convergence of rank type equations abstract convergence results are presented for rank type difference equations whose evolution rule is defined at each step as the k th largest of p univariate difference equations if the univariate equations are individually contractive then the equation converges to a fixed point equal to the k th largest of the individual fixed points of the univariate equations examples are max type equations for k 1 and the median of an odd number p of equations for k p 1 2 in the non hyperbolic case conjectures are stated about the eventual periodicity of the equations generalizing long standing conjectures of g ladas

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

the analytical solution of the additive constant problem ifd is a measure of dissimilarity on a finite set withn elements the smallest positive constantc such thatd c has an euclidean representation for allc c is shown to be the largest eigen value of a matrix of size 2n

density based geodesic distance for identifying the noisy and nonlinear clusters we propose a density based geodesic distance that resists noise for clustering the proposed distance is calculated based on the neighborhood graph for nonlinearity simulation and real case data demonstrate the accuracy of the proposed distance clustering analysis can facilitate the extraction of implicit patterns in a dataset and elicit its natural groupings without requiring prior classification information for superior clustering analysis results a number of distance measures have been proposed recently geodesic distance has been widely applied to clustering algorithms for nonlinear groupings however geodesic distance is sensitive to noise and hence geodesic distance based clustering may fail to discover nonlinear clusters in the region of the noise in this study we propose a density based geodesic distance that can identify clusters in nonlinear and noisy situations experiments on various simulation and benchmark datasets are conducted to examine the properties of the proposed geodesic distance and to compare its performance with that of existing distance measures the experimental results confirm that a clustering algorithm with the proposed distance measure demonstrated superior performance compared to the competitors this was especially true when the cluster structures in the data were inherently noisy and nonlinearly patterned

impulse output rapid stabilization for heat equations abstract the main aim of this paper is to provide a new feedback law for the heat equations in a bounded domain with dirichlet boundary condition two constraints will be compulsory first the controls are active in a subdomain of and at discrete time points second the observations are made in another subdomain and at different discrete time points our strategy consists in linking an observation estimate at one time minimal norm impulse control approximate inverse source problem and rapid output stabilization

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

uniform controllability of semidiscrete approximations of parabolic control systems controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model in the present paper under the main assumptions that the discretized semigroup is uniformly analytic and that the control operator is mildly unbounded we prove that the semidiscrete approximation models are uniformly controllable moreover we provide a computationally efficient way to compute the approximation controls an example of application is implemented for the one and two dimensional heat equation with neumann boundary control

uniform controllability properties for space time discretized parabolic equations this article is concerned with the analysis of semi discrete in space and fully discrete approximations of the null controllability and controllability to the trajectories for parabolic equations we propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters for time discretization we use schemes with theta in frac 1 2 1 for the proofs of controllability we rely on the strategy introduced by lebeau and robbiano comm partial differ equ 20 335 356 1995 for the null controllability of the heat equation which is based on a spectral inequality we obtain relaxed uniform observability estimates in both the semi discrete and fully discrete frameworks and associated uniform controllability properties for the practical computation of the control functions we follow j l lions hilbert uniqueness method strategy exploiting the relaxed uniform observability estimate algorithms for the computation of the controls are proposed and analysed in the semi discrete and fully discrete cases additionally we prove an error bound between the fully discrete and the semi discrete control functions this bound is however not uniform with respect to the space discretization the theoretical results are illustrated through numerical experimentations

hybrid dimensionality reduction method based on support vector machine and independent component analysis this paper presents a new hybrid dimensionality reduction method to seek projection through optimization of both structural risk supervised criterion and data independence unsupervised criterion classification accuracy is used as a metric to evaluate the performance of the method by minimizing the structural risk projection originated from the decision boundaries directly improves the classification performance from a supervised perspective from an unsupervised perspective projection can also be obtained based on maximum independence among features or attributes in data to indirectly achieve better classification accuracy over more intrinsic representation of the data orthogonality interrelates the two sets of projections such that minimum redundancy exists between the projections leading to more effective dimensionality reduction experimental results show that the proposed hybrid dimensionality reduction method that satisfies both criteria simultaneously provides higher classification performance especially for noisy data sets in relatively lower dimensional space than various existing methods

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

impulse and sampled data optimal control of heat equations and error estimates we consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multidimensional heat equation with a periodic terminal state constraint this problem has a unique optimal solution which can be characterized by an optimality system derived from the pontryagin maximum principle we define two approximations of this optimal control problem the first one is an impulse approximation and consists of considering a system of linear heat equations with impulse control the second one is obtained by the sample and hold procedure applied to the control resulting in a sampled data approximation of the controlled heat equation we prove that both problems have a unique optimal solution and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled data approximations

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

eventual periodicity of some systems of max type difference equations the periodicity of the next system of max type difference equations xn 1 max1 i1 m1f1i1xn ki1 1 1 1 xn ki1 2 1 2 xn ki1 l 1 l n xn t1s 1 xn 2 max1 i2 m2f2i2xn ki2 1 2 1 xn ki2 2 2 2 xn ki2 l 2 l n xn t2s 2 xn l max1 il mlflilxn kil 1 l 1 xn kil 2 l 2 xn kil l l l n xn tls l n n0n n0 where s l mj tj kij h j n j h 1 l j h 1 l 1 2 l 1 2 l is a permutation of 1 2 l 1 2 l and fjij 0 l n0 0 fjij 0 l n0 0 j 1 l j 1 l ij 1 mj ij 1 mj is studied it is shown that under some conditions posed on functions fjijfjij all positive solutions of the system are eventually periodic with period ts for some t nt n

controllability of impulse controlled systems of heat equations coupled by constant matrices abstract this paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair a b of constant matrices we present a necessary and sufficient condition for the approximate controllability which is exactly kalman s controllability rank condition of a b we prove that when such a system is approximately controllable the approximate controllability over an interval 0 t can be realized by adding controls at arbitrary q a b different control instants 0 1 2 q a b t provided that q a b 1 d a where d a min im a and q a b n we also show that in general such systems are not null controllable

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

finite element approximations of impulsive optimal control problems for heat equations abstract in this paper we study the priori error analysis for semidiscrete galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint we first derive pontryagin s maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem then the error estimates for optimal control u the related state y and the adjoint state will be obtained finally an efficient algorithm for our numerical experiment is designed which verifies the theoretical results obtained in this paper

using locally estimated geodesic distance to optimize neighborhood graph for isometric data embedding to deal with the highly twisted and folded manifold this paper propose a geodesic distance based approach to build the neighborhood graph for isometric embedding this approach assumes that the neighborhood of a point located at the highly twisted place of the manifold may not be linear so that its neighbors should be determined by geodesic distance this approach firstly determines the neighborhood for each point using euclidean distance and then applies the locally estimated geodesic distances to optimize the neighborhood it increases only linear time complexity furthermore the optimized neighborhood can speed up the subsequent embedding process the proposed approach is simple general and easy to deal with a wider range of data the conducted experiments on both synthetic and real data sets validate the approach

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

regularized discriminant entropy analysis in this paper we propose the regularized discriminant entropy rde which considers both class information and scatter information on original data based on the results of maximizing the rde we develop a supervised feature extraction algorithm called regularized discriminant entropy analysis rdea rdea is quite simple and requires no approximation in theoretical derivation the experiments with several publicly available data sets show the feasibility and effectiveness of the proposed algorithm with encouraging results we propose the regularized discriminant entropy rde the rde is based on the within class entropy and robust estimation we design a supervised algorithm regularized discriminant entropy analysis rdea rdea can be regarded as a framework for supervised feature extraction

on a system of difference equations with maximum in this paper we study the periodic character of the solutions of the system of difference equations with maximum x n 1 max a y n x n 1 y n 1 max b x n y n 1 where a b are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers we prove that every solution is eventually periodic

