positive constrained approximation via rbf based partition of unity method in this paper we discuss the problem of constructing radial basis function rbf based partition of unity pu interpolants that are positive if data values are positive more specifically we compute positive local approximants by adding up several constraints to the interpolation conditions this approach considering a global approximation problem and compactly supported rbfs csrbfs has been previously proposed in wu etal 2010 here the use of the pu technique enables us to intervene only locally and as a consequence to reach a better accuracy this is also due to the fact that we select the optimal number of positive constraints by means of an a priori error estimate and we do not restrict to the use of csrbfs numerical experiments and applications to population dynamics are provided to illustrate the effectiveness of the method in applied sciences

linear and sub linear growth and the clt for hitting times of a random walk in random environment on a strip the main goal of this work is to study the asymptotic behaviour of hitting times of a random walk rw in a quenched random environment re on a strip we introduce enlarged random environments in which the traditional hitting time can be presented as a sum of independent random variables whose distribution functions form a stationary random sequence this allows us to obtain conditions stated in terms of properties of random environments for a linear growth of hitting times of relevant random walks in some important cases e g independent random environments these conditions are also necessary for this type of behaviour we also prove the quenched central limit theorem clt for hitting times in the general ergodic setting a particular feature of these ballistic laws in random environment is that whenever they hold under standard normalization the convergence is a convergence with a speed the latter is due to certain properties of moments of hitting times which are also studied in this paper the asymptotic properties of the position of the walk are stated but are not proved in this work since this has been done in goldhseid probab theory relat fields 139 1 41 64 2007

root properties and convergence rates of median filters median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to additional passes of the median filter in this paper the theory is developed both for determining the cardinality of the root signal space of arbitrary window width filters applied to signals with any number of quantization levels and for counting or estimating the number of passes required to produce a root for binary signals

the finite difference streamline diffusion methods for sobolev equations with convection dominated term in this paper we give two finite difference streamline diffusion fdsd schemes for solving linear sobolev equations with convection dominated term stability and optimal error estimates for them are derived in suitable norms

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

a training framework for stack and boolean filtering fast optimal design procedures and robustness case study a training framework is developed in this paper to design optimal nonlinear filters for various signal and image processing tasks the targeted families of nonlinear filters are the boolean filters and stack filters the main merit of this framework at the implementation level is perhaps the absence of constraining models making it nearly universal in terms of application areas we develop fast procedures to design optimal or close to optimal filters based on some representative training set furthermore the training framework shows explicitly the essential part of the initial specification and how it affects the resulting optimal solution symmetry constraints are imposed on the data and consequently on the resulting optimal solutions for improved performance and ease of implementation the case study is dedicated to natural images the properties of optimal boolean and stack filters when the desired signal in the training set is the image of a natural scene are analyzed specifically the effect of changing the desired signal using various natural images and the characteristics of the noise the probability distribution function the mean and the variance is analyzed elaborate experimental conditions were selected to investigate the robustness of the optimal solutions using a sensitivity measure computed on data sets a remarkably low sensitivity and consequently a good generalization power of boolean and stack filters are revealed boolean based filters are thus shown to be not only suitable for image restoration but also robust making it possible to build libraries of optimal filters which are suitable for a set of applications

two dimensional root structures and convergence properties of the separable median filter the root signals invariant to filtering structures of the two dimensional separable median filter are derived and presented in addition it is proved that with rare exception after repetitive passes of the separable median filter any two dimensional signal will be reduced to a signal containing only root structures

optimal median type filtering under structural constraints this paper presents a method for the design of median type filters that achieve the maximum noise attenuation under structural constraints imposed by the requirement of preserving certain signal or image features as compared with the design of optimal weighted median wm filters that calls for solving a set of linear inequalities this method is extremely simple yet general enough the filter is obtained by modifying directly the boolean function of a median filter and an analytic closed form representation of its boolean function can be obtained furthermore it is proven theoretically that under the same set of structural constraints the filter designed in this way will never do worse in removing i i d noise with any distribution than the optimal wm filter improvements as high as 41 46 and 52 have been achieved in simulations in a 2 d case for the uniform gaussian and double exponential distributions respectively also the improvement in the impulsive noise environment is very significant as is demonstrated by an image enhancement application

