frames reproducing kernels regularization and learning this work deals with a method for building a reproducing kernel hilbert space rkhs from a hilbert space with frame elements having special properties conditions on existence and a method of construction are given then these rkhs are used within the framework of regularization theory for function approximation implications on semiparametric estimation are discussed and a multiscale scheme of regularization is also proposed results on toy and real world approximation problems illustrate the effectiveness of such methods

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

pattern formation in the brusselator system in the paper we deal with a reaction diffusion system well known as the brusselator model and some improved results for the steady states of this model are presented we first give an a priori estimates positive upper and lower bounds of positive steady states then we obtain the non existence and existence of positive non constant steady states as the parameters and b are varied which means some certain conditions under which the pattern formation occurs or not

partially linear least squares regularized regression for system identification in this technical note we propose a partially linear least squares regularized regression pl lsrr method for system identification this method identifies a general nonlinear function as a sum of two functions which come from a linear and a nonlinear function space respectively both the linear and nonlinear functions can involve all regressors therefore the pl lsrr can make use of the partially linear structure of a given system to reduce prediction errors more efficiently than exiting partially linear identification methods two examples show that the pl lsrr can reduce prediction errors and estimate the true linear expansion of the system well

learning the kernel with hyperkernels this paper addresses the problem of choosing a kernel suitable for estimation with a support vector machine hence further automating machine learning this goal is achieved by defining a reproducing kernel hilbert space on the space of kernels itself such a formulation leads to a statistical estimation problem similar to the problem of minimizing a regularized risk functional we state the equivalent representer theorem for the choice of kernels and present a semidefinite programming formulation of the resulting optimization problem several recipes for constructing hyperkernels are provided as well as the details of common machine learning problems experimental results for classification regression and novelty detection on uci data show the feasibility of our approach

strange attractors in coupled reaction diffusion cells abstract a numerical study of two identical reaction cells with diffusion coupling has shown that the structure of motion in the system in principle agrees with results reported for variety of dynamic systems when the characteristic parameter is varied alternating sub intervals of stable periodic p k and stable aperiodic a l solutions appear the sub intervals are connected by intervals where tangent bifurcations and infinite sequences of subharmonic bifurcations occur feigenbaum relation holds for the studied sequence of subharmonic bifurcations aperiodic chaotic states are characterized by a complete set of one dimensional lyapunov exponents by power spectra and by corresponding poincare maps the spectra of lyapunov exponents are of the type 0 and show that the topological dimension of the chaotic attractor is two the power spectra are of two different types a the spectra with sharp peaks located above the broad band noise showing statistical phase coherence of the attractor b the flat spectra showing only broad band noise corresponding to phase incoherent attractor the phase coherence is present close after every point of accumulation phase incoherence arises when the strange attractors contain unstable periodic orbits with different topology the relations of bifurcated stable and unstable periodic solutions computed by means of continuation techniques to the structure of the strange attractor is discussed and poincare maps are used to illustrate the dependence of the structure of the attractors on the value of the characteristic parameter

a connection between quantum hilbert space and classical phase space operators we develop a general classicalization procedure that links hilbert space and phase space operators using weyl s operator then we transform the time dependent schrodinger equation into a phase space picture using free parameters they include position q and momentum p we expand the phase space hamiltonian in an taylor series and fix parameters with the condition that coefficients of 0 i1 q and i1 p vanish this condition results in generalized hamilton equations and a natural link between classical and quantum dynamics while the quantum motion equation remains exact in this picture the schrodinger equation reduces in the classical limit to a generalized liouville equation for the quantum mechanical system state we modify glauber s coherent states with a suitable phase factor s q p t and use them to obtain phase space representations of quantum dynamics and quantum mechanical quantities

