some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

a theorem on lower semicontinuity of integral functionals a general lower semicontinuity theorem in which not only mappings u m and p m but also the integrands f m depend on m is proved for integrands f f m under certain general hypotheses these include thai f x u p is convex with respect to p and f m converge to f locally uniformly but f m x u p are not required to be convex with respect to p and f m x do not even need to be lower semicontintious some more usable criteria for lower semicontinuity of integral funetionals are also given as corollaries of the main theorem

analysis of input output clustering for determining centers of rbfn the key point in design of radial basis function networks is to specify the number and the locations of the centers several heuristic hybrid learning methods which apply a clustering algorithm for locating the centers and subsequently a linear least squares method for the linear weights have been previously suggested these hybrid methods can be put into two groups which will be called as input clustering ic and input output clustering ioc depending on whether the output vector is also involved in the clustering process the idea of concatenating the output vector to the input vector in the clustering process has independently been proposed by several papers in the literature although none of them presented a theoretical analysis on such procedures but rather demonstrated their effectiveness in several applications the main contribution of this paper is to present an approach for investigating the relationship between clustering process on input output training samples and the mean squared output error in the context of a radial basis function network rbfn we may summarize our investigations in that matter as follows 1 a weighted mean squared input output quantization error which is to be minimized by ioc yields an upper bound to the mean squared output error 2 this upper bound and consequently the output error can be made arbitrarily small zero in the limit case by decreasing the quantization error which can be accomplished through increasing the number of hidden units

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

a condition for the holder regularity of local minimizers of a nonlinear elastic energy in two dimensions we prove the local holder continuity of strong local minimizers of the stored energy functional e u int omega lambda nabla u 2 h rm det nabla u rm d x subject to a condition of positive twist the latter turns out to be equivalent to requiring that u maps circles to suitably star shaped sets the convex function h s grows logarithmically as s to 0 linearly as s to infty and satisfies h s infty if s leqq 0 these properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers the main innovation is to prove that if a local minimizer has positive twist a e on a ball then an euler lagrange type inequality holds and a caccioppoli inequality can be derived from it the claimed holder continuity then follows by adapting some well known elliptic regularity theory we also demonstrate the regularizing effect that the term int omega h rm det nabla u rm d x can have by analysing the regularity of local minimizers in the class of shear maps in this setting a more easily verifiable condition than that of positive twist is imposed with the result that local minimizers are holder continuous

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

mean tracking clustering algorithm for radial basis function centre selection radial basis functions can be combined into a network structure that has several advantages over conventional neural network solutions however to operate effectively the number and positions of the basis function centres must be carefully selected although no rigorous algorithm exists for this purpose several heuristic methods have been suggested in this paper a new method is proposed in which radial basis function centres are selected by the mean tracking clustering algorithm the mean tracking algorithm is compared with k means clustering and it is shown that it achieves significantly better results in terms of radial basis function performance as well as being computationally simpler the mean tracking algorithm in general selects better centre positions thus providing the radial basis functions with better modelling accuracy

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

a mesh transformation method for computing microstructures a numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition numerical examples are given to show that the method especially when applied together with a continuation method and some other numerical techniques is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures

phase locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies abstract we study phase locking in a network of coupled nonlinear oscillators with local interactions and random intrinsic frequencies the oscillators are located at the vertices of a graph and interact along the edges they are coupled by sinusoidal functions of the phase differences across the edges and their intrinsic frequencies are independent and identically distributed with finite mean and variance we derive an exact expression for the probability of phase locking in a linear chain of such oscillators and prove that this probability tends to zero as the number of oscillators grows without bound however if the coupling strength increases as the square root of the number of oscillators the probability of phase locking tends to a limiting distribution the kolmogorov smirnov distribution this latter result is obtained by showing that the phase locking problem is equivalent to a discretization of pinned brownian motion the results on chains of oscillators are extended to more general graphs in particular for a hypercubic lattice of any dimension the probability of phase locking tends to zero exponentially fast as the number of oscillators grows without bound we also consider a less stringent type of synchronization characterized by large clusters of oscillators mutually entrained at the same average frequency it is shown that if such clusters exist they necessarily have a sponge like geometry

