impulsive control of sticking motion in van der pol one sided constraint system in this paper the impulsive control method is used to dominate sticking motion in van der pol one sided constraint system we find that the method can restrain sticking motion effectively and the method can also induce sticking motion for special parameters in addition numerical results show that the method is stable even for high levels of multiplicative noise or additive noise

an example of unitary equivalence between self adjoint extensions and their parameters abstract the spectral problem for self adjoint extensions is studied using the machinery of boundary triplets for a class of symmetric operators having weyl functions of a special type we calculate explicitly the spectral projections in the form of operator valued integrals this allows one to give a constructive proof of the fact that in certain intervals the resulting self adjoint extensions are unitarily equivalent to the parameterizing boundary operator acting in a smaller space and one is able to provide an explicit form for the associated unitary transform applications to differential operators on metric graphs and to direct sums are discussed

an index of topological preservation for feature extraction abstract this paper is about the ability of principal components analysis the sammon algorithm and an extension of the kohonen self organizing feature map to preserve spatial order during feature extraction on unlabeled data transformations to q space that preserve the order of all pairwise distances in any set of vectors in p space are defined as metric topology preserving mtp transformations we give a necessary and sufficient condition for this new property in terms of the spearman rank correlation coefficient unlike many other measures of extracted feature quality the mtp index is independent of the extraction method a modification of the kohonen self organizing feature map algorithm that extracts vectors in q space from data in p space is developed the extent to which principal components sammon s algorithm and our extension of the self organizing feature map sofm preserve the mtp property is discussed our mtp index shows that the first two methods preserve distance ranks on seven data sets much more effectively than extended sofm

quantifying the neighborhood preservation of self organizing feature maps it is shown that a topographic product p first introduced in nonlinear dynamics is an appropriate measure of the preservation or violation of neighborhood relations it is sensitive to large scale violations of the neighborhood ordering but does not account for neighborhood ordering distortions caused by varying areal magnification factors a vanishing value of the topographic product indicates a perfect neighborhood preservation negative positive values indicate a too small too large output space dimensionality in a simple example of maps from a 2d input space onto 1d 2d and 3d output spaces it is demonstrated how the topographic product picks the correct output space dimensionality in a second example 19d speech data are mapped onto various output spaces and it is found that a 3d output space instead of 2d seems to be optimally suited to the data this is an agreement with a recent speech recognition experiment on the same data set

explicit magnification control of self organizing maps for forbidden data in this paper we examine the scope of validity of the explicit self organizing map som magnification control scheme of bauer 1996 on data for which the theory does not guarantee success namely data that are n dimensional nges2 and whose components in the different dimensions are not statistically independent the bauer algorithm is very attractive for the possibility of faithful representation of the probability density function pdf of a data manifold or for discovery of rare events among other properties since theoretically unsupported data of higher dimensionality and higher complexity would benefit most from the power of explicit magnification control we conduct systematic simulations on forbidden data for the unsupported n 2 cases that we investigate the simulations show that even though the magnification exponent alphaachieved achieved by magnification control is not the same as the desired alphadesired alphaachieved systematically follows alphadesired with a slowly increasing positive offset we show that for simple synthetic higher dimensional data information theoretically optimum pdf matching alphaachieved 1 can be achieved and that negative magnification has the desired effect of improving the detectability of rare classes in addition we further study theoretically unsupported cases with real data

development of oriented ocular dominance bands as a consequence of areal geometry it has been hypothesized that the different appearance of ocular dominance bands in the cat and the monkey is a consequence of the different mapping geometries in these species levay et al 1985 anderson et al 1988 here i investigate the impact of areal geometries on the preferred direction of ocular dominance bands in two adaptive map formation models the self organizing feature map and the elastic net algorithm in the case of the self organizing feature map the occurrence of instabilities that correspond to ocular dominance bands can be analytically investigated the instabilities automatically yield stripes of correct orientation these analytic results are complemented by simulations in the case of the elastic net algorithm simulations reveal two different parameter regimes of the algorithm only one of which leads to stripes of correct orientation the results suggest that neighborhood preservation in visual maps is enforced in the backward direction such that neighboring cells in the cortex have neighboring receptive fields and not vice versa

