spaces of paths and the path topology the natural topology on the space of causal paths of a space time depends on the topology chosen on the space time itself here we consider the effect of using the path topology on space time instead of the manifold topology and its consequences for how properties of space time are reflected in the structure of the space of causal paths

the phase plane picture for a class of fourth order conservative differential equations we study the bounded solutions of a class of fourth order equations u u f u 0 0 we show that when is not too large then the paths in the u u plane of two bounded solutions do not cross moreover the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase plane and a continuation theorem shows that they fill up part of the phase plane we apply these results to the extended fisher kolmogorov efk equation a fourth order model equation for bi stable systems the uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary efk equation correspond exactly to those of the classical second order fisher kolmogorov equation besides we establish the asymptotic stability of the heteroclinic solution of the efk equation

non imprisonment conditions on spacetime the non imprisonment conditions on spacetimes are studied it is proved that the non partial imprisonment property implies the distinction property moreover it is proven that feeble distinction a property which stays between weak distinction and causality implies non total imprisonment as a result the non imprisonment conditions can be included in the causal ladder of spacetimes finally totally imprisoned causal curves are studied in detail and results concerning the existence and properties of minimal invariant sets are obtained

the disappearance of causality at small scale in almost commutative manifolds this paper continues the investigations of noncommutative ordered spaces put forward by one of the authors these metaphoric spaces are defined dually by the so called isocones which generalize to the noncommutative setting the convex cones of order preserving functions in this paper we will consider the case of isocones inside almost commutative algebras of the form c m af with m a compact metrizable space we will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family we conjecture that this family is in fact a complete classification a hypothesis related with the noncommutative stone weierstrass conjecture we also obtain that every isocone in c m af with af noncommutative induces an order relation on m with the property that every point in m lies in a neighbourhood of incomparable points thus if the causal order relation on spacetime is induced by an isocone in an almost commutative but not commutative alg

the integration and application of fuzzy and grey modeling methods abstract an integrated fuzzy and grey model and its applications to the prediction control problems are presented the basic grey model gm 1 1 is accompanied with the adaptive fuzzy method to improve its prediction capability the gradient descent scheme is applied to the fuzzy rules to determine whether the predicted results from the grey model should be adjusted the quantity of adjustment is judged from the degrees of the correlation between the past data and the current input we select a few most correlated patterns to decide the direction of adjustment due to the simplicity of the structure and its fast learning characteristics this model is good as a real time controller under the proposed methodology the simulation results are shown to be superior to those systems which exploit complicated control variables and rules a well known difference equation and the weather forecast prediction problems are depicted to verify the superiority of the proposed method

totally vicious space times the concepts of symmetry and local reflectability in space times are introduced which are weaker conditions than the reflecting condition using these concepts some characterizations for total viciousness are given it is shown that any locally reflecting space time with a closed timelike curve is totally vicious also it is shown that any compact homogeneous space time is totally vicious

shock wave interactions and the riemann flat condition the geometry behind metric smoothing and the existence of locally inertial frames in general relativity we prove that the essential smoothness of the gravitational metric at shock waves in gr a pde regularity issue for weak solutions of the einstein equations is determined by a geometrical condition which we introduce and name the riemann flat condition the riemann flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the riemann curvature tensor this provides a geometric framework for the open problem as to whether regularity singularities points where the curvature is in l infty but the essential smoothness of the gravitational metric is only lipschitz continuous can be created by shock wave interaction in gr or whether metrics lipschitz at shocks can always be smoothed one level to c 1 1 by coordinate transformation as a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes independent of whether the metric itself can be smoothed to c 1 1 locally this latter result yields an explicit procedure analogous to riemann normal coordinates in smooth spacetimes for constructing locally inertial coordinates for lipschitz metrics and is a new regularity result for gr solutions constructed by the glimm scheme