algorithms for minimal model structure detection in nonlinear dynamic system identification the minimal model structure detection mmsd problem in nonlinear dynamic system identification is formulated as a search for the optimal orthogonalization path while an exhaustive search for a model with 20 candidate terms would involve 2 43 1018 possible paths it is shown that this can typically be reduced to 2 103 by augmenting the orthogonal estimation algorithm with genetic search procedures the mmsd algorithm provides the first practical solution for optimal structure detection in narmax modelling training neural networks and fuzzy systems modelling based on the mmsd algorithm a refined forward regression orthogonal rfro algorithm is developed the rfro algorithm initially detects a parsimonious model structure using the forward regression orthogonal algorithm and then refines the model structure by applying the mmsd algorithm to the reduced model term set the rfro algorithm cannot guarantee to find the minimal model structure but it is computationally more efficient than the mmsd algori

shared kernel models for class conditional density estimation we present probabilistic models which are suitable for class conditional density estimation and can be regarded as shared kernel models where sharing means that each kernel may contribute to the estimation of the conditional densities of an classes we first propose a model that constitutes an adaptation of the classical radial basis function rbf network with full sharing of kernels among classes where the outputs represent class conditional densities in the opposite direction is the approach of separate mixtures model where the density of each class is estimated using a separate mixture density no sharing of kernels among classes we present a general model that allows for the expression of intermediate cases where the degree of kernel sharing can be specified through an extra model parameter this general model encompasses both the above mentioned models as special cases in all proposed models the training process is treated as a maximum likelihood problem and expectation maximization algorithms have been derived for adjusting the model parameters

genetic evolution of radial basis function coverage using orthogonal niches a well performing set of radial basis functions rbfs can emerge from genetic competition among individual rbfs genetic selection of the individual rbfs is based on credit sharing which localizes competition within orthogonal niches these orthogonal niches are derived using singular value decomposition and are used to apportion credit for the overall performance of the rbf network among individual nonorthogonal rbfs niche based credit apportionment facilitates competition to fill each niche and hence to cover the training data the resulting genetic algorithm yields rbf networks with better prediction performance on the mackey glass chaotic time series than rbf networks produced by the orthogonal least squares method and by k means clustering

single clustering synchronization in a ring of kuramoto oscillators multiple clustering synchronization is a common scenario of global phase synchronization however a novel single clustering type of phase synchronization in a ring of kuramoto oscillators has been recently reported in studying the influence of the permutation of the natural frequencies of oscillators on the synchronization efficiency wu et al 2012 europhys lett 97 40005 it was found that it occurs for a particular spatial frequency distribution and gives rise to a very small critical coupling strength kc even if the oscillator number is large here we focus on this particular type of synchronization and study its generality we provide some solid evidence for the convergence of kc to a small constant in the thermodynamic limit based on the finite size analysis further we demonstrate that it is robust in the sense of either switching the natural frequencies of any two oscillators or randomly perturbing the frequencies of all coupled oscillators all these findings prove that the single clustering synchronization is indeed generically observable with merit for potential engineering applications

analysis of augmented input layer rbfnn in this paper we present and analyze a new structure for designing a radial basis function neural network rbfnn in the training phase input layer of rbfnn is augmented with desired output vector generalization phase involves the following steps 1 identify the cluster to which a previously unseen input vector belongs 2 augment the input layer with an average of the targets of the input vectors in the identified cluster and 3 use the augmented network to estimate the unknown target it is shown that under some reasonable assumptions the generalization error function admits an upper bound in terms of the quantization errors minimized when determining the centers of the proposed method over the training set and the difference between training samples and generalization samples in a deterministic setting when the difference between the training and generalization samples goes to zero the upper bound can be made arbitrarily small by increasing the number of hidden neurons computer simulations verified the effectiveness of the proposed method