on a difference equation with maximum abstract this paper studies the asymptotic behavior of positive solutions to the difference equation x n max 1 x n 1 a x n 2 n 0 1 where 0 0 we prove that every positive solution of this equation approaches x 1 or is eventually periodic with period 4 according as a 1 or a 1

robust kernel isomap isomap is one of widely used low dimensional embedding methods where geodesic distances on a weighted graph are incorporated with the classical scaling metric multidimensional scaling in this paper we pay our attention to two critical issues that were not considered in isomap such as 1 generalization property projection property 2 topological stability then we present a robust kernel isomap method armed with such two properties we present a method which relates the isomap to mercer kernel machines so that the generalization property naturally emerges through kernel principal component analysis for topological stability we investigate the network flow in a graph providing a method for eliminating critical outliers the useful behavior of the robust kernel isomap is confirmed through numerical experiments with several data sets

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

optimal cluster preserving embedding of nonmetric proximity data for several major applications of data analysis objects are often not represented as feature vectors in a vector space but rather by a matrix gathering pairwise proximities such pairwise data often violates metricity and therefore cannot be naturally embedded in a vector space concerning the problem of unsupervised structure detection or clustering in this paper a new embedding method for pairwise data into euclidean vector spaces is introduced we show that all clustering methods which are invariant under additive shifts of the pairwise proximities can be reformulated as grouping problems in euclidian spaces the most prominent property of this constant shift embedding framework is the complete preservation of the cluster structure in the embedding space restating pairwise clustering problems in vector spaces has several important consequences such as the statistical description of the clusters by way of cluster prototypes the generic extension of the grouping procedure to a discriminative prediction rule and the applicability of standard preprocessing methods like denoising or dimensionality reduction

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

on the continuous dependence with respect to sampling of the linear quadratic regulator problem for distributed parameter systems the convergence of solutions to the discrete or sampled time linear quadratic regulator problem and associated riccati equation for infinite dimensional systems to the solutions to the corresponding continuous time problem and equation as the length of the sampling interval the sampling rate tends toward zero infinity is established both the finite and infinite time horizon problems are studied in the finite time horizon case strong continuity of the operators that define the control system and performance index together with a stability and consistency condition on the sampling scheme are required for the infinite time horizon problem in addition the sampled systems must be stabilizable and detectable uniformly with respect to the sampling rate classes of systems for which this condition can be verified are discussed results of numerical studies involving the control of a heat diffusion equation a hereditary or delay system and a flexible beam are presented and discussed

an approximation theorem for the algebraic riccati equation for an infinite dimensional linear quadratic control problem in hilbert space approximation of the solution of the algebraic riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems as an application strong onvergence of the approximating riccati operators in case of a previously developed spline approximation scheme for delay systems is established finally convergence of the transfer functions of the approximating systems is investigated

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

a difference equation with eventually periodic solutions we show that every solution of the difference equation xn 1 max 1xn axn 1 n 0 1 where the parameter a and the initial conditions x 1 and x0 are nonzero real numbers is eventually periodic with period 2 3 or 4 we also give a precise description of the period in terms of a and the initial conditions

a convex matrix optimization for the additive constant problem in multidimensional scaling with application to locally linear embedding the additive constant problem has a long history in multidimensional scaling and it has recently been used to resolve the issue of indefiniteness of the geodesic distance matrix in isomap but it would lead to a large positive constant being added to all eigenvalues of the centered geodesic distance matrix often causing significant distortion of the original distances in this paper we reformulate the problem as a convex optimization of almost negative semidefinite matrix so as to achieve minimal variation of the original distances we then develop a newton cg method and further prove its quadratic convergence finally we include a novel application to the famous lle locally linear embedding in nonlinear dimensionality reduction addressing the issue when the input of lle has missing values we justify the use of the developed method to tackle this issue by establishing that the local gram matrix used in lle can be obtained through a local euclidean distance matrix the effectiveness of our method is

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied