discrete spectral triples and their symmetries we classify zero dimensional spectral triples over complex and real algebras and provide some general statements about their differential structure we investigate also whether such spectral triples admit a symmetry arising from the hopf algebra structure of the finite algebra we discuss examples of commutative algebras and group algebras

differential calculi and linear connections a method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra as an example the generalized quantum plane is studied it is found that there is a strong correlation but not a one to one correspondence between the module structure of the 1 forms and the metric torsion free connections on it in the commutative limit the connection remains as a shadow of the algebraic structure of the 1 forms

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

on boundaries on the predual of the lorentz sequence space let x be the canonical predual of the lorentz sequence space and let au bx be the banach algebra of all complex valued functions defined on the closed unit ball bx of x which are uniformly continuous on bx and holomorphic on the interior of bx endowed with the sup norm a characterization of the boundaries for au bx is given in terms of the distance to the strong peak sets of this algebra

multistage order statistic filters for image sequence processing the application of multistage order statistic filters mos to the task of noise suppression in time varying imagery is studied it is shown that mos filters efficiently preserve image structures under motion without motion compensation preprocessing in particular the families of multistage median and weighted median filters are considered motion preservation and statistical smoothing measures are derived it is shown that spatiotemporal filtering allows for a significant improvement over both spatial and temporal filtering in terms of output image resolution and noise suppression

numerical boundaries for some classical banach spaces abstract globevnik gave the definition of boundary for a subspace a c b this is a subset of that is a norming set for a we introduce the concept of numerical boundary for a banach space x a subset b x is a numerical boundary for a subspace a c b b x x if the numerical radius of f is the supremum of the modulus of all the evaluations of f at b for every f in a we give examples of numerical boundaries for the complex spaces x c 0 c k and d w 1 the predual of the lorentz sequence space d w 1 in all these cases if k is infinite we show that there are closed and disjoint numerical boundaries for the space of the functions from b x to x which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary in the case of c 0 we characterize the numerical boundaries for that space of holomorphic functions

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

discrete riemannian geometry within a framework of noncommutative geometry we develop an analog of pseudo riemannian geometry on finite and discrete sets on a finite set there is a counterpart of the continuum metric tensor with a simple geometric interpretation the latter is based on a correspondence between first order differential calculi and digraphs the vertices of the latter are given by the elements of the finite set arrows originating from a vertex span its co tangent space if the metric is to measure length and angles at some point it has to be taken as an element of the left linear tensor product of the space of 1 forms with itself and not as an element of the nonlocal tensor product over the algebra of functions as considered previously by several authors it turns out that linear connections can always be extended to this left tensor product so that metric compatibility can be defined in the same way as in continuum riemannian geometry in particular in the case of the universal differential calculus on a finite set the euclidean geometry of polyhedra is recovered from conditions of metric compatibility and vanishing torsion in our rather general framework which also comprises structures which are far away from continuum differential geometry there is in general nothing like a ricci tensor or a curvature scalar because of the nonlocality of tensor products over the algebra of functions of forms corresponding components with respect to some module basis turn out to be rather nonlocal objects but one can make use of the parallel transport associated with a connection to localize such objects and in certain cases there is a distinguished way to achieve this in particular this leads to covariant components of the curvature tensor which allow a contraction to a ricci tensor several examples are worked out to illustrate the procedure furthermore in the case of a differential calculus associated with a hypercubic lattice we propose a new discrete analogue of the vacuum einstein equations

some statistical properties of median filters some statistical properties of median filters are analyzed it is shown that median filters can remove impulsive plus gaussian white noise better than hanning filters when the amplitude of the impulses is large or the energy of the gaussian noise is relatively low it is also shown that unlike linear filters median filters preserve sharp changes in signals when the noise energy is not too high

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

moduli of quantum riemannian geometries on 4 points we classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously s majid commun math phys 225 131 2002 the full moduli space is found for 3 points and a restricted moduli space for 4 points generalized levi civita connections and their curvatures are found for a variety of models including models of a discrete torus the topological part of the moduli space is found for 9 points based on the known atlas of regular graphs we also remark on aspects of quantum gravity in this approach

mixed noise removal by weighted encoding with sparse nonlocal regularization mixed noise removal from natural images is a challenging task since the noise distribution usually does not have a parametric model and has a heavy tail one typical kind of mixed noise is additive white gaussian noise awgn coupled with impulse noise in many mixed noise removal methods are detection based methods they first detect the locations of in pixels and then remove the mixed noise however such methods tend to generate many artifacts when the mixed noise is strong in this paper we propose a simple yet effective method namely weighted encoding with sparse nonlocal regularization wesnr for mixed noise removal in wesnr there is not an explicit step of impulse pixel detection instead soft impulse pixel detection via weighted encoding is used to deal with in and awgn simultaneously meanwhile the image sparsity prior and nonlocal self similarity prior are integrated into a regularization term and introduced into the variational encoding framework experimental results show that the proposed wesnr method achieves leading mixed noise removal performance in terms of both quantitative measures and visual quality