dimensions of random average conformal repellers abstract this paper defines random average conformal repellers which are a generalization of random conformal repellers using the subadditive thermodynamic formalism the hausdorff dimension of random average conformal repellers is obtained as the root of the random pressure function

an alternative local collocation strategy for high convergence meshless pde solutions using radial basis functions this work proposes an alternative decomposition for local scalable meshless rbf collocation the proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary forming a small rbf collocation system around each internal node unlike other meshless local rbf formulations that are based on a generalised finite difference rbf fd principle in the proposed finite collocation method the solution of the pde is driven entirely by collocation of pde governing and boundary operators within the local systems a sparse global collocation system is obtained not by enforcing the pde governing operator but by assembling the value of the field variable in terms of the field value at neighbouring nodes in analogy to full domain rbf collocation systems communication between stencils occurs only over the stencil periphery allowing the pde governing operator to be collocated in an uninterrupted manner within the stencil interior the local collocation of the pde governing operator allows the method to operate on centred stencils in the presence of strong convective fields the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics this implicit upwinding effect mitigates the need for ad hoc upwinding stencils in convective dominant problems boundary conditions are also enforced within the local collocation systems allowing arbitrary boundary operators to be imposed naturally within the solution construction the performance of the method is assessed using a large number of numerical examples with two steady pdes the convection diffusion equation and the lame navier equations for linear elasticity the method exhibits high order convergence in each case tested greater than sixth order and the use of centred stencils is demonstrated for convective dominant problems in the case of linear elasticity the stress fields are reproduced to the same degree of accuracy as the displacement field and exhibit the same order of convergence the method is also highly stable towards variations in basis function flatness demonstrating significantly improved stability in comparison to finite difference type rbf collocation methods

explicit and implicit meshless methods for linear advection diffusion type partial differential equations simple mesh grid free explicit and implicit numerical schemes for the solution of linear advection diffusion problems is developed and validated herein unlike the mesh or grid based methods these schemes use well distributed quasi random points and approximate the solution using global radial basis functions the schemes can be seen as generalized finite differences with random points instead of a regular grid system this allows the computation of problems with complex shaped boundaries in higher dimensions with no need for complex mesh grid structure and with no extra implementation difficulties

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

output distributions of stack filters based on mirrored threshold decomposition the output distribution formula for stack filters based on mirrored threshold decomposition is derived this formula allows one to compute the cumulative distribution function of the output of a stack filter for a given independent and identically distributed i i d input noise distribution mirrored threshold decomposition permits us to analyze the input output characteristics of the stack filter in the binary domain the sliding window operation of the stack filter is modeled by a deterministic finite automaton the output distribution of the filter is obtained by interpreting the automaton as a markov chain whose transition probabilities depend on the probabilistic description of the binary input signal an example illustrating the use of the output distribution formula is provided

on the dimension of deterministic and random cantor like sets symbolic dynamics and the eckmann ruelle conjecture in this paper we unify and extend many of the known results on the dimension of deterministic and random cantor like sets in n and apply these results to study some problems in dynamical systems in particular we verify the eckmann ruelle conjecture for equilibrium measures for holder continuous conformal expanding maps and conformal axiom a topologically hyperbolic homeomorphims we also construct a holder continuous axiom a homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere this example shows that the non conformal holder continuous version of the eckmann ruelle conjecture is false

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

a super accurate shifted tau method for numerical computation of the sobolev type differential equation with nonlocal boundary conditions in this article we propose a super accurate numerical scheme to solve the one dimensional sobolev type partial differential equation with an initial and two nonlocal integral boundary conditions our proposed methods are based on the shifted standard and shifted chebyshev tau method firstly we convert the model of partial differential equation to a linear algebraic equation and then we solve this system shifted standard and shifted chebyshev polynomials are applied for giving the computational results numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach the method is easy to apply and produces very accurate numerical results

a new explicit method for the diffusion convention equation abstract we consider here the finite difference approximation to the diffusion convection equation from which new explicit formulation are obtained which are asymmetric these explicit schemes can then be used to develop a new class of methods called group explicit as introduced in 2 theoretical aspects of the stability consistency convergence and truncation errors of this new class of methods is briefly discussed and numerical evidence presented to confirm our recomendations

weak galerkin finite element methods for sobolev equation we present some numerical schemes based on the weak galerkin finite element method for one class of sobolev equations in which differential operators are approximated by weak forms through the usual integration by parts in particular the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element the proposed schemes will be proved to have good numerical stability and high order accuracy when time variable is continuous also an optimal error estimate is obtained for the fully discrete scheme finally some numerical results are given to verify our analysis for the scheme

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

meshless rbfs method for numerical solutions of two dimensional high order fractional sobolev equations abstract in this paper meshless rbfs method is proposed to solve two dimensional time fractional sobolev equations the proposed method uses rbfs for approximation of spatial operator finite difference formula of o t 2 0 1 is used for time fractional derivative approximation while rule 0 1 as time stepping scheme for the advancement of solution in time validation of the proposed method is made by considering various test examples from literature simulated results are found in very good agreement with available exact solutions a rigorous comparative analysis made with other methods testifies proposed method s superiority in higher dimensions efficiency and accuracy of rbfs method are examined by varying number of nodes n in the domain of influence time step size t as well as l 2 l and l rms error norms linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis

convergence behavior and n roots of stack filters the convergence behavior of two types of stack filters is investigated both types are shown to possess the convergence property and to exhibit nontrivial behavior the first type of stack filter has the erosive property it erodes any input signal to a root after a sufficient number of passes the second type of stack filter has the dilative property it dilates any input signal to a root after a sufficient number of passes for each type of stack filter an algorithm is presented which can determine a filter that has any specific signal or set of signals as roots these two algorithms are efficient in that their execution time is a linear function of the length of the input signal the width of the filter window and the number of signals to be preserved since some stack filters have the phenomenon of oscillations when they filter some input signals successively a partial ordering is defined over the set of stack filters which makes it possible to determine upper and lower bounds for these oscillations

a new class of nonlinear filters neural filters a class of nonlinear filters based on threshold decomposition and neural networks is defined it is shown that these neural filters include all filters defined either by continuous functions such as linear finite impulse response fir filters or by boolean functions such as generalized stack filters adaptive least mean absolute error and adaptive least mean square error algorithms are derived for determining optimal neural filters as special cases adaptive generalized stack and adaptive generalized weighted order statistic filtering algorithms under both error criteria are derived experimental results in 1d and 2d signal processing are presented to compare the performances of the adaptive neural filters and other widely used filters

minimum mean absolute error stack filtering with structural constraint and goals a theory for the structural behavior of stack filters is developed this theory provides a test which can determine if a given stack filter has any root signals a method for classifying the root signal behavior of any stack filter found to have roots and perhaps most important a method for designing stack filters with specific root signals or other structural behavior this theory of root signals for stack filters is then combined with the theory of minimum mean absolute error stack filtering this unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior

local rbf fd solutions for steady convection diffusion problems this paper describes the application of radial basis function rbf based finite difference type scheme rbf fd for solving steady convection diffusion equations numerical studies are made using multiquadric mq rbf by varying the shape parameter in mq the accuracy of the solution is seen to be highly improved for large values of reynolds numbers the developed scheme has been compared with the corresponding finite difference scheme and found that the solutions obtained using the former are non oscillatory copyright 2007 john wiley sons ltd

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

a rbf partition of unity collocation method based on finite difference for initial boundary value problems abstract meshfree radial basis function rbf methods are popular tools used to numerically solve partial differential equations pdes they take advantage of being flexible with respect to geometry easy to implement in higher dimensions and can also provide high order convergence since one of the main disadvantages of global rbf based methods is generally the computational cost associated with the solution of large linear systems in this paper we focus on a localizing rbf partition of unity method rbf pum based on a finite difference fd scheme specifically we propose a new rbf pum fd collocation method which can successfully be applied to solve time dependent pdes this approach allows to significantly decrease ill conditioning of traditional rbf based methods moreover the rbf pum fd scheme results in a sparse matrix system reducing the computational effort but maintaining at the same time a high level of accuracy numerical experiments show performances of our collocation scheme on two benchmark problems involving unsteady convection diffusion and pseudo parabolic equations

the discrete duality finite volume method for convection diffusion problems in this paper we extend the discrete duality finite volume ddfv formulation to the steady convection diffusion equation the discrete gradients defined in ddfv are used to define a cell based gradient for the control volumes of both the primal and dual meshes in order to achieve a higher order accurate numerical flux for the convection term a priori analysis is carried out to show convergence of the approximation and a global first order convergence rate is derived the theoretical results are confirmed by some numerical experiments

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

random subshifts of finite type let x be an irreducible shift of finite type sft of positive entropy and let bn x be its set of words of length n define a random subset of bn x by independently choosing each word from bn x with some probability let x be the random sft built from the set for each 0 1 and n tending to infinity we compute the limit of the likelihood that x is empty as well as the limiting distribution of entropy for x for near 1 and n tending to infinity we show that the likelihood that x contains a unique irreducible component of positive entropy converges exponentially to 1 these results are obtained by studying certain sequences of random directed graphs this version of random sft differs significantly from a previous notion by the same name which has appeared in the context of random dynamical systems and bundled dynamical systems

on the topological pressure of random bundle transformations in sub additive case in this paper we define the topological pressure for sub additive potential via separated sets in random dynamical systems and give a proof of the relativized variational principle for the topological pressure

a trivariate interpolation algorithm using a cube partition searching procedure in this paper we propose a fast algorithm for trivariate interpolation which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube partition searching procedure more precisely we construct a cube structure which partitions the domain and strictly depends on the size of its subdomains so that the new searching procedure and accordingly the resulting algorithm enable us to efficiently deal with a large number of nodes complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm

state description for the root signal set of median filters median filtering is a simple digital technique for smoothing signals one main characteristic of the filter is that it maps the input signal space into a root signal space where signals invariant to median filters are called roots of the signal in this paper we develop the theory for the root signal set of median filters a tree structure for the root signal set is obtained for binary signals the number of roots r n for a signal of length n and window size filter 2s 1 is exactly represented by the difference equation r n r n 1 r n s a general solution is obtained in a z domain approach finally a method for faster one dimensional median filter operation is introduced

a unifying view for stack filter design based on graph search methods stack filters are operators that commute with the thresholding operation i e thresholding a signal applying the binary filter on each thresholded binary signals and then summing up stacking the results yields the same result as applying the multi level gray scale filter on the original signal several approaches for designing optimal stack filters from training data where optimality is characterized in terms of costs based on input output joint observations have been proposed this work considers stack filter design from training data under a general statistical framework developed in the context of morphological image operator design this framework 1 provides a common point of view for distinct design approaches being useful for comparative analysis or for emphasizing differences 2 clearly answers the issue of why binary signals from different threshold levels although following distinct distributions can be pooled together in the cost estimation process and 3 helps to show that several stack filter design approaches based on lattice diagrams search methods share a common underlying formulation

efficient computation of partition of unity interpolants through a block based searching technique in this paper we propose a new efficient interpolation tool extremely suitable for large scattered data sets the partition of unity method is used and performed by blending radial basis functions rbfs as local approximants and using locally supported weight functions in particular we present a new space partitioning data structure based on a partition of the underlying generic domain in blocks this approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points leading to an optimized searching routine complexity analysis and numerical experiments in two and three dimensional interpolation support our findings some applications to geometric modelling are also considered moreover the associated software package written in matlab is here discussed and made available to the scientific community

unexpected distribution phenomenon resulting from cantor series expansions abstract we explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their cantor series expansions as a direct consequence of our main theorem we deduce numerous new results as well as strengthen the known ones

superconvergence analysis of an h 1 galerkin mixed finite element method for sobolev equations an h 1 galerkin mixed finite element method mfem is discussed for the sobolev equations with the bilinear element and zero order raviart thomas element q 11 q 10 q 01 the existence and uniqueness of the solutions about the approximation scheme are proved two new important lemmas are given by using the properties of the integral identity and the bramble hilbert lemma which lead to the superclose results of order o h 2 for original variable u in h 1 norm and flux q in h div norm under semi discrete scheme furthermore two new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained on the other hand a second order fully discrete scheme with superclose property o h 2 2 is also proposed at last numerical experiment is included to illustrate the feasibility of the proposed method here h is the subdivision parameter and is the time step

digital filters as absolute norm regularizers a variational formulation of the digital filtering problem based on a certain regularizing functional is presented in particular the special case where the regularizing functional is defined by means of l sub 1 norm is considered and a characterization of its minima is presented it is shown that for a certain class of signals one solution to the regularization problem can be obtained by means of a suitable recursive median filter some examples and a possible extension of the variational approach are also discussed

the classification and associative memory capability of stack filters the notion of the on set of a positive boolean function is used to classify stack filters into three different types called decreasing increasing and mixed the associative memory capability of each of these three types of stack filters is then investigated the associative memory of a stack filter is defined to be the set of root signals of that filter in a class of stack filters in which each filter s root set contains a desired set of patterns those filters whose root sets have the smallest cardinality are said to be minimal among all filters in that class for that set of patterns some learning schemes are proposed to find minimal decreasing and increasing stack filters it is also shown that for any specified set of patterns there is always a mixed stack filter which is minimal when one considers all stack filters which preserve those patterns in this sense mixed stack filters are always at least as good as decreasing or increasing stack filters

a property of the minimum vectors of a regularizing functional defined by means of the absolute norm we consider a regularizing functional defined by means of the l sub 1 norm where the regularization is obtained using first differences as is well known such a functional can be put in relation with recursive median filters of appropriate window length we show that at least one of the minima is reached at a vector whose components have values over the same discrete set of the given signal this suggests a simple method to refine the approximate solution to the regularization problem which can be obtained with recursive median filters of increasing order we also report an example of application where the refinement method is employed for a signal detection problem

optimal parallel stack filtering under the mean absolute error criterion the authors extend the configuration of stack filtering to develop a new class of stack type filters called parallel stack filters psfs as a basis for the parallel stack filtering the block threshold decomposition btd is introduced and its properties are investigated the design of optimal pshs under the mean absolute error mae criterion is shown to be similar to the minimum mae stack filtering theory the only difference is that one needs now to design more than one stack filter that together construct an optimal psf as a result while reviewing briefly the optimal stack filtering theory they will put more efforts to demonstrate via several examples the improvement by switching from stack filtering to parallel stack filtering for the task of image noise removal

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

topological median filters this paper describes the definition and testing of a new type of median filter for images the topological median filter implements some existing ideas and some new ideas on fuzzy connectedness to improve over a conventional median filter the extraction of edges in noise the concept of spl alpha connectivity is defined and used to create an algorithm for computing the degree of connectedness of a pixel to all the other pixels in an arbitrary neighborhood the resulting connectivity map of the neighborhood effectively disconnects peaks in the neighborhood that are separated from the center pixel by a valley in the brightness topology the median of the connectivity map is an estimate of the median of the peak or plateau to which the center pixel belongs unlike the conventional median filter the topological median is relatively unaffected by disconnected features in the neighborhood of the center pixel four topological median filters are defined qualitative and statistical analyses of the four filters are presented it is demonstrated that edge detection can be more accurate on topologically median filtered images than on conventionally median filtered images

intersecting random translates of invariant cantor sets given two cantor setsx andy in 0 1 invariant under the mapx b x mod 1 the hausdorff dimension of x t y is constant almost everywhere whenx y are defined by admissible digits in baseb and more generally by sofic systems we compute this dimension in terms of the largest lyapunov exponent of a random product of matrices the results are extended to higher dimensions and multiple intersections

parallel algorithms and vlsi architectures for stack filtering using fibonacci p codes a parallel decompositional algorithm and vlsi architecture is proposed for computation of the output of a stack filter over a single window of input samples using fibonacci p codes for a subclass of positive boolean functions a more efficient parallel algorithm and vlsi architecture for running stack filtering is also presented the area time complexities of the proposed designs are estimated

an adaptive orthogonal improved interpolating moving least square method and a new boundary element free method abstract the improved interpolating moving least square iimls method has been widely used in data fitting and meshfree methods and the obtained shape functions have the property of the delta function compared with those obtained by the moving least square mls method however the moment matrix in iimls may be singular or ill conditioned because of the ill quality of the point sets used in this paper the weighted orthogonal basis functions are applied in iimls to obtain a diagonal moment matrix which can overcome the difficulty caused by directly inversing singular or ill conditioned matrices however the weighted orthogonal basis functions cannot change the nature of the singular or ill conditioned moment matrix since the diagonal elements of the new moment matrix may be zero or close to zero thus an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix combined with shifted and scaled polynomial basis functions a stabilized adaptive orthogonal iimls sao iimls approximation is obtained based on this approximation a new boundary element free method is proposed for solving elasticity problems numerical results for curve fitting surface fitting and the new boundary element free method have shown that the proposed sao iimls approximation is accurate stable and performs well for ill quality point sets

on the dimensions of conformal repellers randomness and parameter dependency bowen s formula relates the hausdorff dimension of a conformai repeller to the zero of a pressure function we present an elementary self contained proof to show that bowen s formula holds for c1 conformai repellers we consider time dependent conformai repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class we show that bowen s formula generalizes to such a repeller and that if the sequence is picked at random then the hausdorff dimension of the repeller almost surely agrees with its upper and lower box dimensions and is given by a natural generalization of bowen s formula for a random uniformly hyperbolic julia set on the riemann sphere we show that if the family of maps and the probability law depend realanalytically on parameters then so does its almost sure hausdorff dimension 1 random julia sets and their dimensions let u djj be an open connected subset of the riemann sphere avoiding at least three points and equipped with a hyperbolic metric let k c u be a compact subset we denote by at u the space of unramified conformai covering maps vf u with the requirement that the covering domain vf c k denote by df vf m the conformai derivative of see equation 2 4 and by df supf ik df the maximal value of this derivative over the set f 1k let t fn c k u be a sequence of such maps the intersection

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

a polynomial time algorithm for computing absolutely normal numbers we give an algorithm to compute an absolutely normal number so that the first n digits in its binary expansion are obtained in time polynomial in n in fact just above quadratic the algorithm uses combinatorial tools to control divergence from normality speed of computation is achieved at the sacrifice of speed of convergence to normality

adaptive stack filtering with application to image processing with the aid of threshold decomposition it is shown that optimal stack filters under the mean absolute error mae criterion are equal to optimal or bayesian classifiers subject to stacking constraints under the mean classification error mce criterion nonadaptive and adaptive constrained least mean absolute lma algorithms are developed for the esti mation of stack filters through the linearization of the unit step function in the objective function the convergence of the al gorithms is proven under certain conditions although the methods do not generally give optimal stack filters under the mae criterion these algorithms have several distinct merits compared to other stack filter optimization methods 1 the op timization problem has a unique solution that approximates the optimal stack filters in the least mean square sense 2 the meth ods can be implemented in the binary and real domains and 3 for stack filters defined by linearly separable positive boolean functions pbf s or weighted order statistic wos filters the number of the parameters to be estimated is reduced to the window width of the filter a comparison between images re stored by the new algorithms and the stack filtering algorithm optimal under the mae criterion confirms the effectiveness of the proposed algorithms

on the asymptotic solutions of a differential equation with multiple singular points abstract a singularly perturbed model problem with multiple distinct regular singular points is studied the uniform approximation of wazwaz and hanson 1986 is effectively extended to many singular points thus establishing a generalized version of that theorem where the classic inner and outer expansions are not employed a leading order general asymptotic solution is correctly represented by a set of matched exponential asymptotic expansions where each approximation contains dominant and recessive terms the resonance criteria due to the influence of multiple singular points are discussed for an even number of singular points the eigenvalues were found to be positive with a minimal eigenvalue however negative eigenvalues with a maximal eigenvalue arise for an odd number of singular points the maximal and the minimal eigenvalue may not be unique and each is 0 1

root signals of rank order processors a rank order processor rop is a signal processor that operates purely by making rank determination on input values and includes as special cases median filters and rank order filters it is shown how the methods of automata theory can be used to show the existence of root signal i e fixed point sets of input sequences and to determine them for specific rops in either algebraic or graphical form observations on graph subgraph relationships are made it is shown how certain earlier methods of determining root signals can be improved and extended to the general case remarks on the synthesis problem are made

minimum mean absolute error estimation over the class of generalized stack filters a class of sliding window operators called generalized stack filters is developed this class of filters which includes all rank order filters stack filters and digital morphological filters is the set of all filters possessing the threshold decomposition architecture and a consistency property called the stacking property conditions under which these filters possess the weak superposition property known as threshold decomposition are determined an algorithm is provided for determining a generalized stack filter which minimizes the mean absolute error mae between the output of the filter and a desired input signal given noisy observations of that signal the algorithm is a linear program whose complexity depends on the window width of the filter and the number of threshold levels observed by each of the filters in the superposition architecture the results show that choosing the generalized stack filter which minimizes the mae is equivalent to massively parallel threshold crossing decisions making when the decision are consistent with each other

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

penalty methods for the numerical solution of american multi asset option problems we derive and analyze a penalty method for solving american multi asset option problems a small non linear penalty term is added to the black scholes equation this approach gives a fixed solution domain removing the free and moving boundary imposed by the early exercise feature of the contract explicit implicit and semi implicit finite difference schemes are derived and in the case of independent assets we prove that the approximate option prices satisfy some basic properties of the american option problem several numerical experiments are carried out in order to investigate the performance of the schemes we give examples indicating that our results are sharp finally the experiments indicate that in the case of correlated underlying assets the same properties are valid as in the independent case

stack filter design a structural approach a new approach is developed for finding the optimal stack filter that minimizes noise subject to constraints on its structural behavior based on the output moments of stack filters it is proven that the optimal stack filter is a combination of the median filter which has the same window width as the stack filter and a set of maximum and minimum filters which are attributed to the structural constraints design examples for 1 d signal processing and image processing are provided

local rbf method for multi dimensional partial differential equations abstract in this paper a local meshless differential quadrature collocation method is utilized to solve multi dimensional reaction convection diffusion pdes numerically in some cases global version of the meshless method is considered as well the meshless methods approximate solution on scattered and uniform nodes in both local and global sense in the case of convection dominated pdes the local meshless method is coupled with an upwind technique to avoid spurious oscillations for this purpose a physically motivated local domain is utilized in the flow direction both regular and irregular geometries are taken into consideration numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains

recursive median filters of increasing order a variational approach for a certain class of signals it is known that the recursive median filters provide a minimum of a related regularizing functional defined by means of the l sub 1 norm we generalize such results and we show that a good approximation to the minima of such a functional can be obtained for any signal by means of successive applications of recursive median filters of increasing order

stack filters and the mean absolute error criterion a method to determine the stack filter which minimizes the mean absolute error between its output and a desired signal given noisy observations of this desired signal is presented specifically an optimal window width b stack filter can be determined with a linear program with o b2 sup b variables this algorithm is efficient since the number of different inputs to a window width b filter is m sup b if the filter has m valued input and the number of stack filters grows faster than 2 raised to the 2 sup b 2 power it is shown that optimal stack filtering under the mean absolute error criterion is analogous to optimal linear filtering under the mean squared error criterion both linear filters and stack filters are defined by superposition properties both classes are implementable and both have tractable procedures for finding the optimal filter under an appropriate error criterion

the viterbi optimal runlength constrained approximation nonlinear filter simple nonlinear filters are often used to enforce hard syntactic constraints while remaining close to the observation data e g in the binary case it is common practice to employ iterations of a suitable median or a one pass recursive median openclose or closeopen filter to impose a minimum symbol runlength constraint while remaining faithful to the observation unfortunately these filters are in general suboptimal motivated by this observation we pose the following optimization given a finite alphabet sequence of finite extent y y n sub n 0 sup n 1 find a sequence x x n sub n 0 sup n 1 that minimizes d x y spl sigma sub n 0 sup n 1 d sub n y n x n subject to the following x is piecewise constant of plateau run length spl ges m we show how a suitable reformulation of the problem naturally leads to a simple and efficient viterbi type optimal algorithmic solution we call the resulting nonlinear input output operator the viterbi optimal runlength constrained approximation vorca filter the method can be easily generalized to handle a variety of local syntactic constraints the vorca is optimal computationally efficient and possesses several desirable properties e g idempotence we therefore propose it as an attractive alternative to standard median stack and morphological filtering we also discuss some applications

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

turing s unpublished algorithm for normal numbers in an unpublished manuscript alan turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have lebesgue measure 1 furthermore he gave an algorithm for computing instances in this set we complete his manuscript by giving full proofs and correcting minor errors while doing this we recreate turing s ideas as accurately as possible one of his original lemmas remained unproved but we have replaced it with a weaker lemma that still allows us to maintain turing s proof idea and obtain his result

resource bounded randomness and weakly complete problems we introduce and study resource bounded random sets based on lutz s concept of resource bounded measure 5 6 we concentrate on nc randomness c 2 which corresponds to the polynomial time bounded p measure of lutz and which is adequate for studying the internal and quantative structure of e dtime 2lin first we show that the class of nc random sets has p measure 1 this provides a new simplified approach to p measure 1 results next we compare randomness with genericity in the sense of 1 2 and we show that nc 1 random sets are nc generic whereas the converse fails from the former we conclude thatnc random sets are not p btt complete for e our technical main results describe the distribution of the nc random sets under p m reducibility we show that every nc random set in e has nk random predecessors in e for any k 1 whereas the amount of randomness of the successors is bounded we apply this result to answer a question raised by lutz 8 we show that the class of weakly complete sets has measure 1 in e and that there are weakly complete problems which are not p btt complete for e

a new space time continuous galerkin method with mesh modification for sobolev equations abstract in this study we first propose the continuous galerkin cg method for the sobolev equations which allows different temporal step sizes and spatial grids in each time step and then we demonstrate the existence and uniqueness of the approximate solutions and derive the optimal rates of convergence of the approximate solutions under the restrictive assumptions that the space time finite element subspaces between two successive time steps are conforming elements finally we provide some numerical examples on unstructured meshes to demonstrate the efficiency and flexibility of this method

analysis of second order finite volume element methods for pseudo parabolic equations in three spatial dimensions abstract in this paper we study a finite volume element approximation of pseudo parabolic equations in three spatial dimensions the triquadratic elements are used as solution space to obtain a high order convergence rate based on some special dual partition we prove the optimal error estimate in h 1 norm of second order accuracy numerical results are also given at the end to demonstrate the efficiency of the method

an example of a computable absolutely normal number the first example of an absolutely normal number was given by sierpinski in 1916 twenty years before the concept of computability was formalized in this note we give a recursive reformulation of sierpinski s construction which produces a computable absolutely normal number

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

on the lp which finds a mmae stack filter two methods are proposed to modify the linear program lp developed by e j coyle and j h lin 1988 to find a stack filter which minimizes the mean absolute error mae in the first approach the number of constraints is substantially reduced at the expense of requiring a zero one lp to solve for an optimal filter this scheme reduces the number of constraints from o n2 sup n to o 28 sup n which is exactly the cardinality of the set of possible binary vectors which can appear in the window of the filter in the second approach the lp is transformed into a max flow problem this guarantees that the problem can be solved in time which is a polynomial function of the number of variables in the lp as opposed to the worst case exponential time that may occur with the simplex method it also allows the many fast algorithms for the max flow problem to be used to find an optimal stack filter recursive algorithms for construction of the window width n constraint matrix for both the original lp and the max flow modification are also provided