on equivalence of fis and elm for interpretable rule based knowledge representation this paper presents a fuzzy extreme learning machine f elm that embeds fuzzy membership functions and rules into the hidden layer of extreme learning machine elm similar to the concept of elm that employed the random initialization technique three parameters of f elm are randomly assigned they are the standard deviation of the membership functions matrix c rule combination matrix and matrix d don t care dc matrix fuzzy if then rules are formulated by the rule combination matrix of f elm and a dc approach is adopted to minimize the number of input attributes in the rules furthermore f elm utilizes the output weights of the elm to form the target class and confidence factor for each of the rules this is to indicate that the corresponding consequent parameters are determined analytically the operations of f elm are equivalent to a fuzzy inference system several benchmark data sets and a real world fault detection and diagnosis problem have been used to empirically evaluate the efficacy of the proposed f elm in handling pattern classification tasks the results show that the accuracy rates of f elm are comparable if not superior to elm with distinctive ability of providing explicit knowledge in the form of interpretable rule base

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in euclidean and minkowski spaces abstract in this paper we investigate the existence of rotationally symmetric entire graphs resp entire spacelike graphs with prescribed k th mean curvature function in euclidean space r n 1 resp minkowski spacetime l n 1 as a previous step we analyze the associated homogeneous dirichlet problem on a ball which is not elliptic for k 1 and then we prove that it is possible to extend the solutions moreover a sufficient condition for uniqueness is given in both cases

lines in space times we construct a complete timelike maximal geodesic line in a timelike geodesically complete spacetimem containing a compact acausal spacelike hypersurfaces which lies in the past of somes ray ans ray is a future complete geodesic starting ons which maximizes lorentzian distance froms to any of its points if the timelike convergence condition strong energy condition holds a line exists only ifm is static i e it splits geometrically as space time so timelike completeness must fail for a nonstatic spacetime with strong energy condition which contains a closed universe s with the above properties

the lorentzian splitting theorem without the completeness assumption on demontre le theoreme de decomposition lorentzien pour des espace temps globalement hyperboliques obeissant a la condition d energie forte sans hypothese de completude geodesique du genre temps

bayesian takagi sugeno kang fuzzy classifier in this paper the takagi sugeno kang tsk fuzzy classifier is casted into the bayesian inference framework and a new fuzzy classifier called bayesian tsk fuzzy classifier b tsk fc is proposed accordingly the proposed classifier can be constructed by learning both the antecedent and consequent parameters of the involved fuzzy rules simultaneously as a result of the introduction of bayesian inference the proposed b tsk fc can be distinguished as follows 1 unlike most existing tsk fuzzy classifiers where the antecedent and consequent parameters of fuzzy rules are learnt in a decoupled manner and the antecedent parameters are learnt only in the input space the antecedent parameters of the fuzzy rules in b tsk fc are learnt by developing a fuzzy clustering method in the input output space and the consequent parameters of fuzzy rules are learnt in accordance with the maximum margin of separation principle thereby resulting to form an intrinsic link between the input and output spaces to achieve improved classification performance and better interpretability 2 with a dirichlet prior assumption about fuzzy memberships in fuzzy clustering a markov chain monte carlo technique is employed to estimate the parameters of the proposed classifier from a sampling perspective 3 rather than being rivals fuzziness and probability in b tsk fc are collaboratively modeled to enhance the performance of tsk fuzzy classifier in terms of classification and interpretability our experimental results in synthetic datasets as well as several real world datasets confirm such merits of the proposed fuzzy classifier

equivalence of the spinor and tensor methods in the positive energy problem we prove that nontrivial solutions of the sen witten equation with asymptotically flat data set on maximal hypersurface does not equal zero at any point of this hypersurface on this basis we ascertain the equivalence of witten s spinor method and nester s tensor method in the positive energy problem in general relativity

remarks on cosmological spacetimes and constant mean curvature surfaces we give a general condition which ensures the existence of constant mean curvature cmc cauchy surfaces in cosmological spacetimes however there is an example of a spacetime which does not satisfy this condition and does not admit any cmc cauchy surfaces we discuss conditions under which cmc surfaces may exist