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

braided lie algebras and bicovariant differential calculi over co quasitriangular hopf algebras we show that if g is the quantum tangent space or quantum lie algebra in the sense of woronowicz of a bicovariant first order differential calculus over a co quasitriangular hopf algebra a r then a certain extension of it is a braided lie algebra in the category of a comodules this is used to show that the woronowicz quantum universal enveloping algebra u g is a bialgebra in the braided category of a comodules we show that this algebra is quadratic when the calculus is inner examples with this unexpected property include finite groups and quantum groups with their standard differential calculi we also find a quantum lie functor for co quasitriangular hopf algebras which has properties analogous to the classical one this functor gives trivial results on standard quantum groups oq g but reasonable ones on examples closer to the classical case such as the cotriangular jordanian deformations in addition we show that split braided lie algebras define generalized lie algebras in a different sense of deforming the adjoint representation we construct these and their enveloping algebras for oq sl n recovering the witten algebra for n 2

fir median hybrid filters with predictive fir substructures a class of finite impulse response fir median hybrid fmh filters that contain linear fir substructures to estimate the current signal value using forward and backward prediction is introduced the output of the overall filter is the median of the predicted values and the actual signal value in the middle of the filter window predictors maximizing the signal to noise ratio on signal sections described by an lth order polynominal are derived the ramp enhancement filters are shown to attenuate the noise on a ramp signal better than the standard median sm filters the new predictive fmh filters are shown to have root signals which do not exist for the sm filters e g triangular waves by combining the level and the ramp enhancement fmh filters a filter is obtained which attenuates noise on constant and ramp signals the noise attenuation on ramp signals is better than with the sm filter and the predictive fmh filter has novel and meaningful root structures the number of arithmetic operations needed to implement the predictive fmh filter grows linearly with the length of the filter

discrete differential calculus graphs topologies and gauge theory differential calculus on discrete sets is developed in the spirit of noncommutative geometry any differential algebra on a discrete set can be regarded as a reduction of the universal differential algebra and this allows a systematic exploration of differential algebras on a given set associated with a differential algebra is a di graph where two vertices are connected by at most two antiparallel arrows the interpretation of such a graph as a hasse diagram determining a locally finite topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure it is shown that field theories and in particular gauge theories can be formulated on a discrete set in close analogy with the continuum case the framework presented generalizes ordinary lattice theory which is recovered from an oriented hypercubic lattice graph it also includes e g the two point space used by connes and lott and others in models of elementary particle physics the formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an internal discrete space a la connes et al to models of dimensionally reduced gauge fields furthermore a symmetric lattice is also studied which in a certain continuum limit turns out to be related to a noncommutative differential calculus on manifolds

differential calculus and gauge theory on finite sets we develop differential calculus and gauge theory on a finite set g an elegant formulation is obtained when g is supplied with a group structure and in particular for a cyclic group connes two point model 1986 which is an essential ingredient of his reformulation of the standard model of elementary particle physics is recovered in our approach reductions of the universal differential calculus to lower dimensional differential calculi are considered the complete reduction leads to a differential calculus on a periodic lattice

iterative truncated arithmetic mean filter and its properties the arithmetic mean and the order statistical median are two fundamental operations in signal and image processing they have their own merits and limitations in noise attenuation and image structure preservation this paper proposes an iterative algorithm that truncates the extreme values of samples in the filter window to a dynamic threshold the resulting nonlinear filter shows some merits of both the fundamental operations some dynamic truncation thresholds are proposed that guarantee the filter output starting from the mean to approach the median of the input samples as a by product this paper unveils some statistics of a finite data set as the upper bounds of the deviation of the median from the mean some stopping criteria are suggested to facilitate edge preservation and noise attenuation for both the long and short tailed distributions although the proposed iterative truncated mean itm algorithm is not aimed at the median it offers a way to estimate the median by simple arithmetic computing some properties of the itm filters are analyzed and experimentally verified on synthetic data and real images