identifying a fuzzy model by using the bipartite membership functions the gradient descent method and genetic algorithms have been widely used to refine fuzzy models constructed for the given data this paper approaches from another viewpoint to adjusting the fuzzy model built for the preprocessed data the membership functions defined for each premise variable are equally distributed and fixed in the transformed domain to better identify the fuzzy model either the transformation functions or consequent parts of fuzzy rules or both need to be optimized since adjusting a rule to satisfy one pattern may deteriorate the others performance and result in a lengthy tuning process we then treat each triangular membership function as two disjoint ones such that each fuzzy rule is divided into mutually independent rules this in turn benefits the refinement of consequent parts in the fuzzy rules since adjusting a rule will not affect the others not only the simulation results from the proposed model will be demonstrated but also the results from the conventional approaches will be given for comparisons we use the least squared method to calculate the desired consequent real numbers for the data located in the same region of transformed domain the conformity of the after tuned consequent parts in the fuzzy rules with the desired values further verifies the effectiveness of the presented methodology

lines in space times we construct a complete timelike maximal geodesic line in a timelike geodesically complete spacetimem containing a compact acausal spacelike hypersurfaces which lies in the past of somes ray ans ray is a future complete geodesic starting ons which maximizes lorentzian distance froms to any of its points if the timelike convergence condition strong energy condition holds a line exists only ifm is static i e it splits geometrically as space time so timelike completeness must fail for a nonstatic spacetime with strong energy condition which contains a closed universe s with the above properties

simplifying fuzzy modeling by both gray relational analysis and data transformation methods abstract instead of following the traditional approaches which utilize original data patterns to construct the fuzzy model this paper proposes to exploit both gray relational analysis and data transformation techniques to simplify the modeling procedures the transformation method allows us to map the original data to other domains such that there is no need to adjust the membership functions and the fuzzification process is simply taking place on the fixed ones since too many system variables involved may complicate the fuzzy modeling the gray relational method is exploited to select the crucial variables from a finite set of candidates based on the calculated relational degrees between the output and the prospective input variables we can decide which are the important premise variables the proposed methods have definite effects on the model s performance therefore the way to systematically adjust the transformation functions is also investigated ease in selecting the premise variables and minimal effort needed to adjust system parameters are the merits of the proposed work simulation results from two different examples are presented to demonstrate the superiority of the proposed model to the conventional methodologies

remarks on cosmological spacetimes and constant mean curvature surfaces we give a general condition which ensures the existence of constant mean curvature cmc cauchy surfaces in cosmological spacetimes however there is an example of a spacetime which does not satisfy this condition and does not admit any cmc cauchy surfaces we discuss conditions under which cmc surfaces may exist

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

periodic phases in second order materials in 1984bernard coleman in consultation with v j mizel constructed a theory of materials for which the free energy density is influenced by both first and second spatial derivatives of the mass density u the goal was to see if such a theory would imply the possibility of occurrence of layered structures with interesting properties related to the spacing of the layers e g periodicity or quasiperiodicity the one dimensional isothermal free energy density for such second order materials was taken to be given by h u u u with h subject to certain inequalities the simplest prototypes for h have the form

k causality coincides with stable causality it is proven that k causality coincides with stable causality and that in a k causal spacetime the relation k coincides with the seifert s relation as a consequence the causal relation the spacetime is strongly causal and the closure of the causal relation is transitive stays between stable causality and causal continuity

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

the splitting theorem for space times with strong energy condition soit m g une variete de lorentz connexe orientee en temps globalement hyperbolique qui est geodesiquement complete du genre temps et satisfait ric v v 0 pour chaque vecteur tangent du genre temps v soit ir m une ligne c est a dire une geodesique du genre temps qui realise la distance entre deux de ses points alors m g est isometrique a r h dt 2 h ou h h est une variete de riemann complete et le facteur r dt 2 est represente par

global hyperbolicity is stable in the interval topology we prove that global hyperbolicity is stable in the interval topology on the spacetime metrics we also prove that every globally hyperbolic spacetime admits a cauchy hypersurface which remains cauchy under small perturbations of the spacetime metric moreover we prove that if the spacetime admits a complete timelike killing field then the light cones can be widened preserving both global hyperbolicity and the killing property of the field

stability and instability of the cauchy horizon for the spherically symmetric einstein maxwell scalar field equations this paper considers a trapped characteristic initial value problem for the spherically symmetric einstein maxwell scalar field equations for an open set of initial data whose closure contains in particular reissner nordstrdata the future boundary of the maximal domain of development is found to be a light like surface along which the curvature blows up and yet the metric can be continuously extended beyond it this result is related to the strong cosmic censorship conjecture of roger penrose the principle of determinism in classical physics is expressed mathemat ically by the uniqueness of solutions to the initial value problem for certain equations of evolution indeed in the context of the einstein equations of general relativity where the unknown is the very structure of space and time uniqueness is equivalent on a fundamental level to the validity of this principle the question of uniqueness may thus be termed the issue of the predictability of the equation the present paper explores the issue of predictability in general relativity since the work of leray it has been known that for the einstein equations contrary to common experience uniqueness for the cauchy problem in the large does not generally hold even within the class of smooth solutions in other words uniqueness may fail without any loss in regularity such failure is thus a global phenomenon the central question is whether this violation of predictability may occur in solutions representing actual physical processes physical phenomena and concepts related to the general theory of relativity namely gravitational collapse black holes angular momentum etc must cer tainly come into play in the study of this problem unfortunately the math ematical analysis of this exciting problem is very difficult at present beyond reach for the vacuum einstein equations in the physical dimension conse

existence of travelling wave front solutions of a two dimensional anisotropic model this paper considers a two dimensional anisotropic email protected email protected 0 email protected k 0 2 email protected k 0 email protected y email protected email protected 0 email protected x email protected y email protected j 3 introduced by pesch and kramer 1986 28 assume that j travels with a speed c in the propagation direction x and is periodic in the transverse direction y this model is formulated as a spatial dynamic system in which the variable x is a time like variable a center manifold reduction technique and a normal form analysis are applied to show that this dynamic system can be reduced to a system of ordinary differential equations a bifurcation analysis yields the persistence of the heteroclinic orbit for the reduced system when higher order terms are added and the speed c is small enough which establishes the existence of travelling wave front solutions in order to overcome the difficulty caused by the irreversibility some appropriate constants are adjusted

time functions as utilities every time function on spacetime gives a continuous total preordering of the spacetime events which respects the notion of causal precedence the problem of the existence of a semi time function on spacetime and the problem of recovering the causal structure starting from the set of time functions are studied it is pointed out that these problems have an analog in the field of microeconomics known as utility theory in a chronological spacetime the semi time functions correspond to the utilities for the chronological relation while in a k causal stably causal spacetime the time functions correspond to the utilities for the k relation seifert s relation by exploiting this analogy we are able to import some mathematical results most notably peleg s and levin s theorems to the spacetime framework as a consequence we prove that a k causal i e stably causal spacetime admits a time function and that the time or temporal functions can be used to recover the k or seifert relation which indeed turns out to be the intersection of the time or temporal orderings this result tells us in which circumstances it is possible to recover the chronological or causal relation starting from the set of time or temporal functions allowed by the spacetime moreover it is proved that a chronological spacetime in which the closure of the causal relation is transitive for instance a reflective spacetime admits a semi time function along the way a new proof avoiding smoothing techniques is given that the existence of a time function implies stable causality and a new short proof of the equivalence between k causality and stable causality is given which takes advantage of levin s theorem and smoothing techniques

geometric analysis of lorentzian distance function on spacelike hypersurfaces some analysis on the lorentzian distance in a spacetime with controlled sectional or ricci curvatures is done in particular we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the omori yau maximum principle as a consequence and under appropriate hypotheses on the sectional or ricci curvatures of the ambient spacetime we obtain sharp estimates for the mean curvature of those hypersurfaces moreover we also give a sufficient condition for its hyperbolicity

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

one dimensional infinite horizon variational problems arising in continuum mechanics in this paper we study a variational problem for real valued functions defined on an infinite semiaxis of the line to wit given x 2 we seek a minimal solution to the problem minimize the functional given by i left w left cdot right right int limits 0 infty f left w left s right ddot w left s right right ds w in ax left upsilon in w loc 2 1 left 0 infty right left upsilon left 0 right dot upsilon left 0 right x right right p

timelike completeness as an obstruction to c 0 extensions the study of low regularity in extendibility of lorentzian manifolds is motivated by the question whether a given solution to the einstein equations can be extended or is maximal as a weak solution in this paper we show that a timelike complete and globally hyperbolic lorentzian manifold is c 0 inextendible for the proof we make use of the result recently established by samann ann henri poincare 17 6 1429 1455 2016 that even for continuous lorentzian manifolds that are globally hyperbolic there exists a length maximizing causal curve between any two causally related points

