gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

nonparametric inference under biased sampling from a finite population we consider a biased sampling model that has been found useful in incorporating size biases inherent in many types of discovery data the model postulates that the data are obtained from a finite population by selecting successively without replacement and with probability proportional to some measure of size unlike the ppswor scheme in survey sampling it is assumed here that the size measure is a function of the unknown population values in this article we consider maximum likelihood estimation of the finite population parameters under this biased sampling model we study the large sample behavior of the mle s and derive a simple asymptotically efficient approximation to the mle the approximate mle is structurally similar to the horvitz thompson estimator we show that information about the order in the sample can be used to make inference even when the population size is unknown which in fact can be estimated small sample behavior of the estimators is investigated through a limited simulation study and the results are used to analyze oil and gas discovery data from the north sea basin

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

asymptotic behavior of the lempel ziv parsing scheme and digital search trees the lempel ziv parsing scheme finds a wide range of applications most notably in data compression and algorithms on words it partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase the parameter of interest is the number mn of phrases that one can construct from a sequence of length n in this paper for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of mn which turns out to be normal this proves a long standing open problem in fact to obtain this result we solved another open problem namely that of establishing the limiting distribution of the internal path length in a digital search tree the latter is a consequence of an asymptotic solution of a multiplicative differential functional equation often arising in the analysis of algorithms on words interestingly enough our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability and analytical techniques such as mellin transform differential functional equations de poissonization and so forth in concluding remarks we indicate a possibility of extending our results to markovian models

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

motion and pinning of discrete interfaces we describe the motion of interfaces in a two dimensional discrete environment by coupling the minimizing movements approach by almgren taylor and wang and a discrete to continuous analysis we show that below a critical ratio of the time and space scalings we have no motion of interfaces pinning while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by almgren and taylor the critical regime is much richer exhibiting a pinning threshold small setsmove large sets are pinned partial pinning portions of interfaces may not move pinning after an initial motion possibly to a non convex limit set quantization of the interface velocity and non uniqueness effects

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

improved inference for respondent driven sampling data with application to hiv prevalence estimation respondent driven sampling is a form of link tracing network sampling which is widely used to study hard to reach populations often to estimate population proportions previous treatments of this process have used a with replacement approximation which we show induces bias in estimates for large sample fractions and differential network connectedness by characteristic of interest we present a treatment of respondent driven sampling as a successive sampling process unlike existing representations our approach respects the essential without replacement feature of the process while converging to an existing with replacement representation for small sample fractions and to the sample mean for a full population sample we present a successive sampling based estimator for population means based on respondent driven sampling data and demonstrate its superior performance when the size of the hidden population is known we present sensitivity analyses for unknown population sizes in addition we note tha

defects in nematic shells a γ convergence discrete to continuum approach in this paper we rigorously investigate the emergence of defects on nematic shells with a genus different from one this phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field to this end we consider a discrete xy system on the shell m described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell defects emerge when we let the mesh size of the triangulation go to zero namely in the discrete to continuum limit in this paper we investigate the discrete to continuum limit in terms of convergence in two different asymptotic regimes the first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell m via the poincare hopf theorem the second scaling produces the so called renormalized energy that governs the equilibrium of the configurations with defects

asymptotic properties of data compression and suffix trees recently wyner and ziv see ibid vol 35 p 1250 8 1989 have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is 1 h log n in probability where h is the entropy of the alphabet this finding was used to obtain several insights into certain universal data compression schemes most notably the lempel ziv data compression algorithm wyner and ziv have also conjectured that their result can be extended to a stronger almost sure convergence in this paper we settle this conjecture in the negative in the so called right domain asymptotic that is during a dynamic phase of expanding the data base we prove under an additional assumption involving mixing conditions that the length of a typical repeated subword oscillates almost surely a s between 1 h sub 1 log n and 1 h sub 2 log n where d

efficient computation of subset selection probabilities with application to cox regression summary an efficient method is presented to compute the probability of selection of a specified subset from the set of all subsets of a fixed size where the subsets are taken from a population whose units have varying individual probabilities of selection the problem is motivated by the computation of the exact marginal likelihood for the cox proportional hazards model

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

estimation of finite population properties when sampling is without replacement and proportional to magnitude abstract geologists often evaluate aggregate volumes of discovered plus undiscovered oil and or gas in a petroleum basin by use of geologic volumetric methods although sophisticated geological reasoning may be employed the essential idea behind these methods is simple estimate a the volume of hydrocarbon bearing sediment in the basin b the amount of hydrocarbons present per unit volume of sediment and c the fraction of hydrocarbons present per unit volume that is technologically recoverable the product of a and b is interpretable as a point estimate of the sum of amounts of oil and gas in place in individual oil and gas deposits fields in the basin the product of a b and c is a point estimate of amounts of oil and gas recoverable from all individual deposits in the basin

variational analysis of the asymptotics of the xy model in this paper we consider the xy n dimensional possibly anisotropic spin type model and by comparison with a ginzburg landau type functional we perform a variational analysis in the limit when the number of particles diverges in particular we show how the appearance of vortex like singularities can be described by properly scaling the energy of the system through a convergence procedure we also address the problem in the case of long range interactions and solve it in two dimensions

digital search trees again revisited the internal path length perspective this paper studies the asymptotics of the variance for the internal path length in a symmetric digital search tree under the bernoulli model this problem has been open until now it is proved that the variance is asymptotically equal to n cdot0 26600 n cdot delta log 2 n where n is the number of stored records and delta x is a periodic function of mean zero and a very small amplitude this result completes a series of studies devoted to the asymptotic analysis of the variances of digital tree parameters in the symmetric case in order to prove the previous result a number of nontrivial problems concerning analytic continuations and some others of a numerical nature had to be solved in fact some of these techniques are motivated by the methodology introduced in an influential paper by flajolet and sedgewick

singularities of energy minimizing maps from the ball to the sphere examples counterexamples and bounds we study maps qp from domains q in r3 to s2 which have prescribed boundary value functions 4 and which minimize dirichlet s integral energy such p s can have isolated singular points and our principal concern is the number and arrangement of such singularities our main result is that the number of singular points is bounded above by a constant times dirichlet s integral of 4 on d q and we show that this linear law is the best possible in two different senses furthermore singularities of minimizing q s sometimes have peculiar and counterintuitive properties i we illustrate by example that the mapping area jacobian integral of 4 can be zero although qp has many singular points ii we also show how to construct uniquely minimizing qp s having many singular points stacked up vertically near d q like bubbles in a pan of water about to boil iii finally we show by example that symmetries of q and 4 do not insure corresponding symmetries of qp and its singularities in particular singularities in p can be unstable under small perturbations of 4

the line tension approximation as the dilute limit of linear elastic dislocations we prove that the classical line tension approximation for dislocations in crystals that is the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length follows as the limit of regularized linear elasticity as the lattice parameter becomes increasingly small or equivalently as the dislocation measure becomes increasingly dilute we consider two regularizations of the theory of linear elastic dislocations a core cutoff and a mollification of the dislocation measure we show that both regularizations give the same energy in the limit namely an energy defined on matrix valued divergence free measures concentrated on lines the corresponding self energy per unit length b t which depends on the local burgers vector and orientation of the dislocation does not however necessarily coincide with the self energy per unit length 0 b t obtained from the classical theory of the prelogarithmic factor of linear elastic straight dislocations indeed microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its h1 elliptic envelope

a variational model for dislocations in the line tension limit we study the interaction of a singularly perturbed multiwell energy with an anisotropic nonlocal regularizing term of h1 2 type and a pinning condition this functional arises in a phase field model for dislocations which was recently proposed by koslowski cuitino and ortiz but it is also of broader mathematical interest in the context of the dislocation model we identify the limit of the energy in all scaling regimes for the number n of obstacles the most interesting regime is n ln where is a nondimensional length scale related to the size of the crystal lattice in this case the limiting model is of line tension type one important feature of our model is that the set of energy wells is periodic and hence not compact thus a key ingredient in the proof is a compactness estimate up to a single translation for finite energy sequences which generalizes earlier results from alberti bouchitte and seppecher for the two well problem with a h1 2 regularization

discrete crystal elasticity and discrete dislocations in crystals this article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals the theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups differential forms and operators and a theory of integration of forms in particular we define the lattice complex of a number of commonly encountered lattices including body centered cubic and face centered cubic lattices we show that material frame indifference naturally leads to discrete notions of stress and strain in lattices lattice defects such as dislocations are introduced by means of locally lattice invariant but globally incompatible eigendeformations the geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations such as the dislocation density tensor the equation of conservation of burgers vector kroner s relation and mura s formula for the stored energy we additionally supply conditions for the existence of equilibrium displacement fields we show that linear elasticity is recovered as the limit of harmonic lattice statics as the lattice parameter becomes vanishingly small we compute the limit of dilute dislocation distributions of dislocations and we show that the theory of continuously distributed linear elastic dislocations is recovered as the limit of the stored energy as the lattice parameter and burgers vectors become vanishingly small

ginzburg landau functionals and renormalized energy a revised γ convergence approach abstract we give short and self contained proofs of convergence results for ginzburg landau energy functionals in two dimensions in the logarithmic energetic regime in particular we derive the renormalized energy by convergence

optimal stopping of sequential size dependent search in many areas of application one searches within finite populations for items of interest where the probability of sampling an item is proportional to a random size attribute from an i i d superpopulation of attributes which may or may not be observable upon discovery here we treat the problem of asymptotically optimal stopping rules for size dependent searches of this type as the size of the underlying population grows where the loss function includes an asymptotically smooth time dependent cost a constant cost per item sampled and a cost per undiscovered item which may depend on the size attribute of the undiscovered item under some regularity and convexity conditions related to the asymptotic expected loss we characterize asymptotically optimal rules even when the initial population size and the distribution of size attributes are unknown we direct especial attention to applications in software reliability where the items of interest are software faults bugs in this setting the size attributes will not be observable when faults are found and in addition our search model allows new bugs to be introduced into the software when faults are detected imperfect debugging our results extend those of dalal and mallows and kramer and starr and are illustrated in the perfect debugging case on a previously analyzed dataset of musa

maximum likelihood estimation under a successive sampling discovery model a common problem encountered in the analysis of discovery data is the size bias phenomenon in which the larger units tend to be discovered first one approach to account for this bias is to model the discovery process as sampling successively from a finite population without replacement and with probability proportional to size we consider in this article a generalized version of this model for analyzing multivariate data with any given measure of size we assume a superpopulation framework and develop procedures for maximum likelihood estimation of the parameters of the distribution the use of the em algorithm for computing the maximum likelihood estimates associated computational issues and relationships to regression estimators in survey sampling are discussed oil discovery data from the rimbey meadowbrook reef play are used to illustrate the techniques

analysis of boundary conditions for crystal defect atomistic simulations numerical simulations of crystal defects are necessarily restricted to finite computational domains supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment this work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed we formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers using this foundation we then present rigorous error estimates for i a truncation method dirichlet boundary conditions ii periodic boundary conditions iii boundary conditions from linear elasticity and iv boundary conditions from nonlinear elasticity numerical results confirm the sharpness of the analysis

approximate counting with m counters a detailed analysis the classical algorithm approximate counting has been recently modified by cichon and macyna instead of one counter m counters are used and the assignment of an incoming item to one of the counters is random the parameter of interest is the sum of the values of all the counters we analyse expectation and variance getting explicit and asymptotic formulae

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

existence and stability of a screw dislocation under anti plane deformation we formulate a variational model for a geometrically necessary screw dislocation in an anti plane lattice model at zero temperature invariance of the energy functional under lattice symmetries renders the problem non coercive nevertheless by establishing coercivity with respect to the elastic strain and a concentration compactness principle we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model

γ convergence analysis of systems of edge dislocations the self energy regime this paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity the dislocations are introduced as point topological defects of the displacement gradient fields following the core radius approach we introduce a parameter varepsilon 0 representing the lattice spacing of the crystal we remove a disc of radius varepsilon around each dislocation and compute the elastic energy stored outside the union of such discs namely outside the core region then we analyze the asymptotic behaviour of the elastic energy as varepsilon rightarrow 0 in terms of convergence we focus on the self energy regime of order log frac 1 varepsilon we show that configurations with logarithmic diverging energy converge up to a subsequence to a finite number of multiple dislocations and we compute the corresponding limit

dynamics for systems of screw dislocations the goal of this paper is the analytical validation of a model of cermelli and gurtin arch ration mech anal 148 1999 pp 3 52 for an evolution law for systems of screw dislocations under the assumption of antiplane shear the motion of the dislocations is restricted to a discrete set of glide directions which are properties of the material the evolution law is given by a maximal dissipation criterion leading to a system of differential inclusions short time existence uniqueness cross slip and fine cross slip of solutions are proved

convergence and non convergence of many particle evolutions with multiple signs we address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity we consider two types of particles called positive and negative same sign particles repel each other and opposite sign particles attract each other the interaction potential is the same for all particles up to the sign and has a logarithmic singularity at zero the central example of such systems is that of dislocations in crystals because of the singularity in the interaction potential the discrete evolution leads to blow up in finite time we remedy this situation by regularising the interaction potential at a length scale delta n 0 which converges to zero as the number of particles n tends to infinity we establish two main results the first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled pdes which describe the evolution of their continuum densities in the setting of dislocations these pdes are known as the groma balogh equations in the proof we rely on both the theory of lambda convex gradient flows to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta n regularised version of the groma balogh equations and a priori estimates for the groma balogh equations to pass to the small regularisation limit in a functional setting based on orlicz spaces in order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta n to converge to zero sufficiently slowly the second result is a counterexample demonstrating that if delta n converges to zero sufficiently fast then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the groma balogh equations these results show how the validity of the groma balogh equations as the limit of many particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised

leaf to leaf distances and their moments in finite and infinite ordered m ary tree graphs we study the leaf to leaf distances on full and complete m ary graphs using a recursive approach in our formulation leaves are ordered along a line we find explicit analytical formulae for the sum of all paths for arbitrary leaf to leaf distance r as well as the average path lengths and the moments thereof we show that the resulting explicit expressions can be recast in terms of hurwitz lerch transcendants results for periodic trees are also given for incomplete random binary trees we provide first results by numerical techniques we find a rapid drop of leaf to leaf distances for large r

metastability and dynamics of discrete topological singularities in two dimensions a γ convergence approach this paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions based on convergence we consider discrete systems described by scalar functions defined on a square lattice and governed by periodic interaction potentials our main motivation comes from xy spin systems described by the phase parameter and screw dislocations described by the displacement function for these systems we introduce a discrete notion of vorticity as the lattice spacing tends to zero we derive the first order limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices as a byproduct of this analysis we show that such systems exhibit increasingly many metastable configurations of singularities therefore we propose a variational approach to the depinning and dynamics of discrete vortices based on minimizing movements we show that letting first the lattice spacing and then the time step of the minimizing movements tend to zero the vortices move according with the gradient flow of the renormalized energy as in the continuous ginzburg landau framework