analysis of boundedness and convergence of online gradient method for two layer feedforward neural networks this paper presents a theoretical boundedness and convergence analysis of online gradient method for the training of two layer feedforward neural networks the well known linear difference equation is extended to apply to the general case of linear or nonlinear activation functions based on this extended difference equation we investigate the boundedness and convergence of the parameter sequence of concern which is trained by finite training samples with a constant learning rate we show that the uniform upper bound of the parameter sequence which is very important in the training procedure is the solution of an inequality regarding the bound it is further verified that for the case of linear activation function a solution always exists and moreover the parameter sequence can be uniformly upper bounded while for the case of nonlinear activation function some simple adjustment methods on the training set or the activation function can be derived to improve the boundedness property then for the convergence analysis it is shown that the parameter sequence can converge into a zone around an optimal solution at which the error function attains its global minimum where the size of the zone is associated with the learning rate particularly for the case of perfect modeling a strong global convergence result where the parameter sequence can always converge to an optimal solution is proved

hypersurfaces of prescribed mean curvature in lorentzian manifolds we give a new existence proof for closed hypersurfaces of prescribed mean curvature in lorentzian manifolds

geometric analysis of lorentzian distance function on spacelike hypersurfaces some analysis on the lorentzian distance in a spacetime with controlled sectional or ricci curvatures is done in particular we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the omori yau maximum principle as a consequence and under appropriate hypotheses on the sectional or ricci curvatures of the ambient spacetime we obtain sharp estimates for the mean curvature of those hypersurfaces moreover we also give a sufficient condition for its hyperbolicity

positive rotationally symmetric solutions for a dirichlet problem involving the higher mean curvature operator in minkowski space abstract in this paper we investigate the global behavior of positive rotationally symmetric solutions for dirichlet problem involving the k th mean curvature operator in minkowski space l n 1 which is not elliptic for k 1 the proofs of our main results are based upon bifurcation techniques

online steepest descent yields weights with nonnormal limiting distribution we study the asymptotic properties of the sequence of iterates of weight vector estimates obtained by training a feedforward neural network with a basic gradient descent method using a fixed learning rate and no batch processing earlier results based on stochastic approximation techniques kuan and hornik 1991 finnoff 1993 bucklew et al 1993 have established the existence of a gaussian limiting distribution for the weights but they apply only in the limiting case of a zero learning rate we here prove from an exact analysis of the one dimensional case and constant learning rate weak convergence to a distribution that is not gaussian in general we also run simulations to compare and contrast the results of our analysis with those of stochastic approximation

parameter convergence and learning curves for neural networks we revisit the oft studied asymptotic in sample size behavior of the parameter or weight estimate returned by any member of a large family of neural network training algorithms by properly accounting for the characteristic property of neural networks that their empirical and generalization errors possess multiple minima we rigorously establish conditions under which the parameter estimate converges strongly into the set of minima of the generalization error convergence of the parameter estimate to a particular value cannot be guaranteed under our assumptions we then evaluate the asymptotic distribution of the distance between the parameter estimate and its nearest neighbor among the set of minima of the generalization error results on this question have appeared numerous times and generally assert asymptotic normality the conclusion expected from familiar statistical arguments concerned with maximum likelihood estimators these conclusions are usually reached on the basis of somewhat informal calculations although we shall see that the situation is somewhat delicate the preceding results then provide a derivation of learning curves for generalization and empirical errors that leads to bounds on rates of convergence

the isometry groups of asymptotically flat asymptotically empty space times with timelike adm four momentum we give a complete classification of all connected isometry groups together with their actions in the asymptotic region in asymptotically flat asymptotically vacuum space times with timelike adm four momentum

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

maximal slicings in spherical symmetry local existence and construction we show that any spherically symmetric spacetime locally admits a maximal space like slicing and we give a procedure allowing its construction the designed construction procedure is based on purely geometrical arguments and in practice leads to solve a decoupled system of first order quasi linear partial differential equations we have explicitly built up maximal foliations in minkowski and friedmann spacetimes our approach admits further generalizations and efficient computational implementation as by product we suggest some applications of our work in the task of calibrating numerical relativity complex codes usually written in cartesian coordinates

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

maximal hypersurfaces in stationary asymptotically flat spacetimes existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike near infinity the first class consists of strongly causal asymptotically flat spacetimes which contain no black hole or white hole but may contain ergoregions where the killing orbits fail to be timelike the second class of spacetimes possess a black hole and a white hole with the black and white hole horizons intersecting in a compact 2 surfaces

convergence of learning algorithms with constant learning rates the behavior of neural network learning algorithms with a small constant learning rate epsilon in stationary random input environments is investigated it is rigorously established that the sequence of weight estimates can be approximated by a certain ordinary differential equation in the sense of weak convergence of random processes as epsilon tends to zero as applications backpropagation in feedforward architectures and some feature extraction algorithms are studied in more detail

interpolation by ridge polynomials and its application in neural networks we consider multivariate interpolation at arbitrary points and provide effective interpolation formulas by using simple ridge polynomials which essentially possess the nature of univariate polynomials in computation and show that ridge polynomials with least degree can be found for interpolation purpose which a computational algorithm is provided correspondingly we then apply the interpolation results to design a computational algorithm for training feedforward neural networks

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

some results about the level sets of lorentzian busemann function and bartnik s conjecture some results about the level sets of busemann functions are obtained for spacetimes and in a special case cosmological spacetime these results will be used to prove the conjecture stated by r bartnik in b under some special conditions in this paper we employ some results of galloway horta and eschenburg

convergence analyses on sparse feedforward neural networks via group lasso regularization in this paper a new variant of feedforward neural networks has been proposed for a class of nonsmooth optimization problems the penalty term of the presented neural networks stems from the group lasso method which selects hidden variables in a grouped manner to deal with the non differentiability of the original penalty term l 1 l 2 norm and avoid oscillations smoothing techniques have been used to approximate the objective function it is assumed that the training samples are supplied to the networks in a specific incremental way during training that is in each cycle samples are supplied in a fixed order then under suitable assumptions on learning rate penalization coefficients and smoothing parameters the weak and strong convergence of the training process for the smoothing neural networks have been proved the convergence analysis shows that the gradient of the smoothing error function approaches zero and the weight sequence converges to a fixed point respectively we demonstrate how the smoothing approximation parameter can be updated in the training procedure so as to guarantee the convergence of the procedure to a clarke stationary point of the original optimization problem in addition we have proved that the original nonsmoothing algorithm with l 1 l 2 norm penalty converges consistently to the same optimum solution with the corresponding smoothed algorithm numerical simulations demonstrate the convergence and effectiveness of the proposed training algorithm

splitting theorems for spatially closed space times a lorentzian splitting theorem is obtained for spatially closed spacetimes the proof employs and extends some recent results of bartnik and gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space times a splitting theorem for spatially closedtime periodic space times which generalizes a result first considered by avez is derived as a corollary

an online gradient method with momentum for two layer feedforward neural networks an online gradient method with momentum for two layer feedforward neural networks is considered the momentum coefficient is chosen in an adaptive manner to accelerate and stabilize the learning procedure of the network weights corresponding convergence results are proved that is the weak convergence result is proved under the uniformly boundedness assumption of the activation function and its derivatives moreover if the number of elements of the stationary point set for the error function is finite then strong convergence result holds

a general existence proof for non linear elliptic equations in semi riemannian spaces we present a general existence proof for a wide class of non linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions as an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient riemannian manifold n without supposing any sign condition on the sectional curvatures k n a curvature flow wouldn t work in this situation neither the method of successive approximation

functional equivalence between radial basis function networks and fuzzy inference systems it is shown that under some minor restrictions the functional behavior of radial basis function networks rbfns and that of fuzzy inference systems are actually equivalent this functional equivalence makes it possible to apply what has been discovered learning rule representational power etc for one of the models to the other and vice versa it is of interest to observe that two models stemming from different origins turn out to be functionally equivalent

on maximal surfaces in asymptotically flat space times existenc of maximal and almost maximal hypersurfaces in asymptotically flat space times is established under boundary conditions weaker than those considered previously we show in particular that every vacuum evolution of asymptotically flat data for the einstein equations can be foliated by slices maximal outside a spatially compact set and that every strictly stationary asymptotically flat space time can be foliated by maximal hypersurfaces amongst other uniqueness results we show that maximal hypersurfaces can be used to partially fix an asymptotic poincare group

the localised bounded l 2 l2 curvature theorem in this paper we prove a localised version of the bounded l 2 curvature theorem of klainerman et al invent math 202 1 91 216 2015 more precisely we consider initial data for the einstein vacuum equations posed on a compact spacelike hypersurface sigma with boundary and show that the time of existence of a classical solution depends only on an l 2 bound on the ricci curvature an l 4 bound on the second fundamental form of partial sigma subset sigma an h 1 bound on the second fundamental form and a lower bound on the volume radius at scale 1 of sigma our localisation is achieved by first proving a localised bounded l 2 curvature theorem for small data posed on b 0 1 and then using the scaling of the einstein equations and a low regularity covering argument on sigma to reduce from large data on sigma to small data on b 0 1 the proof uses the author s previous works and the bounded l 2 curvature theorem as black boxes

when does online bp training converge the backpropogation bp neural networks have been widely applied in scientific research and engineering the success of the application however relies upon the convergence of the training procedure involved in the neural network learning we settle down the convergence analysis issue through proving two fundamental theorems on the convergence of the online bp training procedure one theorem claims that under mild conditions the gradient sequence of the error function will converge to zero the weak convergence and another theorem concludes the convergence of the weight sequence defined by the procedure to a fixed value at which the error function attains its minimum the strong convergence the weak convergence theorem sharpens and generalizes the existing convergence analysis conducted before while the strong convergence theorem provides new analysis results on convergence of the online bp training procedure the results obtained reveal that with any analytic sigmoid activation function the online bp training procedure is always convergent which then underlies successful application of the bp neural networks

rotationally symmetric spacelike translating solitons for the mean curvature flow in minkowski space abstract the rotationally symmetric spacelike translating solitons for the mcf in the minkowski space are considered in this paper to be specific the complete classification for such translating solitons according to their rotation axes which are timelike spacelike and lightlike is provided in particular the explicit parametrization of the rotationally symmetric spacelike hypersurface of the lightlike rotation axis with the prescribed mean curvature is obtained

regularity of lorentzian busemann functions a general theory of regularity for lorentzian busemann functions in future timelike geodesically complete spacetimes is presented this treatment simplifies and extends the local regularity developed by eschenburg galloway and newman to prove the lorentzian splitting theorem criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of bartnik

spacelike hypersurfaces with prescribed boundary values and mean curvature we consider the boundary value problem for the mean curvature operator in minkowski space and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions our main results hold for non constant mean curvature and make no assumptions about the smoothness of the boundary or boundary data

stable maximal hypersurfaces in lorentzian spacetimes abstract we study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions for instance the timelike convergence condition tcc we characterize stability when the target space has constant sectional curvature as well as give sufficient conditions on the geometry of the ambient spacetime e g the validity of tcc to ensure stability some rigidity results and height estimates are also proven in grw spacetimes in the last part of the paper we consider k stability of spacelike hypersurfaces a concept related to mean curvatures of higher orders

decomposition theorems for lorentzian manifolds with nonpositive curvature soit m g un espace temps globalement hyperbolique de dimension 2 avec partout des courbures sectionnelles du genre temps non positives k 0 qui contient une ligne du genre temps complete m g alors m g est isometrique a un produit r h dt 2 h ou h h est une variete de riemann complete le facteur r dt 2 est represente par et h h est represente par un ensemble de niveau d une fonction de busemann associee a

neuro fuzzy rule generation survey in soft computing framework the present article is a novel attempt in providing an exhaustive survey of neuro fuzzy rule generation algorithms rule generation from artificial neural networks is gaining in popularity in recent times due to its capability of providing some insight to the user about the symbolic knowledge embedded within the network fuzzy sets are an aid in providing this information in a more human comprehensible or natural form and can handle uncertainties at various levels the neuro fuzzy approach symbiotically combining the merits of connectionist and fuzzy approaches constitutes a key component of soft computing at this stage to date there has been no detailed and integrated categorization of the various neuro fuzzy models used for rule generation we propose to bring these together under a unified soft computing framework moreover we include both rule extraction and rule refinement in the broader perspective of rule generation rules learned and generated for fuzzy reasoning and fuzzy control are also considered from this wider viewpoint models are grouped on the basis of their level of neuro fuzzy synthesis use of other soft computing tools like genetic algorithms and rough sets are emphasized rule generation from fuzzy knowledge based networks which initially encode some crude domain knowledge are found to result in more refined rules finally real life application to medical diagnosis is provided

shadowing property in analysis of neural networks dynamics we show that the shadowing and the inverse shadowing the notions considered in the theory of dynamical systems may be successfully applied in the analysis of a multilayer neural networks learning process our main result is that generically any such a process is robust implementation implications are discussed as well

deterministic convergence of an online gradient method for neural networks the online gradient method has been widely used as a learning algorithm for neural networks we establish a deterministic convergence of online gradient methods for the training of a class of nonlinear feedforward neural networks when the training examples are linearly independent we choose the learning rate to be a constant during the training procedure the monotonicity of the error function in the iteration is proved a criterion for choosing the learning rate is also provided to guarantee the convergence under certain conditions similar to those for the classical gradient methods an optimal convergence rate for our online gradient methods is proved

convergence of an online gradient method for feedforward neural networks with stochastic inputs in this paper we study the convergence of an online gradient method for feed forward neural networks the input training examples are permuted stochastically in each cycle of iteration a monotonicity and a weak convergence of deterministic nature are proved

curvature estimates for stable marginally trapped surfaces we derive local integral and sup estimates for the curvature of stable marginally outer trapped surfaces in a sliced space time the estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface these estimates are well adapted to situations of physical interest such as dynamical horizons