mesa type patterns in the one dimensional brusselator and their stability the brusselator is a generic reaction diffusion model for a tri molecular chemical reaction we consider the case when the input and output reactions are slow in this limit we show the existence of k periodic spatially bi stable structures mesas and study their stability using singular perturbation techniques we find a threshold for the stability of k mesas this threshold occurs in the regime where the exponentially small tails of the localized structures start to interact by comparing our results with turing analysis we show that in the generic case a turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state we also study a breather type instability of a mesa which occurs due to a hopf bifurcation full numerical simulations are shown to confirm the analytical results in 1952 turing proposed that the formation of spatial patterns during morphogenesis could be explained in terms of the instability of a homogeneous steady state solution of a reaction diffusion network describing the evolution of a set of morphogens 31 turing himself illustrated his ideas on two chemical models turing s original work is primarily concerned with the stability analysis of the homogeneous steady state solution of the rate equations for the interacting morphogens 22 the main point of biological interest however is whether stable spatial structures may be generated beyond the instability i e whether the rate equations admit stable and positive inhomogeneous solutions exhibiting the most characteristic features of morphogenetic patterns this point has been taken up seriously in the early seventies more systematic numerical studies of turing s model have been performed showing irregular spatial structures 4 this led to serious reservations about the relevance of turing s theory in developmental biology particularly its ability to generate regular patterns but these criticisms originate from a somewhat unfortunate choice of turing s example and they do not touch the essential points of turing s theory more specifically the obvious requirement that the rate equations must admit positive and bounded solutions is not satisfied in turing s example 11

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

poincare lie algebra and relativistic phase space in this work we investigate representations of the poincare group taking as the representation space the hilbert space of thermofield dynamics a real time formalism developed in quantum field theory at finite temperature we concentrate our study on those representations that give rise to the notion of phase space with direct application in kinetic theory as a result we show an alternative way to derive a relativistic boltzmann equation based on the notion of a propagator defined in phase space the quantum counterpart of the approach is discussed through the notion of the wigner function

capacity of reproducing kernel spaces in learning theory the capacity of reproducing kernel hilbert spaces rkhs plays an essential role in the analysis of learning theory covering numbers and packing numbers of balls of these reproducing kernel spaces are important measurements of this capacity we first present lower bound estimates for the packing numbers by means of nodal functions then we show that if a mercer kernel is c sup s for some s 0 being not an even integer the rkhs associated with this kernel can be embedded into c sup s 2 this gives upper bound estimates for the covering number concerning sobolev smooth kernels examples and applications to v sub spl gamma dimension and tikhonov 1977 regularization are presented to illustrate the upper and lower bound estimates

coexistence states of a holling type ii predator prey system abstract this paper characterize the existence of coexistence states to a reaction diffusion predator prey model with holling type ii functional response subject to dirichlet boundary conditions we find the necessary and sufficient conditions for existence of coexistence states by fixed point index theory and bifurcation theory

stripes or spots nonlinear effects in bifurcation of reaction diffusion equations on the square bifurcation to spatial patterns in a two dimensional reaction diffusion medium is considered the selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model the absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns examples that include neural nets and more general pattern formation equations are considered

coexistence of a diffusive predator prey model with holling type ii functional response and density dependent mortality in this paper we consider a two competitor one prey diffusive model in which both competitors exhibit holling type ii functional response and one of the competitors exhibits density dependent mortality rate first we study the local and global existence of strong solution by using the c0 analytic semigroup then we consider the local and global stability of the positive constant equilibrium by using the linearization method and laypunov functional method respectively furthermore we derive some results for the existence and non existence of non constant stationary solutions when the diffusion rate of a certain species is small or large the existence of non constant stationary solutions implies the possibility of pattern formation in this system

on the symbiotic lotka volterra model with diffusion and transport effects abstract in this work we analyze the existence stability and multiplicity of coexistence states for a symbiotic lotka volterra model with general diffusivities and transport effects global bifurcation theory blowing up arguments for a priori bounds singular perturbation results singularity theory and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species our methodology works out to treat much more general classes of symbiotic models

learning rates of lq coefficient regularization learning with gaussian kernel regularization is a well recognized powerful strategy to improve the performance of a learning machine and lq regularization schemes with are central in use it is known that different q leads to different properties of the deduced estimators say l2 regularization leads to a smooth estimator while l1 regularization leads to a sparse estimator then how the generalization capability of lq regularization learning varies with q is worthy of investigation in this letter we study this problem in the framework of statistical learning theory our main results show that implementing lq coefficient regularization schemes in the sample dependent hypothesis space associated with a gaussian kernel can attain the same almost optimal learning rates for all that is the upper and lower bounds of learning rates for lq regularization learning are asymptotically identical for all our finding tentatively reveals that in some modeling contexts the choice of q might not have a strong impact on the generalization capability from this perspective q can be arbitrarily specified or specified merely by other nongeneralization criteria like smoothness computational complexity or sparsity

pattern formation in a general two cell brusselator model abstract in this article we are concerned with the following general coupled two cell brusselator type system d 1 u a b 1 u f u v c w u in d 2 v b u f u v in d 1 w a b 1 w f w z c u w in d 2 z b w f w z in u v w z 0 on here r n n 1 is a smooth and bounded domain a b c d 1 d 2 are positive constants and f c 1 0 c 0 is a nonnegative and nondecreasing function such that f 0 in 0 when f u u 2 this system corresponds to the coupled two cell brusselator model in the present work we exhibit the crucial role played by the nonlinearity f in generating the stationary patterns our conclusions show that if f has a sublinear growth then no stationary patterns occur while if f has a superlinear growth stationary patterns may exist

svm soft margin classifiers linear programming versus quadratic programming support vector machine svm soft margin classifiers are important learning algorithms for classification problems they can be stated as convex optimization problems and are suitable for a large data setting linear programming svm classifiers are especially efficient for very large size samples but little is known about their convergence compared with the well understood quadratic programming svm classifier in this article we point out the difficulty and provide an error analysis our analysis shows that the convergence behavior of the linear programming svm is almost the same as that of the quadratic programming svm this is implemented by setting a stepping stone between the linear programming svm and the classical 1 norm soft margin classifier an upper bound for the misclassification error is presented for general probability distributions explicit learning rates are derived for deterministic and weakly separable distributions and for distributions satisfying some tsybakov noise condition

symmetry groups density matrix equations and covariant wigner functions abstract a representation theory for lie groups is developed taking the hilbert space say h w of the w algebra standard representation as the representation space in this context the states describing physical systems are amplitude wave functions but closely connected with the notion of the density matrix then based on symmetry properties a general physical interpretation for the dual variables of thermal theories in particular the thermofield dynamics tfd formalism is introduced the kinematic symmetries galilei and poincare are studied and density amplitude matrix equations are derived for both of these cases in the same context of group theory the notion of phase space in quantum theory is analysed thus in the non relativistic situation the concept of density amplitude is introduced and as an example a spin half system is algebraically studied wigner function representations for the amplitude density matrices are derived and the connection of tfd and the usual wigner function methods are analysed for the poincare symmetries the relativistic density matrix equations are studied for the scalar and spinorial fields the relativistic phase space is built following the lines of the non relativistic case so for the scalar field the kinetic theory is introduced via the klein gordon density matrix equation and a derivation of the juttiner distribution is presented as an example thus making it possible to compare with the standard approaches the analysis of the phase space for the dirac field is carried out in connection with the dual spinor structure induced by the dirac field density matrix equation with the physical content relying on the symmetry groups gauge invariance is considered and as a basic result it is shown that the heinz density operator which has been used to develope a gauge covariant kinetic theory is a particular solution for the klein gordon and dirac density matrix equation

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

unified approach to coefficient based regularized regression abstract in this paper we consider the coefficient based regularized least squares regression problem with the l q regularizer 1 q 2 and data dependent hypothesis spaces algorithms in data dependent hypothesis spaces perform well with the property of flexibility we conduct a unified error analysis by a stepping stone technique an empirical covering number technique is also employed in our study to improve sample error comparing with existing results we make a few improvements first we obtain a significantly sharper learning rate that can be arbitrarily close to o m 1 under reasonable conditions which is regarded as the best learning rate in learning theory second our results cover the case q 1 which is novel finally our results hold under very general conditions

positive steady state solutions of the noyes field model for belousov zhabotinskii reaction abstract in the paper we investigate the noyes field model for belousov zhabotinskii reaction and study positive steady state solutions of this model with the homogeneous neumann boundary condition we obtain the existence and non existence of non constant positive steady state solutions

the learning rate of lp coefficient regularized shannon sampling algorithm in the present paper we provide an investigation on the learning rate of the shannon sampling algorithms with l p coefficient regularization we give the upper bounds for the sample error and the regularization error with the convex inequality in l p spaces and show the approximation error by a k functional whose convergence rate can be sum up to the translation network approximation basing on these estimates we show an explicit learning rate in possibility

learning with sample dependent hypothesis spaces many learning algorithms use hypothesis spaces which are trained from samples but little theoretical work has been devoted to the study of these algorithms in this paper we show that mathematical analysis for these algorithms is essentially different from that for algorithms with hypothesis spaces independent of the sample or depending only on the sample size the difficulty lies in the lack of a proper characterization of approximation error to overcome this difficulty we propose an idea of using a larger function class not necessarily linear space containing the union of all possible hypothesis spaces varying with the sample to measure the approximation ability of the algorithm we show how this idea provides error analysis for two particular classes of learning algorithms in kernel methods learning the kernel via regularization and coefficient based regularization we demonstrate the power of this approach by its wide applicability

on the connection between quantisation schemes and coherent states the authors analyse a possible connection between different quantisation schemes and the bargman segal realisation of the heisenberg algebra h they show that only a one parameter subfamily of the family of heisenberg algebras hq subduced from h h can be rewritten in the bargman segal form

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

least squares regression with l1 regularizer in sum space in this paper we propose a least squares regularized regression algorithm withl 1 regularizer in a sum space of some base hypothesis spaces this sum space contains more functions than single base hypothesis space and therefore has stronger approximation capability we establish an excess error bound for this algorithm under some assumptions on the kernels the input space the marginal distribution and the regression function for error analysis the excess error is decomposed into the sample error hypothesis error and regularization error which are estimated respectively from the excess error bound convergency and a learning rate can be derived by choosing a suitable value of the regularization parameter the utility of this method is illustrated with two simulated data sets and one real life database

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

a system of resource based growth models with two resources in the unstirred chemostat abstract models of single species growth in the unstirred chemostat on two growth limiting nonreproducing resources are considered for the case of two complementary resources the existence and uniqueness of a positive steady state solution is established it is also proved that the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values for the case of two substitutable resources the existence of a positive steady state solution is determined for a range of the parameter m n techniques include the maximum principle monotone method and global bifurcation theory the longtime behavior of the corresponding limiting system is given for a range of m n in the special case of m n the uniqueness and global attractivity of the positive steady state solution of the original system is established

on the relationship between generalization error hypothesis complexity and sample complexity for radial basis functions feedforward networks together with their training algorithms are a class of regression techniques that can be used to learn to perform some task from a set of examples the question of generalization of network performance from a finite training set to unseen data is clearly of crucial importance in this article we first show that the generalization error can be decomposed into two terms the approximation error due to the insufficient representational capacity of a finite sized network and the estimation error due to insufficient information about the target function because of the finite number of samples we then consider the problem of learning functions belonging to certain sobolev spaces with gaussian radial basis functions using the above mentioned decomposition we bound the generalization error in terms of the number of basis functions and number of examples while the bound that we derive is specific for radial basis functions a number of observations deriving from it apply to any approximation technique our result also sheds light on ways to choose an appropriate network architecture for a particular problem and the kinds of problems that can be effectively solved with finite resources i e with a finite number of parameters and finite amounts of data

pattern formation in the three dimensional reaction diffusion systems existing group theoretic analysis of pattern formation in three dimensions t k callahan e knobloch symmetry breaking bifurcations on cubic lattices nonlinearity 10 1997 1179 1216 is used to make specific predictions about the formation of three dimensional patterns in two models of the turing instability the brusselator model and the lengyel epstein model spatially periodic patterns having the periodicity of the simple cubic sc face centered cubic fcc or body centered cubic bcc lattices are considered an efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae sc fcc double diamond hexagonal prism bcc and bcci states both models possess a special wavenumber k at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams this property is generic for two species chemical reaction diffusion models with a single activator and inhibitor 1999 elsevier science b v all rights reserved

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

error analysis of coefficient based regularized algorithm for density level detection in this letter we consider a density level detection dld problem by a coefficient based classification framework with regularizer and data dependent hypothesis spaces although the data dependent characteristic of the algorithm provides flexibility and adaptivity for dld it leads to difficulty in generalization error analysis to overcome this difficulty an error decomposition is introduced from an established classification framework on the basis of this decomposition the estimate of the learning rate is obtained by using rademacher average and stepping stone techniques in particular the estimate is independent of the capacity assumption used in the previous literature

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

reproducing kernel banach spaces with the l 1 norm ii error analysis for regularized least square regression a typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error the hypothesis error and the regularization error using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically following this direction we illustrate how reproducing kernel banach spaces with the 1 norm can be applied to improve the learning rate estimate of 1 regularization in machine learning

least square regression with lp coefficient regularization the selection of the penalty functional is critical for the performance of a regularized learning algorithm and thus it deserves special attention in this article we present a least square regression algorithm based on lp coefficient regularization comparing with the classical regularized least square regression the new algorithm is different in the regularization term our primary focus is on the error analysis of the algorithm an explicit learning rate is derived under some ordinary assumptions

existence and instability of neumann layer solutions for a 3 component lotka volterra model with diffusion abstract we consider a 3 component lotka volterra model with diffusion which describes the dynamics of the population of two competing prey and one predator and we discuss the existence of positive stationary solutions and their stability property to do this the singular perturbation method and the associated singular limit spectral analysis are employed

is extreme learning machine feasible a theoretical assessment part i an extreme learning machine elm is a feedforward neural network fnn like learning system whose connections with output neurons are adjustable while the connections with and within hidden neurons are randomly fixed numerous applications have demonstrated the feasibility and high efficiency of elm like systems it has however been open if this is true for any general applications in this two part paper we conduct a comprehensive feasibility analysis of elm in part i we provide an answer to the question by theoretically justifying the following 1 for some suitable activation functions such as polynomials nadaraya watson and sigmoid functions the elm like systems can attain the theoretical generalization bound of the fnns with all connections adjusted i e they do not degrade the generalization capability of the fnns even when the connections with and within hidden neurons are randomly fixed 2 the number of hidden neurons needed for an elm like system to achieve the theoretical bound can be estimated and 3 whenever the activation function is taken as polynomial the deduced hidden layer output matrix is of full column rank therefore the generalized inverse technique can be efficiently applied to yield the solution of an elm like system and furthermore for the nonpolynomial case the tikhonov regularization can be applied to guarantee the weak regularity while not sacrificing the generalization capability in part ii however we reveal a different aspect of the feasibility of elm there also exists some activation functions which makes the corresponding elm degrade the generalization capability the obtained results underlie the feasibility and efficiency of elm like systems and yield various generalizations and improvements of the systems as well

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied

efficient sparse generalized multiple kernel learning kernel methods have been successfully applied in various applications to succeed in these applications it is crucial to learn a good kernel representation whose objective is to reveal the data similarity precisely in this paper we address the problem of multiple kernel learning mkl searching for the optimal kernel combination weights through maximizing a generalized performance measure most mkl methods employ the norm simplex constraints on the kernel combination weights which therefore involve a sparse but non smooth solution for the kernel weights despite the success of their efficiency they tend to discard informative complementary or orthogonal base kernels and yield degenerated generalization performance alternatively imposing the norm constraint on the kernel weights will keep all the information in the base kernels this leads to non sparse solutions and brings the risk of being sensitive to noise and incorporating redundant information to tackle these problems we propose a generalized mkl gmkl model by introducing an elastic net type constraint on the kernel weights more specifically it is an mkl model with a constraint on a linear combination of the norm and the squared norm on the kernel weights to seek the optimal kernel combination weights therefore previous mkl problems based on the norm or the norm constraints can be regarded as special cases furthermore our gmkl enjoys the favorable sparsity property on the solution and also facilitates the grouping effect moreover the optimization of our gmkl is a convex optimization problem where a local solution is the global optimal solution we further derive a level method to efficiently solve the optimization problem a series of experiments on both synthetic and real world datasets have been conducted to show the effectiveness and efficiency of our gmkl