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

a new class of detail preserving filters for image processing a new class of median type filters for image processing is proposed in the filters linear fir substructures are used in conjunction with the median operation the root signals and noise attenuation properties of the fir median hybrid filters are analyzed and compared to representative edge preserving filtering operations the concept of multilevel median operation is introduced to improve the detail preserving property of conventional median and the fir median hybrid filters in the multilevel filters there exists a tradeoff between noise attenuation and detail preservation the analysis and examples indicate that fir median hybrid filters preserve details better and are computationally much more efficient than the conventional median and the k nearest neighbor averaging filters

interframe dpcm with robust median based predictors for transmission of image sequences over noisy channels a new image sequence coding technique based on robust median based predictors is presented for the transmission of image sequences over noisy channels we analyze the robustness of median based predictors against channel errors a heuristic algorithm for the design of a robust predictor from a given median based predictor is presented it is shown that with small modifications in terms of a necessary requirement for a median based predictor to be robust against channel errors the robustness of a given median based predictor can be considerably improved simulations on a real image sequence show significant improvement over the conventional differential pulse code modulation dpcm at high bit error rate ber using this new technique the technique does not increase the transmission rate it is shown that the quality of reconstructed images obtained by robust median based predictors can be further improved by postprocessing the image using a nonlinear detail preserving noise smoothing filter

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

noncommutative differential calculus and lattice gauge theory the authors study consistent deformations of the classical differential calculus on algebras of functions and more generally commutative algebras such that differentials and functions satisfy nontrivial commutation relations for a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice these differential calculi generate a lattice on a space continuum the whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus in this framework one just has to express the lagrangian for the continuum theory in terms of differential forms this expression then also makes sense for the deformed differential calculus there is a natural integral associated with the latter integration of the lagrangian over a space continuum then produces the correct lattice action for a large class of theories this is explicitly shown for the scalar field action and the action for su m gauge theory

spin geometry on quantum groups via covariant differential calculi abstract let a be a cosemisimple hopf algebra with antipode s and let be a left covariant first order differential calculus over a such that is self dual see section 2 and invariant under the hopf algebra automorphism s2 a quantum clifford algebra cl g is introduced which acts on woronowicz external algebra a minimal left ideal of cl g which is an a bimodule is called a spinor module metrics on spinor modules are investigated the usual notion of a linear left connection on is extended to quantum clifford algebras and also to spinor modules the corresponding dirac operator and connection laplacian are defined for the quantum group slq 2 and its bicovariant 4d calculi these concepts are studied in detail a generalization of bochner s theorem is given all invariant differential operators over a given spinor module are determined the eigenvalues of the dirac operator are computed

a separable median filter for image noise smoothing this paper investigates some properties of the separable filter resulting from successive applications of a one dimensional median filter on the rows and columns of an image although the output of this separable filter is not identical to the corresponding nonseparable two dimensional median filter with a square window its performance in image noise smoothing is close in particular its effectiveness in smoothing noise and its behavior with edges are characterized and compared with those of the two dimensional median filter it is shown that the separable filter has a much simpler implementation in real time hardware at video rates for example

boundaries for algebras of holomorphic functions on marcinkiewicz sequence spaces abstract let a b e be the banach algebra of all complex valued bounded continuous functions on the closed unit ball b e of a complex banach space e and holomorphic in the interior of b e and let a u e be the closed subalgebra of those functions which are uniformly continuous on b e for the case e m w 0 whose bidual is a marcinkiewicz sequence space m w we describe some sufficient conditions for a set to be a boundary of either a b e or a u e moreover we consider some analogous problems on m w 0 to those which were studied on the gowers space g p of characteristic p by grados and moraes l r grados l a moraes boundaries for algebras of holomorphic functions j math anal appl 281 2003 575 586 l r grados l a moraes boundaries for an algebra of bounded holomorphic functions j korean math soc 41 1 2004 231 242

discrete differential manifolds and dynamics on networks a discrete differential manifold is a countable set together with an algebraic differential calculus on it this structure has already been explored in previous work and provides a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time several examples are presented and a notion of differentiability of maps between discrete differential manifolds is introduced particular attention is given to differentiable curves in such spaces every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity

quasitriangular and differential structures on bicrossproduct hopf algebras abstract let x gm be a finite group factorisation it is shown that the quantum double d h of the associated bicrossproduct hopf algebra h km k g is itself a bicrossproduct kx k y associated to a group yx where y g m op this provides a class of bicrossproduct hopf algebras which are quasitriangular we also construct a subgroup y x associated to every order reversing automorphism of x the corresponding hopf algebra kx k y has the same coalgebra as h using related results we classify the first order bicovariant differential calculi on h in terms of orbits in a certain quotient space of x

