linearly stable orbits in 3 dimensional billiards we construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart it shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing lyapunov exponents cannot be easily extended to 3 dimensions

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

survival probability for the stadium billiard abstract we consider the open stadium billiard consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides due to the hyperbolic nature of the stadium billiard the initial decay of trajectories due to loss through the hole appears exponential however some trajectories bouncing ball orbits persist and survive for long times and therefore form the main contribution to the survival probability function at long times using both numerical and analytical methods we concur with previous studies that the long time survival probability for a reasonably small hole drops like c o n s t a n t t i m e 1 here we obtain an explicit expression for the constant

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

an estimate from above of the number of periodic orbits for semi dispersed billiards for a large class of semi dispersed billiards an exponential estimate from above is found for the number of periodic points of the billiard ball map

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

vanishing cross diffusion limit in a keller segel system with additional cross diffusion abstract keller segel systems in two and three space dimensions with an additional cross diffusion term in the equation for the chemical concentration are analyzed the cross diffusion term has a stabilizing effect and leads to the global in time existence of weak solutions the limit of vanishing cross diffusion parameter is proved rigorously in the parabolic elliptic and parabolic parabolic cases when the signal production is sublinear the existence of global in time weak solutions as well as the convergence of the solutions to those of the classical parabolic elliptic keller segel equations are proved the proof is based on a reformulation of the equations eliminating the additional cross diffusion term but making the equation for the cell density quasilinear for superlinear signal production terms convergence rates in the cross diffusion parameter are proved for local in time smooth solutions since finite time blow up is possible the proof is based on careful h s estimates and a variant of the gronwall lemma numerical experiments in two space dimensions illustrate the theoretical results and quantify the shape of the cell aggregation bumps as a function of the cross diffusion parameter

global solvability of chemotaxis fluid systems with nonlinear diffusion and matrix valued sensitivities in three dimensions abstract in this work we extend a recent result to chemotaxis fluid systems which include matrix valued sensitivity functions s x n c 0 2 r 3 3 in addition to the porous medium type diffusion which were discussed in the previous work namely we will consider the system n t u n n m n s x n c c x t 0 c t u c c c n x t 0 u t u u u p n x t 0 u 0 x t 0 in a bounded domain r 3 with smooth boundary assuming that m 1 0 satisfy m 4 3 that the matrix valued function s x n c 0 2 r 3 3 satisfies s x n c s 0 1 n for some s 0 0 and suitably regular nonnegative initial data we show that the corresponding no flux dirichlet boundary value problem emits at least one global very weak solution upon comparison with results for the fluid free system this condition appears to be optimal moreover imposing a stronger condition for the exponents m and i e m 2 5 3 we will establish the existence of at least one global weak solution in the standard sense

hard chaos in magnetic billiards on the hyperbolic plane in this paper some results on the local and global stability analysis of magnetic billiard systems established on two dimensional riemannian manifolds of constant curvature are presented with particular emphasis on the hyperbolic plane for special billiards possessing a discrete group of rotational or translational symmetry a geometrical theorem illustrated by numerical simulations is given on the stability of trajectories with the same symmetry we also present sufficient criteria for the global hyperbolicity of the dynamics hard chaos and give lower estimations for the lyapunov exponent in terms of the shape of the billiard

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

the ergodicity of a hamiltonian system of two particles in an external field abstract for a hamiltonian system of two one dimensional balls in an external field we prove the ergodicity if the total energy of the system is fixed our proof is an extension of that applied earlier to dispersing billiard systems

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

the l average value and the fuzzy expectation of a fuzzy random variable abstract the concepts of fuzzy random variable and the associated fuzzy expected value have been introduced by puri and ralescu as an extension of measurable set valued functions random sets and of the aumann integral of these functions respectively on the other hand the average function has been suggested by campos and gonzalez as an appropriate function to rank fuzzy numbers in this paper we are going to analyze some useful properties concerning the average value of the expectation of a fuzzy random variable and some practical implications of these properties are also commented on

boundedness in a chemotaxis model with oxygen consumption by bacteria abstract this paper deals with the keller segel model u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 with nonnegative initial data u 0 v 0 w 1 r 2 for some r n u 0 l 1 0 and v 0 l 0 this model describes bacteria movement toward the concentration gradient of the oxygen that is consumed by the bacteria it is proved that if 0 1 6 n 1 v 0 l then the corresponding initial boundary value problem possesses a unique global solution that is uniformly bounded