self tuning fuzzy modeling with adaptive membership function rules and hierarchical structure based on genetic algorithm abstract recently fuzzy systems have been used in many fields and places in order to apply the fuzzy system to the various fields the tuning and optimizing method of the fuzzy system is the key issue some self tuning methods have been proposed so far however these conventional self tuning methods do not have sufficient capability of learning in this paper we propose a new supervised self tuning fuzzy modeling which consist of some membership function expressed by the radial basis function with insensitive region learning is carried out by the genetic algorithms the descent method is also utilized for tuning the shapes of the membership function and consequent parts the effectiveness of the proposed methods is shown by some numerical examples

divergence measure fields sets of finite perimeter and conservation laws divergence measure fields in l over sets of finite perimeter are analyzed a notion of normal traces over boundaries of sets of finite perimeter is introduced and the gauss green formula over sets of finite perimeter is established for divergence measure fields in l the normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak star limit of the normal traces introduced in chen frid 6 over the lipschitz deformation surfaces which implies their consistency as a corollary an extension theorem of divergence measure fields in l over sets of finite perimeter is also established then we apply the theory to the initial boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter

symmetries of cosmological cauchy horizons with exceptional orbits we show here that if an analytic space time satisfying the vacuum or electrovacuum einstein equations contains a compact null hypersurface with closed generators then the space time must have a nontrivial killing vector field this result is an extension of an earlier theorem which required that the generators of the hypersurface must not only be closed but in addition must satisfy a local product bundle lpb condition this lpb condition which is known to be violated in certain of the kerr taub nut space time models is equivalent to the requirement that the generators must all be ordinary fibers of a seifert fibration we prove here that the lpb condition can be dropped thus we have stronger support for our conjecture that causality violations as evidenced by cauchy horizons in cosmological solutions of the einstein equations are essentially an artifact of symmetry and are therefore nongeneric

t topology on the n dimensional minkowski space in this paper a topological study of the n dimensional minkowski space n 1 with t topology denoted by mt has been carried out this topology unlike the usual euclidean one is more physically appealing being defined by means of the lorentzian metric it shares many topological properties with similar candidate topologies and it has the advantage of being first countable compact sets of mt and continuous maps into mt are studied using the notion of zeno sequences besides characterizing those sets that have the same subspace topologies induced from the euclidean and t topologies on n dimensional minkowski space a necessary and sufficient condition for a compact set in the euclidean n space to be compact in mt is obtained thereby proving that the n cube n 1 as a subspace of mt is not compact while a segment on a timelike line is compact in mt this study leads to the nonsimply connectedness of mt for n 2 further minkowski space with s topology has also been dealt with

a strong maximum principle for weak solutions of quasi linear elliptic equations with applications to lorentzian and riemannian geometry the strong maximum principle is proved to hold for weak in the sense of support functions sub and supersolutions to a class of quasi linear elliptic equations that includes the mean curvature equation for c0 space like hypersurfaces in a lorentzian manifold as one application a lorentzian warped product splitting theorem is given 1998 john wiley sons inc

the geometry of the space of null geodesics the topology and geometry of the space of null geodesics n of a space time m are used to study the causal structure of the space time itself in particular the question of whether the topology of n is hausdorff or admits a compatible manifold structure carries information on the global structure of m and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic

the hawking penrose singularity theorem for c 1 1 lorentzian metrics we show that the hawking penrose singularity theorem and the generalisation of this theorem due to galloway and senovilla continue to hold for lorentzian metrics that are of c 1 1 regularity we formulate appropriate weak versions of the strong energy condition and genericity condition for c 1 1 metrics and of c 0 trapped submanifolds by regularisation we show that under these weak conditions causal geodesics necessarily become non maximising this requires a detailed analysis of the matrix riccati equation for the approximating metrics which may be of independent interest

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

completeness of cauchy horizon generators it is proved that every compactly generated future cauchy horizon has past complete generators and dually no condition on the differentiability of the horizon is imposed

some results about the level sets of lorentzian busemann function and bartnik s conjecture some results about the level sets of busemann functions are obtained for spacetimes and in a special case cosmological spacetime these results will be used to prove the conjecture stated by r bartnik in b under some special conditions in this paper we employ some results of galloway horta and eschenburg

splitting theorems for spatially closed space times a lorentzian splitting theorem is obtained for spatially closed spacetimes the proof employs and extends some recent results of bartnik and gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space times a splitting theorem for spatially closedtime periodic space times which generalizes a result first considered by avez is derived as a corollary

on almost causality the almost causal precedence concept proposed by woodhouse and a causality axiom based on it are analyzed properties of the almost future are examined and the above causality axiom is shown to be deducible for a causally continuous space time it is shown to be coinciding with known causality conditions for a reflecting space time also the equivalence of various causality conditions for a reflecting space time and a reduction of the causal continuity axiom are obtained it is next shown that the above causality axiom is not stable under metric perturbations then the seifert future concept has been analyzed and the woodhouse causality is proved to be strictly weaker than the stable causality condition it is concluded that as far as the uncertainties in the form of metric perturbations are concerned the seifert future is useful but not the almost future

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

implications of energy conditions on standard static space times abstract in the framework of standard static space times we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories we concentrate our study on questions about the sub harmonicity of the warping function the scalar curvature map conformal hyperbolicity conjugate points and the time like diameter of this class of space times

cauchy horizon end points and differentiability cauchy horizons are shown to be differentiable at end points where only a single null generator leaves the horizon a cauchy horizon fails to have any null generator end points on a given open subset iff it is differentiable on the open subset and also iff the horizon is at least of class c1 on the open subset given the null convergence condition a compact horizon which is of class c2 almost everywhere has no end points and is at least of class c1 at all points

on the differentiability of cauchy horizons chru ciel and galloway constructed a cauchy horizon that is nondifferentiable on a dense set we prove that in a certain class of cauchy horizons densely nondifferentiable cauchy horizons form a dense set we show that our class of densely nondifferentiable cauchy horizons implies the existence of densely nondifferentiable cauchy horizons arising from partial cauchy surfaces and also the existence of densely nondifferentiable black hole event horizons

singularity theorems for c 1 lorentzian metrics continuing recent efforts in extending the classical singularity theorems of general relativity to low regularity metrics we give a complete proof of both the hawking and the penrose singularity theorem for c 1 lorentzian metrics a regularity where one still has existence but not uniqueness for solutions of the geodesic equation the proofs make use of careful estimates of the curvature of approximating smooth metrics and certain stability properties of long existence times for causal geodesics on the way we also prove that for globally hyperbolic spacetimes with a c 1 metric causal geodesic completeness is c 1 fine stable this improves a similar older stability result of beem and ehrlich where they also used the c 1 fine topology to measure closeness but still required smoothness of all metrics lastly we include a brief appendix where we use some of the same techniques in the riemannian case to give a proof of the classical myers theorem for c 1 metrics

some results on the zeeman topology in a 1967 paper e c zeeman the topology of minkowski space topology 6 161 170 1967 zeeman proposed a new topology for minkowski space time physically motivated but much more complicated than the standard one here a detailed study is given of some properties of the zeeman topology which had not been considered at the time the general setting refers to minkowski space time of any dimension k 1 in the special case k 1 a full characterization is obtained for the compact subsets of space time moreover the first homotopy group is shown to be nontrivial

chronological spacetimes without lightlike lines are stably causal the statement of the title is proved it implies that under physically reasonable conditions spacetimes which are free from singularities are necessarily stably causal and hence admit a time function read as a singularity theorem it states that if there is some form of causality violation on spacetime then either it is the worst possible namely violation of chronology or there is a singularity the analogous result non totally vicious spacetimes without lightlike rays are globally hyperbolic is also proved and its physical consequences are explored

generalized homoclinic solutions for the swift hohenberg equation abstract this paper considers the swift hohenberg equation u u u 3 u 0 0 applying a perturbation method and adjusting some phase shift constants we prove that it has a homoclinic solution connecting a periodic solution called generalized homoclinic solution thereafter near the origin for each positive constant which is a new result

regularity of lorentzian busemann functions a general theory of regularity for lorentzian busemann functions in future timelike geodesically complete spacetimes is presented this treatment simplifies and extends the local regularity developed by eschenburg galloway and newman to prove the lorentzian splitting theorem criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of bartnik