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

median radial basis function neural network radial basis functions rbfs consist of a two layer neural network where each hidden unit implements a kernel function each kernel is associated with an activation region from the input space and its output is fed to an output unit in order to find the parameters of a neural network which embeds this structure we take into consideration two different statistical approaches the first approach uses classical estimation in the learning stage and it is based on the learning vector quantization algorithm and its second order statistics extension after the presentation of this approach we introduce the median radial basis function mrbf algorithm based on robust estimation of the hidden unit parameters the proposed algorithm employs the marginal median for kernel location estimation and the median of the absolute deviations for the scale parameter estimation a histogram based fast implementation is provided for the mrbf algorithm the theoretical performance of the two training algorithms is comparatively evaluated when estimating the network weights the network is applied in pattern classification problems and in optical flow segmentation

layered neural networks with gaussian hidden units as universal approximations a neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real valued maps defined on convex compact sets of rn

on the efficiency of the orthogonal least squares training method for radial basis function networks the efficiency of the orthogonal least squares ols method for training approximation networks is examined using the criterion of energy compaction we show that the selection of basis vectors produced by the procedure is not the most compact when the approximation is performed using a nonorthogonal basis hence the algorithm does not produce the smallest possible networks for a given approximation error specific examples are given using the gaussian radial basis functions type of approximation networks

computation of the lower semicontinuous envelope of integral functionals and non homogeneous microstructures abstract a numerical method is established for computing the weakly lower semicontinuous envelope of integral functionals with non quasiconvex integrands the convergence of the method is proved and it is shown that the method is capable of capturing curved and non homogeneous microstructures numerical examples are given to show the effectiveness of the method for capturing curved and non homogeneous laminated microstructures

collective phase slips and phase synchronizations in coupled oscillator systems phase synchronization dynamics in coupled limit cycles with distributed natural frequencies are explored a synchronization tree from free oscillations to local clustering and global phase locking is found a desynchronization induced transition to chaos is shown near the onset of various phase synchronization points a simultaneous quantized stick slip feature of the phases of oscillators is observed on the desynchronization side and heuristically interpreted in terms of a heteroclinic path instability

laminated microstructure in a variational problem with a non rank one connected double well potential a variational problem with a non rank one connected double well potential is considered it is proved that the problem has a laminated microstructure solution which is uniquely determined by the two potential wells yet certainly not characterized by the fine oscillations between them because of the lack of rank one connection the laminated microstructure is explicitly worked out it is also shown that an application of a nonconforming finite element method can cause overrelaxation and fail to approximate the right microstructure

on rank one convex and polyconvex conformal energy functions with slow growth we make some remarks about rank one convex and polyconvex functions on the set of all real n n matrices that vanish on the subset k n consisting of all conformal matrices and grow like a power function at infinity we prove that every non negative rank one convex function that vanishes on k n and grows below a power of degree n 2 must vanish identically in odd dimensions n 3 we prove that every non negative polyconvex function that vanishes on k n must vanish identically if it grows below a power of degree n while in even dimensions such polyconvex functions can exist that also grow like a power of half dimension degree

simultaneous numerical approximation of microstructures and relaxed minimizers the problem of minimizing multiple integral functionals with nonquasiconvex integrands is considered a numerical method which is based on an alternative minimizing problem to the relaxed problem and thus uses no quasiconvex envelope of the integrands nor its numerical approximation in the computation is introduced to approximate simultaneously the highly oscillating minimizing sequences or in other words microstructures and the minimizers of the corresponding relaxed problem existence and convergence of the discrete solutions are proved and an error estimate is obtained a numerical example is given