global existence and asymptotic behavior in a two species chemotaxis system with logistic source abstract in this work we consider the two species chemotaxis system with logistic source we present the global existence of generalized solutions under appropriate regularity assumptions on the initial data in addition the asymptotic behavior of the solutions is studied and our results generalize and improve some well known results in the literature

generalized convexity in fuzzy vector optimization through a linear ordering in this article we study efficiency and weakly efficiency in fuzzy vector optimization after formulating the problem we introduce two new concepts of generalized convexity for fuzzy vector mappings based on the generalized hukuhara differentiability pseudoinvexity i and pseudoinvexity ii we prove that pseudoinvexity is the necessary and sufficient condition for a stationary point to be a solution of a fuzzy vector optimization problem we give conditions to insure that a fuzzy vector mapping is invex and pseudoinvex i and ii moreover we present some examples to illustrate the results lastly we use these results to study the class of problems which have uncertainty and inaccuracies in the objective function coefficients of mathematical programming models

power law scaling of the top lyapunov exponent of a product of random matrices a sequence of i i d matrix valued random variables left x n right cdot x n left begin array 20 c 1 d 0 1 end array right with probabilityp and x n left begin array 20 c 1 a varepsilon b varepsilon c varepsilon 1 a varepsilon end array right with probability 1 p is considered leta e a0e o e c e c0e o e lime 0b e oa0 c0 e 0 andb e 0 for all e 0 it is shown show that the top lyapunov exponent of the matrix productxnxn 1 x1 limn 1 n n xnxn 1 x1 satisfies a power law with an exponent 1 2 that is lime 0 ln ln e 1 2

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

big islands in dispersing billiard like potentials abstract we derive a rigorous estimate of the size of islands in both phase space and parameter space appearing in smooth hamiltonian approximations of scattering billiards the derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials thus universality classes for the local behavior are found moreover for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter 0 islands of polynomial size in appear this suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling for example scattering molecules may be of physically noticeable effect

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

on another edge of defocusing hyperbolicity of asymmetric lemon billiards defocusing mechanism provides a way to construct chaotic hyperbolic billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component if all focusing components of the boundary of the billiard table are circular arcs then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table in the present paper we demonstrate that a class of convex tables asymmetric lemons whose boundary consists of two circular arcs generate hyperbolic billiards this result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other and because these tables are perturbations of the first convex ergodic billiard constructed more than 40 years ago

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

stability in high dimensional steep repelling potentials the appearance of elliptic periodic orbits in families of n dimensional smooth repelling billiard like potentials that are arbitrarily steep and limit to sinai billiards is established for any finite n for typical potentials the stability regions in the parameter space scale as a power law in 1 n and in the steepness parameter thus it is shown that even though these systems have a uniformly hyperbolic albeit singular limit the ergodicity of this limit system is destroyed in the more realistic smooth setting the considered example is highly symmetric and is not directly linked to the smooth many particle problem nonetheless the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

repulsion effects on boundedness in the higher dimensional fully parabolic attraction repulsion chemotaxis system abstract this paper deals with an attraction repulsion chemotaxis system u t d u u u v u w x t 0 1 v t v u v x t 0 2 w t w u w x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 where parameters i i 1 2 and are positive and diffusion coefficient d u c 2 0 satisfies d u 0 for u 0 d u d u m 1 with d 0 and m 1 for all u 0 it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m 2 2 n in particular in the case 1 2 and the solution is globally bounded if m 2 2 n n 2 n 2 n 2 therefore due to the inhibition of repulsion to the attraction the range of m 2 2 n of boundedness is enlarged and the results of 21 is thus extended to the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion

global generalized solutions to a keller segel system with nonlinear diffusion and singular sensitivity abstract we consider the chemotaxis system with nonlinear diffusion and singular sensitivity u t u m u v v x t 0 v t v u v x t 0 u v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x in a smooth bounded domain r n n 2 in this work it is shown that for all reasonably regular initial data u 0 0 and v 0 0 the corresponding neumann initial boundary value problem possesses a global generalized solution provided that m 1 n 2 2 n

symbolic dynamics and the discrete variational principle we show how to construct symbolic dynamics for the class of 2d dimensional twist mappings generated by piecewise strictly convex concave generating functions the method is constructive and gives an efficient way to find all periodic orbits of these high dimensional symplectic mappings it is illustrated with the cardioid and the stadium billiard

ergodicity of classical billiard balls abstract sinai in 1970 proved the ergodicity of two discs on the 2 torus further essential progress was only reached in 1987 when chernov and sinai established the ergodicity of two billiard balls on the v torus then by applying a strategy suggested by kramli simanyi and myself in 1989 we obtained the ergodicity of three and four balls on the torus 3 in the latter case in 1991 and 1992 while recently simanyi proved that of n balls on the torus whenever n after a survey of this progress the study of toric billiards with cylindric scatterers is initiated and the k property of a general class of these is claimed

transition from poissonian to gaussian orthogonal ensemble level statistics in a modified artin s billiard one wall of an artin s billiard on the poincar e half plane is replaced by a one parameter mathit c mathit p family of nongeodetic walls a brief description of the classical phase space of this system is given in the quantum domain the continuous and gradual transition from the poisson like to gaussian orthogonal ensemble goe level statistics due to the small perturbations breaking the symmetry responsible for the arithmetic chaos at mathit c mathit p 1 is studied another goe ensuremath rightarrow poisson transition due to the mixed phase space for large perturbations is also investigated a satisfactory description of the intermediate level statistics by the brody distribution was found in both cases the study supports the existence of a scaling region around mathit c mathit p 1 a finite size scaling relation for the brody parameter as a function of 1 mathit c mathit p and the number of levels considered can be established

design of hyperbolic billiards we formulate a general framework for the construction of hyperbolic billiards spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions other examples include the papenbrock stadium

analysis of a chemotaxis model with indirect signal absorption abstract we consider the chemotaxis model u t u u v v t v v w w t w u in smooth bounded domains r n n n where 0 is a given parameter if either n 2 or v 0 l 1 3 n we show the existence of a unique global classical solution u v w and convergence of u t v t w t towards a spatially constant equilibrium as t the proof of global existence for the case n 2 relies on a bootstrap procedure as a starting point we derive a functional inequality for a functional being sublinear in u which appears to be novel in this context

an example of pet computation of the hausdorff dimension of the aperiodic set we introduce a family of piecewise isometries this family is similar to the ones studied by hooper and schwartz we prove that a renormalization scheme exists inside this family and compute the hausdorff dimension of the discontinuity set the methods use some cocycles and a continued fraction algorithm

on the k property of some planar hyperbolic billiards thek property is demonstrated for a class of planar billiards satisfying wojtkowski s principles their boundary may consist of convex scattering concave and linear pieces earlier wojtkowski showed that these billards had non zero lyapunov exponents

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

a discrete method for studying indifference and order relations between fuzzy numbers abstract this paper proposes a generalization of the ranking function approach to fuzzy numbers permitting the definition of different kinds of dominance conferring a degree on the order relation our generalization enables the subjectivity of the decisionmaker to be taken into account unlike the other methods for ranking fuzzy numbers our model provides an improved indifference relation in comparison with previous models finally some properties of the ranking function are described

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

boundedness in a fully parabolic chemotaxis system with singular sensitivity abstract this paper deals with a fully parabolic chemotaxis system u t u u v v v t v v u with singular sensitivity v 0 on a bounded domain r n n 2 the main result solves the open problem of uniform in time boundedness of solutions for 2 n which was conjectured by winkler 16

a subjective approach for ranking fuzzy numbers abstract starting from a subjective assignation of weights related to the relative importance of the different level sets we define a new method of comparison of fuzzy numbers this procedure is an extension of some well known indices adamo tsumura et al yager some properties of our index are studied in this paper as well as its behaviour on several particular cases

on statistical properties of hyperbolic systems with singularities we study hyperbolic systems with singularities and prove the coupling lemma and exponential decay of correlations under weaker assumptions than previously adopted in similar studies our new approach allows us to study the mixing rates of the reduced map for certain billiard models that could not be handled by the traditional techniques these models include modified bunimovich stadia which are bounded by minor arcs and flower type regions that are bounded by major arcs

the polygon quantum billiard problem the present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates the polygon ground state theorem is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the elemental polygons defined in the text the polygon first excited state theorem is established which states that for every n sided regular polygon n equivalent first excited states exist each of which contains a nodal curve that is a line of mirror symmetry of the related polygon a vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the n sided regular polygon is spanned by any two of these n nodal diagonal eigenstates i e the first excited state is twofold degenerate at various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations thus for example correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive nonoverlapping classes of polygons stemming from the preceding conclusions a discussion is included on the possibility of employing transverse magnetic tm modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic

open mushrooms stickiness revisited we investigate mushroom billiards a class of dynamical systems with sharply divided phase space for typical values of the control parameter of the system an infinite number of marginally unstable periodic orbits mupos exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit quasi regular behaviour for long periods of time the problem of finding these mupos is expressed as the well known problem of finding optimal rational approximations of a real number subject to some system specific constraints by introducing a generalized mushroom and using properties of continued fractions we describe for the first time a zero measure set of control parameter values 0 1 for which all mupos are destroyed and therefore the system is less sticky leading to a different power law exponent for the poincar recurrence time distribution statistics the open mushroom billiard with a hole is then considered in order to quantify the stickiness exhibited due to mupos and exact leading order expressions for the algebraic decay of the survival probability function are calculated for mushrooms with triangular and rectangular stems numerical simulations are also performed which confirm our predictions for both sticky and less sticky mushrooms

stochastic modeling of a billiard in a gravitational field power law behavior of lyapunov exponents we consider the motion of a point particle billiard in a uniform gravitational field constrained to move in a symmetric wedge shaped region the billiard is reflected at the wedge boundary the phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic it is known that the system is integrable for two values of the wedge half angle 1 and 2 and chaotic for 1 2 we study the system at three levels of approximation first where the deterministic dynamics is replaced by a random evolution second where in addition the tangent map in each region is replaced by its average and third where the tangent map is replaced by a single global average we show that at all three levels the lyapunov exponent exhibits power law behavior near 1 and 2 with exponents 1 2 and 1 respectively we indicate the origin of the exponent 1 which has not been observed in unaccelerated billiards

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

perturbations of elliptic billiards abstract the dynamical system arising from the problem of billiards is a classical example where the theory of twist maps can be applied in the case of an elliptic billiard table the corresponding twist map is integrable and has a saddle connection between two hyperbolic period two points using a discrete analog to the melnikov method we are able to show that this saddle connection can be deformed into a transversal heteroclinic connection under certain analytic perturbations of the table from the formulas that we get we can show that the splitting of the separatrices is exponentially small as a function of the eccentricity of the original unperturbed elliptic table in addition we also include a characterization of the period two periodic points for any billiard table

locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion abstract we show the existence of locally bounded global solutions to the chemotaxis system u t d u u u v v in 0 v t v u v in 0 u v 0 in 0 u 0 u 0 v 0 v 0 in in smooth bounded domains r n n 2 for d u u m 1 with some 0 provided that m 1 n 4

mathematical theory of lyapunov exponents this paper reviews some basic mathematical results on lyapunov exponents one of the most fundamental concepts in dynamical systems the first few sectionscontainsomeverygeneralresultsinnonuniformhyperbolictheory we consider f where f is an arbitrary dynamical system and is an arbitrary invariant measure and discuss relations between lyapunov exponents and several dynamical quantities of interest including entropy fractal dimension and rates of escape the second half of this review focuses on observable chaos characterized by positive lyapunov exponents on positive lebesgue measure sets much attention is given to srb measures a very special kind of invariantmeasuresthatofferawaytounderstandobservablechaosindissipative systems paradoxical as it may seem given a concrete system it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied case studies will be discussed the final section is on noisy or stochastically perturbed systems for which we present a dynamical picture simpler than that for purely deterministic systems in this short review we have elected to limit ourselves to finite dimensional systems and to discrete time the phase space which is assumed to be r d or a riemannian manifold is denoted by m throughout the lebesgue or the riemannian measure on m is denoted by m and the dynamics are generated by iterating a self map of m written f m for flows the reviewed results

renormalized radial large data solutions to the higher dimensional keller segel system with singular sensitivity and signal absorption abstract the chemotaxis system u t u u v v v t v u v is considered under homogeneous neumann boundary conditions in the ball b r 0 r n where r 0 and n 2 despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria since its introduction by keller and segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well posedness global existence results in the literature are restricted to spatially one or two dimensional cases so far or alternatively require certain smallness hypotheses on the initial data for all suitably regular and radially symmetric initial data u 0 v 0 satisfying u 0 0 and v 0 0 the present paper establishes the existence of a globally defined pair u v of radially symmetric functions which are continuous in 0 0 and smooth in 0 0 and which solve the corresponding initial boundary value problem for with u 0 v 0 u 0 v 0 in an appropriate generalized sense to the best of our knowledge this in particular provides the first result on global existence for the three dimensional version of involving arbitrarily large initial data

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

a study of the ranking function approach through mean values abstract the average value was introduced to help in the ordering of fuzzy numbers and it was defined by means of an integrating process of a parametric function representing the position of every cut in the real line by considering the relation between a particular case of the average value and the notion of expectation for fuzzy numbers we interpret the general ranking function approach as a ranking process using a mean value and an interval relation some well known indices are included in this schema using the general average value a more general mean value for a fuzzy number is defined finally some properties of the general mean value are studied as well as its behaviour on several particular cases

decay of correlations and invariance principles for dispersing billiards with cusps and related planar billiard flows following recent work of chernov markarian and zhang it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1 n since the collisions inside a cusp occur in quick succession it is reasonable to expect a much faster decay rate in continuous time in this paper we prove that the flow is rapid mixing correlations decay faster than any polynomial rate a consequence is that the flow admits strong statistical properties such as the almost sure invariance principle even though the billiard map does not the techniques in this paper yield new results for other standard examples in planar billiards including bunimovich flowers and stadia

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

fuzzy quadratic minimum spanning tree problem in this paper a fuzzy quadratic minimum spanning tree problem is formulated as expected value model chance constrained programming and dependent chance programming according to different decision criteria then the crisp equivalents are derived when the fuzzy costs are characterized by trapezoidal fuzzy numbers furthermore a simulation based genetic algorithm using prufer number representation is designed for solving the proposed fuzzy programming models as well as their crisp equivalents finally a numerical example is provided for illustrating the effectiveness of the genetic algorithm

power law behavior of lyapunov exponents in some conservative dynamical systems abstract three billiards whose border depends on a parameter e are considered for e 0 they are integrable while for e 0 they are known to be k systems the maximal lyapunov exponent 1 is numerically computed and a power law behavior 1 e is found with 1 2 for all billiards a related abstract problem is then considered precisely the case of an infinite product of conservative 2 2 random matrices which are perturbations of commuting parabolic ones analogous computations give here two power laws with 1 2 or 2 3 depending on the probability law used to construct random matrices all these phenomena seem to have an universal character a heuristic mean field theory is also proposed based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition

asymptotic profile of a parabolic hyperbolic system with boundary effect arising from tumor angiogenesis this paper concerns a parabolic hyperbolic system on the half space r with boundary effect the system is derived from a singular chemotaxis model describing the initiation of tumor angiogenesis we show that the solution of the system subject to appropriate boundary conditions converges to a traveling wave profile as time tends to infinity if the initial data is a small perturbation around the wave which is shifted far away from the boundary but its amplitude can be arbitrarily large

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

on the bernoulli property of planar hyperbolic billiards we consider billiards in non polygonal domains of the plane with boundary consisting of curves of three different types straight segments strictly convex inward curves and strictly convex outward curves of a special kind the billiard map for these domains is known to have non vanishing lyapunov exponents a e provided that the distance between the curved components of the boundary is sufficiently large and the set of orbits having collisions only with the flat part of the boundary has zero measure under a few additional conditions we prove that there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained up to a zero measure set in one bernoulli component of the billiard map using this result we show that there exists a large class of planar hyperbolic billiards that have the bernoulli property this class includes the billiards in convex domains bounded by straight segments and strictly convex inward arcs constructed by donnay

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

focusing components in typical chaotic billiards should be absolutely focusing we demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing more precisely we construct billiard tables with arbitrary long free path away from a non absolutely focusing component such that a nonlinearly stable periodic orbit exists therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing

hyperbolic billiards with nearly flat focusing boundaries i abstract the standard wojtkowski markarian donnay bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary this is due to the physical principle that is used in the proofs the so called defocusing mechanism of geometrical optics in this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms

global classical solvability and generic infinite time blow up in quasilinear keller segel systems with bounded sensitivities abstract the chemotaxis system u t d u v u s u v v v t v v u is considered under homogeneous neumann boundary conditions in a bounded domain r n n 2 along with initial conditions involving suitably regular and nonnegative data it is firstly asserted that if the positive smooth function d decays at most algebraically with respect to u then for any smooth nonnegative and bounded s fulfilling a further mild assumption especially satisfied when s s u with s 0 0 possesses a globally defined classical solution if is a ball then under appropriate assumptions on d and s generalizing the prototypical choices in d u v u 1 m 1 and s u v u u 1 1 u 0 v 0 with m r and r such that m n 2 n 0 the phenomenon of infinite time blow up is next shown to occur for all initial data within a set b of functions which inter alia is found to be dense in the set of all radially symmetric and suitably regular positive functions on up to equality in thereby covering the largest possible range of nonpositive for the appearance of unbounded solutions this extends previous findings on blow up in infinite time which in the context of were limited to a smaller parameter region and which were restricted to mere existence results without information on the richness of b

comparison of clusters from fuzzy numbers fuzzy numbers are used for representation of numerical quantities in a vague environment their comparison or ranking is important for application purposes a new index for comparing of fuzzy numbers based on their geometrical properties is suggested in this paper this geometrical index is tested on a group of selected examples and compared with the other well known indexes a method for comparison of m tuples of fuzzy numbers and an algorithm for comparison of subsets clusters of similar closed m tuples of trapezoidal fuzzy numbers are presented

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

hard chaos in magnetic billiards on the euclidean plane this work presents local and global results on the stability of the dynamics of classical magnetic billiard systems with homogeneous magnetic field established on the euclidean plane in the first part of the paper our previous results concerning the properties of the stability matrices on curved riemannian manifolds are rederived in a simpler elementary way in the euclidean case as applications the stability regions for special symmetric orbits are determined analytically and numerically using a new technique necessary conditions for hard chaos and lower estimations for the lyapunov exponent are given for planar magnetic billiards with dispersing and focusing boundary segments too it is also shown that in the investigated billiard types hard chaos is structurally stable below a certain threshold magnetic field

soft billiards with corners we develop a framework for dealing with smooth approximations to billiards with corners in the two dimensional setting let a polygonal trajectory in a billiard start and end up at the same billiard s corner point we prove that smooth hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are acceptable the criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners which is expressed in terms of a scattering function we define such an asymptotic scattering function and prove the existence of it explain how can it be calculated and predict some of its properties in particular we show that it is non monotone for some potentials in some phase space regions we prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there we then prove that if the scattering function is extremal the smooth system has elliptic periodic orbits limiting to the corner polygon and furthermore that the return map near these periodic orbits is conjugate to a small perturbation of the henon map and therefore has elliptic islands we find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit

global existence and stabilization in a degenerate chemotaxis stokes system with mildly strong diffusion enhancement abstract a class of chemotaxis stokes systems generalizing the prototype n t u n n m 1 n n c c t u c c n c u t p u n u 0 is considered in bounded convex three dimensional domains where w 2 is given the paper develops an analytical approach which consists in a combination of energy based arguments and maximal sobolev regularity theory and which allows for the construction of global bounded weak solutions to an associated initial boundary value problem under the assumption that 0 1 m 9 8 moreover the obtained solutions are shown to approach the spatially homogeneous steady state 1 n 0 0 0 in the large time limit this extends previous results which either relied on different and apparently less significant energy type structures or on completely alternative approaches and thereby exclusively achieved comparable results under hypotheses stronger than 0 1

diffusion speed in piecewise smooth billiards a classical consequence of kam theory is that if is a smooth strictly convex curve then neither inner nor outer billiard orbits can accumulate on on the other hand several authors observed that this result is false in general if is not smooth or if it is not strictly convex in this paper we consider piecewise smooth strictly convex curves with curvature jump discontinuities and estimate the diffusion speed for those curves namely we show that both inner and outer billiard maps for have orbits which approach with speed 1 m 2

static and time dependent perturbations of the classical elliptical billiard the elliptical billiard problem defines a two dimensional integrable discrete dynamical system integrability not being a robust property we study some static and time dependent perturbations of this problem for the static case we observe the transition from integrability to chaos on some perturbations of the ellipse then we study time dependent perturbations supposing that the boundary deforms periodically with the time remaining always an ellipse we investigate numerically the now four dimensional phase space looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely

ergodic properties of plane billiards with symmetric potentials we study chaotic behaviour of the motion of a particle moving like in a billiard table outside some disks where a symmetric potential acts quadratic forms introduced in markarian 1988 to study non vanishing lyapunov exponents are used

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically