conformally flat metrics on 4 manifolds we prove that for each closed smooth spin 4 manifold m there exists a closed smooth 4 manifold n such that m n admits a conformally flat riemannian metric

geometrically finite groups patterson sullivan measures and ratner s ridigity theorem the rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated ratner s theorem generalizes to these foliations with respect to the patterson sullivan measure in the spirit of mostow we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups finally we show that the frame flow on geometrically finite hyperbolic manifolds is bernoulli

explicit recursive and adaptive filtering in reproducing kernel hilbert spaces this brief presents a methodology to develop recursive filters in reproducing kernel hilbert spaces unlike previous approaches that exploit the kernel trick on filtered and then mapped samples we explicitly define the model recursivity in the hilbert space for that we exploit some properties of functional analysis and recursive computation of dot products without the need of preimaging or a training dataset we illustrate the feasibility of the methodology in the particular case of the gamma filter which is an infinite impulse response filter with controlled stability and memory depth different algorithmic formulations emerge from the signal model experiments in chaotic and electroencephalographic time series prediction complex nonlinear system identification and adaptive antenna array processing demonstrate the potential of the approach for scenarios where recursivity and nonlinearity have to be readily combined

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces olshausen and field 1996 applied the principle of independence maximization by sparse coding to extract features from natural images this leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency thus resembling gabor functions and simple cell receptive fields in this article we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features similar to those found in complex cells this new kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces instead of the independence of simple linear filter outputs the norms of the projections on such independent feature subspaces then indicate the values of invariant features

second order directional derivatives of spectral functions a spectral function of a symmetric matrix x is a function which depends only on the eigenvalues of x l 1 x l 2 x l n x and may be written as l 1 x l 2 x l n x for some symmetric function in this paper we assume that is a c 1 1 function and discuss second order directional derivatives of such a spectral function we obtain an explicit expression of second order directional derivative for the spectral function

regularized nonparametric volterra kernel estimation abstract in this paper the regularization approach introduced recently for nonparametric estimation of linear systems is extended to the estimation of nonlinear systems modeled as volterra series the kernels of order higher than one representing higher dimensional impulse responses in the series are considered to be realizations of multidimensional gaussian processes based on this prior information about the structure of the volterra kernel is introduced via an appropriate penalization term in the least squares cost function it is shown that the proposed method is able to deliver accurate estimates of the volterra kernels even in the case of a small amount of data points

second order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization abstract in this paper we introduce a new second order directional derivative and a second order subdifferential of hadamard type for an arbitrary nondifferentiable function we derive second order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second order in unconstrained optimization in particular we obtain two results with strongly pseudoconvex functions we also compare our conditions with the results of the recently published paper bednarik and pastor 2008 and a lot of other works published in high level journals and prove that they are particular cases of our necessary and sufficient ones we prove that the necessary optimality conditions concern more functions than the conditions in terms of lower dini directional derivative even the optimality conditions with the last derivative can be applied to a function which does not belong to some special class at last we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone constrained vector optimization

on scalar and vector l stable functions abstract the notion of a scalar function that is l stable at a point is introduced in d bedna ik k pastor on second order conditions in unconstrained optimization math program ser a 113 2008 283 298 in the present paper a characterization of the l stable functions is obtained further the notion of an l stable function is generalized from scalar to vector functions in an application optimality conditions for constrained vector problems with l stable data are established

a statistical framework to infer delay and direction of information flow from measurements of complex systems in neuroscience data are typically generated from neural network activity the resulting time series represent measurements from spatially distributed subsystems with complex interactions weakly coupled to a high dimensional global system we present a statistical framework to estimate the direction of information flow and its delay in measurements from systems of this type informed by differential topology gaussian process regression is employed to reconstruct measurements of putative driving systems from measurements of the driven systems these reconstructions serve to estimate the delay of the interaction by means of an analytical criterion developed for this purpose the model accounts for a range of possible sources of uncertainty including temporally evolving intrinsic noise while assuming complex nonlinear dependencies furthermore we show that if information flow is delayed this approach also allows for inference in strong coupling scenarios of systems exhibiting synchronization phenomena the validity of the method is demonstrated with a variety of delay coupled chaotic oscillators in addition we show that these results seamlessly transfer to local field potentials in cat visual cortex

a technical note concerning one example abstract our aim is to correct one mistake that appeared in nonlinear anal theory methods appl 26 1996 55 example 2 1 and example 5 1 and nonlinear anal theory methods appl 36 1999 595 example 2 2 in calculation of certain cominetti correa generalized second order derivative introduced in siam j control optim 28 1990 789