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

an integral representation theorem for lower semicontinuous envelopes of integral functionals in this paper an integral representation theorem on sequentially weakly lower semicontinuous envelope of multiple integral functionals is proved for integrands which satisfy growth conditions of order p and have lower compactness property the result generalizes the standard results in the area

a new computational framework for electro activation in cardiac mechanics abstract this paper presents a novel computational framework for the numerical simulation of the electromechanical response of the myocardium during the cardiac cycle the paper presents the following main novelties 1 two new mixed formulations tailor made for active stress and active strain coupling approaches have been developed and used in conjunction with two different ionic models namely bueno orovio et al 2008 and ten tusscher et al 2004 taking as a reference the mixed formulations introduced by bonet et al 2015 in the context of nonlinear elasticity the proposed formulations include as unknown fields the geometry and the transmembrane potential and possibly a lagrange multiplier enforcing weakly the incompressibility constraint as well as the deformation gradient tensor its cofactor its determinant the gradient of the transmembrane potential and their respective work conjugates the finite element implementation of these formulations is shown in this paper where a static condensation procedure is presented in order to yield an extremely competitive computational approach 2 a comprehensive and rigorous study of different ionic models i e bueno orovio and ten tusscher and electromechanical activation couplings i e active strain and active stress has been carried out 3 an analytical and numerical analysis of the possible loss of ellipticity and polyconvexity of one of the most widely used constitutive models in the context of cardiac mechanics is carried out in this paper putting forward possible polyconvexifications of the existing model 4 in addition an invariant representation of guccione s constitutive model is proposed finally a series of numerical examples are included in order to demonstrate the applicability and robustness of the proposed formulations

reformulated radial basis neural networks trained by gradient descent this paper presents an axiomatic approach for constructing radial basis function rbf neural networks this approach results in a broad variety of admissible rbf models including those employing gaussian rbfs the form of the rbfs is determined by a generator function new rbf models can be developed according to the proposed approach by selecting generator functions other than exponential ones which lead to gaussian rbfs this paper also proposes a supervised learning algorithm based on gradient descent for training reformulated rbf neural networks constructed using the proposed approach a sensitivity analysis of the proposed algorithm relates the properties of rbfs with the convergence of gradient descent learning experiments involving a variety of reformulated rbf networks generated by linear and exponential generator functions indicate that gradient descent learning is simple easily implementable and produces rbf networks that perform considerably better than conventional rbf models trained by existing algorithms

synchronization in the kuramoto model a dynamical gradient network approach we propose a dynamical gradient network approach to consider the synchronization in the kuramoto model our scheme to adaptively adjust couplings is based on the dynamical gradient networks where the number of links in each time interval is the same as the number of oscillators but the links in different time intervals are also different the gradient network in the n 1 th time interval is decided by the oscillator dynamics in the nth time interval according to the gradient network in the n 1 th time interval only one inlink s coupling for each oscillator is adjusted by a small incremental coupling during the transition to synchronization the intensities for all oscillators are identical direct numerical simulations fully verify that the synchronization in the kuramoto model is realized effectively even if there exist delayed couplings and external noise

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

existence of minimizers and microstructure in nonlinear elasticity in this paper the existence of a solution in the form of a minimizer or microstructure is established for the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of st venant kirchhoff type materials necessary and sufficient conditions for minimizing sequences of the potential energy to converge to a minimizer or to microstructure are given

some properties of the degree for a class of sobolev maps we prove that a map u w loc1 1 m n whose differential and adjoint are integrable with exponents n 1 p n and n n 1 q respectively possesses an integer valued degree with the usual properties here m c 0 1 and n c 1 1 are n dimensional oriented riemannian manifolds with m possibly non empty

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

fast orthogonal identification of nonlinear stochastic models and radial basis function neural networks a new fast orthogonal estimation algorithm is derived for a wide class of nonlinear stochastic models including training radial basis function neural networks the selection of significant regressors and the estimation of unknown parameters in the presence of nonlinear noise sources are considered and simulated examples are included to demonstrate the efficiency of the new procedure

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer