fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

fuzzy geometry perpendicular to fuzzy line segment abstract in this study we define the perpendicular to fuzzy line segment pfls which intersects a given fuzzy line segment fls normally at a specified fuzzy point in order to construct pfls some spatial transformations of fuzzy points such as translation rotation expansion contraction and reflection trecore are needed and have also been defined here the concept of pfls has been justified and formulated in three different ways with the help of same and inverse point theory of fuzzy points their properties and interrelations are also investigated the area of a fuzzy triangle has been calculated with the proposed methodology

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

connectedness of coherent fuzzy pictures abstract a fuzzy digital picture is called connected if it has only one connected component of constant fuzzy level that is maximal i e not adjacent to any component of higher fuzzy level in this paper we propose two acceptors defined over fuzzy digital pictures one is a one pebble acceptor and another is a deterministic one way nonwriting parallel sequential acceptor we also prove that both acceptors can determine whether a given coherent fuzzy digital picture is connected

stabilizing controllers for bilinear systems the problem of finding stabilizing feedback controllers for continuous bilinear systems where the controls act additively and multiplicatively simultaneously is treated the applicability of the quadratic control law of jacobson 5 and others is extended to the case when the a matrix has arbitary eigenvalues under certain conditions a class of bilinear systems common among biochemical flow systems is defined the dyadic bilinear systems a control scheme the so called division controller for dyadic bilinear systems is suggested the practicality of the control schemes is demonstrated on the problem of controlling the neutron level in a fission reactor

principal superconnected fuzzy topological spaces abstract in this note principal superconnected fuzzy topological spaces are introduced and it is proved that such a space is path connected

data imprecision under λ geometry model introducing a model called geometry for handling a dynamic form of imprecision which allows the precision changes applications of the geometry model in facility locations and designing robust algorithms are discussed proposing efficient algorithms for solving the problem of axis aligned bounding box under geometry model proposing efficient algorithms for solving the proximity problems under the geometry model an essential assumption in most algorithms in computational geometry is that the input data is precise which is not always true in real world problems in this paper we introduce a model called geometry model for handling a dynamic form of imprecision which allows the precision changes in the input data of geometric problems also we study two families of basic computational geometry problems under this model axis aligned bounding box aabb and proximity problems for a set of n imprecise points where each point is modeled by a convex polygonal region with o k representation complexity we propose an o n log n log k m time algorithm to solve the largest aabb problem where m is the maximum number of corner defining functions when the largest aabb is defined by points on its corners also we propose an o n log n nk k2 log k k time algorithm for finding the smallest aabb under the assumption that the regions are disjoint we show that the algorithms are optimal when k is o log n further we propose o n2k log nk nk time algorithms for finding the closest and furthest pair problems and present a lower bound of n2 for the closest pair problem when k is constant

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

preprocessing imprecise points and splitting triangulations traditional algorithms in computational geometry assume that the input points are given precisely in practice data is usually imprecise but information about the imprecision is often available in this context we investigate what the value of this information is we show here how to preprocess a set of disjoint regions in the plane of total complexity n in o n log n time so that if one point per set is specified with precise coordinates a triangulation of the points can be computed in linear time in our solution we solve another problem which we believe to be of independent interest given a triangulation with red and blue vertices we show how to compute a triangulation of only the blue vertices in linear time

fuzzy cell hough transform for curve detection in this paper a new variation of hough transform is proposed it can be used to detect shapes or contours in an image with better accuracy especially in noisy images the parameter space of hough transform is split into fuzzy cells which are defined as fuzzy numbers this fuzzy split provides the advantage to use the uncertainty of the contour point location which is increased when noisy images are used by using fuzzy cells each contour point in the spatial domain contributes in more than one fuzzy cell in the parameter space the array that is created after the fuzzy voting process is smoother than in the crisp case and the effect of noise is reduced the curves can now be detected with better accuracy the computation time that is slightly increased by this method can be minimized in comparison with classical hough transform by using recursively the fuzzy voting process in a roughly split parameter space to create a multiresolution fuzzily split parameter space

the diameter of a fuzzy set abstract in pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image this paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp we also define height and width for a fuzzy subset and show how they relate to the area i e integral of membership for convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter but it can be smaller finally for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter but this need not be true in the nonconvex case

a finiteness criterion for nonlinear input output differential systems we solve the following problem which has been dealt with in several publications when does a system have the same input output behaviour as a system depending linearly on the state variables precise definitions of this question are supplied in the text a particular case of the problem is the equivalence of a system with a bilinear one at the end we illustrate our methods and the scope of our results by a brief study of several examples

degree of adjacency or surroundedness abstract definitions of the degree of adjacency of two regions in the plane and the degree of surroundedness of one region by another are proposed some elementary properties of these concepts are established and it is also shown that they have natural generalizations to fuzzy subsets of the plane applications of the proposed measures to digital polygons are demonstrated and fast algorithms for computing these measures are given

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

a medial axis transformation for grayscale pictures blum s medial axis transformation mat for binary pictures yields medial axis points that lie midway between opposite borders of a region or along angle bisectors this note discusses a generalization of the mat in which a score is computed for each point p of a grayscale picture based on the gradient magnitudes at pairs of points that have p as their midpoint these scores are high at points that lie midway between pairs of antiparallel edges or along angle bisectors so that they define a mat like skeleton which we may call the gradmat however this skeleton is rather sensitive to the presence of noise edges or to irregularities in the region edges and it also is subject to artifacts created by pairs of edges belonging to different objects

a min max medial axis transformation blum s medial axis transformation mat of the set s of 1 s in a binary picture can be defined by an iterative shrinking and reexpanding process which detects corners on the contours of constant distance from s and thereby yields a skeleton of s for unsegmented gray level pictures one can use an analogous definition in which local min and max operations play the roles of shrinking and expanding to compute a mmmat value at each point of the picture the set of points having high values defines a good skeleton for the set of high gray level points in the given picture

on a fuzzy theory based computer aided particle shape description abstract a central problem of particle shape analysis consists in the characterization of the shape of any natural of produced particle by a small number of parameters which are to be evaluated from a two dimensional picture of the particle e g from its projection we suggest using the grey tone picture obtained by a camera to compute a sympathy function with respect to a suitable family of planar shapes the computation is performed by means of the image processing equipment robotron a 6471 the sympathy function defines a fuzzy set on the parameter space the procedure is applied to a quartz particle

three dimensional connected pictures are not recognizable by finite state acceptors abstract it is proved that two dimensional connected pictures over 0 1 are not recognizable by finite state acceptors

a note on fuzzy starshaped fuzzy sets fuzzy starshaped sets have starshaped levels sets and generalise fuzzy convex sets they have similar metric properties and lp metrics may be defined using gauge functions the induced metric spaces are shown to be locally compact for 1 p and compact sets are characterised

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

global exact quadratization of continuous time nonlinear control systems we consider a class denoted by cal q of the nonlinear control systems which can be densely represented as a subsystem of a certain kind of quadratic system namely a quadratic target we say that a system in cal q undergoes a globally exact quadratization here globally adds up to a slight extension of the notion of c infty immersion of systems namely a dense immersion which amounts to saying that it is defined on the whole manifold of the system states except possibly a zero measure set it is proven that the class cal q includes all systems characterized by vector fields whose components are analytic integral closed form functions icffs the result is first proven for algebraic system functions by means of a constructive proof and next extended up to analytic icffs for nonanalytic icffs a weaker result is proven as well also the case of a partially observed system is considered as well as the internal structure of every quadratic representation which is proven to be alwa

a note on the quantitative measure of image enhancement through fuzziness the index of fuzziness and entropy of an image reflect a kind of quantitative measure of its enhancement quality their values are found to decrease with enhancement of an image when different sets of s type membership functions with appropriate crossover points were considered for extracting the fuzzy property plane from the spatial domain of the image

on the perimeter and area of fuzzy sets abstract the note of area comes naturally as a measure of a fuzzy set recently a rosenfeld defined the permeter diameter height and width of a fuzzy set athough these do not generalize the corresponding notions for crisp sets they lack inner conformity in that they do not satisfy simple interrelations we are used to in the case of crisp sets in the present note we modify definitions and show that our perimeter diameter and height fit more comfortably with the notion of area we also define a projection operator that should be used along with the multiplicative conjunction in the process of fuzzy reasoning

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

the controllability of planar bilinear systems recent results concerning the controllability of planar bilinear systems are summarized the results are then used to characterize the controllability behavior of the homogeneous and general bilinear system in r 2

analytical fuzzy plane geometry iii in this study we attempt to construct fuzzy circles in a fuzzy geometrical plane we provide a comprehensive study where we find a fuzzy number with a predetermined fuzzy distance from a given fuzzy number we formulate fuzzy circles using the conventional basic definitions of crisp circles two different formulations of fuzzy circles are proposed which depend on the information known about a fuzzy circle the interrelations between the evaluated fuzzy circles are investigated we show that the center of a fuzzy circle may not be a fuzzy point however the radius of a fuzzy circle that passes through three given fuzzy points is always a fuzzy number the concepts of same and inverse points and a fuzzy number along a line are used to define and analyze the proposed ideas our discussions and studies are supported by suitable numerical and pictorial examples

finite determination of accessibility and singular points of nonlinear systems an algebraic approach abstract exploiting tools from algebraic geometry the problem of determination of accessibility strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems the results are constructive and algorithms are given to find the maximum depth of lie brackets necessary for deciding accessibility strong accessibility of the system at any point called here accessibility strong accessibility index of the system and known as the degree of non holonomy in the literature alternatively upper bounds on the accessibility strong accessibility index are obtained which can be computed easier in each approach the entire set of accessibility strong accessibility singular points are obtained as a limiting algebraic set of a strictly increasing chain of ideals that stabilizes in finite time several examples demonstrate the applicability of the results using computer algebra tools

image segmentation using fuzzy correlation a concept of correlation between two properties fuzzy representations of an image is introduced a set of algorithms for image segmentation both fuzzy and nonfuzzy has been formulated the spatial information is taken care of by the following measures transitional correlation and within class correlation a relation between the correlation coefficient and the index of fuzziness is theoretically established and experimentally verified the effectiveness of the algorithms is illustrated on images having different types of histograms

on the solution calculation of nonlinear ordinary differential equations via exact quadratization abstract we show a general method allowing the solution calculation in the form of a power series for a very large class of nonlinear ordinary differential equations odes namely the real analytic odes and more in general the real analytic reducible odes in many indeterminates characterized by an ode function given by generalized polynomials of the indeterminates and their derivatives i e functions formally polynomial with exponents though the exponent can be any real number and whose coefficients are analytic time functions the solution method consists in reducing the ode to a certain canonical homogeneous quadratic ode named driver type riccati equation with a larger number of indeterminates whose solutions include as sub solutions the original solutions and for which a recursive formula is shown giving all coefficients of the solution power series directly from the ode parameters the reduction method is named exact quadratization and was formerly introduced in another our article where we considered explicit odes only in the present paper which is self contained to a large extent we review and complete the theory of exact quadratization by solving issues such as for instance the existence of a piecewise quadratization that had remained open and also extend it to the more general case of an implicitly defined ode finally we argue that the result can be seen as a partial solution of a differential version of the 22nd hilbert s problem

investigations on fuzzy thresholding based on fuzzy clustering thresholding the problem of pixel classification is attempted here using fuzzy clustering algorithms the segmented regions are fuzzy subsets with soft partitions characterizing the region boundaries the validity of the assumptions and thresholding schemes are investigated in the presence of distinct region proportions the hard k means and fuzzy c means algorithms have been found useful when object and background regions are well balanced fuzzy thresholding is also formulated as extraction of normal densities to provide optimal partitions regional imbalances in gray distributions are taken care of in region normalized histograms