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

second order moderate derivatives f x f x it is the purpose of the present paper to introduce a notion of second order generalized derivative for arbitrary finite functions which retains the preceding properties in particular which coincides with the usual second order derivative when it exists and enjoys useful calculus rules our proposal is close to the proposal of 3 cf also 4 the variant we use is similar in spirit to the first order variant of the clarke circa derivatives introduced in l 21 here however the concept of fan plays a key role as in 3 51 let us note that the absence of built in upper semicontinuity is intimately linked with the accuracy of the concept under added assumptions of semicontinuity one recovers the notions of 3 51 our results can be found in 6

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

second order subdifferentials and convexity of real valued functions abstract in this paper we show that the positive semi definiteness psd of the frechet and or mordukhovich second order subdifferentials can recognize the convexity of c 1 functions however the psd is insufficient for ensuring the convexity of a locally lipschitz function in general a complete characterization of strong convexity via the second order subdifferentials is also given

a unifying view of wiener and volterra theory and polynomial kernel regression volterra and wiener series are perhaps the best understood nonlinear system representations in signal processing although both approaches have enjoyed a certain popularity in the past their application has been limited to rather low dimensional and weakly nonlinear systems due to the exponential growth of the number of terms that have to be estimated we show that volterra and wiener series can be represented implicitly as elements of a reproducing kernel hilbert space by using polynomial kernels the estimation complexity of the implicit representation is linear in the input dimensionality and independent of the degree of nonlinearity experiments show performance advantages in terms of convergence interpretability and system sizes that can be handled

penalized least squares estimation of volterra filters and higher order statistics volterra filters vfs and higher order statistics hos are important tools for nonlinear analysis processing and modeling despite their highly desirable properties the transfer of vfs and hos to real world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates the identification of vfs and the estimation of hos both fall into the category of ill posed estimation problems we develop penalized least squares pls estimation methods for vfs and hos it is shown that pls is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure hence pls produces much more reliable estimates the main contributions of this paper are the development of appropriate penalizing functionals and cross validation procedures for pls based vf identification and hos estimation

some applications of variational inequalities in nonsmooth analysis abstract in this paper we apply some results on variational inequalities to the clarke generalized gradient and the generalized second order directional derivative f x u v and obtain some variational inequalities of them

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

first and second order epi differentiability in eigenvalue optimization abstract the subject of this work is the first and second order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi differentiability we show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi differentiable and we derive an explicit expression of its second order epi derivative we also prove that the mth largest eigenvalue function is twice epi differentiable if and only if it ranks first in a group of equal eigenvalues finally we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

dissipative control system for the stochastic nonlinear h problems abstract we characterize functions satisfying a dissipative inequality associated with a stochastic control problem such a characterization is provided in terms of an upper generalized gaussian solution or a viscosity supersolution to a partial differential equation called hamilton jacobi equation h j links between upper generalized gaussian solutions and viscosity supersolutions to hamilton jacobi equation are studied finally it shows that generalized gaussian solutions is identical to viscosity solutions to hamilton jacobi equation

