra modulation term with a parameter m

partially invariant solutions of nonlinear klein gordon and laplace equations the concept of partially invariant solutions is discussed in the framework of the group analysis of systems of partial differential equations using a general and systematic approach based on subgroup classification methods nontrivial partially invariant solutions for specific equations are explicitly obtained belonging to the nlkge and nlle classes in two dimensions it is proven that the obtained partially invariant solutions are distinct from the invariant ones and from solutions obtained by other methods of reducing pdes to odes

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

classes of hypercomplex polynomials of discrete variable based on the quasi monomiality principle with the aim of derive a quasi monomiality formulation in the context of discrete hypercomplex variables one will amalgamate through a clifford algebraic structure of signature 0 n the umbral calculus framework with lie algebraic symmetries the exponential generating function egf carrying the continuum dirac operator d j 1 n e j x j together with the lie algebraic representation of raising and lowering operators acting on the lattice h z n is used to derive the corresponding hypercomplex polynomials of discrete variable as appell sets with membership on the space clifford vector valued polynomials some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the poisson charlier polynomials are introduced certain applications from the view of interpolation theory and integral transforms are also discussed