pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

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

fixed points selections and common fixed points for nonexpansive type mappings we study the existence of fixed points in the context of uniformly convex geodesic metric spaces hyperconvex spaces and banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity besides we use the fixed point theorems proved here to give common fixed point results for commuting mappings

fuzzy plane geometry i points and lines abstract we introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry we consider the fuzzy distance between fuzzy points and show it is a weak fuzzy metric we study various definitions of a fuzzy line develop their basic properties and investigate parallel fuzzy lines

composite radiation dose representation using fuzzy set theory composite plans created from different image sets are generated through deformable image registration dir and present a challenge in accurately presenting uncertainties which vary with anatomy our effort focuses on the application of fuzzy set theory to provide an accurate dose representation of such a composite treatment plan the accuracy of the dir is generally verified through geometrical visual checks including the confirmation of the corresponding anatomies with edge features such as bone or organ boundaries however the remaining volume of the image mostly soft tissues has few significant image features and therefore greater uncertainty we fuzzified the deformation vector and derived a fuzzy composite dose the fuzzification was implemented using gaussian functions based on the varying uncertainties in the dir after establishing the theoretical basis for this new approach we present two and three dimensional examples as proof of concept using fuzzy set theory composite dose plans displaying locality based uncertainties were successfully created providing information previously unavailable to clinicians previous to fuzzy set dose presentations clinicians had no measure of confidence in the accuracy of a composite dose plan using fuzzified composite dose presentations clinicians can determine a safe additional dose to previously treated anatomy this will possibly increase the treatment success rate and reduce the rate of complications

the maximal distance between imprecise point objects in this paper we propose a new mathematical model of imprecise point objects and a new algorithm for determining the maximal distance between imprecise point objects results obtained with our model can be used in various applications such as gis imprecise spatial object modelling robotics environment models and image analysis imprecise feature extraction imprecise point objects are modelled as fuzzy points in linear fuzzy space linear fuzzy space is defined over r n where the membership function is convex upper semi continuous symmetric w r t the core and linearly decreasing w r t the core distance where the core is a single point a fuzzy convex hull is defined in linear fuzzy space the distance between objects points is modelled as an imprecise point object in linear fuzzy space defined over r 1 fuzzy ordering relations are defined in distance space the maximal distance is a fuzzy set over a set of distances the maximal distance between a set of imprecise point objects is then determined as the maximal distance of fuzzy convex hull edge points an algorithm for determining the maximal distance between imprecise points belonging to one and two dimensional linear fuzzy space is given for an exact calculation of this maximal distance computational complexity of the algorithm is of polynomial growth of the fourth degree however if the maximal distance is determined approximately then the algorithms computational complexity is of power growth of the quadratic degree with respect to the cardinality of the convex edge

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

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

fixed points selections and best approximation for multivalued mappings in r trees abstract in an r tree we consider multivalued mappings with convex values and develop fixed point selection and best approximation theorems for nonexpansive and almost lower semicontinuous mappings

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

the essential numerical range for unbounded linear operators abstract we introduce the concept of essential numerical range w e t for unbounded hilbert space operators t and study its fundamental properties including possible equivalent characterizations and perturbation results many of the properties known for the bounded case do not carry over to the unbounded case and new interesting phenomena arise which we illustrate by some striking examples a key feature of the essential numerical range w e t is that it captures spectral pollution in a unified and minimal way when approximating t by projection methods or domain truncation methods for pdes

porosity results on fixed points for nonexpansive set valued maps in hyperbolic spaces abstract we establish the porosity results on the existence and the topological structure of fixed points for nonexpansive set valued maps in hyperbolic spaces which in particular extend and or improve some known results in this direction

best approximation coincidence and fixed point theorems for set valued maps in r trees abstract suppose x is a closed convex and geodesically bounded subset of a complete r tree m and suppose f x m is an almost lower semicontinuous set valued map whose values are nonempty closed convex suppose also g x x is a continuous onto quasiconvex set valued map with compact convex values then there exists x 0 x such that d g x 0 f x 0 inf x x d x f x 0 as applications we give some coincidence and fixed point results for weakly inward set valued maps our results generalize some well known results in literature