non commutative geometry of finite groups a finite set can be supplied with a group structure which can then be used to select classes of differential calculi on it via the notions of left right and bicovariance a corresponding framework has been developed by woronowicz more generally for hopf algebras including quantum groups a differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and moreover for the formulation of field theories and dynamics on finite sets associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures for a covariant calculus there are notions of invariance for linear connections and tensors all these concepts are explored for finite groups and illustrated with examples some results are formulated more generally for arbitrary associative hopf algebras in particular the problem of extension of a connection on a bimodule over an associative algebra to tensor products is investigated leading to the class of extensible connections it is shown that invariance properties of an extensible connection on a bimodule over a hopf algebra are carried over to the extension furthermore an invariance property of a connection is also shared by a dual connection which exists on the dual bimodule as defined in this work

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

riemannian geometry of bicovariant group lattices group lattices cayley digraphs of a discrete group are in natural correspondence with differential calculi on the group on such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry despite the noncommutativity between functions and generalized differential forms for the subclass of bicovariant group lattices considered in this work it is possible to understand central geometric objects like metric torsion and curvature as tensors with left covariance properties this ensures that tensor components with respect to a basis of the space of 1 forms transform in the familiar homogeneous way under a change of basis there is a natural compatibility condition for a metric and a linear connection the resulting pseudo riemannian geometry is explored in this work it is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles a simple g

rank conditioned rank selection filters for signal restoration a class of nonlinear filters called rank conditioned rank selection rcrs filters is developed and analyzed in this paper the rcrs filters are developed within the general framework of rank selection rs filters which are filters constrained to output an order statistic from the observation set many previously proposed rank order based filters can be formulated as rs filters the only difference between such filters is in the information used in deciding which order statistic to output the information used by rcrs filters is the ranks of selected input samples hence the name rank conditioned rank selection filters the number of input sample ranks used is referred to as the order of the rcrs filter the order can range from zero to the number of samples in the observation window giving the filters valuable flexibility low order filters can give good performance and are relatively simple to optimize and implement if improved performance is demanded the order can be increased but at the expense of filter simplicity in this paper many statistical and deterministic properties of the rcrs filters are presented a procedure for optimizing over the class of rcrs filters is also presented finally extensive computer simulation results that illustrate the performance of rcrs filters in comparison with other techniques in image restoration applications are presented

fir median hybrid filters this paper introduces a new generalized class of median type filters which we call fir median hybrid fmh filters the input signal x n is filtered with m linear phase fir filters and the output of the fmh filter is the median of the outputs of the fir filters the statistical properties of the fmh filters are analyzed for input signals with gaussian double exponential and uniform density functions it is shown that fmh filters have essentially the same statistical properties and same type of root signals as the standard median sm filters fmh filters preserve corner points and ramps and attenuate impulsive type noise components effectively an interesting subclass of the fmh filters requiring only one scaling multiplier two additions and three compare swap operations irrespective of the filter length is introduced in the examples speed improvement by a factor of 450 is obtained over an otherwise approximately equivalent sm filter

gravity in non commutative geometry we study general relativity in the framework of non commutative differential geometry as a prerequisite we develop the basic notions of non commutative riemannian geometry including analogues of riemannian metric curvature and scalar curvature this enables us to introduce a generalized einstein hilbert action for non commutative riemannian spaces as an example we study a space time which is the product of a four dimensional manifold by a two point space using the tools of non commutative riemannian geometry and derive its generalized einstein hilbert action in the simplest situation where the riemannian metric is taken to be the same on the two copies of the manifold one obtains a model of a scalar field coupled to einstein gravity this field is geometrically interpreted as describing the distance between the two points in the internal space

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

differential geometry of group lattices in a series of publications we developed differential geometry on discrete sets based on concepts of noncommutative geometry in particular it turned out that first order differential calculi over the algebra of functions on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set a particular class of digraphs are cayley graphs also known as group lattices they are determined by a discrete group g and a finite subset s there is a distinguished subclass of bicovariant cayley graphs with the property ad s s s we explore the properties of differential calculi which arise from cayley graphs via the above correspondence the first order calculi extend to higher orders and then allow us to introduce further differential geometric structures furthermore we explore the properties of discrete vector fields which describe deterministic flows on group lattices a lie derivative with respect to a discrete vector field and an inner

spl lscr m s filters for image restoration applications a new filtering architecture is proposed generalizing some previously introduced multilevel median filters an efficient design procedure for the new filtering architecture is demonstrated for image restoration application simulation results show a good noise rejection performance combined with a fine detail preservation capability

