multivariate rational approximants for multiclass closed queuing networks closed markovian networks of queues with multiclass customers and having a product form equilibrium state probability distribution are useful in the performance evaluation and design of computer and telecommunication systems therefore the efficient computation of the normalizing function the key element of the solution in product form has attracted considerable effort we consider a network that consists of one infinite server is station and one processor sharing ps or fcfs single server station we use multivariate newton pade approximants computed from data for small numbers of customers in each class to estimate the normalizing function for a larger population in the network the effectiveness and tremendous gain in computing time of this procedure are illustrated through various numerical experiments

slow feature analysis a theoretical analysis of optimal free responses temporal slowness is a learning principle that allows learning of invariant representations by extracting slowly varying features from quickly varying input signals slow feature analysis sfa is an efficient algorithm based on this principle and has been applied to the learning of translation scale and other invariances in a simple model of the visual system here a theoretical analysis of the optimization problem solved by sfa is presented which provides a deeper understanding of the simulation results obtained in previous studies

a lower bound for nodal count on discrete and metric graphs we study the number of nodal domains maximal connected regions on which a function has constant sign of the eigenfunctions of schrodinger operators on graphs under a certain genericity condition we show that the number of nodal domains of the nth eigenfunction is bounded below by n l where l is the number of links that distinguish the graph from a tree

computing the region of convergence for power series in many real variables a ratio like test abstract we give an elementary proof that the region of convergence for a power series in many real variables is a star convex domain but not in general a convex domain in doing so we deduce a natural higher dimensional analog of the so called ratio test from univariate power series from the constructive proof of this result we arrive at a method to approximate the region of convergence up to a desired accuracy while most results in the literature are for rather specialized classes of multivariate power series the method devised here is general as far as applications are concerned note that while theorems such as the cauchy kowalevski theorem and its generalizations to many variables grant the existence of a region of convergence for a multivariate taylor series to certain pdes under appropriate restrictions they do not give the actual region of convergence the determination of the maximal region of convergence for such a series solution to a pde is one application of our result

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

slow feature analysis unsupervised learning of invariances invariant features of temporally varying signals are useful for analysis and classification slow feature analysis sfa is a new method for learning invariant or slowly varying features from a vectorial input signal it is based on a nonlinear expansion of the input signal and application of principal component analysis to this expanded signal and its time derivative it is guaranteed to find the optimal solution within a family of functions directly and can learn to extract a large number of decorrelated features which are ordered by their degree of invariance sfa can be applied hierarchically to process high dimensional input signals and extract complex features sfa is applied first to complex cell tuning properties based on simple cell output including disparity and motion then more complicated input output functions are learned by repeated application of sfa finally a hierarchical network of sfa modules is presented as a simple model of the visual system the same unstructured network can learn translation size rotation contrast or to a lesser degree illumination invariance for one dimensional objects depending on only the training stimulus surprisingly only a few training objects suffice to achieve good generalization to new objects the generated representation is suitable for object recognition performance degrades if the network is trained to learn multiple invariances simultaneously

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

how well can the concept of pade approximant be generalized to the multivariate case what we know about multivariate pade approximation has been developed in the last 25 years in the next sections we compare and discuss many of these results it will become clear that simple properties or requirements such as the uniqueness of the pade approximant and consequently its consistency property can play a crucial role in the development of the multivariate theory a separate section is devoted to a discussion of the convergence properties at the end we include an extensive reference list on the topic

learning invariance from transformation sequences the visual system can reliably identify objects even when the retinal image is transformed considerably by commonly occurring changes in the environment a local learning rule is proposed which allows a network to learn to generalize across such transformations during the learning phase the network is exposed to temporal sequences of patterns undergoing the transformation an application of the algorithm is presented in which the network learns invariance to shift in retinal position such a principle may be involved in the development of the characteristic shift invariance property of complex cells in the primary visual cortex and also in the development of more complicated invariance properties of neurons in higher visual areas

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

sparse interpolation of multivariate rational functions consider the black box interpolation of a sparse n variate rational function f where is the maximum number of terms in either numerator or denominator when numerator and denominator are at most of degree d then the number of possible terms in f is o dn and explodes exponentially as the number of variables increases the complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore it still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree we remove the exponent n and introduce the sparsity in the complexity by reconstructing the auxiliary function s coefficients via sparse multivariate interpolation the approach is new and builds on the normalization of the rational function s representation our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment the latter is illustrated with several examples running from exact finite field arithmetic to noisy floating point evaluations in general the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation if the early termination ben or tiwari algorithm is used our method achieves rational interpolation in o d black box evaluations and thus is sensitive to the sparsity of the multivariate f

a recursive computation scheme for multivariate rational interpolants we derive here a recursive computation scheme for the rational interpolation method introduced in 7 explicit formulas for these multivariate rational interpolants are repeated in 2 while the recursive algorithm is described in 3 a number of interesting special cases such as the univariate rational interpolation problem and the multivariate pade approximants introduced in 6 and 10 are dealt with in 4 for some of these rational approximants other recursive schemes were described previously finally 5 contains the numerical results the multivariate rational interpolants described here are compared with multivariate polynomial interpolants interpolating branched continued fractions introduced by cuyt and verdonk 8 interpolating branched continued fractions introduced by siemaszko 12 and several multivariate pade approximants 6 3 10 besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in ord