fixed point theorems for some generalized multivalued nonexpansive mappings abstract in this paper we introduce a condition on multivalued mappings which is a multivalued version of condition c defined by garcia falset et al 2011 3 it is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition our results generalize the results in lim 1974 lami dozo 1973 kirk and massa 1990 garcia falset et al 2011 dhompongsa et al 2009 and abkar and eslamian 2010 4 5 6 3 7 8 and many others

numerical aspects of searching convective absolute instability transition an overview of various numerical techniques used to determine the spatio temporal character of instabilities in fluid flows is given the advantageous features of various previously known individual techniques are discussed and a practical procedure combining them is suggested for a specific task of determining the complete boundary between linearly convectively and absolutely unstable regimes in a multi parameter space in problems with a fully numerical dispersion relation special attention is paid to aspects of automatization of computations as this is a crucial condition for their efficiency the suggested procedure is successfully used and is shown to provide a high degree of automatism in the physical example of non boussinesq mixed convection in a vertical channel this example comprises most of the major numerical difficulties found in various spatio temporal instability studies of two dimensional fluid flows which previously could not be handled without frequent human intervention and visual inspection of intermediate results this paper focuses on the general numerical aspects of the computations leaving the detailed discussion of the obtained physical results for a separate publication

a selection theorem for quasi lower semicontinuous mappings in hyperconvex spaces we prove a continuous selection theorem for quasi lower semicontinuous mappings with values that are closed sub admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces

best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces abstract in hyperconvex metric spaces we consider best approximation invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive

fixed points for generalized nonexpansive mappings in r trees we prove the existence of a fixed point for multivalued generalized nonexpansive mappings in an r tree in the case where these mappings commute with a quasi nonexpansive mapping we also prove existence of a common fixed point

the effects of obstacle size on periodic travelling waves in oscillatory reaction diffusion equations many natural populations undergo multi year cycles and field studies have shown that these can be organized into periodic travelling waves ptws mathematical studies have shown that large scale landscape obstacles represent a natural mechanism for wave generation here we investigate how the amplitude and wavelength of the selected waves depend on the obstacle size we firstly consider a large circular obstacle in an infinite domain for a reaction diffusion system of l u type we use perturbation theory to derive a leading order approximation to the wave generated by the obstacle this shows the dependence of the wave properties on both parameter values and obstacle size we find that the limiting values of the amplitude and wavelength are approached algebraically with distance from the obstacle edge rather than exponentially in the case of a flat boundary we use our results to predict the properties of waves generated by a large circular obstacle for an oscillatory predator prey system via a reduction of the predator prey model to normal form close to hopf bifurcation our predictions compare well with numerical simulations we also discuss the implications of these results for wave stability and briefly investigate the effects of obstacles with elliptical geometries

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number

hyperconvexity and nonexpansive multifunctions it is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection this implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps the result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex

instability of periodic traveling wave solutions in a modified fitzhugh nagumo model for excitable media we introduce a two variable system of reaction diffusion equations for excitable media we numerically investigate the existence and stability of periodic traveling wave solutions in a two dimensional parameter plane our results based on the method of continuation show a stability change of eckhaus type there are two families of periodic traveling waves fast and slow the fast family is stable in the case of standard fitzhugh nagumo excitable system however we observe that the fast family becomes unstable in our model consequently it bifurcates to an oscillating wave we explain this phenomenon by numerically calculating the essential spectra of the periodic traveling wave solutions moreover we study the stability of the periodic traveling wave solutions for the aliev panfilov excitable system and compare its results with the proposed model we show that the stability boundary of our model agrees qualitatively with that of the aliev panfilov model whereas it is different in the fitzhugh nagumo model in addition our results based on the stability boundary of the models show oscillating periodic traveling waves except for the fitzhugh nagumo model

remarks on approximate fixed points abstract a mapping f from a topological space x to a metric space m d is said to be r continuous r 0 if for any x x there exists a neighborhood u x of x such that diam f u x r in 1961 victor klee proved that an r continuous self mapping of a compact convex subset of a banach space always has a invariant point for any r recently h x phu extended klee s result to mappings that are almost r continuous in this note we extend phu s result to compact hyperconvex spaces and to geodesically bounded complete r trees we also extend klee s stability result to certain geodesic spaces

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

approximate selections of almost lower semicontinuous multimaps in c spaces abstract we study the relations of almost lower semicontinuity lower semicontinuity and other generalized lower semicontinuity then we establish a new approximate selection theorem for almost lower semicontinuous multimaps with the generalized zima type condition in c spaces our result unify and extend the approximate selection theorems in many published works

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

continuation of spiral waves abstract we describe a new numerical method of computing rigidly rotating spiral waves which is based on solving the neumann boundary value problem for the radius dependent angular fourier modes utilizing the established continuation engine auto our method is simple in implementation and can be easily modified to suit a particular reaction diffusion system since the method does not involve direct simulations of the reaction diffusion system unstable branches of rigidly rotating spiral waves can be computed as well we illustrate our method by computing single and multi armed spirals in the barkley model continuation of single armed spirals displays nearly identical results with the barkley s continuation code steady the dependence of spiral waves on the geometry of the medium reproduces the results of numerical simulations reported before in a m pertsov e a ermakova a v panfilov rotating spiral waves in a modified fitz hugh nagumo model physica d 14 1 1984 117 124 revealing however some subtle details like non monotonous dependence of the rotation frequency on the disc radius and the existence of an unstable rotating solution that separates coexisting free and pinned spirals we demonstrate that on bounded discs spiral waves are accompanied by boundary spots slowly rotating solutions which are localized near the outer boundary of the disc boundary spots are shown to be closely related to one and two dimensional unstable critical solutions such as unstable pulses in one dimension and critical fingers in two dimensions which separate spiral waves from shrinking wave segments

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

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

hopf bifurcation from fronts in the cahn hilliard equation we study hopf bifurcation from traveling front solutions in the cahn hilliard equation the primary front is induced by a moving source term models of this form have been used to study a variety of physical phenomena including pattern formation in chemical deposition and precipitation processes technically we study bifurcation in the presence of an essential spectrum we contribute a simple and direct functional analytic method and determine bifurcation coefficients explicitly our approach uses exponential weights to recover fredholm properties and spectral flow ideas to compute fredholm indices simple mass conservation helps compensate for negative indices we also construct an explicit prototypical example prove the existence of a bifurcating front and determine the direction of bifurcation

state selection in the noisy stabilized kuramoto sivashinsky equation in this work we study the one dimensional stabilized kuramoto sivashinsky equation with additive uncorrelated stochastic noise the eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band this is consistent with the behavior of the phase diffusion constants of these states some connections to the phenomenon of state selection in driven out of equilibrium systems are made

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

fuzzy plane geometry ii circles and polygons abstract this paper continues our research in fuzzy plane geometry previously we studied fuzzy points and fuzzy lines now we investigate fuzzy circles and fuzzy polygons we show that the fuzzy area of a fuzzy circle or a fuzzy polygon is a fuzzy number we also argue that the fuzzy perimeter of a fuzzy circle or a fuzzy polygon is a fuzzy number

an experimental study on fuzzy distances for skull face overlay in craniofacial superimposition abstract skull face overlay is the most time consuming and error prone stage in craniofacial superimposition an important skeleton based forensic identification technique this task focuses on achieving the best possible overlay of an unknown skull found and a single ante mortem image of a candidate missing person the process is influenced by some sources of uncertainty since two objects of different nature are involved i e a skull and a face in previous works we have developed a computer aided craniofacial superimposition system aimed to assist forensic anthropologists in obtaining the best possible skull and face overlay the system has successfully allowed us to reduce the processing time simplify the forensic anthropologists work and make the process more objective and reproducible our approach is based on automatically overlaying a skull three dimensional model onto a facial photograph by minimizing the distance between two subsets of corresponding cranial and facial landmarks the proposed method properly deals with the inherent uncertainty sources to the skull face overlay process by considering fuzzy sets to model imprecise landmark location and imprecise cranial and facial landmarks spatial correspondence resulting from the presence of soft tissues in the face accordingly our methodology requires computing two kinds of distance metrics between a point and a fuzzy set and between two fuzzy sets this contribution is devoted to study the performance and influence of the most significant and suitable fuzzy distances proposed in the specialized literature as well as other new ones proposed on our skull face overlay system in particular we have tested the behavior of our automatic method when considering eight different distance measurements the system performance has been objectively evaluated considering 18 case studies resulting from a ground truth dataset following a rigorous statistical experimental setup the fact that the choice of a good distance metric is crucial to our method has been demonstrated since it significantly affects the quality of the final solutions it has been shown that our skull face overlay approach presents the best performance using the weighted mean distance in most of the cases and that the results are both more accurate and robust than the other studied metrics

generation of periodic travelling waves in cyclic populations by hostile boundaries many recent datasets on cyclic populations reveal spatial patterns with the form of periodic travelling waves wavetrains mathematical modelling has identified a number of potential causes of this spatial organization one of which is a hostile habitat boundary in this paper the author investigates the member of the periodic travelling wave family selected by such a boundary in models of reaction diffusion type using a predator prey model as a case study the author presents numerical evidence that the wave generated by a hostile zero dirichlet boundary condition is the same as that generated by fixing the population densities at their coexistence steady state levels the author then presents analysis showing that the two waves are the same in general for oscillatory reaction diffusion models with scalar diffusion close to hopf bifurcation this calculation yields a general formula for the amplitude speed and wavelength of these waves by combining this formula with established results on periodic travelling wave stability the author presents a division of parameter space into regions in which a hostile boundary will generate periodic travelling waves spatio temporal disorder or a mixture of the two

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

notes on fuzzy plane geometry i ii abstract in this paper we give membership functions of some concepts in buckley and eslami fuzzy sets and systems 86 1997 179 187 87 1997 79 85 such as fuzzy distance fuzzy lines fuzzy circles fuzzy area and fuzzy circumference of a fuzzy circle by the use of formulas of those membership functions one can define general fuzzy concepts in plane geometry

lim s theorems for multivalued mappings in cat 0 spaces let x be a complete cat 0 space we prove that if e is a nonempty bounded closed convex subset of x and t e k x a nonexpansive mapping satisfying the weakly inward condition i e there exists p e such that p 1 tx ie x x e 0 1 then t has a fixed point in banach spaces this is a result of lim on asymptotic centers and fixed points of nonexpansive mappings canad j math 32 1980 421 430 the related result for unbounded r trees is given

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

a novel visual codebook model based on fuzzy geometry for large scale image classification the codebook model has been developed as an effective means for image classification however the inherent operation of assigning visual words to image feature vectors in traditional codebook approaches causes serious ambiguities in image classification in particular the nearest word may not be the best fit to a feature and multiple words may be equally appropriate for one specific feature to resolve these ambiguities we propose a novel visual codebook model based on the n dimensional fuzzy geometry n d fg theory where all visual words and features are modeled as fuzzy points in the n d fg space and appropriate uncertainty is introduced to each fuzzy point to enhance the representation capacity this n d fg codebook model not only inherits advantages from the fuzzy set theory but also facilitates the analysis and determination of the relationship between visual words and features in geometric form by explicitly taking into account the ambiguities we propose a novel measure of similarity between the visual words and fuzzy features following the proposed codebook model and the novel similarity measure we develop two useful image classification algorithms by modifying popular image coding algorithms i e spm and llc finally experimental results demonstrate that the classification accuracy of the proposed algorithms is dramatically improved for a standard large scale image database for example with a codebook size of 256 the proposed algorithms achieve similar performance as traditional algorithms with a codebook size of 1024 indicating that the proposed algorithms reduce the computational cost by 75 while achieving almost identical classification accuracy to traditional algorithms thus the proposed algorithms represent a more efficient and appropriate scheme for big image data this paper aims to overcome the drawbacks of traditional visual words fuzzy visual word and fuzzy feature are defined using n dimensional fuzzy geometry a new similarity measure between fuzzy features and fuzzy visual words is designed two modified image classification frameworks based on fuzzy codebook are proposed experimental results demonstrated their advantages against traditional algorithms

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

invariant approximations for commuting mappings in cat 0 and hyperconvex spaces abstract in this paper for a commuting pair consisting of a point valued nonexpansive self mapping t and a set valued nonexpansive self mapping t of a hyperconvex metric space or a cat 0 space x we look for a solution to the problem of existence of z e x such that d z y d y e and z t z t z

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

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future