multiplicity results for double phase problems in rn we consider a double phase problem in rn with a q 1 superlinear subcritical caratheodory reaction term which does not satisfy the ambrosetti rabinowitz condition using variational methods based on the topological degree and critical point theory together with the nehari manifold method and deformation lemma we find three nontrivial solutions among them the first one is the positive ground state solution the second one is the negative ground state solution and the third one is the least energy sign changing solution which changes sign only once we also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions

alien attractors and memory annihilation of structured sets in hopfield networks this paper considers the encoding of structured sets into hopfield associative memories a structured set is a set of vectors with equal hamming distance h from one another and its centroid is an external vector that has distance h 2 from every vector of the set structured sets having centroids are not infrequent when such a set is encoded into a noiseless hopfield associative memory using a bipolar outer product connection matrix and the network operates with synchronous neuronal update the memory of all encoded vectors is annihilated even for sets with as few as three vectors in dimension n 5 four for n 5 in such self annihilating structured sets the centroid emerges as a stable attractor we call it an alien attractor for canonical structured sets self annihilation takes place only if h n 2 self annihilation does not occur and alien attractors do not emerge in dimensions less than five

a theoretical investigation into the performance of the hopfield model an analysis is made of the behavior of the hopfield model as a content addressable memory cam and as a method of solving the traveling salesman problem tsp the analysis is based on the geometry of the subspace set up by the degenerate eigenvalues of the connection matrix the dynamic equation is shown to be equivalent to a projection of the input vector onto this subspace in the case of content addressable memory it is shown that spurious fixed points can occur at any corner of the hypercube that is on or near the subspace spanned by the memory vectors analysed is why the network can frequently converge to an invalid solution when applied to the traveling salesman problem energy function with these expressions the network can be made robust and can reliably solve the traveling salesman problem with tour sizes of 50 cities or more

calderon zygmund estimates for generalized double phase problems abstract we prove calderon zygmund type estimates for distributional solutions to non uniformly elliptic equations of generalized double phase type in divergence form in particular we provide sharp conditions on the nonlinear operators to establish the calderon zygmund type estimates

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

optimal hopfield network for combinatorial optimization with linear cost function an optimal hopfield network is presented for combinatorial optimization problems with linear cost function it is proved that a vertex of the network state hypercube is asymptotically stable if and only if it is an optimal solution to the problem that is one can always obtain an optimal solution whenever the network converges to a vertex in this sense this network can be called the optimal hopfield network it is also shown through simulations of assignment problems that this network obtains optimal or nearly optimal solutions more frequently than other familiar hopfield networks

a review of hopfield neural networks for solving mathematical programming problems the hopfield neural network hnn is one major neural network nn for solving optimization or mathematical programming mp problems the major advantage of hnn is in its structure can be realized on an electronic circuit possibly on a vlsi very large scale integration circuit for an on line solver with a parallel distributed process the structure of hnn utilizes three common methods penalty functions lagrange multipliers and primal and dual methods to construct an energy function when the function reaches a steady state an approximate solution of the problem is obtained under the classes of these methods we further organize hnns by three types of mp problems linear non linear and mixed integer the essentials of each method are also discussed in details some remarks for utilizing hnn and difficulties are then addressed for the benefit of successive investigations finally conclusions are drawn and directions for future study are provided

harmonic analysis of homogeneous networks we introduce a generalization of mutually inhibitory networks called homogeneous networks such networks have symmetric connection strength matrices that are circulant one dimensional case or block circulant with circulant blocks two dimensional case fourier harmonics provide universal eigenvectors and we apply them to several homogeneous examples k wta k cluster on center off surround and the assignment problem we also analyze one nonhomogeneous case the subset sum problem we present the results of 10000 trials on a 50 node k cluster problem and 100 trials on a 25 node subset sum problem