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

kernel based structural equation models for topology identification of directed networks structural equation models sems have been widely adopted for inference of causal interactions in complex networks recent examples include unveiling topologies of hidden causal networks over which processes such as spreading diseases or rumors propagate the appeal of sems in these settings stems from their simplicity and tractability since they typically assume linear dependencies among observable variables acknowledging the limitations inherent to adopting linear models the present paper put forth nonlinear sems which account for possible nonlinear dependencies among network nodes the advocated approach leverages kernels as a powerful encompassing framework for nonlinear modeling and an efficient estimator with affordable tradeoffs is put forth interestingly pursuit of the novel kernel based approach yields a convex regularized estimator that promotes edge sparsity a property exhibited by most real world networks and the resulting optimization problem is amenable to proximal splitting optimization methods to this end solvers with complementary merits are developed by leveraging the alternating direction method of multipliers and proximal gradient iterations experiments conducted on simulated data demonstrate that the novel approach outperforms linear sems with respect to edge detection errors furthermore tests on a real gene expression dataset unveil interesting new edges that were not revealed by linear sems which could shed more light on regulatory behavior of human genes

flat twistor spaces conformally flat manifolds and four dimensional field theory a definition is proposed of four dimensional conformal field theory in which the riemann surfaces of two dimensional cft are replaced by riemannian conformally flat four manifolds and the holomorphic functions are replaced by solutions of the dirac equation the definition is investigated from the point of view of twistor theory allowing methods from complex analysis to be employed the paper fills in the main mathematical details omitted from the preliminary announcement 15

approximate jacobian matrices for nonsmooth continuous maps and c 1 optimization the notion of approximate jacobian matrices is introduced for a continuous vector valued map it is shown for instance that the clarke generalized jacobian is an approximate jacobian for a locally lipschitz map the approach is based on the idea of convexificators of real valued functions mean value conditions for continuous vector valued maps and taylor s expansions for continuously g teaux differentiable functions i e c1 functions are presented in terms of approximate jacobians and approximate hessians respectively second order necessary and sufficient conditions for optimality and convexity of c1 functions are also given

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

the hausdorff dimension of the limit set of a geometrically finite kleinian group we consider in this paper groups of mobius transformations of l r tj zc such a group is called kleinian if it acts discontinuously somewhere in 1 the action of g extends to the n d dimensional hyperbolic space h i r zc and g is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of g in h t for a more precise definition see 15 1 b we prove in this paper that the hausdor f dimension dim l g of the limit set l g of a geometrically finite kleinian group g of 1 is less than n theorem d our proof of this theorem is based on the following obser ation assume that g is of compact t3 pe i e if izi i h iui then i l g g is compact then there is an integer q such that if we divide any n cube q of r into q equal subcubes then at least one of these subcubes does not touch l g let l4 q be the family of these subcubes which touch l g then the n measures of q e le q do not add up to the n measure of q and we get

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

on lower bounds of the second order directional derivatives of ben tal zowe and chaney let f be a regular locally lipschitz real valued function defined on an open convex subset of a normed space we show that at any unit direction u the upper second order derivative d 2f u 0 in the sense of dem yanov and pevnyi dem yanov v f a b pevnyi 1974 expansion with respect to a parameter of the extremal values of game problems ussr computational math and math phys 14 33 45 ben tal and zowe ben tal a j zowe 1982 necessary and sufficient optimality conditions for a class of nonsmooth minimization problems math programming24 70 91 has the same lower bounds as the lower second order derivatives d 2f u 0 consequently one can characterize the convexity of f in terms of these derivatives we also obtain the corresponding results in terms of chaney s second order directional derivatives

support functions of the clarke generalized jacobian and of its plenary hull this paper studies two important mathematical objects which are useful in tackling the first order behaviour of vector valued locally lipschitz functions in a finite dimensional setting the clarke generalized jacobian and its plenary hull we aim at giving analytical expressions of the support functions of these compact convex sets of matrices our study was motivated by earlier works by j b hiriart urruty and recent papers by zs pales and v zeidan the expressions of the support functions are applied for instance to provide a new proof of a chain rule on generalized jacobians of composed locally lipschitz functions without further assumptions applications of our results to the second order behaviour of c 1 functions with locally lipschitz gradients are considered

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

on partial smoothness tilt stability and the mathcal vu decomposition under the assumption of prox regularity and the presence of a tilt stable local minimum we are able to show that a mathcal vu like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly