comments on alpha plane representation for type 2 fuzzy sets theory and applications this comment points out a misnomer and two errors in a previous paper by the author ieee trans fuzzy syst vol 17 no 5 pp 1189 1207 oct 2009 and because one of the errors relates the term plane to the term z slice it also connects these two terms more correctly

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

the center problem on a center manifold in r 3 abstract fix a collection of polynomial vector fields on r 3 with a singularity at the origin for every one of which the linear part at the origin has two pure imaginary and one non zero eigenvalue we show that for each fixed value of the non zero real eigenvalue the set of such systems having a center on the local center manifold at the origin corresponds to a variety in the space of admissible coefficients we explicitly compute it for several families of systems with quadratic higher order terms

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

comparing the performance potentials of interval and general type 2 rule based fuzzy systems in terms of sculpting the state space this paper provides application independent perspectives on why improved performance usually occurs as one goes from an interval type 2 it2 fuzzy system to a general type 2 gt2 fuzzy system this is achieved by using the horizontal slice representation of a gt2 fuzzy set and gt2 fuzzy system and by examining first and second order rule partitions as well as novelty partitions for the horizontal slices it demonstrates that for triangle and trapezoid secondary membership functions the numbers of first and second order rule partitions are exactly the same for it2 and gt2 fuzzy systems but that a maximum amount of change always occurs in every second order rule partition of a gt2 fuzzy system this does not always occur in such partitions of an it2 fuzzy system furthermore when type reduction tr is used in a gt2 fuzzy system the total number of novelty partitions is directly proportional to the number of horizontal slices consequently there are many more such partitions in a gt2 fuzzy system that uses tr than occur in an it2 fuzzy system that also uses tr it is the author s conjecture that it is the maximum changes that occur in every second order rule partition as well as the greater number of novelty partitions when tr is used that provide a gt2 fuzzy system with the potential to outperform an it2 fuzzy system

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

well posedness for the navier slip thin film equation in the case of partial wetting we are interested in the thin lm equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid lm driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourthorder parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with p 13 1 4 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l 2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

existence of inverse jacobi multipliers around hopf points in r 3 emphasis on the center problem abstract we consider the center problem at hopf points of analytic systems in r 3 that has a classical solution in the lyapunov center theorem which is given in terms of an analytic first integral here we give a new solution in terms of an analytic inverse jacobi multiplier v the existence of a smooth and non flat inverse jacobi multiplier around a hopf point of saddle focus type is also proved when studying these problems we needed to discuss the relation between inverse jacobi multipliers and center manifolds w c in particular to know under what conditions w c v 1 0 to illustrate our results we solve the center problem for the lu system

well posedness for a class of thin film equations with general mobility in the regime of partial wetting we establish well posedness for the family of thin film equations left begin array ll h t h n h xxx x 0 quad rm in h 0 h 0 h x 1 quad quad rm on partial h 0 end array right 1 with n in 0 frac 14 5 backslash 1 2 the model 1 with n in 0 3 has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile h geqq 0 the family of thin film equations 1 provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions the parabolicity of the fourth order parabolic problem degenerates at the free boundary which leads to a loss of regularity at the moving contact point our solutions are regular in terms of the two variables d x and d x 3 n where d x is the distance to the free boundary the main technical difficulty in the analysis of 1 is related to the loss of regularity at the contact points

stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem abstract consider the thin film equation h t h h y y y y 0 with a zero contact angle at the free boundary that is at the triple junction where liquid gas and solid meet previous results on stability and well posedness of this equation have focused on perturbations of equilibrium stationary or self similar profiles the latter eventually wetting the whole surface these solutions have their counterparts for the second order porous medium equation h t h m y y 0 where m 1 is a free parameter both porous medium and thin film equation degenerate as h 0 but the porous medium equation additionally fulfills a comparison principle while the thin film equation does not in this note we consider traveling waves h v 6 x 3 x 2 for x 0 where x y v t and v 0 are free parameters these traveling waves are receding and therefore describe de wetting a phenomenon genuinely linked to the fourth order nature of the thin film equation and not encountered in the porous medium case as it violates the comparison principle the linear stability analysis leads to a linear fourth order degenerate parabolic operator for which we prove maximal regularity estimates to arbitrary orders of the expansion in x in a right neighborhood of the contact line x 0 this leads to a well posedness and stability result for the corresponding nonlinear equation as the linearized evolution has different scaling as x 0 and x the analysis is more intricate than in related previous works we anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated such as droplet rupture

on uniqueness of weak solutions for the thin film equation abstract in any number of space variables we study the cauchy problem related to the fourth order degenerate diffusion equation s h h h 0 this equation derived from a lubrication approximation also models the surface tension dominated flow of a thin viscous film in the hele shaw cell our focus is on uniqueness of weak solutions in the complete wetting regime when a zero contact angle between liquid and solid is imposed in this case we transform the problem by zooming into the free boundary and look at small lipschitz perturbations of a quadratic stationary solution in the perturbational setting the main difficulty is to construct scale invariant function spaces in such a way that they are compatible with the structure of the nonlinear equation here we rely on the theory of singular integrals in spaces of homogeneous type to obtain linear estimates in these functions spaces which provide optimal conditions on the initial data under which a unique solution exists in fact this solution can be used to define a class of functions in which the original initial value problem has a unique weak solution moreover we show that the interface between empty and wetted regions is an analytic hypersurface in time and space

the navier slip thin film equation for 3d fluid films existence and uniqueness abstract we consider the thin film equation t h h 2 h 0 in physical space dimensions i e one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension which corresponds to the lubrication approximation of the navier stokes equations of a three dimensional viscous thin fluid film with navier slip at the substrate this equation can have a free boundary the contact line moving with finite speed at which we assume a zero contact angle condition complete wetting regime previous results have focused on the 1 1 dimensional version where it has been found that solutions are not smooth as a function of the distance to the free boundary in particular a well posedness and regularity theory is more intricate than for the second order counterpart the porous medium equation or the thin film equation with linear mobility corresponding to darcy dynamics in the hele shaw cell here we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling wave profile this leads to control on the free boundary and in particular its velocity

volumes of hyperbolic haken manifolds ii in 14 a program was initiated for using the topological theory of 3 manifolds to obtain lower bounds for volumes of hyperbolic 3 manifolds in 1 by a combination of new geometric ideas with relatively standard but specifically 3 dimensional topological techniques we showed that every closed orientable hyperbolic 3 manifold whose first betti number is at least 3 has volume exceeding 0 92 by contrast the best known lower bound 10 5 for the volume of an arbitrary closed hyperbolic 3 manifold is approximately 0 0012 in 3 we showed that every closed orientable hyperbolic 3 manifold whose first betti number is 2 has volume exceeding 0 34 the proof depended on supplementing the results and techniques of 1 with ingenious elementary arguments due to zagier 11 and numerical computations in the present paper we shall show that if one excludes certain special manifolds such as fiber bundles over s 1 then the lower bound of 0 34 also holds for hyperbolic 3 manifolds with betti number 1 the proof depends heavily on the results of 1 and 3 but it involves much deeper topological ideas than these papers the new topological results needed for the proof occupy most of the present paper to some extent these results have the flavor of general topology but the proofs make use of such specifically low dimensional techniques as the characteristic submanifold theory 9 8 the interaction between trees and incompressible surfaces and scott s theorem 12 that surface groups are locally extended residually finite before giving a precise statement of our main result we must review a few elementary notions from 3 manifold theory

hankel norm model reduction of discrete time interval type 2 t s fuzzy systems with state delay this article focuses on the model reduction problem of discrete time time delay interval type 2 takagi sugeno t s fuzzy systems compared with the type 1 t s fuzzy system the interval type 2 t s fuzzy system has more advantages in expressing nonlinearity and capturing uncertainties in addition in order to simplify the analysis process complex high order systems can be approximated as low order systems which is called model reduction in previous studies there are few researches on model reduction of the interval type 2 t s fuzzy system with time delay hankel norm is adopted to limit the error after model reduction based on jensen s inequality a linear matrix inequality lmi condition for the hankel norm performance of the error system is obtained a membership function dependent method based on piecewise linear membership functions is utilized to deal with mismatched membership functions where information of membership functions will be used for relaxing analysis results next by a convex linearization design the model reduction problem is formulated as a convex lmi feasibility optimization condition numerical examples are given to verify the validity of the analysis