unbalanced p 2 fractional problems with critical growth abstract we study the existence multiplicity and regularity results of non negative solutions of following doubly nonlocal problem p s 1 u p s 2 u a x u q 2 u u y r x y d y u r 2 u in u 0 in r n where r n is a bounded domain with c 2 boundary 0 s 2 s 1 1 n 2 s 1 1 q p 2 1 r 2 with 2 2 n n 2 s 1 0 and a l d d q for some q d 2 s 1 2 n n 2 s 1 is a sign changing function we prove that each nonnegative weak solution of p is bounded furthermore we obtain some existence and multiplicity results using nehari manifold method

regularity of relaxed minimizers of quasiconvex variational integrals with p q growth we consider autonomous integrals f u int omega f du rm d x quad rm for u mathbb r n supset omega to mathbb r n in the multidimensional calculus of variations where the integrand f is a strictly quasiconvex c2 function satisfying the p q growth conditions gamma a p leqq f a leqq gamma 1 a q quad rm for every a in mathbb r nn with exponents 1 p leqq q infty

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

multiplicity of positive solutions for a fractional p q laplacian problem in rn abstract in this paper we deal with the following fractional p q laplacian problem p s u q s u v e x u p 2 u u q 2 u f u in r n u w s p r n w s q r n u 0 in r n where s 0 1 e 0 is a small parameter 2 p q n s t s with t p q is the fractional s t laplacian operator v r n r is a continuous function satisfying the global rabinowitz condition and f r r is a continuous function with subcritical growth using suitable variational arguments and ljusternik schnirelmann category theory we prove that the above problem admits multiple solutions for e 0 small enough

growth conditions and regularity for weak solutions to nonlinear elliptic pdes abstract we describe some aspects of the process approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form as well as of minimizers of integrals of the calculus of variations in particular with respect to the growth conditions we emphasize some differences between solving equations and minimizing energy integrals and also between x dependence or x independence

a donsker theorem for levy measures abstract given n equidistant realisations of a levy process l t t 0 a natural estimator n n for the distribution function n of the levy measure is constructed under a polynomial decay restriction on the characteristic function a donsker type theorem is proved that is a functional central limit theorem for the process n n n n in the space of bounded functions away from zero the limit distribution is a generalised brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the fourier integral operator f 1 1 the class of levy processes covered includes several relevant examples such as compound poisson gamma and self decomposable processes main ideas in the proof include establishing pseudo locality of the fourier integral operator and recent techniques from smoothed empirical processes

the singular set of lipschitzian minima of multiple integrals the singular set of any lipschitzian minimizer of a general quasiconvex functional is uniformly porous and hence its hausdorff dimension is strictly smaller than the space dimension

boundary regularity of minimizers of double phase functionals abstract in this paper we treat the functional of double phase with variable exponents d u g p x a x d u g q x d x where a x is a non negative holder continuous function with 0 1 p x and q x holder continuous functions with 1 p x q x p x and g i j g x i j 1 2 for a continuous positive definite matrix valued function g g we prove that the minimizer of the above functional with suitable dirichlet boundary condition is holder continuous up to the boundary when g is holder continuous we see also that du is locally holder continuous

interior gradient regularity for bv minimizers of singular variational problems we consider a class of vectorial integrals with linear growth where as a key feature some degenerate singular behavior is allowed for generalized minimizers of these integrals in bv we establish interior gradient regularity and as a consequence uniqueness up to constants in particular these results apply for 1 p 2 to the singular model integrals z 1 jrw x j p 1 p dx

regularity results for nonlinear parabolic obstacle problems with subquadratic growth abstract in 26 it was shown that the spatial gradient of the solution u to the parabolic obstacle problem with superquadratic growth is local holder continuous provided the obstacle is regular enough in this paper we extend this regularity result to the subquadratic case this means we establish the local holder continuity of the spatial gradient of the solution u to the parabolic obstacle problem with subquadratic growth more precisely we prove that d u c loc 0 2 for some 0 1 provided the coefficients and the obstacle are regular enough moreover we use the local holder continuity to prove the local lipschitz continuity of the solution u i e u c loc 0 1 1 2

borderline gradient continuity for nonlinear parabolic systems we consider the evolutionary p laplacean system begin aligned partial t u triangle p u f qquad p frac 2n n 2 end aligned in cylindrical domains of mathbb r n times mathbb r and prove the continuity of the spatial gradient du under the lorentz space assumption f in l n 2 1 when f is time independent the condition improves in f in l n 1 this is the limiting case of a result of dibenedetto claiming that du is holder continuous when f in l q for q n 2 at the same time this is the natural nonlinear parabolic analog of a linear result of stein claiming the gradient continuity of solutions to the linear elliptic system triangle u in l n 1 is continuous new potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions

decompounding an estimation problem for poisson random sums given a sample from a compound poisson distribution we consider estimation of the corresponding rate parameter and base distribution this has applications in insurance mathematics and queueing theory we propose a plug in type estimator that is based on a suitable inversion of the compounding operation asymptotic results for this estimator are obtained via a local analysis of the decompounding functional

finding nicer mappings with lower or equal energy in nonlinear elasticity abstract in the framework of nonlinear elasticity we consider the model energy f u d u x p h det d u x d x where u r n r n with det d u 0 and h 0 0 is convex moreover h t blows up when t 0 we study the problem fix the mapping u with finite energy f u and find a mapping v with the same boundary values with det d v 0 and energy f v not higher than f u such that every component v of v enjoys a kind of maximum and minimum principle due to the constraint det d v 0 a truncation argument does not work on the contrary the constraint det d v 0 makes the problem easy when p n since it is known that every component is weakly monotone so we can take v u in the present work we address the case 2 p n and we give a positive answer if the blow up of h t is of logarithmic type

regularity results for a class of non autonomous obstacle problems with p q growth abstract we study some regularity issues for solutions of non autonomous obstacle problems with p q growth under suitable assumptions our analysis covers the main models available in the literature

morrey estimates for some classes of elliptic equations with a lower order term abstract we will browse a series of results on elliptic equations and systems with particular nonlinear lower order terms and measure right hand side obtained in the last years in the papers cirmi and leonardi 2014 cianci et al 2017 cirmi et al 2014 2017 2018 namely we take into account equations whose prototypes are u u d u 2 f or u d u 2 u f 0 1 where the right hand side f belongs to a suitable morrey space for instance to l 1 0 n 2 and we prove corresponding morrey estimates for the gradient of a solution

rotation and scale independent pattern recognition through optimization this paper presents a two dimensional pattern recognition paradigm which is independent of rotation and scale it uses a hopfield neural network with a configuration similar to that employed for the traveling salesman problem tsp enhancements to the original hopfield design are included which are based on the eigenvalues of the connection matrix and on modifications suggested by simulations the goal of the method is to determine the underlying linear transformation between a test image and an observed input pattern by minimizing the energy of a hopfield network this procedure is easy to implement and is well suited for sparse binary images e g skeletalized representations or the output from vertex detection algorithms in this work the effects of noise are studied both analytically and through monte carlo techniques yielding some useful performance guidelines and greatly increasing the utility of the method experiments show that this technique performs successfully on a variety of rotated and scaled images and is robust against additive noise

regularity results for quasilinear elliptic equations in the heisenberg group we prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the heisenberg group mathbb h n the model case is the non degenerate p laplacean operator sum i 1 2n x i left left mu 2 left mathfrak x u right 2 right frac p 2 2 x i u right 0 where mu 0 and p is not too far from 2

the singular set of ω minima we consider minima open image in new window of convex variational integrals in the vectorial case n n 2 and we provide estimates for the hausdorff dimension of their singular sets

cluster based blind nonlinear channel estimation a cluster based maximum likelihood sequence estimator mlse for nonlinear channels was described which consists of a clustering network and an mlse implemented by the viterbi algorithm the cluster based mlse can be used for digital communication through nonlinear finite length channels because channel mapping estimation is used instead of channel estimation in the conventional mlse the clustering network of the cluster based mlse which estimates the channel mapping between the signal input vectors and the noiseless channel outputs is a supervised network and requires a training sequence we propose a blind channel mapping estimator to estimate the channel mapping without using the training sequence the blind channel mapping estimator has a clustering block and a mapping block the clustering block estimates the channel outputs which represent the channel mapping subject to an unknown permutation operation because no training sequence is utilized that permutation operation is resolved by the mapping block and therefore the channel mapping is obtained introducing the blind channel mapping estimator into the cluster based mlse a blind cluster based mlse for nonlinear channels can be done computer simulations of the blind channel mapping estimator and the blind mlse for nonlinear channels are presented

approximating maximum clique with a hopfield network in a graph a clique is a set of vertices such that every pair is connected by an edge max clique is the optimization problem of finding the largest clique in a given graph and is np hard even to approximate well several real world and theory problems can be modeled as max clique in this paper we efficiently approximate max clique in a special case of the hopfield network whose stable states are maximal cliques we present several energy descent optimizing dynamics both discrete deterministic and stochastic and continuous one of these emulates as special cases two well known greedy algorithms for approximating max clique we report on detailed empirical comparisons on random graphs and on harder ones mean field annealing an efficient approximation to simulated annealing and a stochastic dynamics are the narrow but clear winners all dynamics approximate much better than one which emulates a naive greedy heuristic

everywhere regularity for a class of vectorial functionals under subquadratic general growth conditions we consider the integral functional of the calculus of variations f du dx where f rnn r satisfies f z g z and g is an n function with subquadratic p q growth we prove that minimizers u rn rn of such a functional are locally lipschitz continuous provided g verifies some additional conditions

a hybrid neural network model for solving optimization problems a hybrid neural network model for solving optimization problems is proposed an energy function which contains the constraints and cost criteria of an optimization problem is derived and then the neural network is used to find the global minimum or maximum of the energy function which corresponds to a solution of the optimization problem the network contains two subnets a constraint network and a goal network the constraint network models the constraints of an optimization problem and computes the gradient updating value of each neuron such that the energy function monotonically converges to satisfy all constraints of the problem the goal network points out the direction of convergence for finding an optimal value for the cost criteria these two subnets ensure that the neural network finds feasible as well as optimal or near optimal solutions the traveling salesman problem and the hamiltonian cycle problem are used to demonstrate the method

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

a note on recovering the distributions from exponential moments the problem of recovering a cumulative distribution function of a positive random variable via the scaled laplace transform inversion is studied the uniform upper bound of proposed approximation is derived the approximation of a compound poisson distribution as well as the estimation of a distribution function of the summands given the sample from a compound poisson distribution are investigated applying the simulation study the question of selecting the optimal scaling parameter of the proposed laplace transform inversion is considered the behavior of the approximants are demonstrated via plots and table

higher order calderon zygmund estimates for the p laplace equation abstract the paper is concerned with higher order calderon zygmund estimates for the p laplace equation div a u div u p 2 u div f 1 p we are able to transfer local interior besov and triebel lizorkin regularity up to first order derivatives from the force term f to the flux a u for p 2 we show that f b q s implies a u b q s for any s 0 1 and all reasonable q 0 in the planar case the result fails for p 2 in case of higher dimensions and systems we have a smallness restriction on s the quasi banach case 0 min q 1 is included since it has important applications in the adaptive finite element analysis as an intermediate step we prove new linear decay estimates for p harmonic functions in the plane for the full range 1 p

partial lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions abstract we consider the partial lipschitz continuity of minimizers of functionals of the form v f x v d v d x where r n is open and bounded the map x u f x u appearing in the above functional is assumed to be holder continuous with respect to its first two arguments of class c 2 in its third argument and asymptotically related to the more regular map x u a x u g as the main novelty is the allowance that even as the map f may retain full dependence on the minimizer u not only its gradient du and the spatial coordinate x

uniform ellipticity and p q growth abstract fix any two numbers p and q with 1 p q we give an example of an integral functional enjoying uniform ellipticity and p q growth

nonparametric estimation of the stationary waiting time distribution function for the gi g 1 queue the gi g 1 queueing model is regarded as a functional that maps the service and interarrival time distribution functions onto the stationary waiting time distribution function by considering the output of the functional when it is applied to nonparametric estimators of the input distribution functions we obtain a nonparametric estimator of the stationary waiting time distribution function using appropriate continuity and differentiability properties of the functional we show that statistical properties of the input estimators carry over to corresponding properties for the stationary waiting time distribution function estimator

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

efficient nonparametric inference for discretely observed compound poisson processes a compound poisson process whose parameters are all unknown is observed at finitely many equispaced times nonparametric estimators of the jump and levy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions the limiting gaussian processes are identified and efficiency of the estimators is established kernel estimators for the mass function the intensity and the drift are also proposed their asymptotic properties including efficiency are analysed and joint asymptotic normality is shown inference tools such as confidence regions and tests are briefly discussed

polyhedral combinatorics and neural networks the often disappointing performance of optimizing neural networks can be partly attributed to the rather ad hoc manner in which problems are mapped onto them for solution in this paper a rigorous mapping is described for quadratic 0 1 programming problems with linear equality and inequality constraints this being the most general class of problem such networks can solve the problem s constraints define a polyhedron p containing all the valid solution points and the mapping guarantees strict confinement of the network s state vector to p however forcing convergence to a 0 1 point within p is shown to be generally intractable rendering the hopfield and similar models inapplicable to the vast majority of problems a modification of the tabu learning technique is presented as a more coherent approach to general problem solving with neural networks when tested on a collection of knapsack problems the modified dynamics produced some very encouraging results

a lagrange multiplier and hopfield type barrier function method for the traveling salesman problem a lagrange multiplier and hopfield type barrier function method is proposed for approximating a solution of the traveling salesman problem the method is derived from applications of lagrange multipliers and a hopfield type barrier function and attempts to produce a solution of high quality by generating a minimum point of a barrier problem for a sequence of descending values of the barrier parameter for any given value of the barrier parameter the method searches for a minimum point of the barrier problem in a feasible descent direction which has a desired property that lower and upper bounds on variables are always satisfied automatically if the step length is a number between zero and one at each iteration the feasible descent direction is found by updating lagrange multipliers with a globally convergent iterative procedure for any given value of the barrier parameter the method converges to a stationary point of the barrier problem without any condition on the objective function theoretical and numerical results show that the method seems more effective and efficient than the softassign algorithm

regularity for minimizers of double phase functionals with mild transition and regular coefficients abstract we prove sharp regularity results for minimizers of the functional p w b x w d w p a x d w p log e d w d x with w w 1 1 p 1 a l a 0 and 0 b l p is a double phase functional with mild transition between d u p and d u p log e d u first under suitable conditions on the moduli of continuity of a and b we prove that local minimizers are of class c 0 for every 0 1 then that they are of class c 1 for some 0 provided the functions a and b are holder continuous

splitting type variational problems with mixed linear superlinear growth conditions abstract splitting type variational problems with mixed linear superlinear growth conditions are considered in the two dimensional case the minimization problem is given by j w f 1 1 w f 2 2 w d x min with respect to a suitable class of comparison functions here f 1 is a convex energy density with linear growth and f 2 is a convex energy density with superlinear growth for instance given by an n function or just bounded from below by an n function one motivation for this kind of problem located between the well known splitting type problems of superlinear growth and the splitting type problems of linear growth recently considered by m bildhauer and m fuchs 2020 8 is the link to mathematical problems in plasticity see r temam 2018 29 here we prove results in the appropriate way of relaxation including approximation procedures duality and existence and uniqueness of solutions as well as some new higher integrability results

calderon zygmund estimates for higher order systems with p x growth for weak solutions u in w m 1 omega mathbb r n of higher order systems of the type int omega left dx int omega left dx for all phi in c infty c omega mathbb r n m 1 with variable growth exponent p 1 we prove that if f p cdot in l q loc omega with 1 0 and in the degenerate case 0

natural discriminant analysis using interactive potts models natural discriminant analysis based on interactive potts models is developed in this work a generative model composed of piece wise multivariate gaussian distributions is used to characterize the input space exploring the embedded clustering and mixing structures and developing proper internal representations of input parameters the maximization of a log likelihood function measuring the fitness of all input parameters to the generative model and the minimization of a design cost summing up square errors between posterior outputs and desired outputs constitutes a mathematical framework for discriminant analysis we apply a hybrid of the mean field annealing and the gradient descent methods to the optimization of this framework and obtain multiple sets of interactive dynamics which realize coupled potts models for discriminant analysis the new learning process is a whole process of component analysis clustering analysis and labeling analysis its major improvement compared to the radial basis function and the support vector machine is described by using some artificial examples and a real world application to breast cancer diagnosis

nonlinear calderon zygmund theory in the limiting case we prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems specifically we deal with the limiting case of the classical theory of calderon and zygmund in the setting of nonlinear possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions a prototype of the results obtained here tells for instance that if rm div du p 2 du mu quad mbox in omega subset mathbb r n with mu being a borel measure with locally finite mass on the open subset omega subset mathbb r n and p 2 1 n then du p 2 du in w sigma 1 rm loc omega quad mbox for every sigma in 0 1 the case sigma 1 is obviously forbidden already in the classical linear case of the poisson equation triangle u mu

the singular set of minima of integral functionals in this paper we provide upper bounds for the hausdorff dimension of the singular set of minima of general variational integrals open image in new window where f is suitably convex with respect to dv and holder continuous with respect to x v in particular we prove that the hausdorff dimension of the singular set is always strictly less than n where open image in new window

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

higher integrability for constrained minimizers of integral functionals with p q growth in low dimension abstract we prove higher summability for the gradient of minimizers of strongly convex integral functionals of the calculus of variations f x d u x d x u r n s n 1 with growth conditions of p q type p f x c q 1 p q in low dimension our procedure is set in the framework of fractional sobolev spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients

energy function analysis of dynamic programming neural networks all analytical examination of the energy function associated with a dynamic programming neural network is presented the analysis is carried out in two steps first the locations and numbers of the minimum states for different components of the energy function are investigated in the extreme cases a clearer insight into the energy function can be gained through the minimum states of different components secondly the locations of the minimum states of the energy function using different parameter values are derived it is shown that the minimum states can reside in regions which are regarded as valid solutions with certain conditions examples and simulation results are given to justify the validity of the theories developed

decompounding on compact lie groups noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound poisson processes on compact lie groups this problem of decompounding is a generalization of a similar classical problem the proposed solution is based on a characteristic function method the treated problem is important to recent models of the physical inverse problem of multiple scattering

gradient regularity for elliptic equations in the heisenberg group abstract we give dimension free regularity conditions for a class of possibly degenerate sub elliptic equations in the heisenberg group exhibiting super quadratic growth in the horizontal gradient this solves an issue raised in j j manfredi g mingione regularity results for quasilinear elliptic equations in the heisenberg group math ann 339 2007 485 544 where only dimension dependent bounds for the growth exponent are given we also obtain explicit a priori local regularity estimates and cover the case of the horizontal p laplacean operator extending some regularity proven in a domokos j j manfredi c 1 regularity for p harmonic functions in the heisenberg group for p near 2 in contemp math vol 370 2005 pp 17 23 in turn using some recent techniques of caffarelli and peral l caffarelli i peral on w 1 p estimates for elliptic equations in divergence form comm pure appl math 51 1998 1 21 the a priori estimates found are shown to imply the suitable local calderon zygmund theory for the related class of non homogeneous possibly degenerate equations involving discontinuous coefficients these last results extend to the sub elliptic setting a few classical non linear euclidean results t iwaniec projections onto gradient fields and l p estimates for degenerated elliptic operators studia math 75 1983 293 312 e dibenedetto j j manfredi on the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems amer j math 115 1993 1107 1134 and to the non linear case estimates of the same nature that were available in the sub elliptic setting only for solutions to linear equations

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

global gradient estimates for a borderline case of double phase problems with measure data abstract we obtain a global calderon zygmund estimate for a borderline case of double phase problems with measure data in terms of the 1 fractional maximal function of the measure

traditional heuristic versus hopfield neural network approaches to a car sequencing problem this paper considers the problem of optimally sequencing different car models along an assembly line according to some contiguity constraints while ensuring that the demands for each of the models are satisfied this car sequencing problem csp is a practical np hard combinatorial optimisation problem the csp is formulated as a nonlinear integer programming problem and it is shown that exact solutions to the problem are difficult to obtain due to the indefinite quadratic form of the csp objective function two traditional heuristics steepest descent and simulated annealing are employed to solve the csp approximately several hopfield neural network hnn approaches are also presented the process of mapping an optimisation problem onto a hnn is demonstrated explicitly and modifications to the existing neural approaches are presented which guarantee feasibility of solutions further modifications are proposed to improve the solution quality by permitting escape from local minima in an attempt to locate the global optimum results from all solutions techniques are compared on a set of instances of the csp and conclusions drawn

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

nonparametric estimation in renewal theory i the empirical renewal function we introduce a nonparametric estimator for the renewal function and discuss its properties including consistency asymptotic normality and asymptotic validity of bootstrap confidence regions the underlying theme is that stochastic models can be regarded as functionals or nonlinear operators this view leads to nonparametric estimators in a natural way and statistical properties of the estimators can be related to the local behaviour of the functionals

nonparametric estimation of the heterogeneity of a random medium using compound poisson process modeling of wave multiple scattering in this paper we present a nonparametric method to estimate the heterogeneity of a random medium from the angular distribution of intensity of waves transmitted through a slab of random material our approach is based on the modeling of forward multiple scattering using compound poisson processes on compact lie groups the estimation technique is validated through numerical simulations based on radiative transfer theory

linear and logarithmic capacities in associative neural networks a model of associate memory incorporating global linearity and pointwise nonlinearities in a state space of n dimensional binary vectors is considered attention is focused on the ability to store a prescribed set of state vectors as attractors within the model within the framework of such associative nets a specific strategy for information storage that utilizes the spectrum of a linear operator is considered in some detail comparisons are made between this spectral strategy and a prior scheme that utilizes the sum of kronecker outer products of the prescribed set of state vectors which are to function nominally as memories the storage capacity of the spectral strategy is linear in n the dimension of the state space under consideration whereas an asymptotic result of n 4 log n holds for the storage capacity of the outer product scheme computer simulated results show that the spectral strategy stores information more efficiently the preprocessing costs incurred in the two algorithms are estimated and recursive strategies are developed for their computation

inference for the limiting cluster size distribution of extreme values any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound poisson under appropriate long range dependence conditions typically exceedances appear in clusters the underlying poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes in the present paper we introduce estimators of the limiting cluster size probabilities which are constructed through a recursive algorithm we derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions we study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data

global lipschitz continuity for minima of degenerate problems we consider the problem of minimizing the lagrangian int f nabla u f u among functions on omega subset mathbb r n with given boundary datum varphi we prove lipschitz regularity up to the boundary for solutions of this problem provided omega is convex and varphi satisfies the bounded slope condition the convex function f is required to satisfy a qualified form of uniform convexity only outside a ball and no growth assumptions are made

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

a novel optimizing network architecture with applications we present a novel optimizing network architecture with applications in vision learning pattern recognition and combinatorial optimization this architecture is constructed by combining the following techniques 1 deterministic annealing 2 self amplification 3 algebraic transformations 4 clocked objectives and 5 softassign deterministic annealing in conjunction with self amplification avoids poor local minima and ensures that a vertex of the hypercube is reached algebraic transformations and clocked objectives help partition the relaxation into distinct phases the problems considered have doubly stochastic matrix constraints or minor variations thereof we introduce a new technique softassign which is used to satisfy this constraint experimental results on different problems are presented and discussed

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians