mean field renormalisation group transformations for geometric phase transitions it is shown that the mean field renormalisation group technique of indekeu et al 1982 may be applied directly to the self avoiding walk saw and percolation problems without reference to the n to 0 limit of interacting spin systems numerical results for the saw and bond and site percolation both directed and undirected problems on the square lattice obtained by using a variety of small cells are reported an isotropic transformation for the percolation problems on the d dimensional hypercubic lattice is also discussed

renormalisation group recursions by mean field approximations within a renormalisation group strategy different rescaled mean field approximations for the magnetisation are combined good qualitative estimates are obtained for the critical properties of classical and quantum spin systems a particularly simple realisation of the method yields recursions for arbitrary dimensionality d the critical couplings are accurate for all d and have the correct asymptotic behaviour for d approaching infinity

surrogate programming and multipliers in quasi convex programming a result due to luenberger on the existence of multipliers in a quasi convex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than slater s condition the existence of solutions is not assumed we point out links with even convexity in the sense of fenchel and quasi subdifferentiability in the sense of greenberg pierskalla and we observe that the couples of primal dual optimal solutions reduce to saddle points of a suitable lagrangian function

smoothing effect for some schrodinger equations abstract we study the cauchy problem for the following schrodinger equation in r n n n i t u 1 2 u v 1 u v 2 u 2 u t x r r n u 0 x r n where v 1 v 1 x 1 x 1 v 2 v 2 x k 2 3 k x k k r 1 k 3 0 1 min 2 n 2 0 2 3 min 2 n we prove the existence uniqueness and smoothing effect of global solutions of with not necessarily in the sobolev space h1 rn

duality for e variational inequalities via the subdifferential calculus abstract based on the properties of the convex e subdifferential calculus we introduce to a general e variational inequality formulated with the help of a set valued operator and a perturbation function a dual one expressed by making use of the fenchel conjugate of the perturbation function under convexity hypotheses we show that the fulfillment of a regularity condition guarantees that the primal e variational inequality is solvable if and only if its dual one is solvable by particularizing the perturbation function we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by kum kim and lee an example justifying this generalization is also provided among the special instances of the general result we rediscover also the duality scheme concerning variational inequalities due to mosco

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

asymptotic closure condition and fenchel duality for dc optimization problems in locally convex spaces a b s t r a c t we consider the dc optimization problem p inf x f1 x f2 x g1 ax g2 ax where f1 f2 g1 and g2 are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear continuous operator from x to y adopting the standard convexification technique a fenchel dual problem of p is given by using properties of the epigraph of conjugate functions some sufficient and necessary conditions for the fenchel duality and for the stable fenchel duality of p are provided

optimality conditions for henig and globally proper efficient solutions with ordering cone has empty interior in this paper we consider the set valued vector optimization problems with constraint in locally convex spaces we present the necessary and sufficient conditions for henig efficient solution pair globally proper efficient solution pair and super efficient solution pair without the ordering cones having the nonempty interior

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

nonexistence of asymptotically free solutions for a nonlinear schrodinger equation let u be a nontrivial smooth solution to iut u u p 1u if n 1 and 2 p 3 then there does not exist any finite energy free solution v such that u t v t 2 0 as t this extends a theorem of strauss in which the same result was proved for 1 p 2

regularity conditions via quasi relative interior in convex programming we give some new regularity conditions for fenchel duality in separated locally convex vector spaces written in terms of the notion of quasi interior and quasi relative interior respectively we provide also an example of a convex optimization problem for which the classical generalized interior point conditions given so far in the literature cannot be applied while the one given by us is applicable by using a technique developed by magnanti we derive some duality results for the optimization problem with cone constraints and its lagrange dual problem and we show that a duality result recently given in the literature for this pair of problems has self contradictory assumptions

modified scattering for the critical nonlinear schrodinger equation abstract we consider the nonlinear schrodinger equation i u t u u 2 n u in all dimensions n 1 where c and 0 we construct a class of initial values for which the corresponding solution is global and decays as t like t n 2 if 0 and like t log t n 2 if 0 moreover we give an asymptotic expansion of those solutions as t we construct solutions that do not vanish so as to avoid any issue related to the lack of regularity of the nonlinearity at u 0 to study the asymptotic behavior we apply the pseudo conformal transformation and estimate the solutions by allowing a certain growth of the sobolev norms which depends on the order of regularity through a cascade of exponents

bounded linear regularity of convex sets in banach spaces and its applications this paper deals with bounded linear regularity linear regularity and the strong conical hull intersection property chip of a collection of finitely many closed convex intersecting sets in banach spaces it is shown that as in finite dimensional space setting see 6 the standard constraint qualification implies bounded linear regularity which in turn yields the strong conical hull intersection property and that the collection of closed convex sets c1 cn is bounded linearly regular if and only if the tangent cones of c1 cn has the chip and the normal cones of c1 cn has the property g uniformly on a neighborhood in the intersection c as applications we study the global error bounds for systems of linear and convex inequalities

a comparison of constraint qualifications in infinite dimensional convex programming in this paper the relationships between various constraint qualifications for infinite dimensional convex programs are investigated using robinson s refinement of the duality result of rockafellar it is demonstrated that the constraint qualification proposed by rockafellar provides a systematic mechanism for comparing many constraint qualifications as well as establishing new results in different topological environments

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

a new mean field like renormalization group transformation a new real space renormalization group transformation combining ideas from mean field and finite size scaling theories is presented application to the two dimensional site directed percolation problem gives better values for the percolation threshold and critical exponent of longitudinal correlation length than those obtained previously with other real space renormalization group approaches when applied to one two and three dimensional ising models the results are comparable to the ones from previous mean field like renormalization group transformations this method can easily be applied to other systems having second order phase transitions

modified scattering operator for the hartree fock equation abstract we study the scattering problem for the hartree fock equation hrf i t u 1 2 u f u t x r r n n 2 where u u 1 u n t is a c n n 2 valued unknown function and f u f 1 u f n u t denotes a nonlinear term whose j th element is defined by f j u r n v x y k 1 n u k y 2 u j x u j y u k y u k x d y where v x x 1 r is called a coulomb potential we show that if 1 2 then the modified scattering operator for the system hrf is well defined from a neighborhood at the origin in the space h 0 to a neighborhood at the origin in the space h 0 where h 0 k l 2 1 x 2 k 2 l 2

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

fenchel duality in infinite dimensional setting and its applications abstract we study fenchel duality problems in infinite dimensional spaces that involve the minimizing of a sum of two proper convex functions where one of which is polyhedral we use a constraint qualification with the notion of the strong quasi interior of a convex set and then deduce duality results and subgradient formula as applications we discuss the strong conical hull intersection property of convex sets finally by using a duality result due to rodriques and simons we establish several duality results for convex optimization over a finite intersection of closed convex sets

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

generic convergence of descent methods in banach spaces given a continuous convex functionf on a banach spacex we consider a complete metric space of vector fieldsv onx with the topology of uniform convergence on bounded subsets with each such vector field we associate two iterative processes we show that for a genericv the values of the functionf tend to its infimum for both processes

a system of quadratic nonlinear klein gordon equations in 2d abstract we consider the system of quadratic nonlinear klein gordon equations t 2 v 1 v 1 m 1 2 v 1 v 1 v 2 t 2 v 2 v 2 m 2 2 v 2 v 1 2 in two space dimensions under the mass condition such that 2 m 1 m 2 we construct the scattering operator in an almost natural weighted sobolev class h 1 1 or the scattering problem in lower order sobolev class h 4 3 1 h 1 where 0 is small

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

on the necessity of the moreau rockafellar robinson qualification condition in banach spaces as well known the moreau rockafellar robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended real valued convex lower semi continuous functions defined on a locally convex space is exact and that the subdifferential of the sum of these functions is the sum of their subdifferentials this note is devoted to proving that this condition is in a certain sense also necessary provided the underlying space is a banach space our result is based upon the existence of a non supporting weak closed hyperplane to any weak closed and convex unbounded linearly bounded subset of the topological dual of a banach space

quasi relative interior type constraint qualifications ensuring strong lagrange duality for optimization problems with cone and affine constraints in this article we provide weak sufficient strong duality conditions for a convex optimization problem with cone and affine constraints stated in infinite dimensional spaces and its lagrange dual problem our results are given by using the notions of quasi relative interior and quasi interior for convex sets the main strong duality theorem is accompanied by several stronger yet easier to verify in practice versions of it as exemplification we treat a problem which is inspired from network equilibrium our results come as corrections and improvements to daniele and giuffre 2007 9

a new geometric condition for fenchel s duality in infinite dimensional spaces in 1951 fenchel discovered a special duality which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions using conjugates fenchel s duality is central to the study of constrained optimization it requires an existence of an interior point of a convex set which often has empty interior in optimization applications the well known relaxations of this requirement in the literature are again weaker forms of the interior point condition avoiding an interior point condition in duality has so far been a difficult problem however a non interior point type condition is essential for the application of fenchel s duality to optimization in this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions we also establish a necessary and sufficient condition for the e subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions our results offer further insight into fenchel s duality

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

an alternative formulation for a new closed cone constraint qualification abstract we give an alternative formulation for the so called closed cone constraint qualification cccq related to a convex optimization problem in banach spaces recently introduced in the literature this new formulation allows to prove in a simple way that cccq is weaker than some generalized interior point constraint qualifications given in the past by means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature

mean field renormalization group approach to two dimensional site directed percolation two dimensional site directed percolation is studied by the mean field renormalization group approach to bulk and surface critical properties extrapolation techniques for the exponents and the percolation threshold are described a very good estimate of the percolation threshold pc 0 7055 or 0 0001 and the first evaluation of the surface exponent yhs 0 653 or 1 are obtained

modulation spaces and non linear hartree type equations abstract we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces m p q r n we establish global well posedness results in m p p r n with 1 p 2 n n when k x x r 0 m i n 2 n 2 in m p q r n with 1 q p 2 when k m 1 r n

on subdifferential calculus for set valued mappings and optimality conditions abstract the aim of this paper is to establish formulas for the subdifferentials of the sum and the composition of convex set valued mappings under the attouch brezis qualification condition an application to a general set valued optimization problem is considered

modified scattering and beating effect for coupled schrodinger systems on product spaces with small initial data in this paper we study a coupled nonlinear schrodinger system with small initial data in a product space we establish a modified scattering of the solutions of this system and we construct a modified wave operator the study of the resonant system which provides the asymptotic dynamics allows us to highlight a control of the sobolev norms and interesting dynamics with the beating effect the proof uses a recent work of hani pausader tzvetkov and visciglia for the modified scattering and a recent work of grebert paturel and thomann for the study of the resonant system