fuzziness measures for fuzzy rectangles abstract taking inspiration from the axioms for shannon entropy and u uncertainty we study fuzziness measures such that the fuzziness of a genuine rectangular set can be computed from the fuzziness of its sides the results found plead in favour of measures based on the max operation rather than measures of the additive type

cubification of σπ sde and exact moment equations abstract for an ito like stochastic differential equation sde system with drift and diffusion that are formal polynomials of the independent variables we show that all moments satisfy an infinite countable set of linear ordinary differential equations this result is achieved by means of the exact cubification of the sde which consists in a set of deterministic transformations of the state variables giving place to a new sde with further finitely many state variables exact cubification can be considered as an extension to the stochastic case of the exact quadratization of deterministic nonlinear systems available in the literature an example is finally shown taken from systems biology in which for a basic chemical reaction network the exact moment equation is written down and an approximate solution is calculated through a moment closure method

primal dual extragradient methods for nonlinear nonsmooth pde constrained optimization we study the extension of the chambolle pock primal dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces local convergence is shown under technical conditions including metric regularity of the corresponding primal dual optimality conditions we also show convergence for a nesterov type accelerated variant provided one part of the functional is strongly convex we show the applicability of the accelerated algorithm to examples of inverse problems with l 1 and l infty fitting terms as well as of state constrained optimal control problems where convergence can be guaranteed after introducing an arbitrarily small still nonsmooth moreau yosida regularization this is verified in numerical examples

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

on connectivity properties of grayscale pictures abstract a grayscale digital picture is called connected if it has only one connected component of constant gray level that is maximal i e not adjacent to any component of higher gray level this note establishes some equivalent conditions for connectedness and also defines a grayscale generalization of the genus in terms of sums of local property values

an adaptive domain decomposition technique for parallelization of the fast marching method abstract the fast marching method fmm is an efficient technique to solve numerically the eikonal equation the parallelization of the fmm is not easy because of its intrinsic sequential nature in this paper we propose a novel approach to parallelize the fmm it leads to an equation dependent domain decomposition and it turns out to be particularly suitable for machines with two or four cores that are in common use today compared to other techniques in the field the proposed method is much simpler to implement and it gives a slightly better computational speed up in order to test the new method on a real world application we solve the shape from shading problem based on a hamilton jacobi equation on a standard four core machine the method confirms the good properties it shows a reasonable speedup factor of about 2 5 and it reveals its potential to good performance if the arithmetic density of the problem is high

flow stability of patchy vector fields and robust feedback stabilization the paper is concerned with patchy vector fields a class of discontinuous piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems we prove the stability of the corresponding solution set w r t a wide class of impulsive perturbations these results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances

connectedness in fuzzy topological spaces abstract we introduce four notions of connectedness for an arbitrary fuzzy set and discuss all the implications that exist between them using these notions various equivalence relations are defined on the set of fuzzy points components and quasi components are formed as the union of certain equivalence classes a fuzzy space is shown to be a dispoint union of its components in all the four cases and non overlapping union of its quasi components

on the linear equivalents of nonlinear systems this paper studies feedback equivalence of nonlinear systems an equivalence relation is established for the systems x f x u a complete characterization of the class of nonlinear systems which are equivalent to controllable linear systems is obtained

fast semi lagrangian schemes for the eikonal equation and applications we introduce and analyze a fast version of the semi lagrangian algorithm for front propagation originally proposed in m falcone the minimum time problem and its applications to front propagation in motion by mean curvature and related topics a visintin and g buttazzo eds de gruyter berlin 1994 pp 70 88 the new algorithm is obtained using the local definition of the approximate solution typical of semi lagrangian schemes and redefining the set of neighboring nodes necessary for fast marching schemes a new proof of convergence is needed since that definition produces a new narrow band centered at the interphase which is larger than the one used in fast marching methods based on finite differences we show that the new algorithm converges to the viscosity solution of the problem and that its complexity is o n log n nb as it is for the fast marching method based on finite difference n and n nb being respectively the total number of nodes and the number of nodes in the narrow band a new sufficient condition for the convergence of the standard finite difference fast marching method is also given we present several tests comparing the two algorithms and other fast methods e g fast sweeping on a series of benchmarks which include the minimum time problem and the shape from shading problem