on the distribution and convergence of feature space in self organizing maps in this paper an analysis of the statistical and the convergence properties of kohonen s self organizing map of any dimension is presented every feature in the map is considered as a sum of a number of random variables we extend the central limit theorem to a particular case which is then applied to prove that the feature space during learning tends to multiple gaussian distributed stochastic processes which will eventually converge in the mean square sense to the probabilistic centers of input subsets to form a quantization mapping with a minimum mean squared distortion either globally or locally the diminishing effect as training progresses of the initial states on the value of the feature map is also shown

on spectral theory for schrodinger operators with operator valued potentials abstract given a complex separable hilbert space h we consider differential expressions of the type d 2 d x 2 v x with x a or x r here v denotes a bounded operator valued potential v b h such that v is weakly measurable and the operator norm v b h is locally integrable we consider self adjoint half line l 2 realizations h in l 2 a d x h associated with assuming a to be a regular endpoint necessitating a boundary condition of the type sin u a cos u a 0 indexed by the self adjoint operator b h in addition we study self adjoint full line l 2 realizations h of in l 2 r d x h in either case we treat in detail basic spectral theory associated with h and h including weyl titchmarsh theory green s function structure eigenfunction expansions diagonalization and a version of the spectral theorem

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

variants of self organizing maps self organizing maps have a bearing on traditional vector quantization a characteristic that makes them more closely resemble certain biological brain maps however is the spatial order of their responses which is formed in the learning process a discussion is presented of the basic algorithms and two innovations dynamic weighting of the input signals at each input of each cell which improves the ordering when very different input signals are used and definition of neighborhoods in the learning algorithm by the minimal spanning tree which provides a far better and faster approximation of prominently structured density functions it is cautioned that if the maps are used for pattern recognition and decision process it is necessary to fine tune the reference vectors so that they directly define the decision borders

kalman filter implementation of self organizing feature maps the self organizing learning algorithm of kohonen and most of its extensions are controlled by two learning parameters the learning coefficient and the width of the neighborhood function which have to be chosen empirically because neither rules nor methods for their calculation exist consequently often time consuming parameter studies precede neighborhood preserving feature maps of the learning data to circumvent those lengthy numerical studies this article describes the learning process by a state space model in order to use the linear kalman filter algorithm training the feature maps then the kalman filter equations calculate the learning coefficient online during the training while the width of the neighborhood function needs to be estimated by a second extended kalman filter for the process of neighborhood preservation the performance of the kalman filter implementation is demonstrated on toy problems as well as on a crab classification problem the results of crab classification are compared to those of generative topographic mapping an alternative method to the self organizing feature map

topology preservation and cooperative learning in identification of multiple model systems the self organizing network son based multiple model system is a recently proposed method for identifying the dynamics of a general nonlinear system it has been observed by researchers that cooperative learning among neighboring regions is sometimes important for the success of identification of a nonlinear system under the multiple model system framework in this paper we intend to formally evaluate the effects of cooperative learning and topology preservation in identification of multiple model system based on son the results of the mathematical analysis supports the heuristic that a good learning strategy for identifying the local model parameters of a son based multiple model system is to choose a neighborhood function whose effective region is initially wider and is reduced gradually during learning an example of nonlinear function approximation is also provided at the end of this paper to demonstrate the results of the mathematical analysis

boundary triples and weyl m functions for powers of the jacobi differential operator abstract the abstract theory of boundary triples is applied to the classical jacobi differential operator and its powers in order to obtain the weyl m function for several self adjoint extensions with interesting boundary conditions separated periodic and those that yield the friedrichs extension these matrix valued nevanlinna herglotz m functions are to the best knowledge of the author the first explicit examples to stem from singular higher order differential equations the creation of the boundary triples involves taking pieces determined in 26 of the principal and non principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space these maps act like quasi derivatives which are usually not well defined for all functions in the maximal domain of singular expressions however well defined regularizations of quasi derivatives are produced by putting the pieces of the non principal solutions through a modified gram schmidt process

corner bifurcations in non smoothly forced impact oscillators abstract we consider an impacting mechanical system in which a particle at position u t impacts with a periodically moving obstacle at position z t the motion of which is non smooth in particular we look at corner events when u impacts with z very close to a point where z loses smoothness we show that this leads through a corner bifurcation to complex dynamics in u which can include periodic orbits of arbitrary period and period adding cascades by analysing associated maps close to the corner event we show that this dynamics can be understood in terms of the iterations of a two dimensional piecewise linear discontinuous map we also show some links between this analysis and the difficult problem of understanding the motion of three objects which may have simultaneous impacts

on the spectral theory of gesztesy seba realizations of 1 d dirac operators with point interactions on a discrete set abstract we investigate spectral properties of gesztesy seba realizations d x and d x of the 1 d dirac differential expression d with point interactions on a discrete set x x n n 1 r here n n 1 and n n 1 r the gesztesy seba realizations d x and d x are the relativistic counterparts of the corresponding schrodinger operators h x and h x with and interactions respectively we define the minimal operator d x as the direct sum of the minimal dirac operators on the intervals x n 1 x n then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator d x in the case d x inf x i x j i j 0 it turns out that the boundary operators b x and b x parameterizing the realizations d x and d x are jacobi matrices these matrices substantially differ from the ones appearing in spectral theory of schrodinger operators with point interactions we show that certain spectral properties of the operators d x and d x correlate with the corresponding spectral properties of the jacobi matrices b x and b x respectively using this connection we investigate spectral properties self adjointness discreteness absolutely continuous and singular spectra of gesztesy seba realizations moreover we investigate the non relativistic limit as the velocity of light c most of our results are new even in the case d x 0

topographic map formation by maximizing unconditional entropy a plausible strategy for online unsupervised competitive learning and nonparametric density estimation an unsupervised competitive learning rule called the vectorial boundary adaptation rule vbar is introduced for topographic map formation since vbar is aimed at producing an equiprobable quantization of the input space it yields a nonparametric model of the input probability density function furthermore since equiprobable quantization is equivalent to unconditional entropy maximization we argue that this is a plausible strategy for maximizing mutual information shannon information rate in the case of online learning we use mutual information as a tool for comparing the performance of our rule with kohonen s self organizing feature map algorithm

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

face recognition a convolutional neural network approach we present a hybrid neural network for human face recognition which compares favourably with other methods the system combines local image sampling a self organizing map som neural network and a convolutional neural network the som provides a quantization of the image samples into a topological space where inputs that are nearby in the original space are also nearby in the output space thereby providing dimensionality reduction and invariance to minor changes in the image sample and the convolutional neural network provides partial invariance to translation rotation scale and deformation the convolutional network extracts successively larger features in a hierarchical set of layers we present results using the karhunen loeve transform in place of the som and a multilayer perceptron mlp in place of the convolutional network for comparison we use a database of 400 images of 40 individuals which contains quite a high degree of variability in expression pose and facial details we analyze the computational complexity and discuss how new classes could be added to the trained recognizer

stochastic responses of vibro impact duffing oscillator excited by additive gaussian noise the stationary responses of vibro impact duffing oscillator excited by additive gaussian white noise are studied by using the quasi conservative averaging method with the help of a non smooth variable transformation and the dirac delta function the response probability density functions pdfs are formulated analytically meanwhile the results are validated numerically using monte carlo simulations