model reduction of discrete time interval type 2 t s fuzzy systems this paper addresses the model reduction problem of discrete time interval type 2 it2 takagi sugeno t s fuzzy systems which represent the discrete time nonlinear systems subject to uncertainty with the use of it2 fuzzy sets the uncertainty of the discrete time nonlinear system can be captured by the lower and upper membership functions for a given high order discrete time it2 t s fuzzy system the purpose is to find a lower dimensional system to approximate the original system to achieve the approximation performance an mathcal h infty norm is used to suppress the error between the original system and its simplified system by introducing a membership functions dependent technique and applying a convex linearization method a membership functions dependent condition which takes the information of membership functions into account is obtained to reduce the dimensions of system matrices and the number of fuzzy rules of the system all the obtained theorems are represented as in the form of linear matrix inequalities finally simulation results are demonstrated to show the effectiveness of the derived results

growth rates _ homology and volumes of hyperbolic 3 manifolds it is shown that if m is a closed orientable irreducible 3 manifold and n is a nonnegative integer and if h 1 m z p has rank n 2 for some prime p then every n generator subgroup of 1 m has infinite index in 1 m and is in fact contained in infinitely many finite index subgroups of 1 m this result is used to estimate the growth rates of the fundamental group of a 3 manifold in terms of the rank of the z p homology

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

stable manifolds to bounded solutions in possibly ill posed pdes abstract we prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations these equations may be ill posed a single specific solution an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions in the later cases which include several orbits we also establish the invariant manifolds of an orbit depend smoothly on the orbit analytically in the case of quasi periodic orbits and finitely differentiably in the case of more general families we first establish a general abstract theorem which under suitable spectral non degeneracy analyticity assumptions on the linearized equation establishes the existence of the desired manifold related results appear in the literature but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds this makes the results in this paper applicable to some several models of current interest that could not be treated otherwise we discuss in detail the boussinesq equation of water waves similar phenomena happen in other long wave approximations and complex ginzburg landau equation more recently we observed 11 that our results also apply to mean field games since the equations we consider may be ill posed part of the requirements for the stable manifold is that one can define the forward dynamics on them note also that the methods that are based in the existence of dynamics such as graph transform do not apply to ill posed equation we use the methods based on integral equations perron method associated with the partial dynamics but we need to take advantage of smoothing properties of the partial dynamics note that even if the families of solutions we started with are finite dimensional the stable manifolds may be infinite dimensional

well posedness for the navier slip thin film equation in the case of complete wetting we are interested in the thin film equation with zero contact angle and quadratic mobility modeling the spreading of a thin liquid film driven by capillarity and limited by viscosity in conjunction with a navier slip condition at the substrate this degenerate fourth order parabolic equation has the contact line as a free boundary from the analysis of the self similar source type solution one expects that the solution is smooth only as a function of two variables x x where x denotes the distance from the contact line with 13 14 0 6514 irrational therefore the well posedness theory is more subtle than in case of linear mobility coming from darcy dynamics or in case of the second order counterpart the porous medium equation here we prove global existence and uniqueness for one dimensional initial data that are close to traveling waves the main ingredients are maximal regularity estimates in weighted l2 spaces for the linearized evolution after suitable subtraction of a t b t x terms

well posedness and uniform bounds for a nonlocal third order evolution operator on an infinite wedge we investigate regularity and well posedness for a fluid evolution model in the presence of a three phase contact point we consider a fluid evolution governed by darcy s law after linearization we obtain a nonlocal third order operator which contains the dirichlet neumann operator on the wedge with opening angle epsilon 0 we show well posedness and regularity for this linear evolution equation in the limit of vanishing opening angle we show the convergence of solutions to a fourth order degenerate parabolic operator related to the thin film equation in the course of the analysis we introduce and characterize a new type of sum of weighted sobolev spaces which are suitable to capture the singular limit as epsilon to 0 in particular the nature of the problem requires the use of techniques that are adapted to the problem in the singular domain as well as the degenerate limit problem