image segmentation using fuzzy homogeneity criterion abstract the approach proposed here is using fuzzy homogeneity vectors and the fuzzy cooccurrence matrix to segment images homogeneity vectors could be used to represent the homogeneous feature between a pixel and its neighbors and the degree of homogeneity could be determined by using a fuzzy membership function combining the homogeneity vectors and the fuzzy membership function we could extract the feature of an image which determines the fuzzy region width with the fuzzy region width we could use the fuzzy cooccurrence matrix to measure the fuzzy geometry properties of an image which is the fuzzy entropy values that can be employed for determinine the thresholds to segment images the advantages of this approach are as follows first homogeneity vectors take account of the spatial gray tone dependence thus using homogeneity vectors has a better noise tolerance second from the extracted feature the fuzzy region width could be determined automatically third because the fuzzy region width is decided by the feature of the image it could be adjusted according to the nature of the image fourth the dimension of the fuzzy cooccurrence matrix representing the image is significantly decreased but the properties of images are faithfully preserved a large number of experiments have been carried out on different kinds of images and good results have been achieved by the proposed method this method will have wide application in image processing

immersion of nonlinear systems into linear systems modulo output injection the problem of the immersion of a siso system into a linear up to an output injection one is studied in order to design luenberger like observers necessary and sufficient conditions are stated within a very general framework effective computations and examples are then provided

quadratic dynamical systems and algebras abstract quadratic dynamical systems come from differential or discrete systems of the form q x or x k 1 q x k where q r n r n is homogeneous of degree 2 i e q x 2 q x for all r x r n defining a bilinear mapping r n r n r n by x y 1 2 q x y q x q y we view xy x y as a multiplication and thus consider a r n to be a commutative nonassociative algebra the quadratic systems are then studied with the general theme that the structure of the algebras helps determine the behavior of the solutions for example semisimple algebras give a decoupling of the original system into systems occurring in simple algebras and solvable algebras give solutions to differential systems via linear differential equations the general three dimensional example of the latter phenomenon is described there are many classical examples and the scope of quadratic systems is large every polynomial system can be embedded into a higher dimensional quadratic system such that solutions of the original system are obtained from the quadratic system for differential systems nilpotents of index 2 n 2 0 are equilibria and idempotents e 2 e give ray solutions the origin is never asymptotically stable and the existence of nonzero idempotents implies that the origin is actually unstable nonzero equilibria are not hyperbolic but can be studied by standard algebra techniques using nondegenerate bilinear forms as lyapunov functions periodic orbits lie on cones they cannot occur in dimension 2 or in power associative algebras no periodic orbit can be an attractor but limit cycles invariant cones can exist automorphisms of the algebra a leave equilibria periodic orbits and domains of attraction invariant also explicit solutions can be given by the action of automorphisms on an initial point the general three dimensional example of this is described thus if there are sufficient automorphisms hilbert s sixteenth problem in r 3 has the following answer if the periodic orbits of fixed period are isolated then there is only one cone of periodic solutions this cone is almost an attractor for discrete systems there are many similarities to the differential systems for example orbits can be given by automorphisms and again the general three dimensional example of this is described however distinctions become more obvious using algebras for example if the algebra a is nilpotent then for the differential system the solutions are unbounded but for the discrete system the trajectories iterate to zero in a also idempotents e 2 e are the fixed points for the discrete system and e is unstable if there exist suitable nilpotents n 2 0 the interplay between algebras and dynamical systems can solve old problems but more importantly it can create new opportunities in both areas

some results on fuzzy digital convexity abstract the definition of fuzzy convexity is reviewed and some results on projections of convex and fuzzy convex sets are established digital fuzzy convexity is defined and relationships among alternative definitions are investigated

a new thinning algorithm for gray scale images by the relaxation technique abstract a new thinning algorithm for gray scale images based on the relaxation technique is proposed thinning is treated as a process of assigning each picture point to either a skeleton or a nonskeleton category so that relaxation can be used to perform thinning contextual information existing in input data is utilized in the thinning process so that line straightness can be preserved five classes are created to classify each point with four classes belonging to the skeleton category for different orientations and one to the nonskeleton category an initial probability value for assigning a point to each class is computed first in the relaxation process interactions take place among the probability values of neighbouring points making the skeleton probability values of the points in compatible orientations to enhance one another the probability value for each point to belong to the nonskeleton class of each point is also refined according to the point type the relaxation process is terminated when it converges to a condition in which points lying on skeletons achieve high skeleton probability values while other points have unity nonskeleton probability values experimental results show that the proposed approach produces good thinning results in keeping line straightness

fuzzy path and fuzzy connectedness abstract this paper discusses fuzzy connectedness in fuzzy topology zadeh s classical paper 1 has introduced the fundamental concept of a fuzzy set upon which the foundation of fuzzy mathematics rests the theory of fuzzy topology is a branch of such mathematics our purpose is to introduce some basic concepts as fuzzy path fuzzy path connected set locally fuzzy path connected set in the fuzzy topological spaces and to prove some fundamental properties related to fuzzy connectedness

optimal bilinear control of an abstract schro dinger equation well posedness of abstract quantum mechanical systems is considered and the existence of optimal control of such systems is proved first order optimality systems are derived convergence of the monotone scheme for the solution of the optimality system is proved

stable numerical schemes for solving hamilton jacobi bellman isaacs equations stable approximation schemes for solving hamilton jacobi bellman isaacs equations in r n are proposed the efficiency of numerical procedures based on such approximations is demonstrated by solving several differential games in two three and four dimensions

convergence of a generalized fast marching method for an eikonal equation with a velocity changing sign we present a new fast marching algorithm for an eikonal equation with a velocity changing sign this first order equation models a front propagation in the normal direction the algorithm is an extension of the fast marching method in two respects the first is that the new scheme can deal with a time dependent velocity and the second is that there is no restriction on its change in sign we analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity solutions finally we present some numerical simulations of fronts propagating in mathbb r 2

local minimization algorithms for dynamic programming equations the numerical realization of the dynamic programming principle for continuous time optimal control leads to nonlinear hamilton jacobi bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls this minimization is often performed by comparison over a finite number of elements of the control set in this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively the considered class of equations includes nonsmooth control problems with ell 1 penalization which lead to sparse controls

an efficient policy iteration algorithm for dynamic programming equations we present an accelerated algorithm for the solution of static hamilton jacobi bellman equations related to optimal control problems our scheme is based on a classic policy iteration procedure which is known to have superlinear convergence in many relevant cases provided the initial guess is sufficiently close to the solution this limitation often degenerates into a behavior similar to a value iteration method with an increased computation time the new scheme circumvents this problem by combining the advantages of both algorithms with an efficient coupling the method starts with a coarse mesh value iteration phase and then switches to a fine mesh policy iteration procedure when a certain error threshold is reached a delicate point is to determine this threshold in order to avoid cumbersome computations with the value iteration and at the same time to ensure the convergence of the policy iteration method to the optimal solution we analyze the methods and efficient coupling in a number of examples

the perimeter of a fuzzy set abstract in pattern recognition one often wants to measure the perimeters of regions in images this is straightforward if the region is crisply defined but if it is fuzzy it is not obvious how its perimeter can be measured this paper proposes a definition of perimeter for fuzzy subsets of the plane and shows that it reduces to the standard definition if the fuzzy subset is an ordinary subset the isoperimetric inequality does not generalize to fuzzy subsets but certain properties of the perimeters of convex sets do generalize to fuzzy perimeters of convex fuzzy subsets