simultaneous design of membership functions and rule sets for fuzzy controllers using genetic algorithms this paper examines the applicability of genetic algorithms ga s in the simultaneous design of membership functions and rule sets for fuzzy logic controllers previous work using genetic algorithms has focused on the development of rule sets or high performance membership functions however the interdependence between these two components suggests a simultaneous design procedure would be a more appropriate methodology when ga s have been used to develop both it has been done serially e g design the membership functions and then use them in the design of the rule set this however means that the membership functions were optimized for the initial rule set and not the rule set designed subsequently ga s are fully capable of creating complete fuzzy controllers given the equations of motion of the system eliminating the need for human input in the design loop this new method has been applied to two problems a cart controller and a truck controller beyond the development of these controllers we also examine the design of a robust controller for the cart problem and its ability to overcome faulty rules

area theorem and smoothness of compact cauchy horizons we obtain an improved version of the area theorem for not necessarily differentiable horizons which in conjunction with a recent result on the completeness of generators allows us to prove that under the null energy condition every compactly generated cauchy horizon is smooth and compact we explore the consequences of this result for time machines topology change black holes and cosmic censorship for instance it is shown that compact cauchy horizons cannot form in a non empty spacetime which satisfies the stable dominant energy condition wherever there is some source content

lyapounov functions of closed cone fields from conley theory to time functions we propose a theory a la conley for cone fields using a notion of relaxed orbits based on cone enlargements in the spirit of space time geometry we work in the setting of closed or equivalently semi continuous cone fields with singularities this setting contains for questions which are parametrization independent such as the existence of lyapounov functions the case of continuous vector fields on manifolds of differential inclusions of lorentzian metrics and of continuous cone fields we generalize to this setting the equivalence between stable causality and the existence of temporal functions we also generalize the equivalence between global hyperbolicity and the existence of a steep temporal function

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

limit curve theorems in lorentzian geometry the subject of limit curve theorems in lorentzian geometry is reviewed a general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one two or at most denumerable some applications are considered it is proved that in chronological spacetimes strong causality is either everywhere verified or everywhere violated on maximizing lightlike segments with open domain as a consequence if in a chronological spacetime two distinct lightlike lines intersect each other then strong causality holds at their points finally it is proved that two distinct components of the chronology violating set have disjoint closures or there is a lightlike line passing through each point of the intersection of the corresponding boundaries

domain patterns in incommensurate systems with the uniaxial real order parameter the basic landau model for the incommensurate commensurate transition to the uniform or dimerized uniaxial ordering is critically reexamined the previous analyses identified only sinusoidal and homogeneous solutions as thermodynamically stable and proposed a simple phase diagram with the first order phase transition between these configurations by performing the numerical analysis of the free energy and the euler lagrange equation we show that the phase diagram is more complex it also contains a set of metastable solutions present in the range of coexistence of homogeneous and sinusoidal solutions these new configurations are periodic patterns of homogeneous domains connected by sinusoidal segments they are lyapunov unstable very probably due to the nonintegrability of the free energy functional we also discuss some other mathematical aspects of the model and compare it with the essentially simpler sine gordon model for the transitions to the states with higher commensurabilities we argue that the present results might be a basis for the explanation of phenomena such as thermal hystereses cascades of phase transitions and memory effects

three dimensional gravity capillary waves on water small surface tension case abstract this paper considers three dimensional gravity capillary waves on water of finite depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction the exact euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time like variable the existence of the solutions of the system is determined by two non dimensional constants the bond number b and the froude number f which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow assume that f 2 c 1 is the curve in the b plane on which the first two eigenvalues for three dimensional waves collide at the imaginary axis and the intersection point of c 1 with 1 is b 1 0 in this paper the case for 0 b b 1 and b near c 1 is considered a center manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations using the existence of a homoclinic orbit connecting to a two dimensional periodic called generalized solitary wave thereafter solution for the reduced system it is shown that such a generalized solitary wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants

smoothness of time functions and the metric splitting of globally hyperbolic spacetimes the folk questions in lorentzian geometry which concerns the smoothness of time functions and slicings by cauchy hypersurfaces are solved by giving simple proofs of a any globally hyperbolic spacetime m g admits a smooth time function open image in new window whose levels are spacelike cauchy hyperfurfaces and thus also a smooth global splitting open image in new window if a spacetime m admits a continuous time function t then it admits a smooth time function open image in new window with timelike gradient open image in new window on all m

conformally embedded spacetimes and the space of null geodesics it is shown that the space of null geodesics of a causally simple spacetime is hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime this provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones