equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

uniform bounds for strongly competing systems the optimal lipschitz case for a class of systems of semi linear elliptic equations including delta u i f i x u i beta u i sum j neq i a ij u j p quad i 1 dots k for p 2 variational type interaction or p 1 symmetric type interaction we prove that uniform l infty boundedness of the solutions implies uniform boundedness of their lipschitz norm as beta to infty that is in the limit of strong competition this extends known quasi optimal regularity results and covers the optimal case for this class of problems the proofs rest on monotonicity formulae of alt caffarelli friedman and almgren type in the variational setting and on the caffarelli jerison kenig almost monotonicity formula in the symmetric one

the two phase membrane problem an intersection comparison approach to the regularity at branch points for the two phase membrane problem delta u lambda 2 chi u 0 lambda 2 chi u 0 and lambda 0 we prove in two dimensions that the free boundary is in a neighborhood of each branch poin

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

storage capacity of a neural network with state dependent synapses the storage capacity of the hopfield network is limited to p n c 0 138 p number of patterns n number of neurons beyond which the contribution of weakly correlated patterns surpasses that of the desired pattern this contribution can be eliminated by introducing a threshold a pattern correlation below this threshold is simply set to zero in the synapses we solve the mean field equations and derive the critical value of the threshold required to stabilize p n patterns with c

a neural network for storing individual patterns in limit cycles a neural network model in which individual memories are stored in limit cycles is studied analytically and numerically in this model there are two kinds of interactions a hopfield like term that tends to stabilize the system in a memorized state and a second term with a time delay that acts to induce transitions between a memorized state and its complement state for a proper choice of the values of the parameters this model exhibits limit cycle behaviour in which the overlap with a target pattern oscillates in time an asymmetrically diluted version of the model is studied analytically in the limit of extreme dilution we find that the model with cycles performs better than a similarly diluted version of the hopfield model the performance of the fully connected model is studied by numerical simulations we find a behaviour qualitatively similar to that of the dilute model the model with cycles is found to perform better than the hopfield model as a pattern classifier if the memory loading level and the degree of corruption of the input patterns are high

singularities in axisymmetric free boundaries for electrohydrodynamic equations we consider singularities in the electrohydrodynamic equations in a regime where we are allowed to neglect surface tension and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non degeneracy condition we prove that either the fluid region or the gas region is asymptotically a cusp our proofs depend on a combination of monotonicity formulas and a non vanishing result by caffarelli and friedman as a by product of our analysis we also obtain a special solution with convex conical air phase which we believe to be new

motion of sets by curvature and derivative of capacity potential abstract we study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential we establish local well posedness and propose two possible weak formulations that exist after singularities

an unstable elliptic free boundary problem arising in solid combustion we prove a regularity result for the unstable elliptic free boundary problem u u 0 0 1 related to traveling waves in a problem arising in solid combustion the maximal solution and every local minimizer of the energy are regular that is u 0 is locally an analytic surface and u u 0 u u 0 are locally analytic functions moreover we prove a partial regularity result for solutions that are non degenerate of second order here u 0 is analytic up to a set of hausdorff dimension n 2 we discuss possible singularities ams classification 35r35 35j60 35b65

hierarchical neural networks for the storage of correlated memories a class of hierarchical neural network models introduced by dotsenko for the storage and associative recall of strongly correlated memories is studied analytically and numerically in these models patterns stored in higher levels of the hierarchy represent generalized categories and those stored in lower levels describe finer details we first show that the models originally proposed by dotsenko have a serious flaw they are not able to detect or correct errors in categorization which may be present in the input we then describe three different models which attempt to overcome this shortcoming of the original models in the first model the interaction between different levels of the hierarchy has the form of an external field conjugate to memories stored in the lower level in the second model a three spin interaction term is included in addition to the usual binary interactions of the hopfield type the third model makes use of a time delay mechanism to induce if necessary transitions between memory states and their complements detailed analytical and numerical studies of the performance of these models are presented our analysis shows that all three models are able to detect and also to correct in varying degrees any error in categorization that may be present in the input pattern

categorization and effective perceptron learning in feed forward neural networks in the initial stages of back propagation learning weights in the first synaptic layer are small and hidden neurons operate in the linear regime so the input output map is then the same as that of a perceptron as the weights increase the hidden neurons begin to saturate each saturation pattern represents a particular category of input vectors for which the neural network nn locally behaves as a perceptron the effective perceptron weights for each category are functions of the synaptic weights of the nn we define an internal temperature as the inverse of the norm of the first synaptic matrix and an entropy function that measures the disorder of the hidden neuron layer the learning curves are recast in the language of a condensation process the entropy drops abruptly when the hidden neurons begin to saturate marking a clear transition from the global perceptron state to an ordered state characterized by different local effective perceptrons in different categories the problem of optimizing the effective perceptrons in different categories is frustrated

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

monotonicity theorems for laplace beltrami operator on riemannian manifolds abstract for free boundary problems on euclidean spaces the monotonicity formulas of alt caffarelli friedman and caffarelli jerison kenig are cornerstones for the regularity theory as well as the existence theory in this article we establish the analogs of these results for the laplace beltrami operator on riemannian manifolds as an application we show that our monotonicity theorems can be employed to prove the lipschitz continuity for the solutions of a general class of two phase free boundary problems on riemannian manifolds

boundary behaviour for a singular perturbation problem abstract in this paper we study the boundary behaviour of the family of solutions u e to singular perturbation problem u e e u e u e 1 in b 1 x n 0 x 1 where a smooth boundary data f is prescribed on the flat portion of b 1 here e 1 e e c 0 0 1 0 0 1 t m 0 is an approximation of identity if f z 0 whenever f z 0 then the level sets u e 0 approach the fixed boundary in tangential fashion with uniform speed the methods we employ here use delicate analysis of local solutions along with elaborated version of the so called monotonicity formulas and classification of global profiles

some monotonicity results for minimizers in the calculus of variations abstract we obtain monotonicity properties for minima and stable solutions of general energy functionals of the type f u u x d x under the assumption that a certain integral grows at most quadratically at infinity as a consequence we obtain several rigidity results of global solutions in low dimensions

a nonlinear free boundary problem with a self driven bernoulli condition abstract we study a bernoulli type free boundary problem with two phases j u u x 2 d x m u m u u u w 0 1 2 where u w 1 2 is a given boundary datum here m 1 and m 2 are weighted volumes of u 0 and u 0 respectively and is a nonnegative function of two real variables we show that for this problem the bernoulli constant which determines the gradient jump condition across the free boundary is of global type and it is indeed determined by the weighted volumes of the phases in particular the bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself and therefore it depends on the minimizer itself and not only on the structural constants of the problem another property of this type of problems is that the minimizer in is not necessarily a minimizer in a smaller subdomain due to the nonlinear structure of the problem due to these features this problem is highly unstable as opposed to the classical case studied by alt caffarelli and friedman it also interpolates the classical case in the sense that the blow up limits of u are minimizers of the alt caffarelli friedman functional namely the energy of the problem somehow linearizes in the blow up limit as a special case we can deal with the energy levels generated by the volume term 0 r 2 r 2 n 1 n which interpolates the athanasopoulos caffarelli kenig salsa energy thanks to the isoperimetric inequality in particular we develop a detailed optimal regularity theory for the minimizers and for their free boundaries

a variational method for multiphase volume preserving interface motions we develop a numerical method for realizing mean curvature motion of interfaces separating multiple phases whose volumes are preserved throughout time the foundation of the method is a thresholding algorithm of the bence merriman osher type the original algorithm is reformulated in a vector setting which allows for a natural inclusion of constraints even in the multiphase case moreover a new method for overcoming the inaccuracy of thresholding methods on non adaptive grids is designed since this inaccuracy becomes especially prominent in volume preserving motions formal analysis of the method and numerical tests are presented

regularity of the free boundary for two phase problems governed by divergence form equations and applications abstract we study a class of two phase inhomogeneous free boundary problems governed by elliptic equations in divergence form in particular we prove that lipschitz or flat free boundaries are c 1 our results apply to the classical prandtl bachelor model in fluiddynamics

optimization problems for an energy functional with mass constraint revisited abstract in this paper we report on a variational problem under a constraint on the mass which is motivated by the torsional rigidity and torsional creep following a device by alt caffarelli and friedman we treat instead a problem without constraint but with a penalty term we will complete some of the results of c bandle a wagner optimization problems for weighted sobolev constants calc var partial differential equations 29 2007 481 507 where the existence of a lipschitz continuous minimizer has been established in particular we prove qualitative properties of the optimal shape

hierarchical associative networks the authors study a modified hopfield model of associative memory with a learning rule proposed by personnaz et al 1985 for the special case of ultrametrically correlated patterns the formula for the synaptic strength tells the teacher how much stress to put on details compared to averages at each level it is still a local rule if one assumes that the ultrametric correlation structure of the patterns is given a priori the result is also given in terms of the parisi function q x in a special limit they get the same result as parga and virasora 1986

universal formulas for percolation thresholds a power law is postulated for both site and bond percolation thresholds the formula is mathit p mathit c mathit p 0 d 1 q 1 mathrm ensuremath mathit a mathit d mathit b where d is the space dimension and q the coordination number all thresholds up to d ensuremath rightarrow ensuremath infty are found to belong to only three universality classes for two classes b 0 for site dilution while b a for bond dilution the remaining class associated with high dimensions is characterized by b 2a 1 for both sites and bonds classes are defined by a set of value for mathit p 0 a deviations from available numerical estimates at d ensuremath le 7 are within ifmmode pm else textpm fi 0 008 and ifmmode pm else textpm fi 0 0004 for high dimensional hypercubic expansions at d ensuremath ge 8 the formula is found to be also valid for ising critical temperatures textcopyright 1996 the american physical society

a quasi exact formula for ising critical temperatures on hypercubic lattices abstract we report a quasi exact power law behavior for ising critical temperatures on hypercubes it reads j k b t c k 0 1 1 d q 1 a where k0 0 8633747 d is the space dimension q the coordination number q 2d j the coupling constant kb the boltzman constant and tc the critical temperature absolute errors from available exact estimates d 2 7 are always less than 0 0005 extension to other lattice is discussed

lipschitz regularity of the eigenfunctions on optimal domains we study the optimal sets omega ast subseteq mathbb r d for spectral functionals of the form f big lambda 1 omega ldots lambda p omega big which are bi lipschitz with respect to each of the eigenvalues lambda 1 omega lambda 2 omega ldots lambda p omega of the dirichlet laplacian on omega a prototype being the problem min big lambda 1 omega cdots lambda p omega omega subseteq mathbb r d omega 1 big we prove the lipschitz regularity of the eigenfunctions u 1 ldots u p of the dirichlet laplacian on the optimal set omega ast and as a corollary we deduce that omega ast is open for functionals depending only on a generic subset of the spectrum as for example lambda k omega our result proves only the existence of a lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved

on the multi neuron interaction model without truncating the interaction a replica symmetric mean field theory approach is presented to the multi neuron interaction model introduced by de almeida and iglesias 1990 phys lett 146a 239 fixed point equations are derived for the relevant order parameters of the model extended to include biased patterns without truncating the interaction the capacity bias and the temperature capacity phase diagrams are discussed compared with the truncated version of the model it is found that the capacity at zero temperature is infinite and that the retrieval states satisfy the de almeida thouless stability condition

variational formulas on lipschitz domains a rigorous treatment is given of variational formulas for solutions of certain dirichlet problems for the laplace operator on lipschitz domains under interior variations in particular we extend well known variational formulas for the torsional rigidity and for capacity from the class of c1 domains to the class of lipschitz domains a motivation for this work comes from the use of variational methods in the study of prandtl batchelor flows in fluid mechanics

existence and nondegeneracy of ground states in critical free boundary problems abstract existence and regularity of minimizers in elliptic free boundary problems have been extensively studied in the literature the corresponding study of higher critical points was recently initiated in jerison and perera 2018 0000 in particular the existence and nondegeneracy of a mountain pass point in a superlinear and subcritical free boundary problem related to plasma confinement was proved in jerison and perera 2018 in this paper we study ground states of a critical free boundary problem related to the brezis nirenberg problem 1983 we extend the results of jerison and perera 2018 to this problem by combining the method introduced there with the concentration compactness principle to overcome the difficulties arising from lack of compactness

almost monotonicity formulas for elliptic and parabolic operators with variable coefficients in this paper we extend the results of caffarelli jerison and kenig ann of math 2 155 2002 and caffarelli and kenig amer j math 120 1998 by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u 0 lu 1 u u 0 in an infinite strip global version or a finite parabolic cylinder localized version where l is a uniformly parabolic operator lu l a b c u div a x s u b x s u c x s u s u with double dini continuous a and uniformly bounded b and c we also prove the elliptic counterpart of this estimate this closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on u in order to obtain an almost monotonicity estimate at the end of the paper we demonstrate how to use this new almost monotonicity formula to prove the optimal c 1 1 regularity in a fairly general class of quasi linear obstacle type free boundary problems

plane like minimizers for an area dirichlet integral given an area dirichlet functional mathcal j which is periodic under integer translations and given three planes in mathbb r d we prove there exists at least one minimizer such that its positive part negative part and zero set remain at a uniform bounded distance from each plane

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

liouville theorems and 1 dimensional symmetry for solutions of an elliptic system modelling phase separation abstract we consider solutions of the competitive elliptic system s u i j i u i u j 2 in r n u i 0 in r n i 1 k we are concerned with the classification of entire solutions according to their growth rate the prototype of our main results is the following there exists a function k n n increasing in k such that if u 1 u k is a solution of s and u 1 x u k x c 1 x d for every x r n then d this means that the number of components k of the solution imposes a lower bound increasing in k on the minimal growth of the solution itself if n 2 the expression of is explicit and optimal while in higher dimension it can be characterised in terms of an optimal partition problem we discuss the sharpness of our results and as a further step for every n 2 we can prove the 1 dimensional symmetry of the solutions of s satisfying suitable assumptions extending known results which are available for k 2 the proofs rest upon a blow down analysis and on some monotonicity formulae

optimal basins of attraction in randomly sparse neural network models the size of the basin of attraction for randomly sparse neural networks with optimal interactions is calculated for all values of the storage ratio alpha p c 2 where p is the number of random uncorrelated patterns and c is the connectivity the basin of attraction is finite while for alpha 0 42 the basin of attraction is almost 100

higher order neural networks polya polynomials and fermi cluster diagrams the problem of controlling higher order interactions in neural networks is addressed with techniques commonly applied in the cluster analysis of quantum many particle systems for multineuron synaptic weights chosen according to a straightforward extension of the standard hebbian learning rule we show that higher order contributions to the stimulus felt by a given neuron can be readily evaluated via polya s combinatoric group theoretical approach or equivalently by exploiting a precise formal analogy with fermion diagrammatics

on the non tangential touch between the free and the fixed boundaries for the two phase obstacle like problem in this paper we consider the following two phase obstacle problem like equation in the unit half ball delta u lambda chi u 0 lambda chi u 0 we classify the angles of touch between the free boundary and the fixed one if the boundary data f satisfies certain conditions these results are connected to the results from andersson et al ark mat 44 1 1 15 2006

a parabolic free boundary problem with double pinning abstract we study the cauchy problem for the equation tue ue e ue in 0 rn as e 0 where the nonlinearity e is assumed to converge to a measure concentrated at 0 in this paper we allow for sign changes of e and ue the solutions are uniformly lipschitz continuous in space and holder continuous in time we show that each limit of ue is a solution of the free boundary problem tu u 0 in u 0 0 rn u 2 u 2 g on u 0 u e the function g in the condition on the free boundary need not be a constant

learning and generalization in multineuron interacting feed forward neural networks we consider learning and generalization in the multi interacting feed forward network model recently proposed by h o carmesin 1994 with an a priori definition of the net architecture based on symmetries presented by the function to be learnt we define a generalized hebb rule extend the maximum stability learning algorithm to multi interactions and obtain training and generalization curves for rules where different orders of synapses are not correlated the results obtained for the simple perceptron concerning the hebb rule and through replica calculations in the space of couplings may be straightforwardly adapted to multi interactions through a simple renormalization of the total number of independent couplings analytical and numerical simulation results are compared and show excellent agreement

chaotic dynamics of high order neural networks the dynamics of an extremely diluted neural network with high order synapses acting as corrections to the hopfield model is investigated the learning rules for the high order connections contain mixing of memories different from all the previous generalizations of the hopfield model the dynamics may display fixed points or periodic and chaotic orbits depending on the weight of the high order connections e the noise levelt and the network load defined as the ratio between the number of stored patterns and the mean connectivity per neuron p c as in the related fully connected case there is an optimal value of the weight e that improves the storage capacity of the system the capacity diverges