darcy s flow with prescribed contact angle well posedness and lubrication approximation we consider the spreading of a thin two dimensional droplet on a solid substrate we use a model for viscous fluids where the evolution is governed by darcy s law at the contact point where air and liquid meet the solid substrate a constant non zero contact angle partial wetting is assumed we show local and global well posedness of this free boundary problem in the presence of the moving contact point our estimates are uniform in the contact angle assumed by the liquid at the contact point in the so called lubrication approximation long wave limit we show that the solutions converge to the solution of a one dimensional degenerate parabolic fourth order equation which belongs to a family of thin film equations the main technical difficulty is to describe the evolution of the non smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter varepsilon

geometric chaos indicators and computations of the spherical hypertube manifolds of the spatial circular restricted three body problem abstract the circular restricted three body problem has five relative equilibria l 1 l 2 l 5 the invariant stable unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria l 1 l 2 have been identified as the separatrices for the motions which transit between the regions of the phase space which are internal or external with respect to the two massive bodies while the stable and unstable manifolds of the planar problem have been extensively studied both theoretically and numerically the spatial case has not been as deeply investigated this paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator the definition of the chaos indicator is not trivial since the mandatory use of the regularizing kustaanheimo stiefel variables may introduce discontinuities in the finite time chaos indicators from the study of such discontinuities we define geometric chaos indicators which are globally defined and smooth and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of l 1 l 2 we illustrate the method for the sun jupiter mass ratio and represent the topology of the asymptotic manifolds using sections and three dimensional representations

approximation by maximal cusps in boundaries of deformation spaces of kleinian groups let m be a compact oriented irreducible atoroidal 3 manifold with nonempty boundary let cc0 m denote the space of convex cocompact kleinian groups uniformizing m we show that any kleinian group in the boundary of cc0 m whose limit set is the whole sphere can be approximated by maximal cusps density of maximal cusps on the boundary of schottky space is derived as a corollary we further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3 manifold with connected conformal boundary

thin film equations with partial wetting energy existence of weak solutions abstract the capillarity driven evolution with slip of a thin liquid film over a dry surface is considered in the regime of partial wetting the focus is on the simplest model case of a constant non zero dynamic contact angle in the lubrication approximation for the analytical treatment of the corresponding free boundary problem a new strategy is proposed based on the introduction of an ad hoc class of disjoining pressures which tend to concentrate at triple junctions a first investigation of this approach yields the existence of weak solutions which satisfy the dissipation relation as an inequality and which are different from those with zero contact angle a heuristic argument is also presented in order to clarify the connection between contact angle and dissipation relation it shows that moving droplets which satisfy the dissipation relation as an equality are forced to have the prescribed contact angle

cyclicity of polynomial nondegenerate centers on center manifolds abstract we consider polynomial families of ordinary differential equations on r 3 parametrized by the admissible coefficients for which the origin is an isolated singularity at which the linear part of the system has one non zero real and two purely imaginary eigenvalues we derive theorems that bound the maximum number of limit cycles within the center manifold that can bifurcate under arbitrarily small perturbation of the coefficients from any center at the origin the bounds are global in that they apply to the system corresponding to any point on any irreducible component of the center variety in the space of parameters we also derive theorems for such bounds when attention is confined to a single irreducible component of the center variety

asymptotic equipartition and long time behavior of solutions of a thin film equation we investigate the large time behavior of classical solutions to the thin film type equation ut uuxxx x it was shown in previous work of carrillo and toscani that for non negative initial data u0 that belongs to h 1 r and also has a finite mass and second moment the strong solutions relax in the l 1 r norm at an explicit rate to the unique self similar source type solution with the same mass the equation itself is gradient flow for an energy functional that controls the h 1 r norm and so it is natural to expect that one should also have convergence in this norm carrillo and toscani raised this question but their methods using a different lyapunov functions that arises in the theory of the porous medium equation do not directly address this since their lyapunov functional does not involve derivatives of u here we show that the solutions do indeed converge in the h 1 r norm at an explicit but slow rate the key to establishing this convergence is an asymptotic equipartition of the excess energy roughly speaking the energy functional whose dissipation drives the evolution through gradient flow consists of two parts one involving derivatives of u and one that does not we show that these must decay at related rates due to the asymptotic equipartition and then use the results of carrillo and toscani to control the rate for the part that does not depend on derivatives from this one gets a rate on the dissipation for all of the excess energy

local existence and regularity for a class of degenerate parabolic equations on etudie des equations paraboliques scalaires a une variable d espace de la forme u t f u u x uu xx ou f f u p s est une fonction lisse de u p s r 3 avec f s strictement positive

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

smooth zero contact angle solutions to a thin film equation around the steady state in the simplest case of a linearly degenerate mobility we view the thin film equation as a classical free boundary problem our focus is on the regularity of solutions and of their free boundary in the complete wetting regime which prescribes zero slope at the free boundary in order to rule out of the analysis possible changes in the topology of the positivity set we zoom into the free boundary by looking at perturbations of the stationary solution our strategy is based on a priori energy type estimates which provide minimal conditions on the initial datum under which a unique global solution exists in fact this solution turns out to be smooth for positive times and to converge to the stationary solution for large times as a consequence we obtain smoothness and large time behavior of the free boundary

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

smooth conjugacy of centre manifolds in this paper we prove that for a system of ordinary differential equations of class c r 1 1 r 0 and two arbitrary c r 1 1 local centre manifolds of a given equilibrium point the equations when restricted to the centre manifolds are c r conjugate the same result is proved for similinear parabolic equations the method is based on the geometric theory of invariant foliations for centre stable and centre unstable manifolds

a generalized type 2 fuzzy granular approach with applications to aerospace in this paper a granular approach for intelligent control using generalized type 2 fuzzy logic is presented granularity is used to divide the design of the global controller into several individual simpler controllers the theory of alpha planes is used to implement the generalized type 2 fuzzy systems the proposed method for control is applied to a non linear control problem to test the advantages of the proposed approach also an optimization method is used to efficiently design the generalized type 2 fuzzy system to improve the control performance

long time asymptotics for strong solutions of the thin film equation in this paper we investigate the large time behavior of strong solutions to the one dimensional fourth order degenerate parabolic equation u t u u xxx x modeling the evolution of the interface of a spreading droplet for nonnegative initial values u 0 x h 1 both compactly supported or of finite second moment we prove explicit and universal algebraic decay in the l 1 norm of the strong solution u x t towards the unique among source type solutions strong source type solution of the equation with the same mass the method we use is based on the study of the time decay of the entropy introduced in 13 for the porous medium equation and uses analogies between the thin film equation and the porous medium equation

a dynamical systems approach for the contact line singularity in thin film flows abstract we are interested in a complete characterization of the contact line singularity of thin film flows for zero and nonzero contact angles by treating the model problem of source type self similar solutions we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system in particular we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points the results are extended to radially symmetric source type solutions in higher dimensions furthermore we give dynamical systems proofs for the existence and uniqueness of self similar droplet solutions in the nonzero dynamic contact angle case

bifurcations under nondegenerated conditions of higher degree and a new simple proof of the hopf neimark sacker bifurcation theorem this paper generalizes the nondegenerated conditions that imply the most common bifurcations in uniparametric families of maps defined on r it also presents a new very simple proof of the hopf bifurcation theorem of maps on r2 see g iooss mathematical studies vol 36 1979 based on one of the results obtained in one dimension and generalizes one of the nondegenerated conditions the hopf condition of the theorem

a complete renormalization group trajectory between two fixed points we give a rigorous nonperturbative construction of a massless discrete trajectory for wilson s exact renormalization group the model is a three dimensional euclidean field theory with a modified free propagator the trajectory realizes the mean field to critical crossover from the ultraviolet gaussian fixed point to an analog recently constructed by brydges mitter and scoppola of the wilson fisher nontrivial fixed point

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given