pade approximation for stochastic discrete event systems we show that pade approximation can be effectively used for approximation of performance functions in discrete event systems the method is 1 obtaining the maclaurin coefficients of the performance function and 2 finding a pade approximant from the maclaurin coefficients and use it to approximate the function we use the method with the expected number of renewals in a random interval gi g 1 systems and inventory systems the results are very good

the role of constraints in hebbian learning models of unsupervised correlation based hebbian synaptic plasticity are typically unstable either all synapses grow until each reaches the maximum allowed strength or all synapses decay to zero strength a common method of avoiding these outcomes is to use a constraint that conserves or limits the total synaptic strength over a cell we study the dynamic effects of such constraints two methods of enforcing a constraint are distinguished multiplicative and subtractive for otherwise linear learning rules multiplicative enforcement of a constraint results in dynamics that converge to the principal eigenvector of the operator determining unconstrained synaptic development subtractive enforcement in contrast typically leads to a final state in which almost all synaptic strengths reach either the maximum or minimum allowed value this final state is often dominated by weight configurations other than the principal eigenvector of the unconstrained operator multiplicative enforcement yields a graded receptive field in which most mutually correlated inputs are represented whereas subtractive enforcement yields a receptive field that is sharpened to a subset of maximally correlated inputs if two equivalent input populations e g two eyes innervate a common target multiplicative enforcement prevents their segregation ocular dominance segregation when the two populations are weakly correlated whereas subtractive enforcement allows segregation under these circumstances these results may be used to understand constraints both over output cells and over input cells a variety of rules that can implement constrained dynamics are discussed

a maximum likelihood interpretation for slow feature analysis the brain extracts useful features from a maelstrom of sensory information and a fundamental goal of theoretical neuroscience is to work out how it does so one proposed feature extraction strategy is motivated by the observation that the meaning of sensory data such as the identity of a moving visual object is often more persistent than the activation of any single sensory receptor this notion is embodied in the slow feature analysis sfa algorithm which uses slowness as a heuristic by which to extract semantic information from multidimensional time series here we develop a probabilistic interpretation of this algorithm showing that inference and learning in the limiting case of a suitable probabilistic model yield exactly the results of sfa similar equivalences have proved useful in interpreting and extending comparable algorithms such as independent component analysis for sfa we use the equivalent probabilistic model as a conceptual springboard with which to motivate several novel extensions to the algorithm

learning perceptually salient visual parameters using spatiotemporal smoothness constraints a model is presented for unsupervised learning of low level vision tasks such as the extraction of surface depth a key assumption is that perceptually salient visual parameters e g surface depth vary smoothly over time this assumption is used to derive a learning rule that maximizes the long term variance of each unit s outputs whilst simultaneously minimizing its short term variance the length of the half life associated with each of these variances is not critical to the success of the algorithm the learning rule involves a linear combination of anti hebbian and hebbian weight changes over short and long time scales respectively this maximizes the information throughput with respect to low frequency parameters implicit in the input sequence the model is used to learn stereo disparity from temporal sequences of random dot and gray level stereograms containing synthetically generated subpixel disparities the presence of temporal discontinuities in disparity does not prevent learning or generalization to previously unseen image sequences the implications of this class of unsupervised methods for learning in perceptual systems are discussed

removing time variation with the anti hebbian differential synapse i describe a local synaptic learning rule that can be used to remove the effects of certain types of systematic temporal variation in the inputs to a unit according to this rule changes in synaptic weight result from a conjunction of short term temporal changes in the inputs and the output formally this is like the differential rule proposed by klopf 1986 and kosko 1986 except for a change of sign which gives it an anti hebbian character by itself this rule is insufficient a weight conservation condition is needed to prevent the weights from collapsing to zero and some further constraint implemented here by a biasing term to select particular sets of weights from the subspace of those which give minimal variation as an example i show that this rule will generate center surround receptive fields that remove temporally varying linear gradients from the inputs

nonlinear dimensionality reduction using a temporal coherence principle temporal coherence principle is an attractive biologically inspired learning rule to extract slowly varying features from quickly varying input data in this paper we develop a new nonlinear neighborhood preserving nnp technique by utilizing the temporal coherence principle to find an optimal low dimensional representation from the original high dimensional data nnp is based on a nonlinear expansion of the original input data such as polynomials of a given degree it can be solved by the eigenvalue problem without using gradient descent and is guaranteed to find the global optimum nnp can be viewed as a nonlinear dimensionality reduction framework which takes into consideration both time series and data sets without an obvious temporal structure according to different situations we introduce three algorithms of nnp named nnp 1 nnp 2 and nnp 3 the objective function of nnp 1 is equal to slow feature analysis sfa and it works well for time series such as image sequences nnp 2 artificially constructs time series consisting of neighboring points for data sets without a clear temporal structure such as image data nnp 3 is proposed for classification tasks which can minimize the distances of neighboring points in the embedding space and ensure that the remaining points are as far apart as possible simultaneously furthermore the kernel extension of nnp is also discussed in this paper the proposed algorithms work very well on some image sequences and image data sets compared to other methods meanwhile we perform the classification task on the mnist handwritten digit database using the supervised nnp algorithms the experimental results demonstrate that nnp is an effective technique for nonlinear dimensionality reduction tasks