two phase anisotropic free boundary problems and applications to the bellman equation in 2d we prove lipschitz continuity of solutions to a class of rather general two phase anisotropic free boundary problems in 2d and we classify global solutions as a consequence we obtain c 2 1 regularity of solutions to the bellman equation in 2d

high dimensional lattice gases we investigate the critical behaviour of hard core lattice gases in four five and six dimensions by means of monte carlo simulations in order to suppress critical slowing down we use a geometrical cluster monte carlo algorithm in particular nearest neighbour exclusion lattice gases on simple hypercubic lattices are investigated these models undergo ising like ordering transitions where the majority of the lattice gas particles settle on one of two sublattices a finite size scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour the results agree with the renormalization predictions at and above the upper critical dimensionality in particular the predicted value of the binder cumulant is confirmed

a free boundary approach to two dimensional steady capillary gravity water waves we prove that suitable weak solutions of the two dimensional steady capillary gravity water wave problem even in the presence of vorticity are smooth on a technical level solutions are closely related to critical points of the mumford shah functional so that our main task is to exclude cusps pointing into the water phase we prove that cusps do not exist and that the free surface is smooth

analysis of a free boundary at contact points with lipschitz data in this paper we consider a minimization problem for the functional j u b 1 u 2 2 u 0 2 u 0 in the upper half ball b 1 n n 2 subject to a lipschitz continuous dirich

a gradient bound for the grad kruskal kulsrud functional ab stract we announce a gradient bound for minimizers of a variational problem arising in the grad kruskal kulsrud model for the equilibrium of a confined plasma the variational problem involves derivatives of the nondecreasing rearrangement of minimizers in the case of a convex domain our results answer in the negative a question raised by grad concerning the possibly singular behavior of the magnetic field at the point of maximum flux it completes a researchprogram begun in 1983 in whichan approach using free boundaries was initiated to handle the nonlinear nonlocal nature of the problem limiting minimizers of the variational problem are shown to have bounded gradient and satisfy a weak equation which for one model problem takes the form where are respectively the nondecreasing rearrangement and distribution function of plasmas and variational problems with global constraints the grad kruskal kulsrud model hereafter gkk ghs kk for the confinement of a thermonuclear plasma is a realistic one in which the nonlinearities are nonlocal nonlinear functionals of the solution here we apply the variational problem to axisymmetric magnetic bottles known as tokamaks ch these devices are at the forefront of experimental research today tftr at princeton jet at culham associated with the plasma in equilibrium is an energy expressed in terms of the magnetic field b 1991 mathematics subject classification primary 35r35 35d10 34jxx secondary 35q35

uniform holder bounds for a strongly coupled elliptic system with strong competition abstract this article is concerned with a strongly coupled elliptic system modeling the steady state of populations that compete in some region we prove that the solutions are uniformly holder bounded as the competition rate tends to infinity the proof relies on the blow up technique and the monotonicity formula

neural network modeling of associative memory beyond the hopfield model a number of neural network models in which fixed point and limit cycle attractors of the underlying dynamics are used to store and associatively recall information are described in the first class of models a hierarchical structure is used to store an exponentially large number of strongly correlated memories the second class of models uses limit cycles to store and retrieve individual memories a neurobiologically plausible network that generates low amplitude periodic variations of activity similar to the oscillations observed in electroencephalographic recordings is also described results obtained from analytic and numerical studies of the properties of these networks are discussed

free transmission problems we study transmission problems with free interfaces from one random medium to another solutions are required to solve distinct partial differential equations rm l and rm l within their positive and negative sets respectively a corresponding flux balance from one phase to another is also imposed we establish existence and l infty bounds of solutions we also prove that variational solutions are non degenerate and develop the regularity theory for solutions of such free boundary problems

on optimal regularity of free boundary problems and a conjecture of de giorgi we present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmission condition on the free boundary as an application we give a full positive answer to a conjecture of de giorgi on the analyticity of local minimizers of the mumford shah functional 2004 wiley periodicals inc

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

three unfinished works on the optimal storage capacity of networks the optimal storage properties of three different neural network models are studied for two of these models the architecture of the network is a perceptron with or j interactions whereas for the third model the output can be an arbitrary function of the inputs analytic bounds and numerical estimates of the optimal capacities and of the minimal fraction of errors are obtained for the first two models the third model can be solved exactly and the exact solution is compared to the bounds and to the results of numerical simulations used for the two other models