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

growing a hypercubical output space in a self organizing feature map neural maps project data from an input space onto a neuron position in a often lower dimensional output space grid in a neighborhood preserving way with neighboring neurons in the output space responding to neighboring data points in the input space a map learning algorithm can achieve an optimal neighborhood preservation only if the output space topology roughly matches the effective structure of the data in the input space we here present a growth algorithm called the gsom or growing self organizing map which enhances a widespread map self organization process kohonen s self organizing feature map sofm by an adaptation of the output space grid during learning the gsom restricts the output space structure to the shape of a general hypercubical shape with the overall dimensionality of the grid and its extensions along the different directions being subject of the adaptation this constraint meets the demands of many larger information processing systems of which the neural map can be a part we apply our gsom algorithm to three examples two of which involve real world data using recently developed methods for measuring the degree of neighborhood preservation in neural maps we find the gsom algorithm to produce maps which preserve neighborhoods in a nearly optimal fashion

conformality in the self organization network abstract the conformality of the self organizing network is studied in this work we use multi dimensional deformation analyses to interpret the self organizing mapping it can be shown that this mapping is quasi conformal with a convergent deformation bound based on analyses a deformation measure and a non conformality measure are derived to indicate the evolution status of the network these measures can serve as new criteria to evolve the network we test these measures with simulations on surface mapping problems

on a generalisation of krein s example abstract we generalise a classical example given by krein in 1953 we compute the difference of the resolvents and the difference of the spectral projections explicitly we further give a full description of the unitary invariants i e of the spectrum and the multiplicity moreover we observe a link between the difference of the spectral projections and hankel operators

local spectral properties of reflectionless jacobi cmv and schrodinger operators we prove that jacobi cmv and schrodinger operators which are reflectionless on a homogeneous set e in the sense of carleson under the assumption of a blaschke type condition on their discrete spectra accumulating at e have purely absolutely continuous spectrum on e

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

the effect of concave and convex weight adjustments on self organizing maps two nonlinear models of weight adjustments of self organizing maps are derived to obtain desirable densities of output units one that approaches the probability distribution p spl xi of the inputs and one that approaches a uniform distribution if a convex model is used to adjust weights the density of output units can be made to approach p spl xi instead of the p spl xi sup 2 3 which results from the linear weight adjustment of kohonen s self organizing maps if a concave model of weight adjustments is used the density approaches a uniform distribution and the winner frequency distribution of output units is proportional to p spl xi the former can provide more efficient data representations for vector quantization while the latter can provide more meaningful measures for cluster analysis numerical demonstrations validate the mathematical derivations the convergence of the concave model is faster than the linear and convex models while the convergence of the convex model is comparable to that of the linear model

spherical schrodinger operators with δ type interactions we investigate spectral properties of spherical schrodinger operators also known as bessel operators with point interactions concentrated on a discrete set we obtain necessary and sufficient conditions for these hamiltonians to be self adjoint lower semibounded and also we investigate their spectra we also extend the classical bargmann estimate to such hamiltonians in certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities we apply our results to schrodinger operators in l2 rn with a singular interaction supported by an infinite family of concentric spheres

on the unitary equivalence of absolutely continuous parts of self adjoint extensions the classical weyl von neumann theorem states that for any self adjoint operator a0 in a separable hilbert space h there exists a non unique hilbert schmidt operator c c such that the perturbed operator a0 c has purely point spectrum we are interesting whether this result remains valid for non additive perturbations by considering the set exta of self adjoint extensions of a given densely defined symmetric operator a in h and some fixed a0 a0 exta we show that the ac parts a ac and a0ac of a a exta and a0 are unitarily equivalent provided that the resolvent difference ka a i 1 a0 i 1 is compact and the weyl function m of the pair a a0 admits weak boundary limits m t w limy 0m t iy for a e t r this result generalizes the classical kato rosenblum theorem moreover it demonstrates that for such pairs a a0 the weyl von neumann theorem is in general not true in the class exta

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space