statistical physics of the freely jointed chain using the method of constraints proposed by s f edwards and a g goodyear j phys a 5 965 1972 5 1188 1972 we do a complete calculation of the canonical partition function of a freely jointed chain fjc from its classical hamiltonian we show how the constraints reduce the phase space of an ideal gas of monomers to the phase space of a fjc and how they permit one to find the canonical partition function by using this function it is possible to study thermodynamical properties of fjc s and to build other thermodynamical ensembles via laplace transforms thus we define a grand canonical ensemble where the monomer number of the fjc can fluctuate in this ensemble the fjc of infinite length is the asymptotic state at low and high temperatures the critical exponents ensuremath gamma and ensuremath nu for fjc s are calculated and found to be equal to the gaussian polymer exponents connections between the properties of fjc s and random walks on regular lattices are also discussed textcopyright 1996 the american physical society

inertia tensor and size of a polymer chain guided by an approach used in the study of membranes we construct the partition function for a discrete gaussian chain model of polymers the hamiltonian is where describes the chain connectivity is the inertia tensor which gives the shape and size of the polymer chain and tr denotes the trace here we consider only the size which is obtained from the partition function by differentiating with respect to the conjugate variable the partition function is expressed in terms of the hypergeometric function an orthogonal polynomial where n is the number of monomers and is the ratio of and the coefficient in the limit corresponds to the random coil while the limit describes a compact object we also study the excluded volume problem by discretizing edwards continuous self avoidance term we obtain dimensionally regularized expressions for the radius of gyration and the end to end distance the short distance cut off dependence of the continuous model is reproduced

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

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

asymptotic bifurcation results for coupled nonlinear wave equations with variable coefficients abstract this paper is concerned with the bifurcation phenomenon of large amplitude periodic solutions of coupled nonlinear wave equations with either constant or variable coefficients such model arises when two waves propagate simultaneously in nonisotropic media the change of degree argument is used to obtain necessary and sufficient conditions for the existence of asymptotic bifurcation points with respect to the matrix spectrum the results we get are also valid for uncoupled systems by changing one of our assumptions on the nonlinear term we also obtain the necessary and sufficient conditions for the existence of the standard bifurcation points

generalizations of slater s constraint qualification for infinite convex programs in this paper we study constraint qualifications and duality results for infinite convex programs p inf f x g x s x c whereg g1 g2 ands s1 s2 si are convex cones i 1 2 c is a convex subset of a vector spacex andf andgi are respectively convex andsi convex i 1 2 in particular we consider the special case whens2 is in afinite dimensional space g2 is affine ands2 is polyhedral we show that a recently introduced simple constraint qualification and the so called quasi relative interior constraint qualification both extend to p from the special case thatg g2 is affine ands s2 is polyhedral in a finite dimensional space the so called partially finite program this provides generalized slater type conditions for p which are much weaker than the standard slater condition we exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong2 and the finite dimensionality assumption ons2 by a local compactness assumption we then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

necessary and sufficient conditions for stable conjugate duality abstract the conjugate duality which states that inf x x x 0 max v y 0 v whenever a regularity condition on is satisfied is a key result in convex analysis and optimization where x y r is a convex function x and y are banach spaces y is the continuous dual space of y and is the fenchel moreau conjugate of in this paper we establish a necessary and sufficient condition for the stable conjugate duality inf x x x 0 x x max v y x v x x and then obtain a new epigraph regularity condition for the conjugate duality the regularity condition is shown to be much more general than the popularly known interior point type conditions as an easy consequence we present an epigraph closure condition which is necessary and sufficient for a stable fenchel rockafellar duality theorem in the case where one of the functions involved is a polyhedral convex function we provide generalized interior point conditions for the epigraph regularity condition moreover we show that a stable fenchel s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

robust conjugate duality for convex optimization under uncertainty with application to data classification in this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization extending the powerful conjugate duality technique we first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition this regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem more importantly we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem as an application we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets the support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences

study of the shape of random walks a comprehensive study of the shape of discrete random walks considering the general case of arbitrary length and any space dimension is presented the probability distributions for several magnitudes such as the principal inertia moments the asphericity and the angle between the principal axis of inertia and the end to end vector are evaluated numerically in most cases and especially in low dimensional spaces these probability distributions spread widely and possess a non zero skewness this implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete this situation does nor vary when random walks of large lengths are considered since the relative fluctuations ratio between the standard deviation and the mean value do not go to zero in that limit on the other hand when the space dimension grows the distributions become more peaked reaching the expected asymptotic limit for infinite dimension the present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi squared distributions analysing the probability distribution for the asphericity it is shown that the distributions for the different moments are not completely independent

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

best approximation from the intersection of a closed convex set and a polyhedron in hilbert space weak slater conditions and the strong conical hull intersection property let x be a real hilbert space c be a closed convex subset and h i x in x mid la x h i ra le b i quad i 1 2 m be a finite collection of half spaces under the assumption that k c cap cap 1 mh i is not empty the problem of characterizing the best approximation from k to any x in x is considered the strong conical hull intersection property strong chip which was introduced by us in 1997 is shown to be both necessary and sufficient for the following perturbation property to hold for each x in x an element x 0 in k satisfies x 0 p k x if and only if x 0 p c x sum 1 m lambda ih i for some scalars lambda i ge 0 with lambda i la x 0 h i ra b i 0 for each i here p d z denotes the unique best approximation from d to z in other words determining the best approximation from the set k to any point is equivalent to the generally easier problem of determining the best approximation from the set c to a perturbation of that point moreover even when the strong chip does not hold the perturbation property still holds except now c must be replaced by a certain convex extremal subset of c we also show that the strong chip is weaker than any of the weak slater conditions that one can naturally impose on the sets in question these results generalize the main results of our 1997 paper j approx theory 90 pp 385 444 and hence those of several other papers as well

on extension of fenchel duality and its application by considering the epigraphs of conjugate functions we extend the fenchel duality applicable to a possibly infinite family of proper lower semicontinuous convex functions on a banach space applications are given in providing fuzzy kkt conditions for semi infinite programming

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

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

global results concerning bifurcation for fredholm maps of index zero with a transversality condition on considere l equation f x ax bx r x 0 ou a b est la derivee de frechet de f a l origine dans x et r un terme d ordre eleve x est un espace de banach et f x r y est un operateur non lineaire continu et y un espace de banach on etudie comment l ensemble solution varie avec le parametre reel

subdifferential calculus using ϵ subdifferentials in applications of convex analysis it is important to be able to calculate the subdifferentials of various combinations of proper and lower semicontinuous convex functions such as the sum of two such functions or their inf convolution epi sum as well as the pre composition of a convex function with an affine map or the marginal function obtained from a convex function and a linear map the classical formulas for such calculations all require additional hypotheses same of which may be difficult to check or are not always satisfied in a given variational problem in this paper we present formulas for the subdifferentials of such combinations without assuming any additional hypotheses by utilizing subdifferentials this wider applicability comes at the price of somewhat more complicated formulas

universal shape ratios for open and closed random walks exact results for all d the mean asphericities ad of open and closed gaussian polymer chains are computed exactly and analytically up to simple quadratures for arbitrary space dimensions d excellent agreement is found with existing simulation data for d 2 3 and 4 the authors technique can also be used to compute other averages of ratios of fluctuating variables as well as extended to include the effects of self avoiding walk interactions

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

duality and infinite dimensional optimization the purpose is to establish various duality theorems including a generalization of the fenchel duality theorem of attouch and brezis to convex processes under relaxed interiority conditions and show the usefulness of the attouch and brezis type conditions in optimization the significance of the interiority conditions is that they provide easily verifiable conditions for a large class of infinite dimensional convex programs whereas the standard slater constraint qualification fails the results are applied to obtain duality theorems for general systems of inequalities and minimax results

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

periodic solutions to nonlinear one dimensional wave equation with dependent coefficients this paper deals with t periodicity and regularity of solutions to the one dimensional nonlinear wave equation with x dependent coefficients

strong duality for generalized monotropic programming in infinite dimensions abstract we establish duality results for the generalized monotropic programming problem in separated locally convex spaces we formulate the generalized monotropic programming gmp as the minimization of a possibly infinite sum of separable proper convex functions restricted to a closed and convex cone we obtain strong duality under a constraint qualification based on the closedness of the sum of the epigraphs of the conjugates of the convex functions when the objective function is the sum of finitely many proper closed convex functions we consider two types of constraint qualifications both of which extend those introduced in the literature the first constraint qualification ensures strong duality and is equivalent to the one introduced by bo and wanka the second constraint qualification is an extension of bertsekas constraint qualification and we use it to prove zero duality gap