extended permutation filters and their application to edge enhancement extended permutation ep filters are defined and analyzed in particular we focus on extended permutation rank selection eprs filters these filters are constrained to output an order statistic from an extended observation vector this extended vector includes n observation samples and k statistics that are functions of the observation samples the rank permutations from selected samples in this extended observation vector are used as the basis for selecting an order statistic output we show that by including the sample mean in the extended observation vector the filters exhibit excellent edge enhancement properties we also show that several previously defined classes of rank order based edge enhancers cs lum and wmmr sharpeners can be formulated as subclasses of eprs filters these sharpening subclasses are in addition to the smoothing subclasses which include rank conditioned rank selection permutation stack and weighted order statistic filters thus this novel class of filters provides a broad framework within which many rank order based smoothers and edge enhancers can be unified edge enhancement properties are developed and an l sub n norm eprs filter optimization procedure is presented finally extensive computer simulation results are presented comparing the performance of eprs and other sharpening filters in edge enhancement applications

fusion of median and bilateral filtering for range image upsampling we present a new upsampling method to enhance the spatial resolution of depth images given a low resolution depth image from an active depth sensor and a potentially high resolution color image from a passive rgb camera we formulate it as an adaptive cost aggregation problem and solve it using the bilateral filter the formulation synergistically combines the median and bilateral filters thus it better preserves the depth edges and is more robust to noise numerical and visual evaluations on a total of 37 middlebury data sets demonstrate the effectiveness of our method a real time high resolution depth capturing system is also developed using commercial active depth sensor based on the proposed upsampling method

differential calculi on commutative algebras a differential calculus on an associative algebra a is an algebraic analogue of the calculus of differential forms on a smooth manifold it supplies a with a structure on which dynamics and field theory can to some extent be formulated in very much the same way we are used to in the geometrical arena underlying classical physical theories and models in previous work certain differential calculi on a commutative algebra exhibited relations with lattice structures stochastics and parametrized quantum theories this motivated the present systematic investigation of differential calculi on commutative and associative algebras various results about their structure are obtained in particular it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of i forms an example of such a product is provided by the ito calculus of stochastic differentials for the case where the algebra a is freely generated by coordinates xi i 1 n we study calculi for which the differentials dxi constitute a basis of the space of 1 forms as a left a module these may be regarded as deformations of the ordinary differential calculus on rn for n or 3 a classification of all orbits under the general linear group of such calculi with constant structure functions is presented we analyse whether these calculi are reducible i e a skew tensor product of lower dimensional calculi or whether they are the extension as defined in this paper of a one dimension lower calculus furthermore generalizations to arbitrary n are obtained for all these calculi

boundaries for algebras of holomorphic functions abstract let a u b g be the banach algebra of all complex valued functions defined on the closed unit ball b g of a complex banach space g which are uniformly continuous on b g and holomorphic in the interior of b g endowed with the sup norm a characterization of the boundaries for a u b g is given in case g belongs to a class of banach spaces that includes the pre dual of a lorentz sequence space studied by gowers in israel j math 69 1990 129 151 the non existence of the shilov boundary for a u b g is also proved

gravity on finite groups gravity theories are constructed on finite groups g a self consistent review of the differential calculi on finite g is given with some new developments the example of a bicovariant differential calculus on the nonabelian finite group s3 is treated in detail and used to build a gravity like field theory on s3

analysis of the properties of median and weighted median filters using threshold logic and stack filter representation the deterministic properties of weighted median wm filters are analyzed threshold decomposition and the stacking property together establish a unique relationship between integer and binary domain filtering the authors present a method to find the weighted median filter which is equivalent to a stack filter defined by a positive boolean function because the cascade of wm filters can always be expressed as a single stack filter this allows expression of the cascade of wm filters as a single wm filter a direct application is the computation of the output distribution of a cascade of wm filters the same method is used to find a nonrecursive expansion of a recursive wm filter as applications of theoretical results several interesting deterministic and statistical properties of wm filters are derived

conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces the conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived for discrete models the conserved charges are constructed explicitly the applications of the general method include equations on quantum plane supersymmetric equations for chiral and antichiral supermultiplets auxiliary equations of integrable models which means various cases of nonlinear toda lattice equations and anomalous diffusion equation

linear connections on graphs in recent years discrete spaces such as graphs have attracted much attention as models for physical spacetime or as models for testing the spirit of noncommutative geometry in this work we construct the differential algebras for graphs by extending the work of dimakis et al and discuss linear connections and curvatures on graphs especially we calculate connections and curvatures explicitly for the general nonzero torsion case there is a metric but no metric compatible connection in general except the complete symmetric graph with two vertices